A Comparison of Structural and Non-Structural Econometric Models in the Toronto Office Market
by
Kimberly Gole
B.S., Mathematics and Statistics and Applied Probability, 2007
University of Alberta
Submitted to the Program in Real Estate Development in Conjunction with the Center for Real Estate in
Partial Fulfillment of the Requirements for the Degree of Master of Science in Real Estate Development
at the
Massachusetts Institute of Technology
September, 2014
©2014 Kimberly Gole
All rights reserved
The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic
copies of this thesis document in whole or in part in any medium now known or hereafter created.
Signature of Author_________________________________________________________
Center for Real Estate
July 30, 2014
Certified by_______________________________________________________________
William C. Wheaton
Professor, Department of Economics
Thesis Supervisor
Accepted by______________________________________________________________
Albert Saiz
Chair, MSRED Committee, Interdepartmental Degree Program in
Real Estate Development
A Comparison of Structural and Non-Structural Econometric Models in the Toronto Office Market
by
Kimberly Gole
Submitted to the Program in Real Estate Development in Conjunction with the Center for Real Estate on
July 30, 2014 in Partial Fulfillment of the Requirements for the Degree of Master of Science in Real Estate
Development
ABSTRACT
This thesis aims to compare five systems of econometric equations to describe the Toronto office market. It
compares four structural systems differing in their demand equations and a non-structural system that does
not require predefined relationships to exist between variables. Within the structural system of equations the
predefined equations require that real rent is estimated solely from vacancy, long-run supply is dependent
upon real rent and changes in employment only affect demand. Demand can be estimated either directly by
estimating occupied stock and obtaining vacancy through an identity or by estimating vacancy; both occupied
stock and vacancy can either be estimated in levels or estimated by an error correction model. Through the
analysis of the structural models it is found that real rent shows significant momentum of real rent one year
previous. As well the long-run supply curve is rising, while the real rent curve is not rising through the
analysis period, as such the long-run supply is estimated in differences as the theoretical relationship between
real rent and long run supply in levels cannot be estimated with the correct sign for the Toronto market. The
structural demand equations show that error correction terms add value to predictions of demand. The nonstructural model is defined as a vector autoregressive model and allows the variables to freely interact
between themselves without the restrictions placed in the structural model. When comparing the structural
systems to the non-structural system in the back test, the non-structural system produces superior estimates
in the system as a whole. The superior results of the VAR agree with the notion that in complicated dynamic
systems by placing restrictions on the interactions of the variables poorer forecasts may result.
Thesis Supervisor: William Wheaton
Title: Professor of Economics
2
Table of Contents
1. Introduction ...................................................................................................................................... 5 2. Previous Literature ......................................................................................................................... 8 Research on Structural Econometric Models in Commercial Rent Analysis ................................ 8 Research on Non-­‐Structural Econometric Models in Commercial Rent Analysis ................... 10 Research on Error Correction Models in Commercial Rental Analysis ...................................... 10 Research on the Comparison of Structural versus Non-­‐Structural Models in Macroeconomics ............................................................................................................................................ 12 3. Exploratory Data Analysis .......................................................................................................... 12 Toronto Office Market Overview ............................................................................................................. 12 Variable Descriptions .................................................................................................................................. 13 Economic Variables ....................................................................................................................................................... 14 Office Stock ........................................................................................................................................................................ 16 Rent and Vacancy ........................................................................................................................................................... 17 4. Structural Econometric Model .................................................................................................. 18 Structural Model Framework ................................................................................................................... 18 Stationarity Tests of Variables ................................................................................................................. 19 Cointegration Tests ...................................................................................................................................... 20 Real Rental Rate Equation .......................................................................................................................................... 21 Long-­‐Run Supply Equation ........................................................................................................................................ 21 Demand Equations ......................................................................................................................................................... 22 Summary of Cointegration Test Results ............................................................................................................... 24 Real Rental Rate Equation .......................................................................................................................... 24 Econometric Model ........................................................................................................................................................ 25 Long-­‐Run Supply Equation ......................................................................................................................... 26 Econometric Model ........................................................................................................................................................ 27 Demand Equations ........................................................................................................................................ 29 Econometric Model – Direct Demand in Levels ................................................................................................ 31 Econometric Model – Direct Demand as an Error Correction Model ...................................................... 33 Econometric Model – Indirect Demand in Levels ............................................................................................ 34 Econometric Model – Indirect Demand as an Error Correction Model .................................................. 36 5. Non-­‐Structural Econometric Model ......................................................................................... 37 Econometric Model ....................................................................................................................................... 38 6. Comparison of the Structural and Non-­‐Structural Framework ..................................... 41 Back-­‐Testing of Models ............................................................................................................................... 41 7. Conclusion ........................................................................................................................................ 46 8. Bibliography .................................................................................................................................... 48 9. Appendices ...................................................................................................................................... 50 Data as Provided by CBRE Econometric Advisors .............................................................................. 50 Stationarity Tests Statistical Output ....................................................................................................... 53 Business Services Employment ............................................................................................................................... 53 Business Services Employment First Difference .............................................................................................. 53 Real Net Effective Rent ................................................................................................................................................ 54 Real Net Effective Rent First Difference ............................................................................................................... 54 Vacancy Rate .................................................................................................................................................................... 55 3
Vacancy Rate First Difference ................................................................................................................................... 55 Occupied Stock ................................................................................................................................................................ 55 Occupied Stock First Difference (Net Absorption) .......................................................................................... 56 Office Stock ........................................................................................................................................................................ 56 Office Stock First Difference (Net Completions) ............................................................................................... 57 Summary of Additional Stationarity Tests ........................................................................................... 57 Cointegration Tests Statistical Results .................................................................................................. 58 Rental Rate Equation Cointegration Test Results ............................................................................................ 58 Additional Cointegration Tests Explored for Rental Rate Equation ........................................................ 58 Supply Equation Cointegration Test Results ...................................................................................................... 59 Additional Cointegration Tests Explored for Supply Equation .................................................................. 59 Direct Demand in Levels Cointegration Test Results ..................................................................................... 60 Additional Cointegration Tests Explored for Direct Demand in Levels Equation ............................. 61 Indirect Demand in Levels Cointegration Test Results ................................................................................. 61 Additional Cointegration Tests Explored for Indirect Demand in Levels Equation .......................... 62 Direct Demand in Differences (ECM) Cointegration Test Results ............................................................ 62 Additional Cointegration Tests Explored for Direct Demand in Differences (ECM) Equation ..... 63 Indirect Demand in Differences (ECM) Cointegration Test Results ......................................................... 63 Additional Cointegration Tests Explored for Indirect Demand in Differences (ECM) Equation . 64 Structural System Econometric Models ................................................................................................. 64 Real Rental Rate Equation .......................................................................................................................................... 64 Additional Models for Real Rental Rate Equation ............................................................................................ 64 Long-­‐Run Supply Equation ........................................................................................................................................ 65 Additional Models for Supply Equation ................................................................................................................ 65 Direct Demand in Levels Equation ......................................................................................................................... 65 Additional Models for Direct Demand in Levels Equation ........................................................................... 66 Direct Demand as an Error Correction Model Equation ............................................................................... 66 Additional Models for Direct Demand as an Error Correction Model Equation ................................. 66 Indirect Demand in Levels Equation ...................................................................................................................... 67 Additional Models for Indirect Demand in Levels Equation ....................................................................... 67 Indirect Demand as an Error Correction Model Equation ........................................................................... 68 Additional Models for Indirect Demand as an Error Correction Model Equation ............................. 68 Non-­‐structural Econometric Models ...................................................................................................... 69 Rental Rate Equation .................................................................................................................................................... 69 Long-­‐Run Supply Equation ........................................................................................................................................ 69 Vacancy (Indirect Demand) Equation ................................................................................................................... 70 4
1. Introduction The ability to predict rental rates of a market into the near future is of crucial importance in
commercial real estate. The typical proforma analysis of an investment or development relies
fundamentally on the future rental forecast to determine if the investment is one that will be
financially feasible.
This thesis endeavors to determine which econometric model is a more suitable fit, a structural or
a non-structural model. It investigates whether rent is impacted by only vacancy, whether only
demand depends upon job growth and whether rent is solely sufficient to generate new supply as
required in the structural framework or if the variables exist together with less stringent
requirements on their relationships.
Within the structural framework this thesis endeavors to determine which econometric model of
modeling demand is a more suitable fit in the Toronto office market. It investigates whether
demand for the amount of space occupied is better estimated by using variables in levels or by
using the changes in variables (differences). It also investigates which methodology of
structuring the system of equations to estimate demand produces better results, either by directly
estimating demand or indirectly estimating demand.
The structure of the econometric models used to forecast the commercial property market has
been dependent upon where the subject city is located. In the United States the research has
revolved around the economic theory that the change in real rents is linked exclusively to
changes of the vacancy rate from the natural vacancy rate. This change in the vacancy rate from
the natural vacancy rate is characterized as a move away from the market equilibrium. The
divergence of vacancy from its natural level causes rents to also move away from their
equilibrium level, which provides a signal to developer. Development then either increases or
decreases in response to the vacancy movement and acts to return the market to equilibrium. This
additional development potentially changes the amount of occupied stock, which by definition
changes the vacancy observed in the market. The structural model requires that the relationships
described above are valid and there hasn’t been faulty inferences made between the variables.
The main theory behind the structural model is that rent is determined through vacancy, and
changes in the vacancy rate drives changes in the whole system.
In Europe structural models have evolved differently, as these markets have limited vacancy rate
data available for analysis. Due to the limited availability of vacancy information reduced form
demand-supply equations have been developed. Instead of estimating occupied stock to
determine the market vacancy as their American counterparts, European models tend to estimate
vacancy. By estimating vacancy the model does not require the stock information that an
estimate of occupied stock would require. A possible limitation of this framework is that there is
no ability to control the vacancy rate within the range of zero to one.
In order to test whether the required variable relationships in the structural system of equations
are necessary or if the variables can interact more than allowed in the economic theory, a nonstructural model will be analyzed. The non-structural model will be predicted using a vector
autoregressive model. In this model the three variables are specified as linear functions of a set
number of their own lags, and the same number of lags of the other two variables. The
5
exogenous variable will be business services employment. The strength of this framework is that
it does not assume any of the variables require relationships but is able to look at each one as a
function of a lagged value of itself and the other variables.
The following charts summarize the relationships between the variables for the direct demand
model, indirect demand model and the non-structural model. In the charts the E represents that
the variable is used in the estimation, the I represents that the variable is part of an identity.
Variable
Completions
Real Rent
Vacancy
Occupied
Stock
Stock
Variable
Completions
Real Rent
Vacancy
Occupied
Stock
Stock
Variable
Completions
Real Rent
Vacancy
Occupied
Stock
Stock
Completions
Direct Demand
Is a Function of:
Real Rent
Vacancy
Occupied
Stock
E
E
I
E
I
Stock
Employment
I
E
I
Completions
Indirect Demand
Is a Function of:
Real Rent
Vacancy
Occupied
Stock
E
E
E
I
I
Stock
Employment
E
I
E
I
Vector Autoregressive Model (VAR)
Is a Function of:
Completions Real Rent
Vacancy
Occupied
(lagged)
(lagged)
(lagged)
Stock
E
E
E
E
E
E
E
E
E
E
I
Stock
Employment
E
E
E
I
A relatively new development in commercial forecasting is the use of error correction models
(ECM). These models are dynamic models that allow long-term effects to be separated from
short-term effects, and are especially useful in determining whether commercial real estate
6
demand is a product of short-term effects or long-run deviations from equilibrium. This type of
model requires that the equation it is applied to show cointegration. The error correction
framework will be applied to the direct and indirect demand equations in differences. This will
allow investigation into whether short-term effects add significant value in predicting demand.
By comparing four systems of structural econometric models and the non-structural econometric
for the Toronto office market this thesis aims to determine which method of forecasting provides
more accurate results. It aims to determine if the error correction model in differences adds any
value to the prediction above that which the typical long-run model in levels does. It also aims to
determine if better results are achieved by estimating the demand directly by estimating occupied
stock or indirectly by estimating vacancy. Along with variable identities, three equations will be
specified to describe the relationships within the structural framework of the variables in the
Toronto market. One model with all of the independent variables will be specified as a vector
autoregressive model for the non-structural system. Once both model forms are created, the
models predictive power will be tested using a portion of the data, from the first quarter of 2008
to the fourth quarter of 2013. As this portion of the data includes both the global recession of
2008-2009 and the recovery it should provide an accurate indication of predictive power.
This thesis will be presented in the following format with the chapters outlined below:
• Chapter 1 introduces the topic and provides a description of the econometric models
being considered.
• Chapter 2 describes the historical development of commercial forecasting. The previous
literature spans several locations and includes research on the development of structural
models, reduced form demand-supply models, and error correction models.
• Chapter 3 provides a description on the Toronto office market and includes exploratory
analysis of the data used in the thesis.
• Chapter 4 creates the four systems structural econometric equations. It begins by
exploring the stationarity of the variables and cointegration between variables. It then
explores the real rental rate equation, the long-run supply equation and the four demand
equations: direct demand in levels, direct demand as an error correction model, indirect
demand in levels, and indirect demand as an error correction model.
• Chapter 5 focuses on the non-structural econometric system. It creates the real rental rate
equation, the long-run supply equation and the demand equation without the necessity of
theoretical relationships between the variables.
• Chapter 6 compares the systems of equations to one another and the actual observed
values through back testing between the first quarter of 2008 to the fourth quarter of
2013.
• Chapter 7 provides the conclusion and summarizes the relevant findings.
7
2. Previous Literature Research on Structural Econometric Models in Commercial Rent Analysis The model of rental adjustment has its origins in labor economics. This model was first applied
to rents and vacancy rates in the rental housing market by Blank and Winnick (1953). This
research was the first to determine that there is an inverse relationship between rents and
vacancy. Following this study several iterations of this idea have been researched.
Early support of the model structure of relating rents to vacancy rates was provided by Smith
(1969) who researched a model of housing starts as a function of the price of houses, the vacancy
rate, the construction and land costs and the cost and availability of mortgage credit. The
research was conducted on quarterly data from 1954-1965 of Canadian housing markets. Smith
(1974) presented support for the traditional view that vacancy rates are very important in
describing housing rents as rents vary inversely with the vacancy rate. Using annual data from
1961-1971 for five Canadian cities, Halifax, Montreal, Toronto, Winnipeg and Vancouver, he
regresses the vacancy, lagged vacancy, property tax change, and lagged property tax change to
determine the change in rents. It was found that the vacancy rate and the rate of change in
property taxes do significantly affect the change in rents. Along with corroborating the original
structure presented by Blank and Winnick (1953), the study also was able to calculate the natural
vacancy rate for the five cities.
Following similar lines to the Smith (1974) study, Eurbank and Sirmans (1979) study the
structural model using operating expenses and vacancy rates using data from 1967-1974 on four
United States cities and four different apartment types. The study found that the operating
expenses were more important in predicting changes in rent levels than the vacancy rate.
Using data from 17 United States cities Rosen and Smith (1983) confirm that the rental price
changes are significantly affected by excess supply or demand in the marketplace. They find that
both the vacancy rate and operating expenses are significant in predicting the percentage increase
in rents.
Rosen (1984) added an additional variable to the previously completed research. He connected
the stock of office space in addition to the vacancy and rent variables previously used. Using
these variables and data from San Francisco from 1961-1983, he created three equations to
determine the changes in rental rates. The first equation estimated the stock of office space as a
function of employment, rent and price. The regression analysis of this equation showed that
there was a strong positive relationship between occupied stock and FIRE employment. It also
showed a negative relationship between real rents and occupied stock. The second equation
estimated the rent as a function of the change in rents, the natural vacancy, the actual vacancy
and the change in the overall price level. The regression analysis of this equation confirmed that
changes in rent are inversely related to changes in vacancy, and that changes in rent are directly
related to changes in the cost of living. The third equation explained occupied office stock as a
function of the stock in the previous period and the new office stock completions. Through the
regression analysis he found that this equation was not important in determining the supply, only
lagged vacancy was significant.
8
Further expanding the research on structural econometric models to analyze the office market,
Wheaton (1987) undertakes research to determine if there is a national real estate cycle. The
national office market is analyzed using five time series: construction, completions, office
employment, absorption, and vacancy rate. Three behavioral equations are specified to describe
the market process. The first, the absorption equation, is specified as a function of the level of
office employment, current real rents, expectations of future space needs, rate of employment
growth, and amount of demand realized each year. The second equation describes supply as a
function of the cost of construction, the cost to finance construction, current rent, current
vacancy, future expectations upon project completion and rate of employment growth. The final
equation describes the rent adjustment process using current rent and vacancy, the previous
period’s realized rent and the natural vacancy level. In addition, three identities are specified: net
absorption which is the difference between the occupied stock this period and last period, the
occupied stock which is the total office stock multiplied by the percentage of occupancy (1vacancy rate), and the total office stock which is made up of the previous period’s office stock
plus the completions in the previous period. Using this system of six equations he is able to
forecast the real estate cycle ahead and finds in the immediate future the oversupply will not
clear as fast as it has in previous cycles.
Schilling, et al. (1987) applied the structural econometric model to 17 United States cities to
determine if the unoccupied space has an impact on what landlords are able to charge for rent in
the local office market. Using the pooled approach on the office data, they find that the operating
expenses are significant in explaining changes in rent in only four of the cities, while vacancy
rate is significant in eleven of the cities. The study finds unrealistic values for the natural
vacancy rates in the cities.
Gabriel and Nothaft (1987) apply the structural model to rental housing in 16 United States cities
using data from 1981-1985. They confirm that the vacancy rate is important in explaining the
rental rate adjustment process in rental housing.
The research on the relationship between vacancy and rental rates is further studied by Wheaton
and Torto (1988) with the examination of the relationship between excess vacancy and changes
in rent. Excess vacancy is defined as the difference between the natural vacancy rate and the
actual vacancy rate. The research finds that the natural vacancy is increasing over time. It also
finds that the relationship between excess vacancy and real rents is statistically significant and
quantitatively meaningful.
Silver and Goode (1990) determined rents for retail properties in Britain derived as a reduced
form equation from supply, demand and equilibrium equations. They find that demand for retail
space is determined to be a function of rents charged, current expenditures in retail outlets, and
an asset demand (investor demand) component.
Wheaton et al. (1997) estimate three structural equations for the London office market using data
from 1970-1995. The three structural equations for office space demand, new supply and rental
movements are accompanied by two identities to determine the stock and vacancy. They find the
increase in construction can be explained with the traditional model but only as long as the
9
previous boom is taken into consideration. The movements in demand and rent are explained by
the office employment growth. Finally, they find that rents react to changes in the vacancy rates
and to leasing activity.
Research on Non-­‐Structural Econometric Models in Commercial Rent Analysis Dobson and Goddard (1992) relate prices, rents and exogenous economic variables in industrial,
retail and office properties in a British time series from 1972-1987. They recognize that
commercial property is a factor of production as well as an asset with stored value. The models
developed take into account this feature of commercial property as well as distinctions between
different types of property users and different types of property owners. The research concludes
that prices and rents are sensitive to employment rates, interest rates and residential property
values.
Using a reduced form demand-supply model D’Arcy, et al. (1997) study 22 European cities with
data from 1982-1994 to explore the impact of national economic conditions, market size,
measures of office growth and changes in the city economy to explain fluctuations in office
rents. They find that national GDP and short-term interest rates are important predictors of office
rent, while market size and city economic growth have an insignificant impact on office rent.
D’Arcy, et al. (1999) expands on their research with the addition of available supply side data in
the Dublin office market. Dublin presented a substantial challenge as it is a small market and
with that large swings in supply can occur in any year due to new completions. Rent is specified
as a function of service sector employment, GDP, and stock of office floor space. They find that
real GDP lagged one period, and changes in office space lagged three periods are significant
determinants of rental rates.
Thompson and Tsolacos (1999) use the reduced form demand-supply model to analyze the
industrial market in the United Kingdom. They specify real rent as a function of lagged changes
in GDP, industrial vacancy rates and lagged values of rental rate changes. They find that real rent
changes are related positively to real GDP and inversely affected by absorption rates.
Taking a global approach De Wit and van Dijk (2003) research the determinants of office returns
for 46 major markets in Asia, Europe and the United States. They find that GDP, inflation,
unemployment, vacancy rate and available office stock all have an impact on real estate returns.
Research on Error Correction Models in Commercial Rental Analysis Research starting in the early 2000’s has explored error correction models (ECM) as a way to
estimate long-run equilibrium relationships and short-term dynamic corrections of commercial
real estate. Error correction models do not require the variables to be stationary, however it does
require that they be cointegrated.
10
Hendershott, et al. (2002a) use London data to estimate both the long-run relationship and the
short-run adjustment equation. The error correction model they specify has been widely used in
the literature that has followed. They conclude that the ECM model has many advantages over
the vacancy gap model. The coefficients in the ECM have useful economic interpretations, the
price and income elasticities of demand. As well it does not require estimates of real interest
rates and risk, and it can be used in studies where vacancy data is unavailable.
Following the research described above Hendershott, et al. (2002b) continues with analysis using
the error correction model framework by examining London retail and office properties. The
Hendershott, et al. (2002b) study concludes that economic drivers may vary, there is no evidence
of differences in the operation of property markets outside of London and they find that
elasticities for retail and office properties are similar. The final models presented support the use
of an error correction model framework.
De Francesco (2008) uses the error correction model framework to analyze the Sydney and
Melbourne office markets. Within the framework vacancy and the work-space ratio are allowed
to be endogenously determined. The research looks at the rental adjustment mechanism and the
demand-employment relationship. In addition whether office market determinants have a
permanent or transitional effect is studied. With the rental adjustment model he finds that in
Sydney the vacancy rate and the real interest rate are long-term determinants, while in
Melbourne only vacancy is a long-term determinant. In the demand-employment relationship he
finds that a variety of macroeconomic factors have a transitory influence while real rent and real
interest are long-term demand drivers only in Sydney.
Using 15 United State’s metropolitan statistical areas, Brounen and Jennen (2009a) adapt the
Hendershott, et al model above to test the asymmetry in rent response to positive changes in
office employment. They find that rents adjust positively with a rise in office employment and
lagged rent rate changes. They also find that office rents react stronger to changes in employment
when the vacancy rate has deviated from the natural level.
Brounen and Jennen (2009b) use the Hendershott, et al. model specified above to determine
whether the long run relationships between premier and second tier cities are similar in ten
European markets. This is to test whether local and national markets move in sync; if they do not
move in sync then a local model based on national economics would be inaccurate. They find
that national and local changes in economic variables to a large extent move in tandem. The
results do not provide evidence that the economic variables defined at the local or national level
produce better estimates.
Again using the Hendershott, et al model above, Ibanez (2013) investigates how rental rates
evolve over time across four different property types, office, industrial, flex and retail. The study
focuses on the speed at which rents move back to equilibrium and it’s found that the office
market is the slowest to return to equilibrium levels.
11
Research on the Comparison of Structural versus Non-­‐Structural Models in Macroeconomics Sims (1980) advocates an alternative class of model, one which there is the opportunity to drop
standard economic assumptions while retaining the ability to test meaningful hypothesis. He
suggests that the vector autoregressive model could be the alternate class of model as it can still
capture the rich inter-relationship between variables in the time series analysis. He comes to the
conclusion that it is unlikely that macroeconomic models have been over identified.
Sims (1989) theorizes that an ideal model is made up of three important features. The model
contains an interpretation of the behavioral interactions for all of the included parameters. It also
connects to the data in detail and takes into account the inherent uncertainty of behavioral
hypothesis. He recommends experimenting with simple behavioral models before moving into a
vector autoregressive model as a safeguard to ensure that interpretations of the vector
autoregressive model are theatrically valid. In the research he uses interest rates, money stock,
output and price level to show how useful interpreting evidence from a vector autoregressive
model can be.
Clements and Mizon (1991) analyze structural and non-structural models by using a vector
autoregressive model to assess the merits of the structural model. They present the two models in
a complementary (as opposed to competing) format and suggest using the vector autoregressive
model to suggest an efficient model development strategy.
3. Exploratory Data Analysis Toronto Office Market Overview Toronto is the capital city of the province of Ontario, and is Canada’s largest city. According to
the 2011 Canadian census, the population of the greater Toronto area is approximately 5.5
million and the population within the city of Toronto is approximately 2.6 million1. Toronto is
the largest Canadian office market with over 150 million total square feet of office stock as of
the fourth quarter of 20132. The key industries occupying office space in Toronto include
business and professional services, financial services, design services, fashion and apparel
services, film and television services, music services, life sciences and tourism.
Toronto is the financial center of Canada and is the location of the Toronto Stock Exchange. As
well the financial district is centered on Bay Street, the Canadian equivalent to Wall Street. The
big five Canadian banks, the Bank of Montreal, Canadian Imperial Bank of Commerce, TorontoDominion Bank, Royal Bank of Canada, and the Bank of Nova Scotia all have their headquarters
in Toronto.
Many large Canadian corporations have their headquarters in Toronto; eighteen of Canada’s fifty
most profitable companies have their headquarters in Toronto. Global headquarters in Toronto
include Brookfield Asset Management, Fairmont Hotels and Resorts, Manulife Financial, Nortel,
1 Census information can be accessed through: http://www12.statcan.gc.ca/census-recensement/2011/as-sa/fogsspg/Facts-cma-eng.cfm?LANG=Eng&GK=CMA&GC=535
2 As per data provided by CBRE
12
the Hudson’s Bay Company, Bell Media and Rogers Corporation. Canadian operation corporate
headquarters located in Toronto include McDonald’s Canada, Coca-Cola Company, Sirius
Canada, and Marriott International. In addition to being the center of the Canadian financial
industry, Toronto is also the center of the Canadian media industry. Focus in the fashion, film
and music industry has given aided in business services occupying a significant portion of the
Toronto marketplace.
Variable Descriptions The dataset is provided by the CBRE Econometric Advisors Canadian Office, and contains
quarterly observations for the Toronto office market. The dataset contains nine variables as
described below:
1. Year – Observation year and quarter
2. FIRE Emp. (Jobs x 1000) – Finance, insurance, and real estate services employment
sector
3. Business Svc. Emp. (Jobs x 1000) – Business services employment sector
4. Office Stock (SF x 1000) – Total office space square footage
5. Completions (SF x 1000) – Office space completions during the observation year and
quarter
6. Net Absorption (SF x 100) – Office space absorbed by market during the observation
year and quarter
7. Vacancy Rate (%) – Vacancy rate
8. Net Effective Rent ($) – Nominal rental rate
9. Net Effective Rent (2013 $) – Real rental rate expressed in 2013 values
The variables used in the final analysis are: Year, Business Svc. Emp., Office Stock, Vacancy
Rate, Net Effective Rent (2013 $). Using these variables the following additional variables have
been computed:
1. Occupied Stock = (1-Vacancy Rate) * Office Stock
2. Occupied Stock First Difference (Computed Net Absorption) = Occupied Stockt –
Occupied Stockt-1
3. Office Stock First Difference (Net Completions) = Office Stockt – Office Stockt-1
4. Real Net Effective Rent First Difference = Real Net Effective Rentt – Real Net Effective
Rentt-1
5. Year-Over-Year Completions = Office Stockt-4 – Office Stockt
6. Year-Over-Year Employment Growth = Business Services Employmentt-4 – Business
Services Employmentt
7. Year-Over-Year Absorption = Occupied Stockt-4 – Occupied Stockt
8. Year-Over-Year Rent Growth = Real Net Effective Rentt-4 –Real Net Effective Rentt
Note that while completions are available in the dataset, a net completions has been computed
using the first difference of occupied stock. Due to redevelopment happening in the office space
market, net completions takes into account the fact that some buildings may have been
demolished and new buildings built in their place. The completions variable available in the
dataset is gross completions and does not take into account this reduction in stock due to
demolition.
A summary of the descriptive statistics of the variables used in the analysis is provided below:
13
Variable
FIRE Emp
Bus Svc Emp
Office Stock
Real Net Effective Rent
Vacancy Rate
Occupied Stock
Occupied Stock First Difference
(Computed Net Absorption)
Office Stock First Difference
(Computed Net Completions)
Real Net Effective Rent First
Difference
Year-Over-Year Completions
Year-Over-Year Employment
Growth
Year-Over-Year Absorption
Year-Over-Year Rent Growth
Descriptive Statistics
Mean
Median Std. Dev.
Min
Max
228.8581
214.8
44.03406
169
321.6
296.4382
279.3
109.5947
135.2
488.9
117,389.7 127,284 28,424.08
56,370
151,440
19.86919
17.515
8.73251
4.42
39.01
0.12367
0.117
0.04392
0.066
0.227
102,810.5 107,109.5 25,915.61 51,522.18 138,086.5
634.6849
598.9375
808.31
-1,644.71 3,549.828
704.2222
490
750.0381
0
4,515
-0.04022
0.15999
1.07785
-3.83
3.49
2799.795
1822.5
2495.91
0
8733
10.4197
8.7
14.97133
-24.3
46.3
2541.654
-0.20667
2412.439
0.30
2285.881
3.6165
-3145.61
-15.09
7090.33
6.3100
Economic Variables The dataset presents two economic indicator variables, FIRE employment and business services
employment. From the above descriptive statistics we can see that while business services
employment starts lower than FIRE employment it ends significantly higher. The following
graph illustrates the patterns in the economic variables from quarter one of 1980 to quarter four
of 2013.
14
Both series show an increasing trend until the late 1980’s and then a stagnant period in the early
1990s during the 1990-1991 recession. The FIRE employment stays stagnant for a considerably
longer period after the recession than business services employment. The business services series
shows a much stronger increase throughout the late 1990s until present than does the FIRE
employment series. Business service employment also shows a stronger recover from the 20082009 global recession.
15
Office Stock The office stock series shows a large increase in stock from 1980 to the early 1990’s. The gain in
stock exhibited during this period is more than the total increase in stock from the end of the
building boom until present. Incorporating the knowledge of the 1990-1991 recession, the
conclusion that the decrease in building output occurs a few years after the recession is
consistent with developers taking building signals from the current economic state. However,
given that office buildings are a product with a long-term build schedule, the additional stock
coming onto the market is consistent with buildings that were started previous to the recession.
16
Rent and Vacancy The real net effective rent peaks at a value of $39.01 in the fourth quarter of 1989. Following this
peak, the rent falls to a low of $4.42 in the fourth quarter of 1993. From this point the rent does
not recover to it’s previous high; the highest real net effective rent achieved after the low is
$23.61 in the first quarter of 2001.
This type of real rental rate pattern is similar to that of cities with significant ability access to
new supply. This is more typical of cities that have weak constraints on supply, either
geographical or political, than Toronto, which is typically considered land constrained. This is
pattern of real rental rates is also characterized by rents that show less sensitivity to vacancy
rates.
17
The two main peaks, the rental high in 1989 and the rental rate low in 1993, correspond to the
same two peaks in the vacancy rate. As vacancy and rents typically have an inverse relationship
to one another the rental high corresponds to a low in the vacancy rate, and the rental low
corresponds to a high in the vacancy rate. The two series exhibit this inverse relationship until
the early 2000s. After this point they both appear to level out and the relationship doesn’t look to
be as strong.
4. Structural Econometric Model Structural Model Framework The economic theory of the structural econometric model is defined by the following
relationships when demand is estimated directly:
1. Rent is determined exclusively through vacancy:
Rent = F(Vacancy)
2. Long-run supply curve is determined by the real rental rate:
Supply = F(Rent)
3. Demand (occupied stock) is determined through the employment variable as an
economic indicator and the rental rate
Occupied Stock = F(Employment, Rent)
4. The vacancy rate is determined as:
18
Vacancy = 1 – (Occupied Stock/Stock)
When demand is estimated indirectly the structural econometric model is estimated through the
following system of equations:
1. Rent is determined exclusively through vacancy:
Rent = F(Vacancy)
2. Long-run supply curve is determined by the real rental rate:
Supply = F(Rent)
3. Demand is determined through estimating vacancy as a function of the employment
variable as an economic indicator, the rental rate, and the total office stock
Vacancy = F(Employment, Rent, Stock)
In addition, to complete the structural model framework two further identities are required:
1. Occupied Stockt = Total Stockt-1 + Absorptiont-1
2. Total Stockt = Total Stockt-1 + Constructiont-1
The following variables required for the structural model have been computed in the dataset as:
Occupied Stockt = (1 - Vacancy Ratet) * Office Stockt
Net Absorptiont = Occupied Stockt – Occupied Stockt-1
Completionst = Office Stockt – Office Stockt-1
When appropriate the structural model is constructed using the variables in levels (using only the
actual variables and the lagged versions of themselves). However, the equations can also be
specified in differences and in the ECM framework, the equations will be specified in
differences. In order to specify any of the equations in levels the variables must be stationary, if
they are not an ordinary least squares regression is not appropriate. If the variables do not show
stationarity and the series are cointegrated, then an error correction model can be used to
estimate the relationships.
Stationarity Tests of Variables To decide if the structural equations can be specified, as preferred by the economic theory, in
levels the variables are tested to determine if they are stationary. The stationarity test results will
provide a starting point to creating the general framework for the econometric equations.
Dickey and Fuller (1979) developed the procedure to test whether a variable has a unit root. The
original test was further developed to allow lags into the autoregressive process by Said and
Dickey (1984). The augmented Dickey-Fuller test has the null hypothesis that the series has a
unit root; the alternative hypothesis is that a stationary process generates the series. The
augmented Dickey-Fuller test fits a model of the form:
Δxt = α + βxt-1 + ζ1Δxt-1 + ζ2Δxt-2 +…+ ζnΔxt-n + εt
where n is the number of lags specified for the test. The augmented Dickey-Fuller tests whether
β=0, which is equivalent to xt having a unit root.
To test for stationarity of the variables, the augmented Dickey-Fuller test is used for each of the
variables. As the test is very sensitive to the number of lags used, the optimal number of lags will
be selected for each variable using the Akaike’s Information Criterion (AIC), and the augmented
Dickey-Fuller test explored for this optimal number of lags. The summary chart below shows the
results of the tests for stationarity:
19
Stationarity Results – Dickey-Fuller and Augmented Dickey-Fuller Test Results
Variable
Optimum Lag
Test Statistic (optimum MacKinnon p-value (optimum
(AIC)
lag)
lag)
Business Service Employment
6
0.259
0.9754
Business Service Employment
5
-4.512
0.0002
First Difference
Net Effective Rent (Real)
5
-2.658
0.0815
Net Effective Rent (Real) First
4
-2.845
0.0521
Difference
Vacancy
4
-2.612
0.0905
Vacancy First Difference
2
-3.510
0.0077
Occupied Stock
2
-1.939
0.3140
Occupied Stock First Difference
1
-5.547
0.0000
(Net Absorption)
Office Stock
4
-2.010
0.2824
Office Stock First Difference
4
-2.361
0.1531
(Net Completions)
The time series for the economic variable, business services employment, is not stationary series.
However, when the first difference the series is stationary at a 1% significance level3. Real net
effective rent and the vacancy rate are stationary at the 10% level for their optimum number of
lags. The occupied stock series is stationary without any lags and the first difference of the
occupied stock series, net absorption, is stationary at the 1% level for without lags and the
optimum number of lags. The total office stock series and its first difference, net completions,
are stationary without any lags.
The variables stationary in levels are real net effective rent and vacancy. All of the variables are
stationary in differences, including the employment variable.
Note that while total employment, FIRE employment and their respective first differences are not
used in the final analysis, they were explored as options for the economic indicator variable
when selecting the optimum variable set. As such they have been included in the stationarity
analysis and the results of these tests are summarized in the Appendix.
A direct regression with non-stationary series would be problematic, as such going forward the
stationarity results as summarized above will be important in defining the econometric equations.
Cointegration Tests Two series are cointegrated if they tend to move together through time, sharing a stochastic drift.
The error correction model framework approach requires that the variables exhibit cointegration.
Engle and Granger (1987) suggest a two-step process to determine cointegration. Using an
ordinary least squares (OLS) a regression equation is first determined between the two variables.
The augmented Dickey-Fuller test is used to test the stationarity of the residuals from this
regression. The augmented Dickey-Fuller test has the null hypothesis that the residuals are
nonstationary. As such a statistically significant result produces the conclusion that the residuals
3 For the Augmented Dickey-Fuller tests the following critical values from MacKinnon (1996) are used: 1% -3.49,
5% -2.89, 10% -2.58, 20% -2.21.
20
are stationary and the series are cointegrated. This process will be used to determine
cointegrating relationships for the rental equation, the supply equation, and the four demand
equations in the structural framework. As with the stationary tests the results are sensitive to the
number of lags used, as such the AIC has been used to select the optimum number of lags.
The cointegration tests that were explored for the econometric equations are summarized
below4.
Real Rental Rate Equation Starting the cointegration tests with the relationship between rent and vacancy, it is determined
that real net effective rent is not cointegrated with vacancy nor with vacancy lagged four, eight
or twelve periods for the optimum number of lags. A summary table of the statistical results is
provided below:
Cointegration with Real Net Effective Rent
R2
Optimum ADF Test
Regression Lag
Statistic
(AIC)
(Optimum Lag)
Vacancy
0.0896
6
-2.164
Vacancy- 4 period lag
0.2379
2
-1.070
Vacancy- 8 period lag
0.2985
2
-1.479
Vacancy- 12 period lag
0.2580
5
-2.194
Variable
MacKinnon pvalue (Optimum
Lag)
0.2194
0.7269
0.5439
0.2083
The relationship between vacancy and the real net effective rental rate is not cointegrated at any
statistical level, even when lagged vacancy has been introduced. As such, the equation where the
rental rate level is determined solely by the vacancy rate cannot be specified as an error
correction model.
The relationship between vacancy and the first difference of the real net effective rental rate
(change in rent) was also explored. The quarterly change in rent shows cointegration with the
vacancy series at 1% when the AIC optimum one lag is included5.
Long-­‐Run Supply Equation Moving into the supply equation and starting with single variable cointegration tests with total
office stock it is determined that none of the employment relationships exhibit cointegration even
at a very generous 20% level.
As the econometric equation specified for supply equation can be multivariate, testing various
combinations of independent variables with dependent variable total office stock have been
included. Using the results from the AIC determination of the optimum lag in the residuals, the
only combinations that do not show cointegration are the two combinations with the rental and
vacancy rate lagged one and four periods and the combination with only business services
employment and real net effective rent. From these results it is determined that the best
4 For the sake of brevity, only the full statistical outputs for the final econometric equations are included in the
Appendix. As well summary charts for the cointegration tests for variable combinations that ultimately were not
selected for the econometric models are also available in the Appendix.
5 For the Augmented Dickey-Fuller tests the following critical values from MacKinnon (1996) are used: 1% -3.49,
5% -2.89, 10% -2.58, 20% -2.21.
21
combination of cointegrating relationships with total office stock is with business services, real
net effective rent lagged one period and the vacancy rate lagged one period.
The supply equation can also be specified as a change in stock instead of using total office stock
as the dependent variable. As such the cointegrating relationships with the first difference of
office stock (net completions) will also be explored. The following chart shows the results of the
cointegration between net completions and real net effective rent, the variables used in
specifying the supply equation.
Cointegration with Total Office Stock First Difference (Net Completions)
Variable
R2
Optimum
ADF Test
MacKinnon pRegression Lag (AIC)
Statistic
value
(Optimum Lag)
(Optimum Lag)
Real Net Effective Rent
0.4033
1
-5.827
0.0000
Net completions show cointegration with business services employment and total employment in
addition to real net effective rent at the 10% level6. Similar to total office stock, the supply
equation can be specified as a multivariate regression with net completions as the dependent
variable. All of the tested combinations of independent variables show cointegration with net
completions at the 10% for the augmented Dickey-Fuller test with the optimum number of lags
as selected using AIC. Net completions show very significant cointegration with all the
combinations of independent variables at the optimum lag length. The only set of variables that
is less significant, business services employment and vacancy rate, are still significant at the 10%
level.
Based on the cointegration relationships of the independent variables with office stock both in
levels and in differences, it is likely that the final choice of variables for the supply equation in
the structural model will be cointegrated. If the final model shows cointegration an error
correction model can be specified for this equation.
Demand Equations The demand variable can be specified either directly or indirectly and either in levels or in
differences; as such four separate combinations of cointegration relationships will be explored.
The first is for the direct specification of demand in levels and will test potential cointegrating
relationships with occupied stock.
The following chart shows the results of the cointegration between occupied stock and the
multivariate selection real net effective rent and business services employment, the variables
used when specifying the demand equation in levels. Occupied stock doesn’t exhibit
cointegration with any of the multivariate combinations of series explored, nor does it exhibit
cointegration individual with real net effective rent, business services employment, FIRE
employment or total employment.
6 For the Augmented Dickey-Fuller tests the following critical values from MacKinnon (1996) are used: 1% -3.49,
5% -2.89, 10% -2.58, 20% -2.21.
22
Multivariate Cointegration with Occupied Stock
R2
Optimum
ADF Test
Regression Lag (AIC)
Statistic
(Optimum Lag)
Real Net Effective Rent, Business
0.9493
6
-2.087
Services Employment
Variable
MacKinnon pvalue
(Optimum Lag)
0.2497
The demand equation can also be indirectly specified in levels by using vacancy as the
dependent variable, and the changes in office stock and business services as explanatory
variables. All of the multivariate combinations tested show cointegration with vacancy, as do the
employment variables; the following table summarizes the results of the cointegration tests for
the combination of variables used in specifying the indirect demand equation in levels:
Multivariate Cointegration with Vacancy
R2
Optimum
ADF Test
Regression Lag (AIC)
Statistic
(Optimum Lag)
Business Services Employment,
Office Stock, and Real Net
0.7988
6
-4.324
Effective Rent
Variable
MacKinnon pvalue
(Optimum Lag)
0.0004
Demand can also be specified both directly and indirectly using differences. These models will
specified with net absorption and the change in vacancy as the dependent variables. Starting with
the relationships with net absorption, all of the variables and variable combinations show
significant cointegration with the net absorption series. The following table summarizes the
results of the cointegration tests for the combination of variables used in specifying the direct
demand equation in differences:
Multivariate Cointegration with Occupied Stock First Difference (Net Absorption)
Variable
R2
Optimum
ADF Test
MacKinnon pRegression Lag (AIC)
Statistic
value
(Optimum Lag)
(Optimum Lag)
Real Net Effective Rent, Business
0.0704
1
-5.942
0.0000
Services Employment
The final specification of demand for the structural systems will be using the change in vacancy.
This is the indirect demand equation specified in differences. Similar to the net absorption series,
all of the variables and variable combinations explored show cointegration with the change in
vacancy series. The following table summarizes the results of the cointegration tests for the
combination of variables used in specifying the indirect demand equation in differences:
Multivariate Cointegration with Vacancy First Difference
R2
Optimum
ADF Test
Regression Lag (AIC)
Statistic
(Optimum Lag)
Real Net Effective Rent, Business
Services Employment, and Office
0.1690
1
-4.642
Stock
Variable
23
MacKinnon pvalue
(Optimum Lag)
0.0001
Both the direct demand and the indirect equations show a stronger cointegrating relationship
with business services employment than they do with total employment. Based on the
cointegration relationships of the independent variables when demand is specified both directly
and indirectly, it is likely that the final choice of variables for the demand equation in the
structural model will be cointegrated. If the final models for demand show cointegration an error
correction model can be specified for the equation.
Summary of Cointegration Test Results For the rental rate equation the series does not show cointegration with the vacancy rate series.
As such an error correction model cannot be specified for this relationship. However, since the
series are individually stationary a regular ordinary least squares regression can be used to
estimate their relationship.
The supply equation explored using net completions shows cointegrating relationships with the
majority of the combinations of independent variables. Once the final equation has been
determined it is likely that an error correction model will be able to be specified, however, a final
check on the cointegration of the specified series will be tested to ensure the error correction
framework is appropriate.
The only demand equation that does not show a cointegration relationship is that with occupied
stock (direct demand in levels). The remaining three options for demand to complete the system
of equations in the structural framework show cointegrating relationships. Both of the equations
specified in differences show strong cointegration, which will be particularly important, as these
equations will be specified as error correction models.
Real Rental Rate Equation The structural model assumes that changes in rent are determined exclusively through deviations
in the vacancy rate from the natural vacancy rate. As office properties tend to have long-term
leases, rents will tend to be slow to adjust to changes in the vacancy rates. While the strength of
the inverse relationship between the change in real rental rates and the vacancy rate in the
Toronto dataset depletes in the early 2000s the series does show the typical relationship: low
(high) vacancy rates correspond to high (low) real rental rates.
Vacancy is the state through which space passes through while waiting to be leased. It can be
quite cumbersome for a tenant to find appropriate space for their needs as vacant space can come
in very different sizes and spatial arrangements. Once an appropriate space has been found
owners and tenants can begin bargaining over the lease agreements, including the rental rate.
Owners set a minimum reserve price for the space at which they are indifferent between renting
the space to the tenant and leaving it vacant. The longer the anticipated lease up time the lower
the reservation price will be. Tenants similarly set a maximum reserve price for the space, the
price at which they are indifferent between leasing the space and continuing to search for a
different vacant space. When there is ample available vacant space and few tenants in the
market, it is easier for the tenant to find space and the maximum reserve price will be lower. The
final agreed upon price should lie between the owner’s reservation price and the tenant’s
reservation price, Wheaton, et al. (1997). Both the owner’s and the tenant’s reservation price
move inversely with vacancy, Wheaton (1990).
24
The modern theory of search and bargaining specifies that a given level of vacancy, either above
or below the natural rate, does not cause rents to rise or fall continuously, Wheaton and Torto
(1993). This deviation from the natural vacancy rate leads to a stable level of rent reflective of
the rental rate that would be an acceptable outcome of bargaining.
Reflecting that rents do not fall or rise continuously for a given level of vacancy, the relationship
between the real rental rate and vacancy will be specified generally as:
Real Rentt = a + b*Vacancy Ratet + c* Real Rentt-n
Econometric Model From the stationarity tests results, we know that both rent and vacancy are stationary series. As
well from the cointegration it was determined that the two series are not cointegrated. An error
correction model cannot be specified for this equation, however, since both series are stationary
an ordinary least squares regression analysis can be specified for the relationship between the
two variables. As such, the econometric model for the real rent at time t will be specified as
(where n≥0):
Real Rentt = a – b*Vacancy Ratet-n + c*Real Rentt-n
Models that were investigated but subsequently discarded are summarized in the Appendix. The
real rent equation as specified using yearly lags using the Toronto data yields7:
Real Rentt = 8.29371 – 43.5531*Vacancy Ratet-4 + 0.8467268*Real Rentt-4
(6.85)
(-6.77)
(26.38)
Adjusted R2 = 0.8799;
N = 132;
F(2,129) = 480.91
The coefficients on the vacancy rate and the lagged real rent are in the correct direction. The
model suggests that rent would have a stable value of approximately $20 using a vacancy of 12%
(the average vacancy rate between 1980-2013 for Toronto). As shown through the vacancy rate
coefficient, vacancy has an inverse relationship with the rental rate. This result is consistent with
conclusions drawn from previous research on the relationship between the vacancy rate and rent.
The coefficient on the lagged real rent of 0.85 is consistent with the rental rate being sticky and
having a slow speed of adjustment to changes in the vacancy rate. The market would move
towards an equilibrium rent at 15% per year. This slow rate of adjustment is expected, as office
leases are generally long-term. As well, the slow rate of adjustment reflects the fact that real rent
and vacancy are poorly correlated and not cointegrated. The model with only vacancy as the
independent variable has an R2 of only 0.0896, and without the series being cointegrated this
relationship cannot be improved upon without additional variables being specified to the model.
7 Full statistical output for regression is in the Appendix
25
The above graph depicts the dynamic forecast based on the model with the coefficients as
specified above. The lagged rent variable is endogenous while the remaining variables are treated
as exogenous to the equation for the purposes of this graph. The predicted rental values show the
same general pattern as the actual rental rates, which are expected with the high R2 of the model
and the lagged rent variable being present in the model. However, the dynamic forecast provides
much more extreme swings in movement than the actual rental rates experienced. The
predictions exhibit overcorrections, most notably becoming negative in the mid-1990s and a
large increase after 2005.
Long-­‐Run Supply Equation The long-run supply is measured as the net completions added to the previous periods total office
stock. New office space is developed when the value of developing the space exceeds the cost to
develop it. The value of an office space asset is directly correlated to the rent that it can generate.
The net effective rental income is used to measure the value typically along with a capitalization
rate. It follows from the asset value calculation that the amount of completions should be related
to the real net effective rent.
The graph below compares the movement of the real rental rate over time and the net
completions delivered to the market. The peak in net completions occurs shortly after the peak in
the rental rate, which would be due to the long-term construction in delivering new office space
into the market. The decision to develop space occurs long before the space is actually delivered
26
to the market suggesting that the completions delivered to the market are dependent on a lagged
rental rate.
The supply equation is specified as a change in supply because the real rental rate does not show
an increasing trend. Supply has an increasing trend throughout the dataset even though the real
rental rate is not increasing. As such it will not be possible to have a correctly signed equation,
which according to economic theory would be increase in supply is due to an increase in real
rent, using supply and real rent.
Reflecting that the decision to deliver office stock based on the prevailing market rental rate
occurs before the completions are delivered to the market, the relationship for the long-run office
supply will be specified generally as:
Net Completionst = a + b* Real Rentt-n
Econometric Model From the stationarity tests results, we know that both net completions and vacancy are stationary
series. As both series are stationary an ordinary least squares regression analysis can be specified
for the relationship between the two variables. As such, the econometric model for the net
completions at time t will be specified as (where n≥0):
Net Completionst = a + b* Real Rentt-n
27
Models that were investigated but subsequently discarded are summarized in the Appendix. The
year-over-year net completions equation as specified using yearly lags using the Toronto data
yields8:
Net CompletionsYoY = -1838.373 + 231.9822*Real Rentt-4
(-5.95)
(16.41)
2
Adjusted R = 0.6718;
N = 132;
F(1,130) = 269.13
The coefficient on the lagged rental value is positive, as expected. Rent increases act as a signal
to developers to increase development resulting in more net completions in one-year time. With
the last observed rent, $15.56, in the dataset (the fourth quarter of 2013) we can anticipate that
there will be approximately 1.85 million square feet of office space added to the market by the
end of 2014. As well the equation suggests that real rent would have to fall to $7.92 in real
dollars in order for development to completely cease.
The above graph shows the predicted values for the yearly changes in supply and the actual
yearly changes in supply. The predicted values follow a much smoother course for reasons
including a smoothing effect. Actual supply tends to be delivered in large amounts when large
buildings are completed and the predicted values do not reflect the times when an atypical
8 Full statistical output for regression is in the Appendix
28
amount of supply is delivered due to a large development’s completion. The predicted values
also unrealistically have negative values. This is a short-coming of the predicted values as in
reality, when real rent decreases space is not demolished to return the system to equilibrium,
rather developments not yet completed are completed and new developments are not started until
market conditions improve.
Demand Equations Previous research indicates that the primary office demand driver is employment in select
economic sectors, DiPasquale and Wheaton (1995). These economic sectors are the main users
of office space and growth in their workforces translates into a need for more office space.
Studies of the economic sectors in the United States submit that greater than 75% of the space in
larger buildings is occupied by the FIRE or the business service sector of the economy, Wheaton
(1987).
Given that the demand depends on the number of space users it follows that it would also be
related to the stock of space. The availability of space and the total supply of space also impact
the demand of space. If every office worker in Toronto were to use the exact same amount of
space a comparison between the change in occupied stock and employment change should
follow the same path, increases in office workers should lead to a proportionate increase in space
to the number of workers added. The graph below shows the relationship between the amount of
occupied stock and the amount of business services employment throughout the data series.
When the absorption and change in office workers diverge the amount of office space per worker
is changing.
29
DiPasquale and Wheaton (1995) attribute the difference in changes in office space and changes
in employment to two explanations. The first is that as the makeup of the workforce changes so
too does the space the changing workforce will occupy. Their example to illustrate this theory is
that clerical services will use less office space than management services will, and if technology
is eliminating clerical jobs then it would be expected that the space per worker would rise over
time. The second explanation for the differing changes in office spaces and changes in
employment are predicated on the fact that office space is a factor of production. Similar to other
factors of production, the amount of space used per worker should vary with the cost of
providing that space. As such, when office space costs rise it is expected that employers will
reduce the amount of space per employee to reduce their costs, and conversely, when office
space costs fall employers will provide more office space per employee.
At this point in the structural system of equations demand can be specified in one of two ways,
directly or indirectly. When estimated directly occupied stock is estimated and then vacancy is
created as an identity based off of this estimation. When estimated indirectly vacancy is
estimated directly and the occupied stock estimation is not necessary. Due to the lack of supply
information in many European countries estimating demand indirectly is preferred. In the US the
preferred estimation of demand is directly with the creation of vacancy through an identity.
The direct demand equation will reflect the impacts that changes in employment, rent and the
stock of space have on the amount of space occupied in the marketplace. The dependent variable
30
will be occupied stock and once estimated, vacancy will be derived using the estimation. This
equation can be specified in both levels and differences. When estimated in levels, the actual
occupied stock is estimated; when estimated in differences, the net absorption is estimated and
occupied stock can then be calculated. The following equations show the general format for the
direct demand equation in levels and then in differences:
Direct Demand Equation – Levels
Occupied Stockt = a + b*Employmentt + c*Real Net Effective Rentt-n
Direct Demand Equation – Differences (Error Correction Model)
Net Absoprtiont = a + b*Employmentt + c*Real Net Effective Rentt-n + d*Occupied
Stockt-n + e*Change in Employmentt + f*Change in Real Rentt
The indirect demand equation will reflect the impact that the above discussion of the impact of
office space drivers has on the demand for office space. Vacancy will be used as the dependent
variable in the equation as when vacancy rises it suggests that demand has lessened for the
product. Reflecting that rental rates are solely dependent upon vacancy rates in the structural
framework, lagged vacancy will be included to capture the change in demand for office space as
a factor of production due to changes in cost. Changes in office space and employment are also
included to reflect the change in demand due to increases in the stock of space and changes in the
number of users of space. Similarly to the direct demand equations, the indirect demand
equations will be estimated both in levels (dependent variable is vacancy) and differences
(dependent variable is change in vacancy).
The general form of the equations will be:
Indirect Demand Equation – Levels
Vacancyt = a + b*Real Net Effective Rentt-n + c*Employmentt + d*Office Stockt
Indirect Demand Equation – Differences (Error Correction Model)
Change in Vacancyt = a + b*Real Net Effective Rentt-n + c*Employmentt + d*Office
Stockt + e*Vacancyt-n + f*Change in Employmentt + g*Change in Real Rentt + h*Change
in Supplyt
Econometric Model – Direct Demand in Levels The cointegration test results showed that occupied stock series is not cointegrated with business
services employment and the real net effective rent. Using the ordinary least squares analysis the
following model has been estimated for the long-run direct demand equation (where n≥0):
Occupied Stockt = a + b*Business Services Employmentt + c*Real Net Effective Rentt-n
Models that were explored but were not used for the direct demand equation in levels are
summarized in the Appendix. The occupied stock equation as specified using yearly lags using
the Toronto data yields9:
9 Full statistical output for regression is in the Appendix
31
Occupied Stockt = 49897.66 + 206.7585*Business Services Employmentt –
(19.06)
(37.82)
394.3987*Real Net Effective Rentt-4
(-5.92)
Adjusted R2 = 0.9434;
N = 132;
F(2,129) = 1092.28
The coefficient on the lagged rental value is negative, as expected. Rent increases in the
marketplace cause space users to decrease the amount of space they wish to occupy in order to
reduce costs. The coefficient on current business services employment is positive, as expected.
An increase in the number of office users causes an increase in demand, as additional space is
needed for these additional workers.
As shown in the above graph the occupied stock as predicted by the direct demand equation
treating both business services and the lagged rent as exogenous variables shows more
fluctuations than the actual occupied stock series. This is partly due to the immediate ability of
the predicted series to react to changes in employment and changes in the real rent. The predicted
series starts higher than the actual series as it does not have any anchor of the amount of stock in
the marketplace. The first prediction point is 71M square feet of occupied space while the market
had only 61M square feet available. Using the direct demand equation a portion of the building
boom in the 1980s can be attributed to the market not having enough supply to support the
market demand.
32
Econometric Model – Direct Demand as an Error Correction Model The cointegration test results showed that the net absorption series is cointegrated with business
services employment and the real rental rate. As such an error correction model can be specified
for the relationship. Using a one-step error correction model the following model has been
estimated for the direct demand equation (where n≥0):
Net Absorptiont = a + b*Business Services Employmentt + c*Real Net Effective Rentt-n +
d*Occupied Stockt-n + e*Change in Employmentt + f*Change in Real Rentt
Models that were explored but were not used for the direct demand equation in levels are
summarized in the Appendix. The net absorption equation as specified using yearly lags using
the Toronto data yields10:
Net AbsorptionYoY = 2752.292 – 11.72436*Business Services Employmentt +
(1.33)
(-1.40)
39.03843*Real Net Effective Rentt-4 + 0.0208025*Occupied Stockt-4
(1.29)
(0.54)
+ 43.26936*Change in EmploymentYoY + 129.8337*Change in Real
(3.40)
(2.07)
RentYoY
Adjusted R2 = 0.2189;
N = 132;
F(5,126) = 8.34
The coefficients in the equation have the expected sign. As this is an error correction model the
change in real rent and change in employment variables represent the short-term effects in the
market while the variables in levels and lagged levels represent the long-term effects in the
market. As the error correction terms are the only variables that are significant in the model, they
suggest that net absorption is affected only by immediate changes in the market, the long-term
effect variables are not having a significant effect on net absorption.
10 Full statistical output for regression is in the Appendix
33
The above graph is the predicted values from the equation with all variables being considered as
exogenous variables with the exception of lagged occupied stock, which is endogenous to the
equation. As the significant variables in the equation are the short-term effects, it is expected that
the predicted values will show changes form period to period. The equation doesn’t capture the
periods of negative absorption that occur in the actual values. As well it does not reflect the
larger values of absorption that can occur when a tenant of a large space footprint either vacates
or moves into a space. While there is period-to-period variability in the predictions the equation
reduces the range of variability that is shown in the actual values.
Econometric Model – Indirect Demand in Levels Using the ordinary least squares analysis the following model has been estimated for the longrun indirect demand equation (where n≥0):
Vacancyt = a + b*Business Services Employmentt + c*Real Net Effective Rentt-n +
d*Stockt
Models that were explored but were not used for the direct demand equation in levels are
summarized in the Appendix. The vacancy equation as specified using yearly lags using the
Toronto data yields11:
11 Full statistical output for regression is in the Appendix
34
Vacancyt = -0.0070384 – 0.0008529*Business Services Employmentt + 0.0006019*Real
(-0.47)
(-22.99)
(2.43)
Net Effective Rentt-4 + 0.00000316*Stockt
(19.26)
Adjusted R2 = 0.8019;
N = 132;
F(3,128) = 177.77
The indirect demand equation in levels shows quite good predictive power with an adjusted R2 of
0.8019. The coefficients on the variables are of the expected sign. An increase in employment
decreases the vacancy rate as there are more workers requiring space. Increasing the rent will
increase the vacancy rate as more expensive rents cause users to want to occupy less space to
keep costs down. As well an increase in space should also increase the vacancy rate as there is
more space total space in the market.
The above graph is the predicted values of the indirect demand equation in levels with all
variables considered exogenous. The predicted values follow the general pattern of the actual
vacancy rate quite closely, which is expected giving the high predictive power of the equation.
The predicted vacancy isn’t as smooth as the actual vacancy rate as the equation has a quicker
ability to capture changes in the market while the actual vacancy rate exhibits a smoother series.
The equation captures the peaks and troughs that occurred in the market.
35
Econometric Model – Indirect Demand as an Error Correction Model The cointegration test results showed that the first difference in vacancy series is cointegrated
with business services employment, the real rental rate and office stock. As such an error
correction model can be specified for the relationship. Using a one-step error correction model
the following model has been estimated for the indirect demand equation (where n≥0):
Change in Vacancyt = a + b*Business Services Employmentt + c*Real Net Effective
Rentt-n + d*Stockt + e*Vacancyt-n + f*Change in Employmentt + g*Change in Real Rentt
+ h*Change in Stockt
Models that were explored but were not used for the direct demand equation in levels are
summarized in the Appendix. The yearly change in vacancy equation as specified using yearly
lags using the Toronto data yields12:
Change in VacancyYoY = 0.0185135 - 0.0002551*Business Services Employmentt +
(1.15)
(-3.36)
0.000415*Real Net Effective Rentt-4 + 0.000000857*Stockt (1.16)
(3.02)
0.4223231*Vacancyt-4 – 0.0000845*Change in EmploymentYoY
(-5.56)
(-0.89)
– 0.0027141*Change in Real RentYoY + 0.000000446*Change in
(-5.39)
(0.39)
StockYoY
Adjusted R2 = 0.5574;
N = 132;
F(7,124) = 24.56
The coefficients of the variables all have the expected sign. As the yearly change in vacancy is
specified as an error correction model the significant coefficients can be separated into long-term
and short-term effects. Current business services employment, current total office stock, and
lagged vacancy have a long-term effect on the change in vacancy. The only error correction term
that is significant is the yearly change in real rent, and as such it is the only short-term effect that
is significant in the model.
12 Full statistical output for regression is in the Appendix
36
The above graph shows the predicted vacancy year-over-year changes using all of the variables
in the equations as exogenous variables, except the lagged vacancy term, which is considered
endogenous. The graph shows the predicted values following the same general pattern as the
actual yearly vacancy changes. The predicted values are both smoother and less varied in total
range than the actual observed yearly vacancy changes.
5. Non-­‐Structural Econometric Model The non-structural econometric model allows the dependent variables from the structural model
to depend not only upon the theoretical predefined economic relationships, but also on each of
the other variables. The strength of the non-structural model is that it does not require any of the
predefined relationships to exist and instead allows the data to determine the interactions
between the variables.
The non-structural model will be specified as a vector autoregressive (VAR) model. The three
dependent variables rent, net completions and vacancy are specified as linear functions of a set of
their own lags and the lags of the other two variables. As well the economic indicator variable,
change in business services employment, will be included in each of the models. The VAR is
capable of explaining the interactions occurring between the variables without imposing
relationships from prior economic theory.
37
Econometric Model As with the structural model, the VAR will use variables that are stationary. As such business
services employment and stock will be specified in differences. The real rental rate and the
vacancy rate were stationary series so will be specified in levels.
The general formats of the equations produced in the non-structural model are as follows:
Rentt = a + b*Rentt-n + c*Vacancyt-n + d*Net Completionst-n + e*Business Service
Employment Growtht
Net Completionst = a + b*Rentt-n + c*Vacancyt-n + d*Net Completionst-n + e*Business
Service Employment Growtht
Vacancyt = a + b*Rentt-n + c*Vacancyt-n + d*Net Completionst-n + e*Business Service
Employment Growtht
In order to make an apple-to-apple comparison between the structural systems of equations and
the VAR, the VAR is specified using yearly lags and changes in the variables to be consistent
with the structural specification. The following set of three equations using yearly lags have been
determined for the VAR model using the Toronto office market data:
Rentt = 5.997348 + 0.9336669*Rentt-4 - 38.02874*Vacancyt-4 - 0.0002504*Net
(3.97)
(15.30)
(-5.59)
(-1.25)
CompletionsYoY + 0.0546646*Business Service Employment GrowthYoY
(2.97)
Net Completionst = -3722.724 + 223.9518*Rentt-4 + 10484.57*Vacancyt-4 +
(-5.65)
(9.92)
(3.15)
0.1579543*Net CompletionsYoY-4 + 21.37944*Business Service
(2.05)
(2.78)
Employment GrowthYoY
Vacancyt = 0.0093584 + 0.0001345*Rentt-4 + 0.8286142*Vacancyt-4 + 4.50E-06*Net
(1.16)
(0.48)
(20.27)
(4.76)
CompletionsYoY-4 - 0.0002504*Business Service Employment GrowthYoY
(-2.65)
As the purpose of the VAR is to estimate the system as a whole the individual coefficients in
each equation and their signs will not be discussed. However it is of note that the rental equation
has significant coefficients outside of the vacancy rate and the lagged rental rate, which were the
only two allowed variables in the structural econometric system of equations. As well net
completions also shows significant variables over and above the predetermined relationship with
the rental rate.
38
39
The above three graphs show the predicted values of the three equations making up the VAR
model. The graphs are generated using only the change in employment as an exogenous variable,
the remaining variables are endogenous to the system.
The rental rate equation shows a significant amount of small fluctuations throughout the year. It
follows the general pattern of the smoother actual real rent series until the early 2000s. At this
point the actual real rent series starts to level out but the predicted values continue to increase
until the end of the data period.
The net completions equation is generally above the actual net completions equation, predicting
more completions than were actually observed. The predicted series follows the patterns of the
actual data fairly closely and shows very similar peaks and troughs.
The predicted vacancy values create a much smoother series than the actual observed values. As
well the predicted vacancy values show a much smaller range of values than the actual observed
values. The prediction completely misses both the rise and the fall in vacancy in the early 2000s
and produces a smooth period at a low vacancy. As well the predicted vacancy values are lower
than the actual observed values for the majority of the data series and reach lows that are not
observed anywhere in the actual vacancy series.
40
6. Comparison of the Structural and Non-­‐Structural Framework To compare the five systems of equations and determine which one provides the best fit for the
Toronto market, back testing will be used. In order to make comparisons on the same variables,
the real rental rate, year-over-year net completions and vacancy will be used. In the systems
where vacancy is not directly estimated it will be calculated using the appropriate identities. The
three variables that are commonly estimated through the five systems will be used to predict
movements in the market from the first quarter in 2008 to the fourth quarter in 2013. This period
of time includes both the global recession of 2008-2009 and the subsequent recovery. As such it
should provide an accurate indication of the predicting power of the five systems.
For the back testing all variables are considered endogenous to the system of equations with the
exception of business services employment, the economic indicator, and the only true exogenous
variable.
Back-­‐Testing of Models The above graph of the observed real net effective rent and the predicted real rental values have
all five variations show rent increasing after 2008. None of the models capture the leveling out
that actually occurred in the market. The following graph shows only the period between the first
quarter of 2008 and the fourth quarter of 2013 to show the five predicted real rent series in more
detail.
41
The predictions of demand in levels (estimated both directly and indirectly) and the VAR all
show a slightly increasing trend through the date range. Both of the models specified as error
correction models show an increase followed by a decrease. The direct error correction model
shows the larger decrease out of the two for the date range. The lack in variation between the
systems, particularly the structural systems, is due to the large amount of momentum in the
lagged rental values. As such, the vacancy rates generated by the structural systems have a lesser
impact than they do in the VAR system causing the structural systems to have very similar rental
predictions.
42
The completions year-over-year predictions show an increasing pattern throughout the back test
period with the exception of the year-over-year completion as predicted by the error correction
model capturing direct demand. The indirect demand specified as an error correction model
system provides estimates that are very smooth in contrast to the uneven actual year-over-year
completions series.
43
The above graph shows the same completions predictions as the previous; it is showing a
restricted date rage to clearly show the details of the five predicted series. The structural system
of equations using the direct demand specified as an error correction model provides year-overyear completions estimates that are quite similar to the actual completions series. The short-term
effects captured in the demand by this model allow the estimates to come quite close to the
actual series. The structural system using indirect demand specified as an error correction model
provides a very smooth increasing series that produces considerably more stock throughout the
back testing data than the other series and considerably more than the actual completions
observed in the market. The remaining three models, the VAR and the two structural systems
specified with demand in levels, show very similar predictions for completions, predicting a
fairly flat slightly increasing amount of stock each year.
44
All of the equations show a very uneven series as compared to the much smoother series of the
actual vacancy rate. This is partly due to the equations having a faster reaction to market
conditions than the slower reaction of the market due to the long-term nature of office leases.
45
The vacancy graph with only the back test date range shows considerably more variation than the
real rental rate graph between the different systems of equations. This is due to the rental
equation for the structural models having considerable momentum in the lagged rental
coefficient. As these rental predictions are so close to one another, the variation in the vacancy
predictions shows the difference in predictions due to the differences between the systems. All of
the systems show troughs in their predicted values where the actual observed series shows a
peak.
While the systems of equations that have demand predicted in levels show high predictive
capability in their adjusted R2s, they do not perform nearly as well in the back testing. This
suggests that the short-term effects introduced through the error correction models have value in
the prediction power of the structural system of equations.
7. Conclusion In general, the year-over-year changes and the yearly lags provide better results than their
quarterly counterparts across all the systems of equations. This leads to the deduction that there
is either something in the way the data is surveyed or in the operation of the market that makes
yearly data a better representation than quarterly.
46
The rental relationship with vacancy shows correlation but not cointegration. In the structural
systems of equations the real rental rate shows significant momentum, showing that rents are less
sensitive to vacancy than they are to their previous levels. Based on the actual observed real
rental values, the Toronto office market acts more as a city with weak constraints on supply than
the land-constrained center it is commonly thought of.
The long-run supply results challenge the notion that there is a long-run supply relationship in
levels. As the real net effective rental rate does not have a long run rising curve, but the total
supply has a long run increasing supply curve the relationship between the two variables would
not provide a correctly signed long run relationship. The market operates differently than the
housing market; it operates more like an industrial model. When the value of developing is
greater than the cost to develop more space is developed in the market.
Within the structural model the demand equations estimated in levels have superior adjusted R2
than the error correction models suggesting that they provide better predictions for the market
variability. However, the poor results of the demand equations in levels in back testing suggest
that there is value added by the error terms in the demand error correction models.
All five systems of equations have poor results in the real rent back test, the predictions all show
increases in the rental rates from 2008-2013 while the actual values remained level for the entire
period. In contrast to the real rent predictions, the predictions for completions and vacancy series
of the five systems share similarities with the actual observed values in the back test. While there
was not a change in the vacancy rate nor anything in the employment or stock series that would
support level rental rates, in the 2008-2013 period the real rent flattened at around $15 in real
dollars. The 2008-2013 recession had little impact on the supply and demand fundamentals but
the real rental rate was steady and showed very little variation throughout the period.
Using back testing to compare the five systems of equations challenges the notion in economic
theory that theoretical relationships between variables are stringently predefined. Complicated
dynamic systems that place restrictions on how variables are able to influence one another may
actually produce poor forecasts when applied in the model. The vector autoregressive model
produces the best results when considering the three equations as a whole system. Removing the
requirements on variable interaction allows the VAR to produce a system of equations for the
Toronto office market that is able to predict the real rental rate with a lessened impact of
momentum of previous real rental rates. While some of the individual equations may produce a
better result for one of the individual equations, the VAR is superior for when the entire system
is taken into consideration.
47
8. Bibliography Blank, D.M., and L. Winnick, (1953). “The Structure of the Housing Market”, Quarterly Journal of Economics, 67,
181-203.
Brounen, D., and M. Jennen, (2009a). “Asymmetric Properties of Office Rent Adjustment”, Journal of Real Estate
Finance and Economics, 39(3), 336–358.
Brounen, D., and M. Jennen, (2009b). “Local Office Rent Dynamics”, Journal of Real Estate Finance and
Economics, 39(4), 385–402.
Clements, M., and G. Mizon, (1991). “Empirical Analysis of Macroeconomic Time Series: VAR and Structural
Models”, European Economic Review, 35(4), 887-917.
D’Arcy, E., T. McGough, and S. Tsolacos, (1997). “National Economic Trends, Market Size and City Growth
Effects on European Office Rents”, Journal of Property Research, 14(4), 297-308.
D’Arcy, E., T. McGough, and S. Tsolacos, (1999). “An Econometric Analysis and Forecasts of the Office Rental
Cycle in the Dublin Area”, Journal of Property Research, 16(4), 309–321.
De Francesco, A. J., (2008). “Time-Series Characteristics and Long-Run Equilibrium for Major Australian Office
Markets”, Real Estate Economics, 36(2), 371–402.
De Wit, I., and R. van Dijk, (2003). “The Global Determinants of Direct Office Real Estate Returns”, Journal of
Real Estate Finance and Economics, 26(1), 27–45.
Dickey, D. and W. Fuller, (1979). “Distribution of the Estimators for Autoregressive Time Series With a Unit Root”,
Journal of the American Statistical Association, 74(366), 427-431.
DiPasquale, D. and W. Wheaton, (1996). “Urban Economics and Real Estate Markets”, Englewood Cliffs, NJ:
Prentice Hall, 23(7).
Dobson, S., and J. Goddard, (1992). “The Determinants of Commercial Property Prices and Rents”, Bulletin of
Economic Research, 44, 301-321.
Engle, R. and Granger, C., (1987). ”Co-integration and Error Correction: Representation, Estimation and Testing”,
Econometrica, 55(2), 252-276.
Eubank, A.A. and C.F. Sirmans, (1979). “The Price Adjustment Mechanism for Rental Housing in the United
States”, Quarterly Journal of Economics, 93, 163-183.
Gabriel, S. A., and F. E. Nothaft, (1988). “Rental Housing Markets and the Natural Vacancy Rate,” Journal of the
American Real Estate and Urban Economics Association, 16, 419-436.
Hendershott, P. H., B. D. MacGregor, and M. J. White, (2002b). “Explaining Real Commercial Rents Using an
Error Correction Model with Panel Data”, Journal of Real Estate Finance and Economics, 24(1–2), 59– 87.
Hendershott, P. H., B. D. MacGregor, and R. Y. C. Tse, (2002a). “Estimation of the Rental Adjustment Process”,
Real Estate Economics, 30(2), 165–183.
Ibanez, M., (2013). “Commercial Property Rent Dynamics in U.S. Metropolitan Areas: An Examination of Office,
Industrial, Flex and Retail Space”, Journal of Real Estate Finance & Economics, 46(2), 232-259.
MacKinnon, J., (1996). “Numerical Distribution Functions for Unit Root and Cointegration Tests”, Journal of
Applied Econometrics, 11(6), 212-243.
48
Rosen, K.T. and L.B. Smith, (1983). “The Price Adjustment Process for Rental Housing and the Natural Vacancy
Rate”, American Economic Review, 73, 779-786.
Rosen, K. T., (1984). “Toward a Model of the Office Building Sector”, ARUENA Journal, 12(3), 261-269.
Said, S. and D. Dickey, (1984). “Testing for Unit Roots in Autoregressive-Moving Average Models of Unknown
Order”, Biometrika, 71(3), 599-607.
Shilling, J. D., C. F. Sirmans, and J. B. Corgel, (1987). “Price Adjustment Process for Rental Office Space”, Journal
of Urban Economics, 22, 90-100.
Silver, M., and M. Goode, (1990). “Econometric Forecasting Model for Rents in the British Retail Property
Market”, International Journal of Management Science, 18(5), 529-539.
Sims, C., (1980). “Macroeconomics and Reality”, Econometrica, 48(1), 1-48.
Sims, C., (1989). “Models and Their Uses”, American Journal of Agricultural Economics, 71(2), 489-494.
Smith, L. B., (1969). “A Model of the Canadian Housing and Mortgage Markets”, The Journal of Political
Economy, 77, 795-816.
Smith, L. B., (1974). “A Note on the Price Adjustment Mechanism for Rental Housing”, American Economic
Review, 64, 478-481.
Thompson, B., and S. Tsolacos, (1999). “Rental Adjustment and Forecasts in the Industrial Market”, Journal of Real
Estate Research, 17(1/2), 151-168.
Wheaton, W., (1987) “The Cyclical Behavior of the National Office Market”, Journal of the American Real Estate
and Urban Economics Association, 15(4), 281-299.
Wheaton, W. C., and R. G. Torto, (1988). “Vacancy Rates and the Future of Office Rents”, Journal of the American
Real Estate and Urban Economics Association, 16(4), 430-436.
Wheaton, W., (1990). “Vacancy, Search and Prices in a Housing Market Matching Model,” Journal of Political
Economy, 98, 1270-1293.
Wheaton, W.C., and R. G. Torto (1993). “Office Rent Indices and Their Behavior Over Time,” Journal of Urban
Economics, 35, 112-139.
Wheaton, W. C., R. G. Torto, and P. Evans, (1997). “The Cyclic Behavior of the Greater London Office Market”,
The Journal of Real Estate Finance and Economics, 15(1), 77-92.
49
9. Appendices Data as Provided by CBRE Econometric Advisors 1980.1
FIRE
Emp.
(Jobs
x
1000)
169.00
Business
Svc.
Emp.
(Jobs x
1000)
135.20
56,370
1980.2
171.80
136.80
57,583
1980.3
174.70
138.40
1980.4
177.50
1981.1
178.30
1981.2
1981.3
Office
Stock
(SF x
1000)
Net
Effective
Rent
($)
Net
Effective
Rent
(2013 $)
Net
Absorption
(SF x 1000)
Vacancy
Rate
(%)
1,097
na
8.60
7.16
20.99
1,213
1,454
8.00
7.75
22.11
58,783
1,200
1,457
7.40
8.34
23.13
140.00
59,970
1,187
1,459
6.80
8.93
24.10
142.90
61,234
1,264
1,239
6.70
9.73
25.40
179.10
145.70
62,498
1,264
1,180
6.70
10.53
26.70
179.90
148.60
63,672
1,174
1,159
6.60
11.33
27.89
1981.4
180.70
151.40
64,756
1,084
1,012
6.60
12.12
29.12
1982.1
181.40
151.10
65,414
658
157
7.30
12.58
29.48
1982.2
182.10
150.70
66,007
593
87
8.00
13.03
29.62
1982.3
182.80
150.40
66,588
581
69
8.70
13.48
30.00
1982.4
183.60
150.00
67,156
568
115
9.30
13.93
30.52
1983.1
181.50
152.90
68,253
1,097
381
10.20
13.76
29.94
1983.2
179.50
155.80
69,311
1,058
396
11.00
13.58
29.16
1983.3
177.50
158.60
70,395
1,084
331
11.90
13.40
28.31
1983.4
175.50
161.40
71,504
1,109
405
12.70
13.23
27.70
1984.1
179.70
161.40
72,381
877
476
13.10
13.22
27.35
1984.2
184.00
161.30
73,271
890
554
13.40
13.22
27.13
1984.3
188.20
161.30
74,110
839
504
13.70
13.22
26.91
1984.4
192.50
161.20
74,910
800
466
14.00
13.22
26.70
1985.1
189.80
162.60
75,323
413
581
13.70
13.70
27.36
1985.2
187.20
164.00
75,646
323
581
13.30
14.18
27.98
1985.3
184.50
165.30
75,994
348
606
12.90
14.65
28.64
1985.4
181.70
166.70
77,890
1,896
1,963
12.50
15.13
29.35
1986.1
185.60
168.20
79,425
1,535
1,422
12.40
15.61
29.89
1986.2
189.40
169.80
81,076
1,651
1,447
12.40
16.10
30.57
1986.3
193.30
171.30
82,766
1,690
1,480
12.40
16.59
31.14
1986.4
197.10
172.80
84,688
1,922
1,768
12.30
17.07
31.72
1987.1
201.30
179.70
86,210
1,522
1,766
11.80
17.63
32.45
1987.2
199.60
192.30
86,881
671
1,026
11.30
18.18
32.99
1987.3
201.70
198.10
87,720
839
1,183
10.80
18.74
33.64
1987.4
203.00
189.60
88,571
851
1,291
10.20
19.29
34.40
1988.1
199.00
196.40
90,248
1,677
1,145
10.60
20.16
35.64
1988.2
210.20
197.90
93,486
3,238
2,521
11.00
21.03
36.71
1988.3
203.00
201.70
95,279
1,793
1,309
11.30
21.90
37.82
1988.4
198.70
212.60
97,214
1,935
1,328
11.70
22.77
39.01
1989.1
199.80
217.50
98,981
1,767
1,065
12.20
22.51
38.09
1989.2
208.10
218.10
101,122
2,141
1,273
12.80
22.24
36.99
1989.3
217.80
221.80
102,902
1,780
935
13.40
21.97
36.03
Year
Completions
(SF x 1000)
50
1989.4
220.10
234.10
105,095
2,193
1,269
14.00
21.71
35.35
1990.1
226.40
231.10
106,411
1,316
174
14.90
20.88
33.51
1990.2
223.20
233.10
107,662
1,251
95
15.80
20.05
31.88
1990.3
219.00
227.30
109,907
2,245
792
16.80
19.22
30.26
1990.4
225.20
218.70
112,552
2,645
1,187
17.70
18.39
28.53
1991.1
222.30
224.70
113,713
1,161
46
18.50
17.51
26.41
1991.2
218.50
230.40
114,848
1,135
6
19.30
16.62
24.88
1991.3
220.30
229.50
116,215
1,367
174
20.10
15.73
23.41
1991.4
219.50
218.80
120,730
4,515
2,762
20.80
14.85
22.13
1992.1
208.70
217.20
121,904
1,174
686
21.00
12.34
18.30
1992.2
206.50
216.50
122,988
1,084
611
21.20
9.83
14.47
1992.3
197.40
218.70
124,188
1,200
697
21.40
7.32
10.74
1992.4
205.10
229.20
124,949
761
348
21.60
4.82
7.04
1993.1
214.60
224.40
125,259
310
-7
21.80
4.38
6.36
1993.2
210.40
210.00
125,491
232
-196
22.10
3.95
5.71
1993.3
200.90
223.30
125,685
194
-225
22.40
3.51
5.06
1993.4
180.00
232.80
125,827
142
-268
22.70
3.08
4.42
1994.1
178.50
216.10
126,033
206
1,168
21.90
3.25
4.69
1994.2
183.70
237.70
126,162
129
984
21.20
3.42
4.95
1994.3
190.90
242.90
126,291
129
985
20.50
3.59
5.17
1994.4
196.00
242.60
126,420
129
609
20.10
3.76
5.39
1995.1
199.10
244.10
126,433
13
516
19.70
3.88
5.51
1995.2
204.80
242.50
126,433
0
1,011
18.90
4.00
5.64
1995.3
215.90
253.60
127,194
761
1,380
18.30
4.12
5.80
1995.4
198.80
268.60
127,207
13
902
17.60
4.24
5.96
1996.1
193.20
272.00
127,207
0
-128
17.70
4.89
6.85
1996.2
191.10
271.20
127,284
77
700
17.20
5.54
7.70
1996.3
200.10
265.00
127,284
0
1,018
16.40
6.19
8.59
1996.4
214.30
266.60
127,284
0
510
16.00
6.84
9.42
1997.1
217.60
286.60
127,284
0
381
15.70
7.14
9.79
1997.2
214.00
299.10
127,284
0
1,782
14.30
7.44
10.18
1997.3
204.20
303.20
127,568
284
2,540
12.50
7.74
10.56
1997.4
200.10
305.30
127,568
0
-510
12.90
8.04
10.96
1998.1
205.10
308.50
127,568
0
1,275
11.90
8.32
11.28
1998.2
210.00
309.80
127,710
142
381
11.70
8.45
11.46
1998.3
208.90
311.60
127,710
0
638
11.20
10.20
13.81
1998.4
211.90
322.20
127,710
0
128
11.10
9.85
13.28
1999.1
216.20
342.10
127,942
232
-49
11.30
10.60
14.25
1999.2
213.90
344.60
128,045
103
219
11.20
11.37
15.17
1999.3
216.30
349.60
128,342
297
264
11.20
12.49
16.56
1999.4
222.80
346.90
129,000
658
842
11.00
12.78
16.83
2000.1
212.50
343.70
129,355
355
3,550
8.50
13.22
17.30
2000.2
208.10
364.50
129,629
274
1,806
7.30
13.50
17.59
2000.3
212.70
381.60
130,673
1,044
445
7.70
14.29
18.45
2000.4
223.10
381.30
131,466
793
864
7.60
15.75
20.12
2001.1
215.00
371.90
132,267
801
-318
8.40
18.53
23.61
2001.2
223.40
369.70
133,409
1,142
112
9.10
18.29
23.00
2001.3
230.30
366.00
133,509
100
-1,645
10.40
16.54
20.80
51
2001.4
229.30
368.90
134,824
1,315
-170
11.40
17.00
21.48
2002.1
229.30
378.40
135,681
857
-597
12.40
17.01
21.34
2002.2
233.10
378.20
135,930
249
-734
13.10
17.43
21.64
2002.3
233.30
367.40
136,971
1,041
768
13.20
16.96
20.82
2002.4
232.20
395.80
137,578
607
-161
13.70
16.78
20.43
2003.1
234.90
391.40
137,653
75
-1,037
14.50
16.49
19.82
2003.2
232.50
389.70
137,691
38
-931
15.20
16.25
19.63
2003.3
231.10
383.40
137,711
20
-672
15.70
16.35
19.65
2003.4
244.30
375.20
137,749
38
32
15.70
16.10
19.27
2004.1
252.40
376.20
138,595
846
1,268
15.30
15.92
18.96
2004.2
255.80
386.00
138,728
133
251
15.20
16.09
19.02
2004.3
257.50
389.10
138,728
0
971
14.50
15.60
18.39
2004.4
253.50
382.40
138,978
250
353
14.40
14.91
17.44
2005.1
259.30
370.80
138,978
0
834
13.80
14.41
16.80
2005.2
263.60
361.70
139,046
68
1,866
12.50
14.46
16.78
2005.3
270.50
386.20
139,251
205
1,015
11.90
14.76
16.96
2005.4
280.50
393.90
139,630
379
2,847
10.10
14.14
16.17
2006.1
275.90
384.30
139,746
116
1,362
9.20
13.53
15.38
2006.2
289.00
382.40
139,746
0
0
9.20
13.90
15.73
2006.3
286.90
362.40
140,078
332
-399
9.70
14.24
16.11
2006.4
281.30
391.90
140,244
166
1,272
8.90
13.27
14.97
2007.1
293.80
400.70
140,617
373
1,324
8.20
13.38
14.93
2007.2
286.40
395.90
141,177
560
514
8.20
13.88
15.38
2007.3
277.30
408.20
141,783
606
1,407
7.60
14.31
15.86
2007.4
278.00
408.50
141,858
75
637
7.20
14.96
16.46
2008.1
277.20
429.50
142,229
371
913
6.80
15.25
16.73
2008.2
282.60
431.70
142,317
88
225
6.70
15.51
16.79
2008.3
292.30
416.10
142,551
234
361
6.60
15.54
16.63
2008.4
285.50
425.30
142,674
123
-171
6.80
15.76
17.03
2009.1
292.50
406.30
142,795
121
-1,172
7.70
16.33
17.70
2009.2
303.40
413.20
142,940
145
-867
8.40
15.66
16.94
2009.3
317.00
429.80
145,506
2,566
1,332
9.10
14.87
16.04
2009.4
321.60
434.40
146,732
1,226
674
9.40
13.38
14.35
2010.1
307.90
447.00
147,263
531
187
9.60
14.05
14.99
2010.2
296.20
459.50
147,792
529
478
9.60
14.75
15.74
2010.3
283.50
467.10
147,792
0
296
9.40
14.63
15.50
2010.4
303.40
463.50
147,792
0
443
9.10
14.50
15.21
2011.1
310.70
469.10
147,932
140
719
8.70
14.94
15.54
2011.2
309.80
465.80
147,932
0
740
8.20
14.94
15.42
2011.3
307.30
443.70
148,058
126
707
7.80
14.94
15.36
2011.4
296.60
467.20
149,029
971
747
7.90
14.97
15.29
2012.1
294.80
472.80
149,029
0
0
7.90
15.04
15.29
2012.2
306.40
467.70
149,402
373
194
8.00
15.49
15.74
2012.3
306.10
471.60
149,604
202
-413
8.40
15.49
15.73
2012.4
307.00
467.10
150,094
490
599
8.30
15.72
15.91
2013.1
316.20
468.30
150,094
0
450
8.00
15.72
15.85
2013.2
313.70
480.00
150,209
115
-495
8.40
15.72
15.85
2013.3
315.30
488.60
150,536
327
-152
8.70
15.72
15.79
52
2013.4
315.30
488.90
151,440
904
-234
9.40
15.56
15.56
Stationarity Tests Statistical Output Business Services Employment Optimum Lag Selection (AIC):
Selection-order criteria
Sample: 1982q1 - 2013q4
Number of obs
=
128
+---------------------------------------------------------------------------+
|lag |
LL
LR
df
p
FPE
AIC
HQIC
SBIC
|
|----+----------------------------------------------------------------------|
| 0 | -777.706
11264.5
12.1673
12.1763
12.1896 |
| 1 | -466.507
622.4
1 0.000 88.4595
7.32042
7.33853
7.36498 |
| 2 | -466.152 .70996
1 0.399
89.356
7.3305
7.35766
7.39734 |
| 3 | -462.791 6.7219
1 0.010 86.1208
7.29361
7.32982
7.38273 |
| 4 | -462.705 .17273
1 0.678 87.3607
7.30788
7.35315
7.41929 |
| 5 | -456.887 11.636
1 0.001 81.0281
7.2326
7.28692
7.36629 |
| 6 | -454.162 5.4502*
1 0.020 78.8764* 7.20565* 7.26902* 7.36162* |
| 7 | -453.745 .83395
1 0.361 79.6025
7.21476
7.28718
7.39301 |
| 8 | -453.456 .57711
1 0.447 80.4979
7.22588
7.30735
7.42641 |
+---------------------------------------------------------------------------+
Endogenous: BusSvcEmp
Exogenous: _cons
Augmented Dickey-Fuller Test (optimum lag):
Augmented Dickey-Fuller test for unit root
Number of obs
=
129
---------- Interpolated Dickey-Fuller --------Test
1% Critical
5% Critical
10% Critical
Statistic
Value
Value
Value
-----------------------------------------------------------------------------Z(t)
0.259
-3.500
-2.888
-2.578
-----------------------------------------------------------------------------MacKinnon approximate p-value for Z(t) = 0.9754
Business Services Employment First Difference Optimum Lag Selection (AIC):
Selection-order criteria
Sample: 1982q2 - 2013q4
Number of obs
=
127
+---------------------------------------------------------------------------+
|lag |
LL
LR
df
p
FPE
AIC
HQIC
SBIC
|
|----+----------------------------------------------------------------------|
| 0 | -463.31
87.7153
7.31197
7.32107
7.33437 |
| 1 | -462.959 .70255
1 0.402 88.6162
7.32219
7.34039
7.36698 |
| 2 | -459.657 6.6041
1 0.010 85.4617
7.28594
7.31323
7.35312 |
| 3 | -459.579 .15497
1 0.694 86.7134
7.30046
7.33686
7.39004 |
| 4 | -453.913 11.333
1 0.001 80.5711
7.22698
7.27247
7.33895 |
| 5 | -451.125 5.5766*
1 0.018
78.336* 7.19881* 7.25341* 7.33318* |
| 6 | -450.691 .86665
1 0.352 79.0415
7.20774
7.27143
7.3645 |
| 7 | -450.392 .59959
1 0.439 79.9223
7.21876
7.29156
7.39793 |
| 8 | -448.859
3.065
1 0.080 79.2605
7.21038
7.29227
7.41194 |
+---------------------------------------------------------------------------+
Endogenous: BusSvcEmpD1
Exogenous: _cons
Augmented Dickey-Fuller Test (optimum lag):
Augmented Dickey-Fuller test for unit root
Number of obs
=
129
---------- Interpolated Dickey-Fuller --------Test
1% Critical
5% Critical
10% Critical
Statistic
Value
Value
Value
-----------------------------------------------------------------------------Z(t)
-4.512
-3.500
-2.888
-2.578
------------------------------------------------------------------------------
53
MacKinnon approximate p-value for Z(t) = 0.0002
Real Net Effective Rent Optimum Lag Selection (AIC):
Selection-order criteria
Sample: 1982q1 - 2013q4
Number of obs
=
128
+---------------------------------------------------------------------------+
|lag |
LL
LR
df
p
FPE
AIC
HQIC
SBIC
|
|----+----------------------------------------------------------------------|
| 0 | -460.683
79.5132
7.2138
7.22285
7.23608 |
| 1 | -189.04 543.29
1 0.000
1.1585
2.985
3.00311
3.02956 |
| 2 | -150.131 77.818
1 0.000 .640697
2.39267
2.41983
2.45952* |
| 3 | -149.264 1.7329
1 0.188 .642043
2.39476
2.43097
2.48388 |
| 4 | -147.146 4.2373
1 0.040 .630931
2.37728
2.42254
2.48869 |
| 5 | -144.943 4.4063*
1 0.036 .619199* 2.35848*
2.4128* 2.49217 |
| 6 | -144.784 .31768
1 0.573 .627416
2.37162
2.43499
2.52759 |
| 7 |
-144.7 .16828
1 0.682 .636494
2.38593
2.45836
2.56419 |
| 8 | -144.699 .00175
1 0.967 .646553
2.40154
2.48302
2.60208 |
+---------------------------------------------------------------------------+
Endogenous: RealNetEffRent
Exogenous: _cons
Augmented Dickey-Fuller Test (optimum lag):
Augmented Dickey-Fuller test for unit root
Number of obs
=
130
---------- Interpolated Dickey-Fuller --------Test
1% Critical
5% Critical
10% Critical
Statistic
Value
Value
Value
-----------------------------------------------------------------------------Z(t)
-2.658
-3.500
-2.888
-2.578
-----------------------------------------------------------------------------MacKinnon approximate p-value for Z(t) = 0.0815
Real Net Effective Rent First Difference Optimum Lag Selection (AIC):
Selection-order criteria
Sample: 1982q2 - 2013q4
Number of obs
=
127
+---------------------------------------------------------------------------+
|lag |
LL
LR
df
p
FPE
AIC
HQIC
SBIC
|
|----+----------------------------------------------------------------------|
| 0 | -188.488
1.15741
2.98407
2.99316
3.00646 |
| 1 |
-150.9 75.175*
1 0.000 .650514
2.40788
2.42608* 2.45267* |
| 2 | -150.338 1.1252
1 0.289 .655014
2.41477
2.44207
2.48196 |
| 3 | -148.871 2.9347
1 0.087 .650219
2.40741
2.44381
2.49699 |
| 4 | -147.596 2.5491
1 0.110 .647426* 2.40309* 2.44858
2.51506 |
| 5 | -147.594 .00488
1 0.944 .657697
2.4188
2.47339
2.55317 |
| 6 | -147.569 .04997
1 0.823 .667901
2.43415
2.49784
2.59092 |
| 7 | -147.346 .44482
1 0.505 .676168
2.4464
2.51919
2.62556 |
| 8 | -146.637
1.419
1 0.234 .679316
2.45097
2.53286
2.65253 |
+---------------------------------------------------------------------------+
Endogenous: RealNetEffRentD1
Exogenous: _cons
Augmented Dickey-Fuller Test (optimum lag):
Augmented Dickey-Fuller test for unit root
Number of obs
=
130
---------- Interpolated Dickey-Fuller --------Test
1% Critical
5% Critical
10% Critical
Statistic
Value
Value
Value
-----------------------------------------------------------------------------Z(t)
-2.845
-3.500
-2.888
-2.578
-----------------------------------------------------------------------------MacKinnon approximate p-value for Z(t) = 0.0521
54
Vacancy Rate Optimum Lag Selection (AIC):
Selection-order criteria
Sample: 1982q1 - 2013q4
Number of obs
=
128
+---------------------------------------------------------------------------+
|lag |
LL
LR
df
p
FPE
AIC
HQIC
SBIC
|
|----+----------------------------------------------------------------------|
| 0 | 221.005
.001882 -3.43758 -3.42852
-3.4153 |
| 1 | 463.827 485.64
1 0.000 .000043 -7.21605 -7.19794 -7.17148 |
| 2 | 500.776 73.899*
1 0.000 .000025 -7.77776
-7.7506 -7.71091* |
| 3 | 502.493 3.4329
1 0.064 .000024 -7.78895 -7.75274 -7.69983 |
| 4 | 504.111
3.237
1 0.072 .000024* -7.79862* -7.75335* -7.68721 |
| 5 |
504.13 .03623
1 0.849 .000024 -7.78327 -7.72896 -7.64959 |
| 6 | 504.188 .11787
1 0.731 .000025 -7.76857
-7.7052
-7.6126 |
| 7 | 504.202 .02623
1 0.871 .000025 -7.75315 -7.68073
-7.5749 |
| 8 | 505.202 2.0014
1 0.157 .000025 -7.75316 -7.67168 -7.55263 |
+---------------------------------------------------------------------------+
Endogenous: Vacancy
Exogenous: _cons
Augmented Dickey-Fuller Test (optimum lag):
Augmented Dickey-Fuller test for unit root
Number of obs
=
131
---------- Interpolated Dickey-Fuller --------Test
1% Critical
5% Critical
10% Critical
Statistic
Value
Value
Value
-----------------------------------------------------------------------------Z(t)
-2.612
-3.500
-2.888
-2.578
-----------------------------------------------------------------------------MacKinnon approximate p-value for Z(t) = 0.0905
Vacancy Rate First Difference Optimum Lag Selection (AIC):
Selection-order criteria
Sample: 1982q2 - 2013q4
Number of obs
=
127
+---------------------------------------------------------------------------+
|lag |
LL
LR
df
p
FPE
AIC
HQIC
SBIC
|
|----+----------------------------------------------------------------------|
| 0 | 459.458
.000043 -7.21981 -7.21072 -7.19742 |
| 1 | 495.108
71.3*
1 0.000 .000025 -7.76548 -7.74728* -7.72069* |
| 2 | 496.226 2.2352
1 0.135 .000025* -7.76733* -7.74004 -7.70015 |
| 3 | 497.035 1.6184
1 0.203 .000025 -7.76433 -7.72793 -7.67475 |
| 4 | 497.293 .51616
1 0.472 .000025 -7.75265 -7.70715 -7.64067 |
| 5 | 497.296 .00607
1 0.938 .000026 -7.73695 -7.68235 -7.60257 |
| 6 | 497.517 .44099
1 0.507 .000026 -7.72467 -7.66098
-7.5679 |
| 7 | 497.886 .73886
1 0.390 .000026 -7.71474 -7.64195 -7.53558 |
| 8 | 498.546 1.3207
1 0.250 .000026 -7.70939
-7.6275 -7.50783 |
+---------------------------------------------------------------------------+
Endogenous: D.Vacancy
Exogenous: _cons
Augmented Dickey-Fuller Test (optimum lag):
Augmented Dickey-Fuller test for unit root
Number of obs
=
132
---------- Interpolated Dickey-Fuller --------Test
1% Critical
5% Critical
10% Critical
Statistic
Value
Value
Value
-----------------------------------------------------------------------------Z(t)
-3.510
-3.499
-2.888
-2.578
-----------------------------------------------------------------------------MacKinnon approximate p-value for Z(t) = 0.0077
Occupied Stock Optimum Lag Selection (AIC):
Selection-order criteria
55
Sample: 1982q1 - 2013q4
Number of obs
=
128
+---------------------------------------------------------------------------+
|lag |
LL
LR
df
p
FPE
AIC
HQIC
SBIC
|
|----+----------------------------------------------------------------------|
| 0 | -1471.26
5.7e+08
23.0041
23.0131
23.0264 |
| 1 | -1035.9 870.73
1 0.000
646476
16.2172
16.2353
16.2617 |
| 2 | -1024.4 22.993*
1 0.000
548690* 16.0532* 16.0803*
16.12* |
| 3 | -1024.2 .40728
1 0.523
555567
16.0656
16.1018
16.1547 |
| 4 | -1024.05 .29042
1 0.590
563048
16.079
16.1242
16.1904 |
| 5 | -1024.03 .03682
1 0.848
571767
16.0943
16.1486
16.228 |
| 6 | -1024.03 .01346
1 0.908
580733
16.1098
16.1732
16.2658 |
| 7 | -1023.49 1.0789
1 0.299
584959
16.117
16.1894
16.2953 |
| 8 | -1023.26 .45673
1 0.499
592095
16.1291
16.2105
16.3296 |
+---------------------------------------------------------------------------+
Endogenous: OccupiedStock
Exogenous: _cons
Augmented Dickey-Fuller Test (optimum lag):
Augmented Dickey-Fuller test for unit root
Number of obs
=
133
---------- Interpolated Dickey-Fuller --------Test
1% Critical
5% Critical
10% Critical
Statistic
Value
Value
Value
-----------------------------------------------------------------------------Z(t)
-1.939
-3.499
-2.888
-2.578
-----------------------------------------------------------------------------MacKinnon approximate p-value for Z(t) = 0.3140
Occupied Stock First Difference (Net Absorption) Optimum Lag Selection (AIC):
Selection-order criteria
Sample: 1982q2 - 2013q4
Number of obs
=
127
+---------------------------------------------------------------------------+
|lag |
LL
LR
df
p
FPE
AIC
HQIC
SBIC
|
|----+----------------------------------------------------------------------|
| 0 | -1031.31
672615
16.2568
16.2659
16.2792 |
| 1 | -1018.17
26.27*
1 0.000
555615* 16.0657* 16.0839* 16.1105* |
| 2 | -1017.81
.7216
1 0.396
561239
16.0758
16.1031
16.143 |
| 3 | -1017.55 .51557
1 0.473
567845
16.0875
16.1239
16.177 |
| 4 | -1017.48 .14311
1 0.705
576220
16.1021
16.1476
16.2141 |
| 5 | -1017.48
.011
1 0.916
585333
16.1177
16.1723
16.2521 |
| 6 | -1017.14 .67348
1 0.412
591503
16.1282
16.1919
16.285 |
| 7 | -1016.73 .81704
1 0.366
597071
16.1375
16.2103
16.3167 |
| 8 | -1016.63 .19274
1 0.661
605671
16.1517
16.2336
16.3533 |
+---------------------------------------------------------------------------+
Endogenous: OccupiedStockD1
Exogenous: _cons
Augmented Dickey-Fuller Test (optimum lag):
Augmented Dickey-Fuller test for unit root
Number of obs
=
133
---------- Interpolated Dickey-Fuller --------Test
1% Critical
5% Critical
10% Critical
Statistic
Value
Value
Value
-----------------------------------------------------------------------------Z(t)
-5.547
-3.499
-2.888
-2.578
-----------------------------------------------------------------------------MacKinnon approximate p-value for Z(t) = 0.0000
Office Stock Optimum Lag Selection (AIC):
Selection-order criteria
Sample: 1982q1 - 2013q4
Number of obs
=
128
+---------------------------------------------------------------------------+
|lag |
LL
LR
df
p
FPE
AIC
HQIC
SBIC
|
|----+----------------------------------------------------------------------|
56
| 0 | -1479.1
6.5e+08
23.1266
23.1357
23.1489 |
| 1 | -1015.78 926.64
1 0.000
472125
15.9029
15.921
15.9474 |
| 2 | -994.497 42.575*
1 0.000
343868
15.5859
15.613
15.6527* |
| 3 | -992.788 3.4162
1 0.065
340089
15.5748
15.611
15.6639 |
| 4 | -991.03 3.5165
1 0.061
336090
15.563
15.6082* 15.6744 |
| 5 | -989.656 2.7481
1 0.097
334141* 15.5571* 15.6114
15.6908 |
| 6 | -989.614 .08451
1 0.771
339193
15.5721
15.6355
15.7281 |
| 7 | -989.56 .10677
1 0.744
344266
15.5869
15.6593
15.7651 |
| 8 | -989.459 .20258
1 0.653
349158
15.6009
15.6824
15.8015 |
+---------------------------------------------------------------------------+
Endogenous: OfficeStock
Exogenous: _cons
Augmented Dickey-Fuller Test (optimum lag):
Augmented Dickey-Fuller test for unit root
Number of obs
=
131
---------- Interpolated Dickey-Fuller --------Test
1% Critical
5% Critical
10% Critical
Statistic
Value
Value
Value
-----------------------------------------------------------------------------Z(t)
-2.010
-3.500
-2.888
-2.578
-----------------------------------------------------------------------------MacKinnon approximate p-value for Z(t) = 0.2824
Office Stock First Difference (Net Completions) Optimum Lag Selection (AIC):
Selection-order criteria
Sample: 1982q2 - 2013q4
Number of obs
=
127
+---------------------------------------------------------------------------+
|lag |
LL
LR
df
p
FPE
AIC
HQIC
SBIC
|
|----+----------------------------------------------------------------------|
| 0 | -1022.81
588403
16.123
16.1321
16.1454 |
| 1 | -991.465 62.696
1 0.000
364851
15.6451
15.6633
15.6899 |
| 2 | -988.607 5.7168
1 0.017
354331
15.6159
15.6432
15.683 |
| 3 | -986.08 5.0541
1 0.025
345916
15.5918
15.6282
15.6814* |
| 4 | -984.258 3.6434
1 0.056
341475* 15.5789* 15.6244* 15.6908 |
| 5 | -984.138 .24019
1 0.624
346250
15.5927
15.6473
15.7271 |
| 6 | -984.01 .25674
1 0.612
351050
15.6064
15.6701
15.7632 |
| 7 | -983.974
.0709
1 0.790
356443
15.6216
15.6944
15.8008 |
| 8 | -981.755 4.4388*
1 0.035
349688
15.6024
15.6843
15.804 |
+---------------------------------------------------------------------------+
Endogenous: OfficeStockD1
Exogenous: _cons
Augmented Dickey-Fuller Test (optimum lag):
Augmented Dickey-Fuller test for unit root
Number of obs
=
130
---------- Interpolated Dickey-Fuller --------Test
1% Critical
5% Critical
10% Critical
Statistic
Value
Value
Value
-----------------------------------------------------------------------------Z(t)
-2.361
-3.500
-2.888
-2.578
-----------------------------------------------------------------------------MacKinnon approximate p-value for Z(t) = 0.1531
Summary of Additional Stationarity Tests Stationarity Results Augmented Dickey-Fuller Test Results
Variable
Optimum Lag
Test Statistic
MacKinnon p-value
(AIC)
(optimum lag)
(optimum lag)
FIRE Employment
5
0.460
0.9836
FIRE Employment First Difference
4
-6.310
0.0000
Total Employment
6
0.719
0.9902
Total Employment First Difference
4
-5.237
0.0000
57
Cointegration Tests Statistical Results Rental Rate Equation Cointegration Test Results OLS Regression for Cointegration
Source |
SS
df
MS
-------------+-----------------------------Model | 991.695516
1 991.695516
Residual | 9302.97375
134 69.4251772
-------------+-----------------------------Total | 10294.6693
135 76.2568093
Number of obs
F( 1,
134)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
136
14.28
0.0002
0.0963
0.0896
8.3322
-----------------------------------------------------------------------------RealNetEff~t |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------Vacancy | -61.70313
16.32588
-3.78
0.000
-93.99287
-29.41339
_cons |
27.50042
2.14181
12.84
0.000
23.26429
31.73654
------------------------------------------------------------------------------
Residuals Optimum Lag Selection (AIC)
Selection-order criteria
Sample: 1982q1 - 2013q4
Number of obs
=
128
+---------------------------------------------------------------------------+
|lag |
LL
LR
df
p
FPE
AIC
HQIC
SBIC
|
|----+----------------------------------------------------------------------|
| 0 | -455.437
73.2558
7.13183
7.14089
7.15412 |
| 1 | -185.38 540.12
1 0.000
1.0941
2.92781
2.94591
2.97237 |
| 2 | -158.715 53.328
1 0.000 .732667
2.5268
2.55396* 2.59365* |
| 3 | -158.476 .47986
1 0.488 .741429
2.53868
2.57489
2.62781 |
| 4 | -156.875 3.2009
1 0.074 .734519
2.5293
2.57456
2.64071 |
| 5 | -156.148 1.4539
1 0.228 .737681
2.53357
2.58788
2.66725 |
| 6 | -153.855 4.5858*
1 0.032 .722958* 2.51336* 2.57674
2.66933 |
| 7 | -153.855 .00065
1 0.980 .734379
2.52898
2.60141
2.70724 |
| 8 | -153.641 .42712
1 0.513
.74351
2.54127
2.62275
2.74181 |
+---------------------------------------------------------------------------+
Endogenous: e
Exogenous: _cons
Augmented-Dickey Fuller Test on Residuals (optimum lag)
Augmented Dickey-Fuller test for unit root
Number of obs
=
129
---------- Interpolated Dickey-Fuller --------Test
1% Critical
5% Critical
10% Critical
Statistic
Value
Value
Value
-----------------------------------------------------------------------------Z(t)
-2.164
-3.500
-2.888
-2.578
-----------------------------------------------------------------------------MacKinnon approximate p-value for Z(t) = 0.2194
Additional Cointegration Tests Explored for Rental Rate Equation Cointegration with Real Net Effective Rent
Variable
R2 Regression Optimum ADF Test Statistic
Lag
(Optimum Lag)
(AIC)
Vacancy- 4 period lag
0.2379
2
-1.070
Vacancy- 8 period lag
0.2985
2
-1.479
Vacancy- 12 period lag
0.2580
5
-2.194
Variable
MacKinnon pvalue (Optimum
Lag)
0.7269
0.5439
0.2083
Cointegration with Real Net Effective Rent First Difference
R2 Regression
Optimum
ADF Test Statistic
MacKinnon p-value
58
Vacancy
0.2176
Lag (AIC)
1
(Optimum Lag)
-4.814
(Optimum Lag)
0.0001
Supply Equation Cointegration Test Results OLS Regression for Cointegration
Source |
SS
df
MS
-------------+-----------------------------Model | 30736226.9
1 30736226.9
Residual | 44646424.4
133 335687.402
-------------+-----------------------------Total | 75382651.3
134
562557.1
Number of obs
F( 1,
133)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
135
91.56
0.0000
0.4077
0.4033
579.39
------------------------------------------------------------------------------OfficeStockD1 |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
--------------+---------------------------------------------------------------RealNetEffR~t |
54.64441
5.710683
9.57
0.000
43.34891
65.93992
_cons | -381.0644
123.8971
-3.08
0.003
-626.1281
-136.0007
-------------------------------------------------------------------------------
Residuals Optimum Lag Selection (AIC)
Selection-order criteria
Sample: 1982q2 - 2013q4
Number of obs
=
127
+---------------------------------------------------------------------------+
|lag |
LL
LR
df
p
FPE
AIC
HQIC
SBIC
|
|----+----------------------------------------------------------------------|
| 0 | -990.014
351037
15.6065
15.6156
15.6289 |
| 1 | -979.717 20.595
1 0.000
303226* 15.4601* 15.4783* 15.5049* |
| 2 | -979.318 .79843
1 0.372
306110
15.4696
15.4969
15.5368 |
| 3 | -978.355 1.9254
1 0.165
306294
15.4702
15.5066
15.5597 |
| 4 | -977.102 2.5065
1 0.113
305081
15.4662
15.5117
15.5781 |
| 5 | -977.063 .07759
1 0.781
309743
15.4813
15.5359
15.6157 |
| 6 | -976.995 .13557
1 0.713
314337
15.496
15.5597
15.6528 |
| 7 | -976.937 .11638
1 0.733
319051
15.5108
15.5836
15.69 |
| 8 | -974.312 5.2502*
1 0.022
311012
15.4852
15.5671
15.6868 |
+---------------------------------------------------------------------------+
Endogenous: e
Exogenous: _cons
Augmented-Dickey Fuller Test on Residuals (optimum lag)
Augmented Dickey-Fuller test for unit root
Number of obs
=
133
---------- Interpolated Dickey-Fuller --------Test
1% Critical
5% Critical
10% Critical
Statistic
Value
Value
Value
-----------------------------------------------------------------------------Z(t)
-5.827
-3.499
-2.888
-2.578
-----------------------------------------------------------------------------MacKinnon approximate p-value for Z(t) = 0.0000
Additional Cointegration Tests Explored for Supply Equation Single and Multivariate Cointegration with Total Office Stock
Variable
R2
Optimum
ADF Test
Regression
Lag
Statistic
(AIC)
(Optimum Lag)
FIRE Employment
0.6043
6
-1.738
Business Services Employment
0.8149
8
-1.895
Total Employment
0.7836
7
-1.885
Real Net Effective Rent
0.4216
5
-1.989
Vacancy Rate
0.0050
5
-3.500
Business Services Employment, Real Net
0.9631
6
-3.191
Effective Rent, and Vacancy Rate
59
MacKinnon pvalue (Optimum
Lag)
0.4117
0.3343
0.3393
0.2915
0.2970
0.0205
Business Services Employment and Real Net
Effective Rent
Business Services Employment and Vacancy
Rate
Real Net Effective Rent and Vacancy Rate
Real Net Effective Rent and Vacancy Rate Both
Lagged 4 Periods
Real Net Effective Rent and Vacancy Rate Both
Lagged 8 Periods
Business Services Employment, Rent Net
Effective Rent and Vacancy Rate (both Lagged
1 Period)
Business Services Employment, Rent Net
Effective Rent and Vacancy Rate (both Lagged
4 Periods)
0.8865
6
-1.950
0.3088
0.9585
6
-3.214
0.0192
0.4437
2
-2.610
0.0909
0.3774
2
-2.418
0.1368
0.3101
5
-2.452
0.1276
0.9634
6
-3.212
0.0193
0.9572
6
-3.019
0.0331
Single and Multivariate Cointegration with Total Office Stock First Difference (Net Completions)
Variable
R2
Optimum
ADF Test
MacKinnon pRegression
Lag
Statistic
value (Optimum
(AIC)
(Optimum Lag)
Lag)
FIRE Employment
0.0652
4
-2.504
0.1144
Business Services Employment
0.1917
4
-2.717
0.0712
Total Employment
0.1558
4
-2.643
0.0844
Vacancy Rate
-0.0042
4
-2.403
0.1408
Business Services Employment, Real Net
0.4685
1
-6.603
0.0000
Effective Rent, and Vacancy Rate
Business Services Employment and Real Net
0.4279
1
-6.040
0.0000
Effective Rent
Business Services Employment and Vacancy
0.1970
4
-2.746
0.0665
Rate
Real Net Effective Rent and Vacancy Rate
0.4723
1
-6.615
0.0000
Real Net Effective Rent and Vacancy Rate Both
0.4743
1
-6.893
0.0000
Lagged 4 Periods
Real Net Effective Rent and Vacancy Rate Both
0.4140
1
-6.010
0.0000
Lagged 8 Periods
Direct Demand in Levels Cointegration Test Results OLS Regression for Cointegration
Source |
SS
df
MS
-------------+-----------------------------Model | 8.6142e+10
2 4.3071e+10
Residual | 4.5269e+09
133 34037007.4
-------------+-----------------------------Total | 9.0669e+10
135
671619092
Number of obs
F( 2,
133)
Prob > F
R-squared
Adj R-squared
Root MSE
=
136
= 1265.41
= 0.0000
= 0.9501
= 0.9493
= 5834.1
------------------------------------------------------------------------------OccupiedStock |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
--------------+---------------------------------------------------------------RealNetEffR~t | -488.1676
64.70653
-7.54
0.000
-616.1546
-360.1806
BusSvcEmp |
210.0433
5.155821
40.74
0.000
199.8453
220.2413
_cons |
50245.12
2457.955
20.44
0.000
45383.38
55106.86
-------------------------------------------------------------------------------
Residuals Optimum Lag Selection (AIC)
Selection-order criteria
Sample: 1982q1 - 2013q4
Number of obs
=
128
+---------------------------------------------------------------------------+
|lag |
LL
LR
df
p
FPE
AIC
HQIC
SBIC
|
|----+----------------------------------------------------------------------|
60
| 0 | -1273.49
2.6e+07
19.9139
19.9229
19.9362 |
| 1 | -1158.01 230.95
1 0.000 4.4e+06
18.1252
18.1433
18.1698* |
| 2 | -1157.95 .12409
1 0.725 4.4e+06
18.1399
18.167
18.2067 |
| 3 | -1155.75 4.4096
1 0.036 4.3e+06
18.121
18.1572
18.2102 |
| 4 | -1155.63 .22489
1 0.635 4.4e+06
18.1349
18.1802
18.2463 |
| 5 | -1153.74 3.7924
1 0.051 4.3e+06
18.1209
18.1752
18.2546 |
| 6 | -1148.1 11.281*
1 0.001 4.0e+06* 18.0484* 18.1118* 18.2044 |
| 7 | -1147.49 1.2194
1 0.269 4.1e+06
18.0545
18.1269
18.2327 |
| 8 | -1147.43 .12368
1 0.725 4.1e+06
18.0691
18.1506
18.2697 |
+---------------------------------------------------------------------------+
Endogenous: e
Exogenous: _cons
Augmented-Dickey Fuller Test on Residuals (optimum lag)
Augmented Dickey-Fuller test for unit root
Number of obs
=
129
---------- Interpolated Dickey-Fuller --------Test
1% Critical
5% Critical
10% Critical
Statistic
Value
Value
Value
-----------------------------------------------------------------------------Z(t)
-2.087
-3.500
-2.888
-2.578
-----------------------------------------------------------------------------MacKinnon approximate p-value for Z(t) = 0.2497
Additional Cointegration Tests Explored for Direct Demand in Levels Equation Single and Multivariate Cointegration with Occupied Stock
Variable
R2 Regression Optimum ADF Test Statistic
Lag
(Optimum Lag)
(AIC)
Real Net Effective Rent
0.3220
5
-1.757
Business Services Employment
0.9282
6
-1.638
FIRE Employment
0.7268
5
-1.556
Total Employment
0.9052
6
-1.301
Real Net Effective Rent, FIRE Employment
0.8374
5
-2.145
Real Net Effective Rent, Total Employment
0.9406
6
-1.880
MacKinnon pvalue (Optimum
Lag)
0.4018
0.4635
0.5056
0.6288
0.2268
0.3417
Indirect Demand in Levels Cointegration Test Results OLS Regression for Cointegration
Source |
SS
df
MS
-------------+-----------------------------Model | .209230593
3 .069743531
Residual | .051243174
132 .000388206
-------------+-----------------------------Total | .260473768
135 .001929435
Number of obs
F( 3,
132)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
136
179.66
0.0000
0.8033
0.7988
.0197
------------------------------------------------------------------------------Vacancy |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
--------------+---------------------------------------------------------------BusSvcEmp | -.0008338
.0000394
-21.14
0.000
-.0009118
-.0007558
OfficeStock |
2.97e-06
1.78e-07
16.65
0.000
2.62e-06
3.32e-06
RealNetEffR~t | -.0000514
.0002801
-0.18
0.855
-.0006054
.0005026
_cons |
.0232038
.0162129
1.43
0.155
-.0088668
.0552745
-------------------------------------------------------------------------------
Residuals Optimum Lag Selection (AIC)
Selection-order criteria
Sample: 1982q1 - 2013q4
Number of obs
=
128
+---------------------------------------------------------------------------+
|lag |
LL
LR
df
p
FPE
AIC
HQIC
SBIC
|
|----+----------------------------------------------------------------------|
| 0 | 321.641
.000391 -5.01002 -5.00097 -4.98774 |
| 1 | 422.175 201.07
1 0.000 .000082 -6.56523 -6.54712 -6.52066 |
61
| 2 | 422.593 .83701
1 0.360 .000083 -6.55614 -6.52898
-6.4893 |
| 3 | 422.643 .10073
1 0.751 .000084
-6.5413 -6.50509 -6.45218 |
| 4 | 426.459 7.6312
1 0.006 .000081
-6.5853 -6.54003 -6.47389 |
| 5 | 426.539 .15941
1 0.690 .000082 -6.57092
-6.5166 -6.43723 |
| 6 | 434.686 16.295*
1 0.000 .000073* -6.68259* -6.61922* -6.52662* |
| 7 |
435.54 1.7072
1 0.191 .000074 -6.68031 -6.60788 -6.50206 |
| 8 | 435.925 .77011
1 0.380 .000074
-6.6707 -6.58922 -6.47017 |
+---------------------------------------------------------------------------+
Endogenous: e
Exogenous: _cons
Augmented-Dickey Fuller Test on Residuals (optimum lag)
Augmented Dickey-Fuller test for unit root
Number of obs
=
129
---------- Interpolated Dickey-Fuller --------Test
1% Critical
5% Critical
10% Critical
Statistic
Value
Value
Value
-----------------------------------------------------------------------------Z(t)
-4.324
-3.500
-2.888
-2.578
-----------------------------------------------------------------------------MacKinnon approximate p-value for Z(t) = 0.0004
Additional Cointegration Tests Explored for Indirect Demand in Levels Equation Single and Multivariate Cointegration with Vacancy
Variable
R2
Optimum ADF Test Statistic
Regression
Lag
(Optimum Lag)
(AIC)
Business Services Employment
0.1085
4
-3.014
Total Employment
0.1225
4
-3.020
Real Net Effective Rent
0.0896
4
-2.141
Office Stock
0.0024
4
-2.552
Business Services Employment and Office
0.8003
6
-4.330
Stock
Total Employment and Office Stock
0.7537
6
-3.495
MacKinnon pvalue (Optimum
Lag)
0.0336
0.0330
0.2282
0.1033
0.0004
0.0081
Direct Demand in Differences (ECM) Cointegration Test Results OLS Regression for Cointegration
Source |
SS
df
MS
-------------+-----------------------------Model |
7378828.7
2 3689414.35
Residual | 80172097.3
132 607364.373
-------------+-----------------------------Total |
87550926
134 653365.119
Number of obs
F( 2,
132)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
135
6.07
0.0030
0.0843
0.0704
779.34
------------------------------------------------------------------------------OccupiedS~kD1 |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
--------------+---------------------------------------------------------------RealNetEffR~t |
5.253494
8.656067
0.61
0.545
-11.86906
22.37605
BusSvcEmp | -1.923282
.695312
-2.77
0.006
-3.298678
-.5478864
_cons |
1102.777
331.1845
3.33
0.001
447.6617
1757.893
-------------------------------------------------------------------------------
Residuals Optimum Lag Selection (AIC)
Selection-order criteria
Sample: 1982q2 - 2013q4
Number of obs
=
127
+---------------------------------------------------------------------------+
|lag |
LL
LR
df
p
FPE
AIC
HQIC
SBIC
|
|----+----------------------------------------------------------------------|
| 0 | -1027.01
628634
16.1892
16.1983
16.2116 |
| 1 | -1016.07 21.893*
1 0.000
537493* 16.0325* 16.0507* 16.0773* |
| 2 | -1015.9 .32869
1 0.566
544617
16.0457
16.073
16.1129 |
62
| 3 | -1015.8 .21122
1 0.646
552348
16.0598
16.0962
16.1494 |
| 4 | -1015.79 .01765
1 0.894
561049
16.0754
16.1209
16.1874 |
| 5 | -1015.77
.0315
1 0.859
569830
16.0909
16.1455
16.2253 |
| 6 | -1015.19 1.1691
1 0.280
573594
16.0974
16.1611
16.2542 |
| 7 | -1014.95 .46885
1 0.494
580583
16.1095
16.1823
16.2887 |
| 8 | -1014.72 .47016
1 0.493
587661
16.1215
16.2034
16.3231 |
+---------------------------------------------------------------------------+
Endogenous: e
Exogenous: _cons
Augmented-Dickey Fuller Test on Residuals (optimum lag)
Augmented Dickey-Fuller test for unit root
Number of obs
=
133
---------- Interpolated Dickey-Fuller --------Test
1% Critical
5% Critical
10% Critical
Statistic
Value
Value
Value
-----------------------------------------------------------------------------Z(t)
-5.942
-3.499
-2.888
-2.578
-----------------------------------------------------------------------------MacKinnon approximate p-value for Z(t) = 0.0000
Additional Cointegration Tests Explored for Direct Demand in Differences (ECM) Equation Single and Multivariate Cointegration with Net Absorption (Occupied Stock First Difference)
Variable
R2 Regression Optimum ADF Test Statistic
MacKinnon pLag
(Optimum Lag)
value (Optimum
(AIC)
Lag)
Real Net Effective Rent
0.0239
1
-5.708
0.0000
Business Services Employment
0.0748
1
-5.922
0.0000
FIRE Employment
0.0365
1
-5.783
0.0000
Total Employment
0.0656
1
-5.890
0.0000
Real Net Effective Rent, FIRE Employment
0.0438
1
-5.859
0.0000
Real Net Effective Rent, Total Employment
0.0634
1
-5.922
0.0000
Indirect Demand in Differences (ECM) Cointegration Test Results OLS Regression for Cointegration
Source |
SS
df
MS
-------------+-----------------------------Model | .001035612
3 .000345204
Residual | .004483914
131 .000034228
-------------+-----------------------------Total | .005519526
134
.00004119
Number of obs
F( 3,
131)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
135
10.09
0.0000
0.1876
0.1690
.00585
------------------------------------------------------------------------------D.Vacancy |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
--------------+---------------------------------------------------------------RealNetEffR~t |
.000438
.000085
5.16
0.000
.0002699
.0006061
BusSvcEmp | -.0000178
.0000119
-1.49
0.138
-.0000413
5.76e-06
OfficeStock |
1.21e-07
5.48e-08
2.21
0.028
1.30e-08
2.30e-07
_cons | -.0176552
.0050227
-3.52
0.001
-.0275914
-.0077191
-------------------------------------------------------------------------------
Residuals Optimum Lag Selection (AIC)
Selection-order criteria
Sample: 1982q2 - 2013q4
Number of obs
=
127
+---------------------------------------------------------------------------+
|lag |
LL
LR
df
p
FPE
AIC
HQIC
SBIC
|
|----+----------------------------------------------------------------------|
| 0 | 471.822
.000035 -7.41451 -7.40541 -7.39212 |
| 1 | 498.075 52.508*
1 0.000 .000024* -7.81221* -7.79401* -7.76742* |
| 2 | 498.779 1.4078
1 0.235 .000024 -7.80755 -7.78025 -7.74036 |
| 3 | 499.469
1.379
1 0.240 .000024 -7.80266 -7.76626 -7.71308 |
| 4 | 499.662 .38754
1 0.534 .000024 -7.78996 -7.74447 -7.67798 |
63
| 5 | 499.675 .02459
1 0.875 .000025 -7.77441 -7.71981 -7.64003 |
| 6 | 499.975 .60037
1 0.438 .000025 -7.76339 -7.69969 -7.60662 |
| 7 | 500.228 .50554
1 0.477 .000025 -7.75162 -7.67883 -7.57246 |
| 8 | 500.744 1.0329
1 0.309 .000025
-7.744 -7.66211 -7.54245 |
+---------------------------------------------------------------------------+
Endogenous: e
Exogenous: _cons
Augmented-Dickey Fuller Test on Residuals (optimum lag)
Augmented Dickey-Fuller test for unit root
Number of obs
=
133
---------- Interpolated Dickey-Fuller --------Test
1% Critical
5% Critical
10% Critical
Statistic
Value
Value
Value
-----------------------------------------------------------------------------Z(t)
-4.642
-3.499
-2.888
-2.578
-----------------------------------------------------------------------------MacKinnon approximate p-value for Z(t) = 0.0001
Additional Cointegration Tests Explored for Indirect Demand in Differences (ECM) Equation Single and Multivariate Cointegration with Vacancy First Difference
Variable
R2
Optimum ADF Test Statistic
Regression
Lag
(Optimum Lag)
(AIC)
Business Services Employment
0.0024
2
-3.548
Total Employment
0.0063
2
-3.537
Real Net Effective Rent
0.1429
1
-4.621
Office Stock
0.0118
2
-3.632
Business Services Employment and Office
0.0080
1
-4.324
Stock
Total Employment and Office Stock
0.0130
1
-4.349
MacKinnon pvalue (Optimum
Lag)
0.0068
0.0071
0.0001
0.0052
0.0004
0.0004
Structural System Econometric Models Real Rental Rate Equation Source |
SS
df
MS
-------------+-----------------------------Model | 9045.73575
2 4522.86787
Residual | 1213.23109
129 9.40489217
-------------+-----------------------------Total | 10258.9668
131
78.312724
Number of obs
F( 2,
129)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
132
480.91
0.0000
0.8817
0.8799
3.0667
------------------------------------------------------------------------------RealNetEffR~t |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
--------------+---------------------------------------------------------------RealNetEffR~4 |
.8467268
.0320966
26.38
0.000
.7832229
.9102307
VacancyL4 |
-43.5531
6.432308
-6.77
0.000
-56.27957
-30.82662
_cons |
8.29371
1.210331
6.85
0.000
5.899041
10.68838
-------------------------------------------------------------------------------
Additional Models for Real Rental Rate Equation 1
Model
Real Rentt = a + b*Vacancy
Ratet-1 + c*Real Rentt-1
2
Real Rentt = a + b*Vacancy
Coefficients (t-value)
a=1.994091 (5.74)
b=-12.7389 (-6.70)
c=0.9771233 (102.30)
a=1.994097 (5.74)
64
Reason For Exclusion
Yearly lags used to keep equation
consistent with other structural
model equations
Yearly lags used to keep equation
Ratet + c*Real Rentt-1
3
Real Rentt = a + b*Vacancy
RateYoY + c*Real RentYoY
4
Real Rentt = a + b*Vacancy
Ratet-4 + c*Real Rentt-4
b=-12.7389 (-6.70)
c=0.9771238 (102.30)
a=19.9512 (31.41)
b=249.5154 (7.48)
c=1.133084 (5.82)
a=8.29371 (6.85)
b=-43.5531 (-6.77)
c=0.8467268 (102.30)
consistent with other structural
model equations
Sign on vacancy coefficient is
positive
High amount of momentum in the
rent coefficient creates a disconnect
between the rental equation and the
rest of the structural model
Long-­‐Run Supply Equation Source |
SS
df
MS
-------------+-----------------------------Model |
550271966
1
550271966
Residual |
265801020
130 2044623.23
-------------+-----------------------------Total |
816072985
131 6229564.77
Number of obs
F( 1,
130)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
132
269.13
0.0000
0.6743
0.6718
1429.9
------------------------------------------------------------------------------OfficeStock~Y |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
--------------+---------------------------------------------------------------RealNetEffR~4 |
231.9822
14.14076
16.41
0.000
204.0064
259.958
_cons | -1838.373
308.9063
-5.95
0.000
-2449.507
-1227.239
-------------------------------------------------------------------------------
Additional Models for Supply Equation Model
Change in Stock Quarterly
= a + b*Real Rentt
Coefficients (t-value)
a=-381.0644(-3.08)
b=54.64441 (9.57)
2
Change in Stock Quarterly
= a + b*Vacancy Ratet-4 +
c*Real Rentt
a=-1117.223 (-4.42)
b=4235.709 (3.30)
c=64.76565 (10.13)
3
Change in Stock Yearly = a
+ b*Vacancy Ratet-4
a=3389.237 (5.18)
b=-4722.685 (-0.95)
4
Change in Stock Yearly = a
+ b*Vacancy Ratet-4 +
c*Real Rentt-4
a=-3689.74 (-6.94)
b=11756.86 (4.16)
c=251.1876 (17.81)
5
Change in Stock Yearly = a
+ b*Vacancy RateYoY
a=2781.783 (15.19)
b=64259.51 (7.38)
1
Reason For Exclusion
Yearly changes used to keep
equation consistent with other
structural model equations
Vacancy coefficient is positive
Model has very little predictive
power and the only independent
variable included is insignificant
Vacancy coefficient is positive
Vacancy coefficient is positive
Direct Demand in Levels Equation Source |
SS
df
MS
-------------+-----------------------------Model | 7.6220e+10
2 3.8110e+10
Residual | 4.5008e+09
129 34890206.2
-------------+-----------------------------Total | 8.0721e+10
131
616188219
65
Number of obs
F( 2,
129)
Prob > F
R-squared
Adj R-squared
Root MSE
=
132
= 1092.28
= 0.0000
= 0.9442
= 0.9434
= 5906.8
------------------------------------------------------------------------------OccupiedStock |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
--------------+---------------------------------------------------------------BusSvcEmp |
206.7585
5.466894
37.82
0.000
195.9421
217.5749
RealNetEffR~4 | -394.3987
66.59941
-5.92
0.000
-526.1672
-262.6301
_cons |
49897.66
2618.458
19.06
0.000
44716.98
55078.35
-------------------------------------------------------------------------------
Additional Models for Direct Demand in Levels Equation 1
2
Model
Occupied Stock = a +
b*Real Rentt-1 + c*Bus Svc
Empt
Coefficients (t-value)
a=50235.73 (20.22)
b=-466.2884 (-7.19)
c=209.0978 (40.15)
Reason For Exclusion
Yearly changes used to keep
equation consistent with other
structural model equations
Occupied Stock = a +
b*Real Rentt-4 + c*Bus Svc
Empt-1
a=51041.42 (19.54)
b=-418.7449 (-6.27)
c=206.3885 (37.52)
Current business services
employment used instead of lagged
value
Direct Demand as an Error Correction Model Equation Source |
SS
df
MS
-------------+-----------------------------Model |
170216265
5
34043253
Residual |
514291569
126 4081679.12
-------------+-----------------------------Total |
684507834
131 5225250.64
Number of obs
F( 5,
126)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
132
8.34
0.0000
0.2487
0.2189
2020.3
------------------------------------------------------------------------------OccupiedSto~Y |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
--------------+---------------------------------------------------------------BusSvcEmp | -11.72436
8.3689
-1.40
0.164
-28.28617
4.837448
RealNetEffR~4 |
39.03843
30.33087
1.29
0.200
-20.98548
99.06234
OccupiedSto~4 |
.0208025
.0383994
0.54
0.589
-.0551888
.0967939
BusSvcEmpYoY |
43.26936
12.73482
3.40
0.001
18.06752
68.47119
RealNetEffR~Y |
129.8337
62.78578
2.07
0.041
5.582474
254.0849
_cons |
2752.292
2063.032
1.33
0.185
-1330.388
6834.972
-------------------------------------------------------------------------------
Additional Models for Direct Demand as an Error Correction Model Equation 1
2
Model
Net AbsorptionQuart = a +
b*Business Services
Employmentt + c*Real Net
Effective Rentt-4 +
d*Occupied Stockt-1 +
e*Change in
EmploymentQuart +
f*Change in Real RentQuart
Coefficients (t-value)
a=258.5885 (0.33)
b=-5.554228 (-1.94)
c=10.82719 (0.95)
d=0.0172784 (1.29)
e=12.02588 (1.59)
f=140.6041 (0.33)
Reason For Exclusion
Yearly changes and lags used to
keep equation consistent with other
structural model equations
Net AbsorptionQuart = a +
b*Business Services
Employmentt + c*Real Net
Effective Rentt-4 +
a=49.71989 (0.06)
b=-6.501609 (-2.22)
c=12.38496 (1.12)
d=0.0216107 (1.58)
Yearly changes and lags used to
keep equation consistent with other
structural model equations
66
d*Occupied Stockt-1 +
e=4.559506 (0.95)
e*Change in
f=52.60924 (2.29)
EmploymentYoY + f*Change
in Real RentYoY
3
Net AbsorptionQuart = a +
b*Business Services
Employmentt-1 + c*Real Net
Effective Rentt-4 +
d*Occupied Stockt-1 +
e*Change in
EmploymentQuart +
f*Change in Real RentQuart
a=258.5885 (0.33)
b=-5.554228 (-1.94)
c=10.82719 (0.95)
d=0.0172784 (1.29)
e=6.471652 (0.85)
f=140.6041 (0.33)
Yearly changes and lags used to
keep equation consistent with other
structural model equations; current
business services employment
instead of lagged value
4
Net AbsorptionQuart = a +
b*Business Services
Employmentt-1 + c*Real Net
Effective Rentt-4 +
d*Occupied Stockt-1 +
e*Change in
EmploymentYoY + f*Change
in Real RentYoY
a=-113.0089 (-0.15)
b=-7.232839 (-2.53)
c=13.98699 (1.27)
d=0.0250117 (1.86)
e=2.641995 (0.55)
f=56.39131 (2.46)
Yearly changes and lags used to
keep equation consistent with other
structural model equations; current
business services employment
instead of lagged value
Indirect Demand in Levels Equation Source |
SS
df
MS
-------------+-----------------------------Model | .202670516
3 .067556839
Residual | .048644397
128 .000380034
-------------+-----------------------------Total | .251314913
131 .001918434
Number of obs
F( 3,
128)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
132
177.77
0.0000
0.8064
0.8019
.01949
------------------------------------------------------------------------------Vacancy |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
--------------+---------------------------------------------------------------BusSvcEmp | -.0008529
.0000371
-22.99
0.000
-.0009263
-.0007795
RealNetEffR~4 |
.0006019
.0002478
2.43
0.017
.0001116
.0010923
OfficeStock |
3.16e-06
1.64e-07
19.26
0.000
2.84e-06
3.49e-06
_cons | -.0070384
.0148687
-0.47
0.637
-.0364587
.0223819
-------------------------------------------------------------------------------
Additional Models for Indirect Demand in Levels Equation 1
2
Model
Vacancyt = a + b*Vacancyt-1
+ c*Growth in Office
StockQuart + d*Growth in
EmploymentQuart
Coefficients (t-value)
a=-0.005314 (-0.33)
b=0.987545 (84.87)
c=3.30e-06 (4.84)
d=-0.0000729 (-1.29)
Vacancyt = a + b*Vacancyt-1 a=0.0008077 (0.49)
+ c*Growth in Office
b=0.9734453 (86.21)
StockYoY + d*Growth in
c=1.21e-06 (6.16)
67
Reason For Exclusion
Yearly changes and lags used to
keep equation consistent with other
structural model equations; business
services employment not in model
Yearly lags used to keep equation
consistent with other structural
model equations; business services
EmploymentYoY
d=-0.0000663 (-2.00)
employment not in model
3
Vacancyt = a + b*Vacancyt-4
+ c*Growth in Office
StockYoY + d*Growth in
EmploymentYoY
a=0.0053992 (1.00)
b=0.9023324 (24.11)
c=0.0000111 (5.07)
d=-0.0002313 (-1.29)
Business services employment not in
model
4
Vacancyt = a + b*Real Net
Eff Rentt-4 + c* Office
Stockt + d*Business
Services Employmentt-1
a=-0.0114215 (-21.66) Yearly changes and lags used to
b=0.0007024 (2.69)
keep equation consistent with other
c=3.15e-06 (18.18)
structural model equations
d=-0.000847 (-21.66)
Indirect Demand as an Error Correction Model Equation Source |
SS
df
MS
-------------+-----------------------------Model | .033906811
7
.00484383
Residual | .024451815
124 .000197192
-------------+-----------------------------Total | .058358626
131 .000445486
Number of obs
F( 7,
124)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
132
24.56
0.0000
0.5810
0.5574
.01404
------------------------------------------------------------------------------VacancyYoY |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
--------------+---------------------------------------------------------------BusSvcEmp | -.0002551
.0000759
-3.36
0.001
-.0004054
-.0001049
RealNetEffR~4 |
.000415
.0003583
1.16
0.249
-.0002943
.0011242
OfficeStock |
8.57e-07
2.84e-07
3.02
0.003
2.95e-07
1.42e-06
VacancyL4 | -.4223231
.0759163
-5.56
0.000
-.5725828
-.2720634
BusSvcEmpYoY | -.0000845
.0000954
-0.89
0.378
-.0002734
.0001044
RealNetEffR~Y | -.0027141
.0005035
-5.39
0.000
-.0037108
-.0017174
OfficeStock~Y |
4.46e-07
1.15e-06
0.39
0.699
-1.83e-06
2.73e-06
_cons |
.0185135
.0160634
1.15
0.251
-.0132805
.0503076
-------------------------------------------------------------------------------
Additional Models for Indirect Demand as an Error Correction Model Equation 1
2
Model
Change in VacancyQuart= a +
b*Business Services
Employmentt + c*Real Net
Effective Rentt-4 + d*Stockt
+ e*Vacancyt-1 + f*Change
in EmploymentQuart +
g*Change in Real RentQuart
+ h*Change in StockQuart
Coefficients (t-value)
a=-0.0051847 (-0.94)
b=-1.73e-06 (-0.07)
c=0.00257 (2.39)
d=2.32e-08 (0.23)
e=-0.0231074 (-0.85)
f=-0.0000052 (-0.96)
g=-0.0011474 (-1.91)
h=1.38e-06 (1.43)
Reason For Exclusion
Yearly changes and lags used to
keep equation consistent with other
structural model equations
Change in VacancyQuart= a +
b*Business Services
Employmentt + c*Real Net
Effective Rentt-4 + d*Stockt
+ e*Vacancyt-1 + f*Change
in EmploymentYoY +
g*Change in Real RentYoY +
a=0.0000326 (0.01)
b=-3.17e-08 (-0.00)
c=0.0001359 (0.97)
d=5.55e-09 (0.05)
e=-0.0395734 (-1.36)
f=-0.0000312 (-0.87)
g=-0.0004537 (-2.13)
Yearly changes and lags used to
keep equation consistent with other
structural model equations
68
h*Change in StockYoY
h=7.03e-07 (1.65)
3
Change in VacancyQuart= a +
b*Business Services
Employmentt-1 + c*Real Net
Effective Rentt-4 + d*Stockt
+ e*Vacancyt-1 + f*Change
in EmploymentQuart +
g*Change in Real RentQuart
+ h*Change in StockQuart
a=-0.0051847 (-0.94)
b=-1.73e-06 (-0.07)
c=0.000257 (2.39)
d=2.32e-08 (0.23)
e=-0.0231074 (-0.85)
f=-0.0000537 (-0.96)
g=-0.0011474 (-1.91)
h=1.38e-06 (-0.94)
Yearly changes and lags used to
keep equation consistent with other
structural model equations
4
Change in VacancyQuart= a +
b*Business Services
Employmentt-1 + c*Real Net
Effective Rentt-4 + d*Stockt
+ e*Vacancyt-1 + f*Change
in EmploymentYoY +
g*Change in Real RentYoY+
h*Change in StockYoY
a=0.000511 (0.08)
b=8.76e-06 (0.33)
c=0.0001158 (0.82)
d=-2.67-08 (-0.26)
e=-0.0319394 (-1.13)
f=-0.0000314 (-0.89)
g=-0.0004489 (-2.11)
h=7.72e-07 (1.80)
Yearly changes and lags used to
keep equation consistent with other
structural model equations
Non-­‐structural Econometric Models Rental Rate Equation Source |
SS
df
MS
-------------+-----------------------------Model | 9127.85346
4 2281.96336
Residual | 1131.11338
127 8.90640456
-------------+-----------------------------Total | 10258.9668
131
78.312724
Number of obs
F( 4,
127)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
132
256.22
0.0000
0.8897
0.8863
2.9844
------------------------------------------------------------------------------RealNetEffR~t |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
--------------+---------------------------------------------------------------RealNetEffR~4 |
.9336669
.0610328
15.30
0.000
.812894
1.05444
VacancyL4 | -38.02874
6.80483
-5.59
0.000
-51.49427
-24.56321
OfficeStock~Y | -.0002504
.0001995
-1.25
0.212
-.0006451
.0001444
BusSvcEmpYoY |
.0546646
.0184143
2.97
0.004
.018226
.0911032
_cons |
5.997348
1.512481
3.97
0.000
3.004421
8.990276
-------------------------------------------------------------------------------
Long-­‐Run Supply Equation Source |
SS
df
MS
-------------+-----------------------------Model |
599213133
4
149803283
Residual |
199282678
123 1620184.37
-------------+-----------------------------Total |
798495811
127 6287368.59
Number of obs
F( 4,
123)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
128
92.46
0.0000
0.7504
0.7423
1272.9
------------------------------------------------------------------------------OfficeStock~Y |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
--------------+---------------------------------------------------------------RealNetEffR~4 |
223.9518
22.58207
9.92
0.000
179.252
268.6516
VacancyL4 |
10484.57
3327.033
3.15
0.002
3898.908
17070.22
|
OfficeStock~Y |
69
L4. |
.1579543
.0769547
2.05
0.042
.0056273
.3102813
|
BusSvcEmpYoY |
21.37944
7.695737
2.78
0.006
6.146202
36.61268
_cons | -3722.724
658.9294
-5.65
0.000
-5027.035
-2418.414
-------------------------------------------------------------------------------
Vacancy (Indirect Demand) Equation Source |
SS
df
MS
-------------+-----------------------------Model | .207080225
4 .051770056
Residual | .030072998
123 .000244496
-------------+-----------------------------Total | .237153223
127 .001867348
Number of obs =
F( 4,
123) =
Prob > F
=
R-squared
=
Adj R-squared =
Root MSE
=
128
211.74
0.0000
0.8732
0.8691
.01564
------------------------------------------------------------------------------Vacancy |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
--------------+---------------------------------------------------------------RealNetEffR~4 |
.0001345
.0002774
0.48
0.629
-.0004146
.0006836
VacancyL4 |
.8286142
.0408706
20.27
0.000
.7477134
.909515
|
OfficeStock~Y |
L4. |
4.50e-06
9.45e-07
4.76
0.000
2.63e-06
6.37e-06
|
BusSvcEmpYoY | -.0002504
.0000945
-2.65
0.009
-.0004376
-.0000633
_cons |
.0093584
.0080945
1.16
0.250
-.0066643
.025381
-------------------------------------------------------------------------------
70