Empirical Identification of Inter-Sectoral General Equilibrium Impacts on Wages
from Changes in Local Composition of Sectoral Employment and National
Sectoral Wage Premia in Mexican Cities 1990-2000*
Jian Mardukhi†
Abstract
Are there in practice general equilibrium (G.E.) mechanisms through which sectors, even
the seemingly unrelated ones, interact with each other? In this study we use Mexican census
data to exploit geographical variation in composition of sectoral employment in Mexican
cities over time to see whether sectoral wages systematically change with the cities‟
composition of sectoral employment and/or national sectoral wage premia. We find ample
evidence on the existence of a G.E. mechanism through which sectoral wages in each city
causally depend on the rents generated by the city as measured by the city‟s employmentshare weighted sum of national sectoral wage premia. The magnitude of this relationship is
substantially larger than the conventional measures of the impact of changes in the
employment composition of a city on the city‟s average wage. According to our estimates
for the Mexican cities during 1990-2000, a compositional change in a city that decreases the
average wages in that city by one dollar keeping the sectoral wage levels unchanged (i.e.,
ignoring the G.E. impact on local wages according to the conventional compositionadjustment approach) will in the long-run result in more than three dollar decrease in the
average wages in that city; an inter-sectoral G.E. impact on wages in local economies that is
at least three times the conventional measures. An impact of this magnitude calls for
excessive caution, especially in the less developed countries, in implementing industrial or
trade policies that may result in severe loss of good jobs or enormous relocation of work
force at the expense of high paying sectors. In addition, this finding emphasizes the need for
a holistic approach in welfare analysis of such policy changes.
*
I am very grateful to my advisor Paul Beaudry for his support and guidance. I am also grateful to David Green,
Patrick Francios, Ben Sand, Matilde Bombardini, and Vadim Marmer for their helpful comments. The usual
disclaimer applies. This paper is part of my PhD thesis at the University of British Columbia.
†
University of British Columbia, Department of Economics, 997 - 1873 East Mall Vancouver, B.C. V6T 1Z1.
jmardukh@interchange.ubc.ca
1
1
Introduction
Are there in practice general equilibrium (G.E.) mechanisms through which sectors, even the
seemingly unrelated ones, interact with each other?
Industrialization and economic growth have been and still are among the main challenges of
the governments in many less developed parts of the world. Efforts for accomplishing these
goals have materialised in forms of various industrial policies. However intricate has achieving
these ends been, forces of globalization and integration into the world economy have made the
tasks even more cumbersome, and sometimes even paradoxical; often the privileges given to
certain sectors have to be removed and the playground has to be levelled to allow for effective
presence and competition of foreign firms within the borders of national economy, all of which
may as well result in significant losses in terms of quantity and quality of the good jobs.
If there are general equilibrium mechanisms through which sectors interact with each other, in
such an environment where the less developed world could end up paying high prices for
integrating with the rest of the world economy at the expense of their long privileged and
protected good jobs, the likely destructive impacts of the transition could in the long-run
multiply and protract beyond the impact industries into other sectors and throughout the whole
economy.
Economic theory has for effortful long been busy with incorporating general equilibrium
mechanisms into the classical models to provide new insights on the working of economies. 1
However, these new insights have brought to view interactions that require empirical
verification. These theoretical possibilities bring to light likely mechanisms that may improve
our understating of how economies work. To become certain of the working of economies,
however, one needs to put the theoretical speculations to empirical tests. Empirical verification
of these theoretical possibilities is not only a crucial step towards scientific progress in
economics but also a vital step on the way to learn how to deal with various economic issues.
Besides the attractions of finding empirical evidence on the existence of general equilibrium
effects from the perspective of scientific progress, the importance of this goal has further swelled
from the need for economic development and the necessity of policy making in many countries
1
See Melitz (2003), Bernard, Eaton, Jensen, and Kortum (2003), Beaudry, Green, and Sand (2007) for a few
examples of the theoretical studies and Coe & Helpman (1995), Keller (1998), MacGarvie (2006), and Beaudry et
al. (2007) for a few examples of the empirical studies.
2
around the world. Some of these mechanisms are at the heart of various economic policies and in
the absence of empirical evidence on the existence of such spill-over mechanisms policy makers
at best tend to either take such mechanisms for granted (without having an understanding of the
magnitudes) or totally ignore them.2 Despite all the attraction and importance, however, the
empirical search for different general equilibrium mechanisms is confined by two major
limitations; lack of suitable data and lack of appropriate empirical strategy for identification.
Whereas the second one is bound only by human wisdom, all of the recent efforts for preparing
and easing the access to high-quality micro-econometric databases have very much made
achieving the goal possible.3
Although the impact of the G.E. mechanisms discussed above is conceived in the context of
the less developed countries, which are the main concern of this paper, it could well take place in
any economy as long as the general equilibrium mechanism in charge of the spill-over is at
work. As an example, a possible disappearance of the good jobs and their associated earnings in
the automobile industry or a relocation of a sizable share of the automobile workers in Canada or
the U.S. due to the international credit crunch crisis in the late 2008 early 2009, not only could
easily extend over to the industries with linkages to the automobile industry, but could in the
long-run as a result of the spill-over mechanism spread out to all the other industries and sectors
(as a collection of industries) throughout the local economies.
In fact, Beaudry et al. (2007) have constructed a theoretical search and bargaining model of a
labour market that incorporates a general equilibrium channel through which changes in
industrial composition of employment causes substantial impacts on wages in all sectors. They
estimate this impact for the U.S. local economies during 1970-2000 and find that sectoral level
wages act as strategic-complements and that the G.E. impact associated with changes in the
fraction of jobs in high paying sectors are very substantial and persistent. The magnitude of their
estimate for the total impact of a change in industrial composition that favours high paying
industries on average wages is about 3.5 times the conventional estimates from a commonly used
composition-adjustment approach which ignores the general equilibrium impact.
2
For example, according to theories in international trade countries opening up to trade tend in the long-run to
specialize in production of the commodities in which they have comparative advantages. This process in essence is
expected to change the sectoral composition of local economies in the long-run through a complex mechanism that
distributes the policy impact across different local geographical regions and is expected to change the wage premia
in different sectors. In the absence of solid empirical evidence, policymakers either take the impacts of these
structural changes for granted without having an understanding of their magnitude or totally ignore them.
3
Among which IPUMS–International, a project in the Minnesota Population Centre Data Projects at
www.ipums.umn.edu is to be highlighted as the provider of the data used in this study.
3
A G.E. mechanism of a similar magnitude could result in a devastating impact on the process
of growth and development in less developed countries in the event of implementing economic
policies that result in significant loss of good jobs. As a result, exploring the existence of such
spill-over mechanisms and empirical identification of this sort of all-sector complementarities
(i.e. G.E. mechanisms) should be of great value and critical importance to the policymakers in
the less developed countries.
The aim of the current study is essentially to explore the existence of such general equilibrium
mechanisms within Mexican cities. Specifically, we address the question that „Do compositional
changes in local sectoral employment or changes in the national sectoral wage premia have in
the long-run important spill-over effects on all sectoral wages in Mexican cities?‟ and exploit
geographical variation in sectoral composition of Mexican cities over time to see whether local
sectoral wages systematically change with the cities‟ distribution of sectoral employment and/or
national sectoral wage premia.
Changes in sectoral wage premia or sectoral composition of employment are both at the heart
of many economic policies and, specifically, could happen as a consequence of industrial or
trade policy changes. As discussed briefly above, these policy changes are the outcome of all the
efforts put to the achievement of the goals of economic development, economic growth, and
integration into the newly and increasingly globalized world economy.
It should also be noted that addressing the question of this research is as well quite important
from the perspective of welfare analysis of economic policies. If such G.E. mechanisms exist,
then the impact of industrial or trade policies does not necessarily die out within the sectors
which the policy is intended for (e.g. manufacturing or financial sector). If so, a proper welfare
analysis of such policies would require a holistic approach which recognizes the interconnectedness of different industries and sectors in the economy and therefore includes in its
analysis a broad definition of the impact area. This modification of definitions seems to be
crucial for an appropriate welfare analysis of different policies and essential for the progress of
economic development as a field.
In this study, we use Mexican Census Data for the period 1990-2000 to verify that in practice
long term changes in the distribution of local sectoral employment shares and/or changes in
national sectoral wage premia generate a general equilibrium mechanism through which all
sectoral wages in each city (metropolitan area) interact with each other, even sectoral wages of
4
the sectors that may have not experienced any changes in their employment share at the city
level or their wage premium at the national level.
Mexico during the 1990s is an interesting case study for the purpose of this paper due to
policy changes and economic events that took place during and just before this decade. As a
developing country, Mexico has a long history in implementing different economic policies. The
macroeconomic, industrial, and trade policies of the import-substitution era of the 1960s gave
place to the export-oriented policies of the 1970s and later to the substantial liberalization of the
economy in the 1980s which continued throughout the 1990s. Mexico entered the General
Agreement on Tariffs and Trade (GATT) in August 1986, which was the starting point of a
major trade and investment liberalization of the economy, and in November 1993 signed the
North American Free Trade Agreement (NAFTA), which came to effect on January 1 st 1994.4 In
addition to the impact of the comprehensive trade liberalization of the late 1980s throughout the
1990s and reinforcement of liberalization policies through NAFTA, in December 1994 as a
result of a sudden devaluation of Peso the country stepped into an economic crisis that although
recovered from it relatively fast (Kose et al., 2004), nevertheless suffered from its impacts.5
These policies, especially the trade liberalizations of the late 1980s and early 1990s, restructured
the Mexican economy with heterogeneous impacts on different regions and different sectors and
industries. These restructurings had a major impact on wages and reallocation of the Mexican
work force across different regions and sectors during the 1990-2000 period (among others see
Hanson, 2003; Tornell, Westermann, and Martinez, 2004; Chiquiar, 2005; Hanson, 2005a and
2005b; Richter, Taylor, and Yúnez-Naude, 2005; Aydemir and Borjas, 2007; Jordaan, 2008),
which has made the Mexico of the 1990s and interesting case study for this study. The choice of
Mexico as well allows a comparison between the results of this study and the ones observed for
the U.S. in Beaudry et al. (2008).
We will employ the theoretical model in Beaudry et al. (2008) to study inter-sectoral wage
spill-over in Mexico. IPUMS-International has provided us with the suitable data to carry out
this project. We find ample evidence on the existence of the general equilibrium mechanism
through which sectoral wages in each city causally depend on the city‟s employment-share
weighted sum of national sectoral wage premia (or the rents each city generates). The magnitude
of this relationship is substantially larger than the conventional existing accounting measures. A
4
5
For details see Ros, J. (1994).
For details see Edwards, Sebastain (1997).
5
pure6 compositional change7 in a city‟s sectoral employment that decreases the average wages in
that city by one dollar while ignoring any impacts on the sectoral wage levels (i.e., ignoring the
general equilibrium impact on wages as measured by the conventional composition-adjustment
approach), will in the long-run result in more than three dollar decrease in average wage in that
city through the general equilibrium impact from city‟s compositional change on wages; an
inter-sectoral general equilibrium impact on wages in a local economy that is more than three
times the conventional measures.
It is essential to clarify that what we are concerned about in this study is a general equilibrium
mechanism, not a demand effect. When jobs disappear or get created one expects to observe
impacts from changes in the overall demand throughout the whole economy. However, we will
show that even controlling for the overall level of demand for labour at city or city-sector level,
we still observe a significant complementarity between sectoral wages.
An impact of this magnitude calls for excessive caution, especially in the less developed
countries, in implementing industrial and trade policies that may result in severe loss of good
jobs or sizable relocation of work force at the expense of high paying sectors. In addition, this
finding implies a holistic approach in welfare analysis of such policy changes.
The paper is structured as follows. Section 2 briefly reviews the theoretical model and takes
away a guideline that forms the empirical strategy discussed in section 3. Section 4 explains the
data, section 5 reports the estimation results, and section 6 concludes.
2
Theoretical Model
In this section we borrow the theoretical model in Beaudry et al. (2008) to show how in a
general equilibrium search and bargaining framework a change in sectoral composition of
employment in a local economy or a change in sectoral wage premia affects all sectoral wages in
the economy, even in sectors that are not part of the change. It is important to note that such
wage effects are different from overall demand effects in the economy. In the event of losing old
jobs or creation of new jobs in a sector in a local economy, one would expect a depreciating or
appreciating impact on local wages due to respective changes in overall demand. However, the
6
„Pure‟ in the sense that, the overall demand for labour does not change. Throughout this paper, we highlight and
emphasize that the general equilibrium impacts we try to identify are different from overall demand effects in the
economy.
7
The initial compositional change can be the result of a change in trade or industrial policy or due to an economic
turmoil.
6
model used here shows that even holding the overall employment rate constant, shifts in the
sectoral composition of employment or changes in sectoral wage premia have an impact on
wages in all sectors. The model provides us with a structural relationship that guides the
empirical strategy in exploring the existence of the kind of general equilibrium mechanism
discussed here.
The economy is characterized by C local economies (cities) in which firms produce goods and
individuals seek employment in I sectors. To produce and make profits, firms create new jobs
and seek to fill the costly vacancies and weight up the discounted costs of keeping those
vacancies versus discounted profits they make by employing workers and paying the city-sector
wages. In the same way as firms, individuals compare the discounted benefits from being
unemployed with being an employee and receiving the city-sector wage. There is a random
matching process that matches workers with firms and as usual, in a steady-state equilibrium of
this economy the value functions satisfy the standard Bellman relationship.
All throughout the model it is assumed that workers are not mobile across cities. It suffices
for the purpose of the current study to refer to the reasoning in Beaudry et al. (2007, p.16-18)
that allowing for inter-city immigration will not cancel the kind of general equilibrium
mechanism we are looking for here.
There is a final good, denoted Y, which is an aggregation of output from a total of I sectors:
1
I
𝜒
𝑎𝑖 𝑍𝑖
𝑌=
𝜒
,
𝜒 < 1.
(1)
𝑖=1
The price of the final good is normalized to 1, while the price of the good produced in sector i is
given by 𝑝𝑖 . The total quantity of each sectoral good produced in the national economy (Zi) is the
sum of local productions of that good.
Filling a vacancy generates a flow of profits for the firm given by:
𝑝𝑖 − 𝑤𝑖𝑐 + 𝜖𝑖𝑐 ,
7
where 𝑤𝑖𝑐 is the wage, 𝜖𝑖𝑐 is a city-sector specific cost advantage, and
𝑐 𝜖𝑖𝑐
= 0. Letting 𝑉 𝑓
denote the discounted value of profits from a filled position and 𝑉 𝑣 the discounted value of a
vacancy, in steady state the value functions must satisfy the standard Bellman relationship:
𝑓
𝑓
𝜌𝑉𝑖𝑐 = 𝑝𝑖 − 𝑤𝑖𝑐 + 𝜖𝑖𝑐 + 𝛿 𝑉𝑖𝑐𝑣 − 𝑉𝑖𝑐 ,
(2)
where 𝜌 is the discount rate and 𝛿 is the exogenous death rate of matches. The discounted value
of profits from a vacant position must satisfy:
𝑓
𝜌𝑉𝑖𝑐𝑣 = 𝜙𝑐 𝑉𝑖𝑐 − 𝑉𝑖𝑐𝑣 ,
(3)
where 𝜙𝑐 is the probability a firm fills a posted vacancy. Here, for simplicity and with no loss of
generality, the periodical cost to maintain the vacancy is assumed to be zero.
The discounted value of being employed in sector i in city c, denoted 𝑈𝑖𝑐𝑒 , must as well satisfy
the Bellman equation:
𝜌𝑈𝑖𝑐𝑒 = 𝑤𝑖𝑐 + 𝛿 𝑈𝑖𝑐𝑢 − 𝑈𝑖𝑐𝑒 ,
(4)
where 𝑈𝑖𝑐𝑢 represents the value associated with being unemployed when the worker‟s previous
job was in sector i.
Representing the probability that an unemployed individual finds a job with 𝜓𝑐 and the
probability that an individual finding a job gets a random draw from jobs in all sectors (including
sector i) rather than being assigned a match in the previous sector with 1 − 𝜇, the value
associated with being unemployed satisfies the Bellman relationship:
𝜌𝑈𝑖𝑐𝑢 = 𝑏 + 𝜏𝑐 + 𝜓𝑐 𝜇𝑈𝑖𝑐𝑒 + 1 − 𝜇
𝑗
𝜂𝑗𝑐 𝑈𝑗𝑐𝑒 − 𝑈𝑖𝑐𝑢 ,
(5)
where 𝑏 is the utility flow of an unemployment benefit, 𝜏𝑐 is a city specific amenity term, and
𝜂𝑗𝑐 represents the fraction of city c’s vacant jobs that are in sector j. As Beaudry et al. (2008)
8
argue, the key assumption for being able to solve the model explicitly is that workers can only
search while being unemployed.
Once a match is made, workers and firms bargain a wage. Assuming that there are always
gains from trade between workers and firms for all jobs created in equilibrium, the bargaining is
set according to the following rule:
𝑓
𝑉𝑖𝑐 − 𝑉𝑖𝑐𝑣 = 𝑈𝑖𝑐𝑒 − 𝑈𝑖𝑐𝑢 × 𝜅 ,
(6)
where 𝜅 indicates the relative bargaining power of workers and firms so that the higher it is, the
lower is the bargaining power of the workers.
The probability a match is made is determined by the matching function:
𝑀 𝐿𝑐 − 𝐸𝑐 , 𝑁𝑐 − 𝐸𝑐 ,
where 𝐿𝑐 is the total number of workers in city c, 𝐸𝑐 is the number of employed workers (or
matches) in city c, and 𝑁𝑐 =
𝑖
𝑁𝑖𝑐 is the number of jobs in city c, with 𝑁𝑖𝑐 being the number of
jobs in sector i in city c. Given the exogenous death rate of matches, 𝛿, and assuming a CobbDouglas form for the match function, the steady state condition is given by:
𝛿𝐸𝑅𝑐 = 𝑀
𝑁𝑐
1 − 𝐸𝑅𝑐 ,
− 𝐸𝑅𝑐
𝐿𝑐
= 1 − 𝐸𝑅𝑐
𝜍
𝑁𝑐
− 𝐸𝑅𝑐
𝐿𝑐
1−𝜍
,
(7)
where 𝐸𝑅𝑐 is the employment rate. It follows that the proportion of filled jobs and vacant jobs in
sector i can be expressed as 𝜂𝑖𝑐 =
𝑁𝑖𝑐
𝑖 𝑁 𝑖𝑐
.
The number of jobs created in sector i in city c, 𝑁𝑖𝑐 , is determined by the free entry condition:
𝑐𝑖𝑐 = 𝑉𝑖𝑐𝑣 ,
(8)
where 𝑐𝑖𝑐 is the cost of creating a marginal job and is necessarily increasing in the number of
new jobs being created locally in that sector to have cities with employment across a wide range
9
of sectors. Cities could also have a comparative advantage in creating certain types of jobs
relative to others. Therefore, it is assumed that 𝑐𝑖𝑐 is a decreasing function of the city-sector
specific measure of advantage denoted 𝛺𝑖𝑐 :
𝑐𝑖𝑐 =
𝑁𝑖𝑐
.
𝛶𝑖 + 𝛺𝑖𝑐
where 𝛶𝑖 reflects any systematic differences in cost of entry across sectors, which allows to
assume that
𝑐
𝛺𝑖𝑐 = 0.
Finally, the probability an unemployed worker finds a match and the probability a firm fills a
vacancy respectively satisfy:
𝛿𝐸𝑅𝑐
1 − 𝐸𝑅𝑐
𝜓𝑐 =
and 𝜙𝑐 =
1 − 𝐸𝑅𝑐
𝛿𝐸𝑅𝑐
𝜍
1−𝜍
.
(9)
A steady state equilibrium in which the price of sectoral output is taken as given, consists of
value of 𝑁𝑖𝑐 , 𝑤𝑖𝑐 , and 𝐸𝑅𝑐 that satisfy equations (6), (7), and (8). These equilibrium values will
depend on (among other things) the city specific productivity parameters 𝛺𝑖𝑐 and 𝜖𝑖𝑐 . An
equilibrium for the entire economy has the additional requirement that the prices for sectoral
goods must ensure that markets for these goods clear.
Solving the model for city-sector wages gives the following relationship:
𝑤𝑖𝑐 = 𝛾𝑐0 + 𝛾𝑐1 𝑝𝑖 + 𝛾𝑐2
𝜂𝑗𝑐 𝑤𝑗𝑐 + 𝛾𝑐1 𝜖𝑖𝑐 ,
8
(10)
𝑗
where the coefficients are:
𝛾𝑐0 =
𝑏 + 𝜏𝑐
𝜓𝑐 1 − 𝜇 𝜌 + 𝜓𝑐
1+
𝜌 𝜌 + 𝜓𝑐 + 𝛿 + 𝛿𝜓𝑐 1 − 𝜇
8
+
𝜌 + 𝜓𝑐 + 𝛿
𝜌 + 𝜙𝑐 + 𝛿 𝜅
The appendix at the end of the paper explains in detail the steps required for deriving the main equations in
Beaudry et al. (2007) following their numbering of equations.
10
𝛾𝑐1 =
𝛾𝑐2 =
1
𝜌 + 𝜓𝑐 1 − 𝜇 𝜌 + 𝜙𝑐 + 𝛿 𝜅
1+
𝜌 𝜌 + 𝜓𝑐 + 𝛿 + 𝛿𝜓𝑐 1 − 𝜇
1
1+
𝜌
𝜌 𝜌 + 𝜓𝑐 + 𝛿 + 𝛿𝜓𝑐 1 − 𝜇
+
𝜓𝑐 1 − 𝜇
𝜌 + 𝜙𝑐 + 𝛿 𝜓𝑐 1 − 𝜇 𝜅
∙ 1+
𝛿
𝜌 + 𝜓𝑐
.
These coefficients are implicitly functions of the employment rate through 𝜓𝑐 and 𝜙𝑐 .
The derived equation for city-sector wages captures the notion that in a search and matching
framework, sectoral wages act as strategic complements; that is, high wages in one sector are
associated with high wages in other sectors. The strength of this strategic complementarity is
captured by 𝛾𝑐2 . Notice that if 𝜇 = 1, that is when workers are immobile across sectors, this
effect disappears and wages are determined solely by value of marginal product. Also, notice
that 𝛾𝑐2 is increasing in 𝜅, implying that sectoral wages are more strongly positively linked the
weaker is worker‟s bargaining power. This suggests that even in environments where workers
have minimal bargaining power, it is possible that 𝛾𝑐2 is high and that wages act as strong
strategic complements.
According to equation (10), a pure sectoral composition shift that causes a one unit increase in
the average city wage,
𝑗
𝜂𝑗𝑐 𝑤𝑗𝑐 , increases the within sector wages by 𝛾𝑐2 in all sectors. But
these increases in all within sector wages cause the average wage to increase by another 𝛾𝑐2
units, inducing a further round of adjustments. The total effect of the pure change in sectoral
composition on the average wage would therefore be
1
1−𝛾𝑐2
.
As Beaudry et al. (2008) indicate, equation (10) has the structure of the classic reflection or
social interaction problem (Manski, 1993; Moffitt, 2001) in that a city-sector wage depends on
the average wages in that city. To make progress toward estimating such a relationship, they
overcome the simultaneity inherent in equation (10) by linking the city-sector wages in
equilibrium to national prices of sectoral products and national averages of city-sector wages
across cities as a result of which equation (10) transforms to:
11
𝑤𝑖𝑐 = 𝑑𝑖𝑐 +
𝛾𝑐1
𝛾𝑐2
𝛾1 1 − 𝛾𝑐2
𝜂𝑐𝑗 𝑤𝑗 − 𝑤1 + 𝛾𝑐1
𝑗
𝛾𝑐2
1 − 𝛾𝑐2
𝜂𝑐𝑗 𝜖𝑗𝑐 + 𝛾𝑐1 𝜖𝑖𝑐 ,
(11)
𝑗
where 𝑤𝑖 − 𝑤1 is the national level wage premium in sector i relative to sector 1, 𝛾1 is the
average of 𝛾𝑐1 across cities, 𝑑𝑖𝑐 = 𝑑𝑖𝑐 −
𝛾𝑐1
1
𝐶
𝛾𝑐2
1−𝛾𝑐2
𝐶
𝑐 =1 𝛾𝑐1
𝛾𝑐1
𝛾𝑐2
𝛾1 1−𝛾𝑐2
𝑗
𝜂𝑐𝑗 𝑑𝑗 , 𝑑𝑖𝑐 = 𝛾𝑐0 1 +
𝛾𝑐2
1−𝛾𝑐2
+ 𝛾𝑐1 𝑝𝑖 +
𝑝1 with 𝑝𝑖 being the price of good i that is the product of sector i, and 𝑑𝑗 =
𝜖𝑗𝑐 − 𝜖1𝑐 is a sector specific constant.
Equation (11) shows how city-sector wage depend on the sectoral composition of a city‟s
employments and the national wage premia of sectors both captured by
𝑗
𝜂𝑐𝑗 𝑤𝑗 − 𝑤1 . We
will denote this term by 𝑅𝑐 and refer to it as average city rent. Notice that a high value for
average city rent indicates that the city‟s employment is concentrated in higher paying sectors.
So far, the employment rate in a city hidden in the 𝛾 parameters and the sectoral composition
were taken as given. To capture the dependence of wages on the city‟s employment rate more
explicitly, Beaudry et al. (2008) take a linear approximation of (11) around the point where cities
1
have identical sectoral composition (𝜂𝑖𝑐 = 𝜂𝑖 = 𝐼 ) and employment rates (𝐸𝑅𝑐 = 𝐸𝑅), which
arises when 𝜖𝑖𝑐 = 0 and 𝛺𝑖𝑐 = 0. Furthermore, to eliminate the city level fixed effects driven by
the amenity term, 𝜏𝑐 , they focus on the differences in wages within a city-sector cell across two
steady state equilibria, denoted ∆𝑤𝑐𝑖 :
∆𝑤𝑖𝑐 = ∆𝑑𝑖 +
𝛾2
∆
1 − 𝛾2
𝜂𝑐𝑗 𝑤𝑗 − 𝑤1 + 𝛾𝑖5 ∆𝐸𝑅𝑐 + ∆𝜉𝑖𝑐 ,
(12)
𝑗
where ∆𝑑𝑖 is a sector specific effect (∆𝑑𝑖 = 𝛾1
𝛾2
1−𝛾2
∆𝑝1 + 𝛾1 ∆𝑝𝑖 ) that can be captured in an
empirical specification by including sector dummies, and ∆𝜉𝑖𝑐 = 𝛾1 ∆𝜖𝑖𝑐 + 𝛾1
𝛾2
1−𝛾2
1
𝑗 𝐼 ∆𝜖𝑗𝑐
is
the error term, with I being the total number of sectors. In (12), the 𝛾 coefficients are the same as
before except that now they are evaluated at common match probabilities, 𝜓 and 𝜙. The added
coefficient, 𝛾𝑖5 , reflects the effect of a change in the employment rate on wage determination; an
effect which depends on all the parameters of the model. As they argue, this coefficient may vary
across sectors since the effects of a tighter labour market may affect the bargaining power of
12
firms in a sector with a high value added product differently from the bargaining power of firms
in a sector with a low value added product.
In this study we are interested in estimating the coefficient on the changes in average city rent
𝛾2
in (12);
1−𝛾2
. If we could estimate this coefficient consistently, we would obtain an estimate of
the extent of (structural) city-level strategic complementarity between wages in different sectors
by backing out 𝛾2 . The coefficient
𝛾2
1−𝛾2
is of interest in its own right as it provides an estimate
of the total – direct and feedback – effect of a one unit increase in average city wages on within
sector wages, as opposed to 𝛾2 , which provides the partial unidirectional effect.
If we examine wages in the same sector in different cities, a positive value for
𝛾2
1−𝛾2
implies
that for example agriculture wages will be higher in cities where employment is more heavily
weighted toward high rent sectors, where high rent sectors are defined in term of national level
wage premia. This arises in the model because the workers in that sector have better outside
option to use when bargaining with firms in a higher rent city.
The conventional accounting measure of the impact of a compositional change in the context
of the model above is captured by 𝐴𝑐 =
𝑗
𝜂𝑗𝑐𝑡 +1 − 𝜂𝑗𝑐𝑡
𝑤𝑗𝑡 − 𝑤1𝑡 =
𝑗
𝜂𝑗𝑐𝑡 +1 − 𝜂𝑗𝑐𝑡 𝜐𝑗𝑡 ;
the change in average wages due to the compositional change while keeping the wages constant.
In the absence of wage complimentarity among sectors in a city, the accounting measure will be
the total impact of this compositional change. However, at the presence of the general
equilibrium mechanism through which the city-sector wages in a city become complementarily
related according to equation (12) or (10), the dynamics of the model imply that the total impact
𝛾
of such a compositional change is equal to 𝐴𝑐 + 1−𝛾2 ∆𝑅𝑐 . The change in city rent can be
2
decomposed as:
∆𝑅𝑐𝑡 =
𝜂𝑐𝑖𝑡 +1 − 𝜂𝑐𝑖𝑡 𝜐𝑖𝑡 +
𝑖
𝜂𝑐𝑖𝑡 +1 𝜐𝑖𝑡 +1 − 𝜐𝑖𝑡 = 𝐴𝑐 +
𝑖
𝜂𝑐𝑖𝑡 +1 𝜐𝑖𝑡 +1 − 𝜐𝑖𝑡 .
𝑖
𝛾
𝛾
Thus, the total impact becomes 𝐴𝑐 + 1−𝛾2 ∆𝑅𝑐 = 1 + 1−𝛾2
2
2
𝛾
𝐴𝑐 + 1−𝛾2
2
𝑖
𝜂𝑐𝑖𝑡 +1 𝜐𝑖𝑡 +1 − 𝜐𝑖𝑡
𝛾
where 1−𝛾2 now clearly indicates the magnitudes of order the general equilibrium impact is larger
2
than the conventional accounting measure in the absence of any changes in wage premia
13
impacted form the compositional change. If
𝛾2
1−𝛾2
is estimated to be zero then the accounting
measure completely captures the effects of the composition shift.
Obviously the endogeneity of the variables in the estimating equation (12) puts the success of
the identification strategy at danger. Beaudry et al. (2008) have discussed the identification
issues of the model in detail. Briefly, the success of the estimation strategy relies upon the
properties of the error term in (12). The requirement for OLS to give consistent estimates of the
coefficients in (12) can be expressed as follows:
11
plim
𝐶,𝐼→∞ 𝐼 𝐶
𝐼
𝐶
𝑖=1 𝑐 =1
11
∆𝑅𝑐 ∆𝜉𝑖𝑐 = plim
𝐶,𝐼→∞ 𝐼 𝐶
Given that this error term is ∆𝜉𝑖𝑐 = 𝛾1 ∆𝜖𝑖𝑐 + 𝛾1
𝛾2
1−𝛾2
𝐶
𝐼
∆𝑅𝑐
𝑐 =1
∆𝜉𝑖𝑐 = 0.
𝑖=1
1
𝑗 𝐼 ∆𝜖𝑗𝑐 ,
this effectively reduces to the
properties of 𝜖𝑖𝑐 . Both the average city rent and employment ratios are endogenous due to the
fact that industrial composition, as captured by 𝜂𝑖𝑐 , is correlated with the 𝜖‟s.
To discuss the identification issues, they express 𝜖𝑖𝑐 as the sum of a common component,
which reflects absolute advantage, and a second component which captures relative advantage.
𝜖
For example, using their notation, let 𝜖𝑐𝑡 represent the common component of the 𝜖‟s and let 𝜐𝑖𝑐𝑡
𝜖
𝜖
represent the relative advantage component, with 𝜖𝑖𝑐𝑡 = 𝜖𝑐𝑡 + 𝜐𝑖𝑐𝑡
, where by definition the 𝜐𝑖𝑐𝑡
‟s
sum to zero across sectors within a city. As they show, identification of the parameters in (12)
depends primarily on properties of the absolute advantage component 𝜖𝑐𝑡 . They show that under
a rather minimal assumption that common component 𝜖𝑐𝑡 behaves as a random walk (that is, that
the increments in 𝜖𝑐𝑡 are independent of the past) a specific IV approach can be devised to
consistently estimate all the coefficients in (12) recognizing the fact that the average city rent
and employment ratios are all endogenous.
They show that consistency of the OLS relies as well (in addition to the assumption that the
absolute advantage component 𝜖𝑐𝑡 be independent of the past) on a more stringent, but still
possible, condition on the 𝜖‟s that the absolute advantage term be independent of all relative
advantage terms (present and past). This implies that whatever drives general city performance is
not related to a particular pattern of sectoral structure. While this assumption is stronger, they
show that the two assumptions together are sufficient for the consistency of OLS estimation of
equation (12). Under the first assumption (random walk behaviour of the absolute advantage
14
terms) and if the IV strategy is successful, by comparing the OLS and IV estimates one would be
able to evaluate whether the data are supportive of the more demanding assumption assuming
that the weaker assumption is valid.
3
Empirical Strategy
The aim of the current study is to explore the existence of general equilibrium impacts from
long term compositional changes in sectoral employment or long term changes in the sectoral
wage premia on wages in all sectors within Mexican cities. Specifically, the question asked here
is that “Do long term compositional changes in sectoral employment and/or long term changes
in the sectoral wage premia have important general equilibrium impacts on wages in all sectors
within Mexican cities?‟ Briefly, to empirically address this question we explore geographical
variation in city rents (as measured by weighted average of national sectoral wage premia using
city-sector employment-share as weights) over two far enough points in time to see whether
changes in sectoral wages are systematically related to changes in city rents that are resulting
from local movements of employment between sectors and/or changes in national sectoral wage
premia. We use the geographical variation in this relationship across different cities to infer
whether there is a systematic co-movement that is statistically significant across all cities. To
ensure about the causality, different instrumental variable (IV) approaches are envisioned and
implemented and a few more potential issues are addressed.
To consider the impact of sectoral composition of employment and/or sectoral wage premia in
the local economies, the measure of city rent derived from the theoretical model is used. The city
rent is measured as local-employment-share weighted average of national sectoral wage premia:
𝑅𝑐𝑡 =
𝑖
𝑒𝑐𝑖𝑡
𝑤𝑖𝑡
∙
−1 ,
𝑤1𝑡
𝑖 𝑒𝑐𝑖𝑡
(13)
where 𝑒𝑐𝑖𝑡 indicates employment in sector i, city c, in year t and 𝑤𝑖𝑡 indicates the national
average wage intrinsic to sector i in year t calculated according to:
1
𝑤𝑖𝑡 = ∙
𝐶
𝐶
𝑤𝑐𝑖𝑡 ,
𝑐=1
15
(14)
with C indicating the total number of local economies and 𝑤𝑐𝑖𝑡 denoting the intrinsic wages to
sector i, city c, at time t. In the same way, 𝑤1𝑡 measures the average intrinsic wage to sector one
across all cities in the national economy:
𝑤1𝑡 =
1
𝐶
𝑤𝑐1𝑡 .
(15)
𝑐
Notice that in (13) we have adjusted the formula by dividing the city rent index by 𝑤1 to allow
for using the logarithm of the city-sector-person wages in estimating the sectoral wage premia.
𝑤 𝑖𝑡
In the construction of the city rent R,
𝑤 1𝑡
− 1 measures the intrinsic wage premium of sector
i in year t choosing sector one as the base for comparison. Since each local geographical region
we consider here is large enough to embody a labour market, and therefore to encompass all
different sectors, choosing a specific sector as the base or changing it is not of any significant
importance. It is simply a way to normalize the calculation of wage premia.
To illustrate how this approach for measuring the city rents is useful, it helps considering the
following situation. In a given year, between two otherwise similar cities, the one with higher
concentration of employment in sectors that intrinsically offer higher wages is expected to have a
higher measure of city rent. This condition is sufficient here because in the construction of the
measure, national wage premia are used. In this way, wage realization in each city has an
insignificant role in the construction of city rents measured here and therefore, a city with
relatively higher wages in all sectors is not necessarily going to have a higher measure of city
rent.
To calculate the city rents from wage and employment data, its formula can be updated in the
following way:
𝑅𝑐𝑡 =
𝑖
𝑒𝑐𝑖𝑡
𝜛𝑖𝑡 ,
𝑖 𝑒𝑐𝑖𝑡
(16)
where 𝜛𝑖𝑡 is the coefficient of the sector dummy in the following estimating equation separately
estimated for different years:
16
𝑙𝑛 𝑊𝑘𝑐𝑖 = 𝛼 + 𝜸𝜿𝑘𝑐𝑖 + 𝜛𝑑𝑖 + 𝜀𝑘𝑐𝑖 ,
(17)
where 𝑊𝑘𝑐𝑖 is the observed wage received by person k in city c working in sector i, 𝜿𝑘𝑐𝑖 denotes
an array of the demographic attributes of individual k in city c who works in sector i, 𝑑𝑖 indicates
a full set of dummies for all sectors other than the base, and ln(.) is the natural logarithm
function. In equation (17), estimates of 𝜛 in each year capture the national level sectoral wage
premia relative to the base sector as an average across cities.
In the theoretical model, the worker is abstracted from all its attributes and the wage
considered in that model is not determined by any of those attributes. It is therefore necessary to
adjust the data on wages for all the attributes for which information is available. The residuals in
the following estimating equation can be considered as regression adjusted wages for the
attributes of the workers:
𝑙𝑛 𝑊𝑘𝑐𝑖 = 𝛼 + 𝜸𝜿𝑘𝑐𝑖 + 𝜔𝑘𝑐𝑖 ,
(18)
Equation (18) can be estimated separately for each year and the weighted (using sampling
weights) average of the residuals (𝜔𝑘𝑐𝑖 ) over k from each round of estimation generates the citysector wages for that year. Later on, these averages form the left-hand-side variable in the main
regressions (𝑤𝑐𝑖𝑡 ).
Having generated the appropriate dependent and explanatory variables, the structural
regression of interest, which closely matches equation (12) derived from the theoretical model,
would be the following:
𝑤𝑐𝑖𝑡 = 𝑑𝑡 + 𝑑𝑖𝑡 + 𝑑𝑐 + 𝛽𝑅𝑐𝑡 + 𝜃𝑖 𝐸𝑅𝑐𝑡 + 𝜉𝑐𝑖𝑡 ,
(19)
where 𝑑𝑡 is a year dummy, 𝑑𝑖𝑡 is a full set of sector-year dummies other than the one associated
to the base sector, 𝑑𝑐 is a full set of city dummies, 𝑅𝑐𝑡 is the city rent measured in year t, 𝐸𝑅𝑐𝑡
denotes the employment ratio in city c in year t, and finally 𝜉𝑐𝑖𝑡 is the error term.
To avoid likely correlation between the residuals and the city fixed effects, equation (19) can
be estimated in the differences. Thus, the actual estimating equation becomes:
17
∆𝑤𝑐𝑖𝑡 = ∆𝑑𝑡 + ∆𝑑𝑖𝑡 + 𝛽∆𝑅𝑐𝑡 + 𝜃𝑖 ∆𝐸𝑅𝑐𝑡 + ∆𝜉𝑐𝑖𝑡 .
(20)
Since the Mexican data suitable for this study is only available for two distinct years, ∆𝑑𝑡
becomes a constant playing the role of an intercept and ∆𝑑𝑖𝑡 becomes nothing but a full set of
sector dummies excluding the base sector.
In estimation of equation (20) and carrying out the inferences, consistency of the estimates is
crucial. If OLS consistently estimates equation (20) the goal would be to test the null hypothesis
that 𝛽 = 0. If the null cannot be rejected, one can disregard the inter-sectoral wage interactions
in the process of wage determination in local economies. On the other hand, a statistically
significant and positive coefficient is indicative of the existence of a general equilibrium
mechanism through which the sectoral-employment-share weighted average of sectoral wage
premia in each local economy has a significant impact on wages in all sectors as predicted by the
model. This would make disregarding the general equilibrium impact on wages costly in terms
of economic policy.
3.1
Endogeneity
As discussed in previous chapters and is reviewed in detail in the appendix (2.H), consistency
of OLS relies on the condition that 𝑝𝑙𝑖𝑚𝐶,𝐼→∞
𝛾
𝛾1 ∆𝜖𝑖𝑐 + 𝛾1 1−𝛾2
2
1
𝑗 𝐼 ∆𝜖𝑗𝑐
11
𝐼𝐶
𝐼
𝑖=1
𝐶
𝑐 =1 ∆𝑅𝑐 ∆𝜉𝑖𝑐
= 0. Knowing that ∆𝜉𝑖𝑐 =
𝜖
and given the decomposition 𝜖𝑖𝑐 = 𝜖𝑐 + 𝜐𝑖𝑐
, where 𝜖𝑐 is the absolute
city advantage and 𝜐𝑖𝑐𝜖 is the comparative city advantage with
𝜖
𝑖 𝜐𝑖𝑐
= 0, OLS will consistently
estimate equation (20) if the two following conditions are satisfied:

The stronger condition that changes in absolute advantage for a city is independent of the
changes in comparative advantage.

The weaker condition that changes in the absolute advantage for a city are independent of
the initial set of comparative advantage factors for that city.
However, obviously these conditions may not hold together, in which case we should consider
using instrumental variables for consistently estimating equation (20). The instruments we have
here are exogenous with respect to ∆𝜉𝑖𝑐 only if the weaker assumption is satisfied; i.e., under the
condition that changes in the absolute advantages for a city are independent of the initial set of
comparative advantage factors for that city.
18
The instruments are constructed based on the two different ways of decomposing ∆𝑅𝑐𝑡 :
∆𝑅𝑐𝑡 =
𝜂𝑐𝑖𝑡 +1 𝜐𝑖𝑡 +1 −
𝑖
=
𝜂𝑐𝑖𝑡 𝜐𝑖𝑡
𝑖
𝜂𝑐𝑖𝑡 +1 𝜐𝑖𝑡 +1 −
𝑖
=
𝜂𝑐𝑖𝑡 +1 𝜐𝑖𝑡 +
𝑖
𝜂𝑐𝑖𝑡 +1 𝜐𝑖𝑡 −
𝑖
𝑖
𝜂𝑐𝑖𝑡 +1 𝜐𝑖𝑡 +1 − 𝜐𝑖𝑡 = ΔR1ct + ΔR2ct ,
𝜂𝑐𝑖𝑡 +1 − 𝜂𝑐𝑖𝑡 𝜐𝑖𝑡 +
𝑖
𝜂𝑐𝑖𝑡 𝜐𝑖𝑡
(21)
𝑖
or,
∆𝑅𝑐𝑡 =
𝜂𝑐𝑖𝑡 +1 𝜐𝑖𝑡 +1 −
𝑖
=
𝜂𝑐𝑖𝑡 𝜐𝑖𝑡
𝑖
𝜂𝑐𝑖𝑡 +1 𝜐𝑖𝑡 +1 −
𝑖
=
𝜂𝑐𝑖𝑡 𝜐𝑖𝑡 +1 +
𝑖
𝜂𝑐𝑖𝑡 𝜐𝑖𝑡 +1 −
𝑖
𝑖
𝜂𝑐𝑖𝑡 𝜐𝑖𝑡 +1 − 𝜐𝑖𝑡 = ΔR3ct + ΔR4ct .
𝜂𝑐𝑖𝑡 +1 − 𝜂𝑐𝑖𝑡 𝜐𝑖𝑡 +1 +
𝑖
𝜂𝑐𝑖𝑡 𝜐𝑖𝑡
(22)
𝑖
The first decomposition in (21) provides us with two instruments that are exogenous under
only the weaker assumption of the two assumptions required for the consistency of OLS as
mentioned above:
𝐼𝑉1 =
𝜂𝑐𝑖𝑡 +1 − 𝜂𝑐𝑖𝑡 𝜐𝑖𝑡 ,
(23)
𝑖
𝐼𝑉2 =
𝜂𝑐𝑖𝑡 +1 𝜐𝑖𝑡 +1 − 𝜐𝑖𝑡 ,
(24)
𝑖
where 𝜂𝑐𝑖𝑡 +1 =
𝑒 𝑐𝑖𝑡 +1
𝑖 𝑒 𝑐𝑖𝑡 +1
and 𝑒𝑐𝑖𝑡 +1 = 𝑒𝑐𝑖𝑡
𝑒 𝑖𝑡 +1
𝑒 𝑖𝑡
= 𝑒𝑐𝑖𝑡
𝑒 𝑖𝑡 +1− 𝑒 𝑖𝑡
𝑒 𝑖𝑡
+ 1 = 𝑒𝑐𝑖𝑡 𝑔𝑖 + 1 with
𝑔𝑖 being the growth rate of employment in industry i at the national level.
19
The second decomposition in (22) can be used to construct two additional instruments that are
also exogenous under the same weaker condition for the consistency of OLS mentioned above:
𝐼𝑉3 =
𝜂𝑐𝑖𝑡 +1 − 𝜂𝑐𝑖𝑡 𝜐𝑖𝑡 +1 ,
(25)
𝑖
𝐼𝑉4 =
𝜂𝑐𝑖𝑡 𝜐𝑖𝑡 +1 − 𝜐𝑖𝑡 .
(26)
𝑖
Notice that all these instruments are constructed based on the initial sectoral composition of
employment in a city being weighted in different fashions in each instrument. Thus, another set
of instruments that could be used is the set of initial employment shares of sectors in cities,
𝜂𝑐𝑖𝑡 ‟s, as they are, and allowing for the estimation process to decide in which fashion each
should be used. That is what we are referring to as IV5. Since there are 14 sectors in the
construction of ∆𝑅𝑐𝑡 , IV5 is a maximum of 14 instruments.9
Assuming that the instruments are valid, the equality between OLS and IV estimates is an
indication of two important points: that OLS is consistently estimating the coefficients in (20)
and that the stronger condition required for consistency of OLS is valid in the data (in other
words, the change in absolute advantage in a city are independent of the changes in comparative
advantage). The instruments are valid under the rather weak assumption that the change in
absolute advantage is independent from the initial set of comparative advantage, as a result of
which the instruments are exogenous, and under the condition that the instruments are strongly
correlated with ∆𝑅𝑐𝑡 according to the first stage statistics, so that we are not dealing with weak
instruments.
After making sure of the consistency of the OLS estimates it only remains to make sure that
the OLS estimates are robust at the presence of other existing alternative explanations for
differences in wages across cities such as those related to city size, education levels (Moretti,
2004; Acemoglu and Angrist, 1999), and diversity of employment in a city (Glaeser, Kallal,
Scheinkman, and Shleifer, 1992). We will add additional variables related to these alternative
explanations to equation (20) to ensure of the robustness of the OLS estimate.
9
In specifications where collinearity is detected, a lower number of instruments in IV5 may be used.
20
Before moving to the estimation results in the next sections, it is important to mention that
similar to Δ𝑅𝑐𝑡 , Δ𝐸𝑅𝑐𝑡 is also likely to be endogenous as explained at the end of section two. To
deal with the endogeneity of this variable, we follow the same approach Beaudry et al. (2007)
are following, which is similar to the Blanchard and Katz (1992). In particular, we use
𝑖
𝜂𝑐𝑖𝑡 𝑔𝑖
as the instrument, were as before 𝑔𝑖 is the growth rate of employment in industry i at the national
level. It can be shown that under the same weaker assumption of the two assumptions required
for the consistency of OLS, under which the other instruments are exogenous, this instrument is
also exogenous. In the same way as well, the validity of this instrument relies upon being highly
correlated with Δ𝐸𝑅𝑐𝑡 in the first stage.
3.2
Selection
In this section we try to address the issue of selection that the empirical strategy may suffer
form. If in practice workers are mobile across cities and choose where to live and work by
comparing different cities in terms of their personal priorities, then individuals currently
observed leaving in a city are not a random sample of the population. An individual‟s wage is
not observed in any city other than the one they choose to be a resident of (born there and not
moved anywhere else or born somewhere else and moved to this city). This will compromise one
of the conditions required for the consistency of OLS estimates of these regressions, being the
required zero mean residual. In practice, in equations (17), (18), and (19) we are actually dealing
with a conditional residual mean term, conditioned on the wage figure being observable. It is not
clear if this conditional error mean term is actually zero in the self selected sample and if this is
not the case and the conditional residual mean term is correlated with other regressors, then OLS
is no more consistent.
To be more explicit, if suddenly a group of individuals move from a city to another city in
expectation of higher wages for reasons not observable to us but related to the structure of wages
(∆𝜉𝑖𝑐 ), the change in city rent in equation (20) will also capture the impact of this sort of
movements and the OLS estimation of this equation may give significantly-different-from-zero
estimates of the relationship we are interested in without it really existing. Thus, it is very
important to adjust the empirical strategy to correct for this possibility. In addressing this issue,
we implement the approach in Dahl (2002).
Dahl (2002) develops an econometric approach to properly answer why high rate of interstate
migration has not led to equalized returns to schooling across different states in the U.S. He
21
develops a multi-market model of mobility and earnings in which individuals choose in which of
the 50 U.S. states to live and work. He models different areas as having different earnings
and
amenity benefits for workers with different schooling level and proposes a semiparametric
methodology to correct for sample selection bias in such a polychotomous choice model. He
shows that the bias correction is an unknown function of a small number of selection
probabilities, which are calculated without making any distributional assumptions simply by
classifying similar individuals into cells and estimating the proportion of movers and stayers for
each place of birth and cell combination. His work essentially shows that in order to correct for
the selection bias, under some sufficiency conditions, the conditional error mean term can be
replaced by an unknown function of the relevant migration probabilities in the outcome
regression, which can then be estimated with a simple OLS.
Following his approach, from equation (17) we can write:
𝐸 𝑙𝑛 𝑊𝑘𝑐𝑖 𝜿𝑘𝑐𝑖 , 𝑑𝑖 , 𝑎𝑛𝑑 𝑊𝑘𝑐𝑖 𝑏𝑒𝑖𝑛𝑔 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑
= 𝛼 + 𝜸𝜿𝑘𝑐𝑖 + 𝜛𝑑𝑖 + 𝐸 𝜀𝑘𝑐𝑖 𝜿𝑘𝑐𝑖 , 𝑑𝑖 , 𝑎𝑛𝑑 𝑊𝑘𝑐𝑖 𝑏𝑒𝑖𝑛𝑔 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 .
According to Dahl (2002), we can identify the mean error term as a function of relevant
migration probabilities:
𝐸 𝜀𝑘𝑐𝑖 𝜿𝑘𝑐𝑖 , 𝑑𝑖 , 𝑎𝑛𝑑 𝑊𝑘𝑐𝑖 𝑏𝑒𝑖𝑛𝑔 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 =
𝑑𝑘𝑏𝑐 ∙ 𝑓𝑏𝑐 𝑃𝑘𝑏𝑐 , 𝑃𝑘𝑏𝑏 + 𝜖𝑘𝑐𝑖 ,
(27)
𝑏
where 𝑑𝑘𝑏𝑐 is an indicator that takes one only if person k born in state b has actually moved to
city c, 𝐸 𝜖𝑘𝑐𝑖 |𝜿𝑘𝑐𝑖 , 𝑑𝑖 , 𝑎𝑛𝑑 𝑊𝑘𝑐𝑖 𝑏𝑒𝑖𝑛𝑔 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 = 0, and 𝑓𝑏𝑐 ∙ is an unknown function of
the probability that person k, born in state b, is observed in city c (𝑃𝑘𝑏𝑐 ) and probability that
person k, born in state b, remains in the same state (𝑃𝑘𝑏𝑏 ). We choose function 𝑓𝑏𝑐 ∙ to be
quadratic in each of the probabilities separately. In this way, equation (17) can be written as:
𝑙𝑛 𝑊𝑘𝑐𝑖 = 𝛼 + 𝜸𝜿𝑘𝑐𝑖 + 𝜛𝑑𝑖 +
𝑑𝑘𝑏𝑐 ∙ 𝑓𝑏𝑐 𝑃𝑘𝑏𝑐 , 𝑃𝑘𝑏𝑏 + 𝜖𝑘𝑐𝑖
𝑏
22
(28)
Notice that for non-movers, the correction terms are only functions of the probability of staying
since for individuals who do not move from their state of birth 𝑐 = 𝑏.
In the same way, equation (18) can also be corrected for selection:
𝑙𝑛 𝑊𝑘𝑐𝑖 = 𝛼 + 𝜸𝜿𝑘𝑐𝑖 +
𝑑𝑘𝑏𝑐 ∙ 𝑓𝑏𝑐 𝑃𝑘𝑏𝑐 , 𝑃𝑘𝑏𝑏 + 𝜎𝑘𝑐𝑖 ,
(29)
𝑏
where 𝐸 𝜎𝑘𝑐𝑖 |𝜿𝑘𝑐𝑖 , 𝑎𝑛𝑑 𝑊𝑘𝑐𝑖 𝑏𝑒𝑖𝑛𝑔 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 = 0.
In a given city c, the identification for the movers (𝑃𝑘𝑏𝑐 ) comes from the variation in the state
of birth. So, here the underlying assumption is that the state of birth is not directly related to the
wage a person receives. In other words, two individuals with exactly similar characteristics,
living and working in the same city, but born in different states, will not receive different
amounts. For the stayers, however, identification comes from the differences in family status and
hence is the assumption that family status is not directly related to the wage the person receives.
4
Data
The data used here are extracted from the eleventh and twelfth Mexican General Population
and Housing Census for years 1990 and 2000, originally produced by the Mexican National
Institute of Statistics, Geography, and Informatics (Instituto Nacional de Estadística, Geografía,
e Informática or INEGI in short) and preserved and harmonized by Minnesota Population Center
(2008). In the original database the sample universe is composed of de jure Mexican citizens,
including Mexican diplomats and their families residing in foreign countries and foreign
residents of Mexico. The census sought to enumerate vagrants, the homeless, and the transient
workers and excludes persons living abroad or in group quarters (buildings used to shelter people
for reasons of assistance, health, education, religion, confinement, or service). Field work
duration for the 1990 census was March 12-16 and for 2000 census was February 7-18 with
enumeration unit being an occupied dwelling and respondent being the householder.
The design of the 1990 census is a systemic sample of dwellings geographically sorted by
population size (municipality of locality) and sampling executed independently for each federal
unity. Dwellings are the sample units, with sample fraction of 10% or sample size of 8,118,242
persons. The design of the 2000 census is a stratified cluster design; stratified geographically by
23
municipality and urban area and clustered by enumeration areas, blocks of dwellings or localities
where all dwellings within a cluster are included in the sample. Sample fraction in 2000 census
depends upon demographic heterogeneity of municipalities. The sample was designed to yield
representative statistics for all localities with 50,000 or more inhabitants. The data includes
weights computed by the census agency that should be used for most types of analysis.
Households and individuals are distinguishable in the microdata.
The data used here are confined to employed males and females aged 16 to 65, who are wage
or salary workers in an identified industry with identified level of education and positive
monthly income, who are living in a metropolitan area. Since it is necessary in this study to
define geographic limits of the local labour markets, the 2004 definition of metropolitan areas in
Mexico (Secretaría de Dessarollo Social, Consejo Nacional de Pabloción, Instituto Nacional de
Estadística, Geografía, e Informática, 2004) is utilised (see Graph 4.1). In 2004, a joint effort
between CONAPO (Consejo Nacional de Pabloción), INEGI (Instituto Nacional de Estadística,
Geografía, e Informática) and the Ministry of Social Development (Secretaría de Dessarollo
Social or SEDESOL) defines the metropolitan areas as:10
˗
A group of two or more municipalities in which a city with a population of at least 50,000
is located, whose urban area extends over the limit of the municipality that originally
contained the core city incorporating either physically or under its area of direct influence
other adjacent predominantly urban municipalities all of which have a high degree of
social and economic integration or are relevant for urban politics and administration; or
˗
a single municipality in which a city with population of at least one million is located and
fully contained (that is, it does not transcend the limits of a single municipality); or
˗
a city with a population of at least 250,000 which forms a metropolitan area with other
cities in the United States.
It should be noted that north-western and south-eastern states are divided into a small number of
large municipalities whereas central states are divided into a large number of smaller
municipalities. As such, metropolitan areas in the northwest usually do not extend over more
than one municipality whereas metropolitan areas in the centre extend over many municipalities.
Few metropolitan areas extend beyond the limits of one state, namely: Greater Mexico City
(Federal District, Mexico and Hidalgo), Puebla-Tlaxcala (Puebla and Tlaxcala, but excludes the
10
See http://en.wikipedia.org/wiki/Metropolitan_areas_of_Mexico#cite_note-CONAPO-0 cited on October 08,
2008.
24
city of Tlaxcala), Comarca Lagunera (Coahuila and Durango), and Tampico (Tamaulipas and
Veracruz). The map of the metropolitan areas is presented at the end of this section, which can
be compared with the population density map of Mexico in year 2000. The list of the states with
their corresponding numerical codes matching the map of the metropolitan areas is presented in
table (4.2).
Confining the sample only to the municipalities that are part of a metropolitan area reduces
the size of the sample in terms of individual observations by 36.05% in 1990 and by 47.20% in
2000. After considering all the necessary constraints for cleaning the data, among the employed,
from total observations of 878,792 for 1990, the females form 30.7% of the sample and from a
total of 947,089 observations in 2000, the females form 34.8% of the sample. Some further
sample statistics are presented in table (4.1).
Among various sorts of individual level information the dataset includes information on age
and gender, nativity, ethnicity and language, education, work, income, and migration. The
information on work specifies numerous variables such as employment status, occupation,
industry and hours of work. The main variable used as the indicator of income is „earned
income‟ (variable INCEARN from the income category), which reports total income from labour
(from wage, a business, or a farm) in previous month. Using this information jointly with
information on class of worker (CLASSWK from the work category) the wage/salary earners
can be distinguished from the pool of labour who earns income by running a business or a farm.
This group, the wage/salary earners, form the sample used in this study. Wage or salary earners
whose industries of work or income or demographic attributes are not identified are dropped
from the sample.
The income variable (INCEARN) is reported in nominal Pesos, the currency of Mexico, and
is top-coded. In order to carry out the inferences using the data for the two years, the 1990 values
of income are converted to constant-year-2000 figures using the 1990 consumer price index
(CPI) taken from International Financial Statistics (IFS). Following Aydemir & Borjas (2007,
p.706), who have worked with the same database, to deal with the top-coded monthly incomes
the multiplier of 1.5 is applied. In addition, in 1993 the „New Peso‟ was coined that is equivalent
to a thousand „Old Pesos‟. The 1990 values of monthly income are adjusted for this deletion of
three zeros from the Peso bills after 1993.
It was possible to carry out the empirical part by adjusting the monthly income using the
hours of work (HRSWRK1) and work with weekly or hourly measures of income. However, in
25
addition to the fact that benefits of doing so or the disadvantages of using monthly income are
not clear, according to IPUMS11 the values recorded for hours of work in years 1990 and 2000
are implausibly high for some observations and this makes the calculation of weekly or hourly
incomes very imprecise. Therefore, we have not used the hours of work to adjust monthly
incomes to incomes for shorter periods.
Using the data on years of schooling (YRSCHL) seven categories of educational attainment
are constructed the list of which is reported in table (4.1). Using the same variable and the age
variable, a measure of experience at work is constructed according to “Experience = Age – 6 –
Years of Schooling,” where „6‟ is the mandatory age for schooling in Mexico. The migration
information contained in the database enables us to distinguish between foreign and domestic
immigration. In addition, the information on whether an individual speaks an indigenous
language allows us to control for being part of an indigenous minority. All these variables are
used in filtering out the impact of workers‟ attributes on the income they received in order to
extract the wages intrinsically paid by sectors (wage premia). Detailed industry groups are not
consistently defined for the two years of the census. Therefore, the recoded general definition of
15 sectors is being used instead the list of which is provided in table (4.1).
5
Estimation Results
This section reports the estimation results. Table (5.1) reports the results of various estimation
techniques applied to equation (20), or equivalently equation (12); the first two columns are the
OLS results and the rest of the columns are for IV estimations using different instruments as
explained in section three and presented here again:
𝐼𝑉1 =
𝜂𝑐𝑖𝑡 +1 − 𝜂𝑐𝑖𝑡 𝜐𝑖𝑡
𝑖
𝐼𝑉2 =
𝜂𝑐𝑖𝑡 +1 𝜐𝑖𝑡 +1 − 𝜐𝑖𝑡
𝑖
𝐼𝑉3 =
𝜂𝑐𝑖𝑡 +1 − 𝜂𝑐𝑖𝑡 𝜐𝑖𝑡 +1
𝑖
11
https://international.ipums.org/international-action/variableDescription.do?mnemonic=HRSWRK1 under
„Comparability – Mexico‟.
26
𝐼𝑉4 =
𝜂𝑐𝑖𝑡 𝜐𝑖𝑡 +1 − 𝜐𝑖𝑡 ,
𝑖
and IV5 is the set of 14 initial employment shares of sectors in cities, 𝜂𝑐𝑖𝑡 , a set of 14
instruments.
The OLS results, which are reported for two different specification of equation (20), one with
city employment ratios as a control variable under OLS (1) and one with city-sector employment
ratios under OLS (2), indicate a positive and statistically highly significant estimates of the
coefficient of ∆𝑅𝑐 . Graph (5.1) depicts scatter diagram of the controlled variation in city-sector
wage changes versus controlled variation in city rent changes.12 At the first look, it appears that
the results of the OLS may be driven by the outlier in the top right corner of the scatter diagram,
which is actually a data point belonging to Real Estate and Business Services sector in San
Francisco del Rico metropolitan area. However, re-estimating the equation after removing the
data point belonging to this city and sector provides a slightly smaller estimate which is still
highly significant. Looking at a similar scatter diagram for this new regression, graph (5.2), there
seems to be another outlier belonging to Mining sector in Nuevo Laredo metropolitan area.
Dropping the associated observations and repeating the estimation will not change the magnitude
of the estimate which is still highly significant. Thus, the OLS results are not driven by any
outlier city-sector observation. If OLS consistently estimates equation (20), which requires a set
of assumptions as reviewed in detail in appendix (2.H), then the fact that the coefficient of ∆𝑅𝑐 is
positive and statistically significant is indicative of very important and interesting points.
First off, as explained in the previous sections, a statistically-different-from-zero estimate
supports the existence of the general equilibrium mechanism through which sectoral composition
of cities have a causal impact on sectoral wages. Secondly, the magnitude of the coefficient of
∆𝑅𝑐 in (20) measures the magnitudes of order by which the general equilibrium impact is larger
from the conventional accounting measure of the direct and once-and-only impact of changes in
employment compositions on average wages in a city.13 An estimate of 𝛽 𝑂𝐿𝑆 = 3.8 measures the
general equilibrium impact to be almost four times in magnitude compared to the conventional
12
Controlled in the sense that it is actually the scatter diagram of the residuals in the regression of changes in citysector wages on all the regressors other than changes in city rents versus the residuals of the regression of changes in
city rents on the rest of the regressors.
𝛾
13
Notice from comparing equations (20) and (16) that = 𝛾2 1 − 𝛾2 , which is the average of 𝑐1 𝛾1 ×
𝛾𝑐21−𝛾𝑐2 over cities, the coefficient of city rent in equation (12).
27
accounting measure. Considering an example will explain this relationship more clearly. With an
estimate of this magnitude, a pure change in sectoral composition14 that brings about a one dollar
direct impact on average wages in a city (the accounting measure of the impact of changes in
sectoral composition) will generate waves of general equilibrium impacts on city-sectoral wages
so that by the time the new steady state establishes, average wages increase by a total of almost
five dollars, letting the general equilibrium impact to be somewhere around 4 dollars. To bring
another example from the real world, during the ten years from 1990 to 2000, city of Tijuana
experienced an increase in its rent of 0.06 units. At the same time, the sectoral wage in
Agriculture, Fishing, and Forestry sector in this city increased by 18 percent. In the absence of
the general equilibrium impacts, wages in agriculture in Tijuana would have decreased by almost
5 percent (0.18 − 0.06 × 3.8 = −0.048).
Of course, this all relies on the assumption that OLS is consistent and our OLS estimates are
not capturing anything other than the general equilibrium impact of our interest. The first pithole is the endogeneity of the regressors. It is also necessary to make sure of the robustness of
the estimates and that our estimates are not picking up the impact of other causes known in the
wage-formation literature not present in the regression.
The results of the instrumental variable approach discussed in previous sections are reported
under the IV columns for different instruments. Before reviewing the results, there are two
important concerns to be addressed again. Based on the discussion in appendix (2.H) on the
conditions under which the OLS is consistent and the IV approach is valid, first we know that if
instruments pass the first stage requirements and the IV estimates are of the same magnitude for
different instruments, this is to be taken as an indication of the validity of the assumptions
required for exogeneity of the instruments simply because each instrument uses different weights
in its structure. In addition, having OLS and IV estimates that are more or less of the same
magnitude should be taken as a sign for validity of the conditions required for consistency of
OLS.
We do not report the results for IV1 and IV3 since they do not pass the first stage
requirements for being valid instruments; they are not strongly correlated with ∆𝑅𝑐 . The
individual partial 𝑅2 for these two instruments in the regression of ∆𝑅𝑐 on all the exogenous
variables and excluded instruments are very low in absolute terms and as compared to other
instruments. The individual partial 𝑅2 for IV1 and IV3 are respectively 0.03 and 0.02, which are
14
Pure in the sense that overall employment does not change.
28
very low as compared to the ones associated with other instruments. In addition, they do not pass
the individual significant F-tests in their associated first stage regression as the p-value
associated with IV1 is 0.45 and the p-value associated with IV3 is 0.48. Thus, these two
instruments are not individually, statistically significant in the associated first stage regressions.
Therefore, at best, even if the exogeneity assumption of the instruments is valid, these two
insturments do not provide any additional information to be used in dealing with the possible
endogeneity of ∆𝑅𝑐 .
On the other hand, the rest of the instruments (IV2, IV4, IV5) perform very well in their
associated first stage regressions. They pass the first stage significance results at any
conventional significance level and show strong co-movement with ∆𝑅𝑐 according to their first
stage associated partial 𝑅2 . In addition, the IV estimates under these instruments are close in
magnitude as compared to the OLS. In fact we have formally tested the equality of the IV
estimates with the OLS results and we have not been able to reject the null hypothesis that the IV
estimates are equal to the OLS estimates15. These tests indicate that the OLS results are
consistent and the estimates do not suffer from endogeneity.
IV5, which is in general a group of 14 instruments as explained before (due to detected
collinearity in different specifications as reported in the caption of table 5.1, IV5 typically
becomes a group of 13 or 12 initial sectoral employment share), passes the over-identification
test at any conventional significance level. The null hypothesis is that excluded instruments are
not correlated with the 2SLS residuals using all the instruments and the test (Hansen‟s J statistic)
is robust to clustering and heteroskedasticity. The p-value of the test is 0.12, indicating that the
over-identification restrictions are satisfied. This is also true if one combines IV2 or IV4 with
IV5. It should be mentioned that the necessary checks for the validity of the instruments for
changes in city employment ratio and city-sector employment ratios are carried out and the IV
approach for these variables are also successful.
The results reported in table (5.1) and the following three graphs indicate that we cannot
reject the claim that OLS is consistently capturing a causal relationship from the city rent to all
of the city-sector wages in a city (and hence to the average city wages). However, since IV2 and
IV4 are constructed based on changes in national wage premia rather than changes in city-sector
employment shares, one might think that the interaction we are capturing through the IV
15
The p-value for the test of equality of the OLS estimate and IV2 estimate is 0.49 and of the equality of the OLS
estimate and IV4 estimate is 0.90.
29
estimation here is different from the interactions resulted from city level movement of labour
among sectors. Therefore, it seems to be interesting to show that in the same way that the
theoretical model does not differentiate between a change in the distribution of sectoral
employment share or a change in the national sectoral wage premia as long as the workers
outside options are concerned, so does the empirics. This is carried out by replacing ∆𝑅𝑐 in the
estimating equation with its decompositions. Table (5.2) reports the results of estimating these
equations. The null hypothesis of the equality of the coefficients on the components of ∆𝑅𝑐
cannot be rejected at conventional levels of significance. In other words, the empirical strategy
used here to deal with the endogeneity of ∆𝑅𝑐 is not leaving out the interactions resulting from
city level movement of labour force among sectors.
In order to address the issue of selection, as discussed in section 3.2, we first have to calculate
the probabilities of migration. To do so, we first divide the sample into “movers” and “stayers”.
Movers are individuals who are now living in a city that is not in their state of birth. Stayers are
individuals who are living in a city that is part of their state of birth. For movers, we define
groups (or “cells”) based on some of the attributes of the individuals. Specifically, we use 5 age
categories, 4 education categories, 2 gender groups, and an indigenous dummy, which in total
generates 80 cells for the movers. For the stayers, as well as these groups, we add some
additional categories based on some family status indicators. We create two groups for being
married or not (spouse being present in the household) and two groups for having or not having
at least one child under the age of five present in the household. In this way we generate a total
of 320 cells for the stayers. The higher number of cells for the stayers is in accordance with their
higher share in the sample. Then, 𝑃𝑘𝑏𝑐 is defined as the fraction of individuals born in state b and
are in the same cell as person k who have moved to city c. 𝑃𝑘𝑏𝑏 is also defined as the fraction of
individuals born in state b that are in the same cell as person k that have stayed in the same state.
After preparing the migration probabilities, we estimate the selection-bias-corrected version
of equation (17) and (18) as presented in equations (28) and (29), respectively, and proceed with
other calculations and estimations. The estimation results of the equations (28) and (29), which
are not presented here, indicate that the correction terms are highly significant and therefore, it is
probable that the previous result actually suffer from the selection bias. Table (5.3) repeats the
results in table (5.1) for the selection-bias-corrected estimations. The results are very similar to
those reported in table (5.1) when we had not corrected for the self selection in the sample. Thus,
it is to be concluded that the OLS estimates are not contaminated by the existing self selection in
30
the sample, indicating that the pattern of self selection is not correlated with the pattern of
changes in the city rents.
It remains to make sure that the estimates of the impact of city rent on city-sector wages are
robust at the presence of other existing alternative explanations for differences in wages across
cities such as those related to labour supply, diversity of employment in a city (Glaeser, Kallal,
Scheinkman, and Shleifer, 1992), education levels (Moretti, 2004; Acemoglu and Angrist, 1999),
and minimum wages (Fairris, Ropli, and Zepeda, 2008). We will add additional variables related
to these alternative explanations to equation (20) to ensure of the robustness of the OLS estimate.
In general, one would expect to observe an inverse relationship between labour supply and
wages in a city. Emigration to the U.S. has played an important role in changing the supply of
workforce in different regions in Mexico (Robertson, 2000; Munshi, 2003; Hanson 2003, 2005a;
Richter et al., 2005; Mishra, 2007; Chiquiar and Hanson, 2007). Hanson (2005a) examines
changes in labour supply and earnings across regions of Mexico during the 1990s focusing on
individuals born on states with different levels of exposure to emigration and finds that the
distribution of male earnings in high-emigration states shifted to the right relative to lowemigration states and that during the 1990s, average hourly earnings in hig-emigration states rose
relative to low migration states by 6 to 9 percent. Mishra (2007) estimates that in Mexico over
the period 1970-2000 the elasticity of wages with respect to the outflow of migrant labour was
0.4 and that emigration raised average wages in the country by 8 percent. To control for the
supply-side impacts on wages and earnings, we add the change in the supply of labour to the
right-hand-side variables (∆log⁡𝐿𝑎𝑏𝑜𝑢𝑟 𝐹𝑟𝑜𝑐𝑒 ). Notice that controlling for the employment
rate at city or city-sector levels already controls for both demand and supply effect on wages to
some extent.
Glaeser et al. (1992) examine predictions of various theories of growth externalities
(knowledge spillovers) within and between industries at city level in the U.S. during 1956 and
1987. They try to verify whether it is the geographic specialization or competition of
geographically proximate industries that promote innovation spillovers and growth in those
industries and cities. One measure of city growth they use is growth in wages. By testing
empirically in which cities industries grow faster, as a function of geographic specialization and
competition, they find that although specialization has no effect on wage growth, diversity in a
city helps wage growth of the industry. Here, we introduce a measure of “fractionalization” of
employment in a city at the start of the decade measured by one minus the Herfindahl index, or
31
one minus the sum of squared sectoral shares in the city. We also control for change in the log of
the size of the city‟s labour force to capture the type of agglomeration effects tested in Glaeser et
al. (1992).
Moretti (2004) examines wages in U.S. cities in the 1980s and finds that cities with greater
increase in the proportion of workers with a BA or higher education have higher wage gains.
Acemoglue and Angrist (1999) find weaker results for the impact of education using average
years of education in a state. Although we have already controlled for the level of education in
estimating the sectoral wage premia and therefore, our measure of city rents does not reflect
cities with higher wages due to having higher levels education, we will also control for both
measures of education discussed in the two studies mentioned above. One measure is the change
in the proportion of workers with a BA or higher education and the other is using average years
of schooling as an alternative measure of the education level of a city.
Fairris et al. (2008) link the observed clustering in wage distributions to minimum wage
multiples in Mexico and show that minimum wages instead of setting a minimum bound on the
wages of formal sector workers, serve as a norm for wage setting throughout the Mexican
economy. They find evidence of clustering around multiples of the minimum wage, and some
evidence suggesting that wage increases over time for certain occupations follow stipulated
increases in the minimum wage. We construct and control for a measure of minimum wage at
the city level and its changes. In Mexico, the minimum wages are different for different
municipalities. We use the data on minimum wages from the Bank of Mexico website and match
it with the geographical divisions that we are using in this study (the metropolitan areas).
Essentially, we use a number-of-worker weighted average of the minimum wages in the
municipalities that form a metropolitan area as the measure of minimum wage at city level.
Table (5.4) reports the results of the robustness tests. As it is clear from the table, the
coefficient of the city rent is fairly stable and remains highly significant after the introduction of
the new variables one at a time or altogether. Among the new controls, only our measure of
fractionalization (1 – Herfindahl Index) is significant, which enters with a negative sign.
6 Conclusion
In this study we addressed the question that „Do compositional changes in local sectoral
employment or changes in the national sectoral wage premia have in the long-run important
spill-over effects on all sectoral wages in Mexican cities?‟ and exploit geographical variation in
32
sectoral composition of Mexican cities over time to see whether local sectoral wages
systematically change with the cities‟ distribution of sectoral employment and/or national
sectoral wage premia. The OLS and IV estimates were indicative of the existence of a general
equilibrium mechanism through which sectoral wages in each city causally depend on the rents
generated by the city as measured by the city‟s employment-share weighted sum of national
sectoral wage premia.
We were concerned about the endogeneity of the regressors and were able to generate
instruments to deal with this issue. The robustness of the estimated causal relationship at the
presence of other alternative explanatory variables used in the literature regarding wage
formation at local economies was addressed. The estimated relationship of the interest is fairly
stable and highly significant at the presence of variables presenting alternative explanations.
The magnitude of this relationship is substantially larger than the conventional existing
accounting measures. A pure16 compositional change17 in a city‟s sectoral employment that
decreases the average wages in that city by one dollar while ignoring any impacts on the sectoral
wage levels (i.e., ignoring the general equilibrium impact on wages as measured by the
conventional composition-adjustment approach), will in the long-run result in more than three
dollar decrease in average wage in that city through the general equilibrium impact from city‟s
compositional change on wages; an inter-sectoral general equilibrium impact on wages in a local
economy that is more than three times the conventional measures.
An impact of this magnitude calls for excessive caution, especially in the less developed
countries, in implementing industrial and trade policies that may result in severe loss of good
jobs or sizable relocation of work force at the expense of high paying sectors. In addition, this
finding implies a holistic approach in welfare analysis of such policy changes.
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17
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33
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35
Table (4.1) – Summary of Sample Statistics
Female
1990
Male
1990
Whole Sample
1990
Female
2000
Male
2000
Whole Sample
2000
270,116
29.7
608,676
32.4
878,792
31.6
329,278
31.8
617,820
33.1
947,098
32.6
0 to 1 [No/Little]
2 to 5 [Some Primary]
6 to 8 [Completed Primary]
9 to 11 [Some Secondary]
12 to 13 [Completed Secondary]
14 to 15 [Some University]
16 or More [University Degree]
23.6
21.9
26.7
37.4
32.1
36.9
33.3
76.4
78.1
73.3
62.6
67.9
63.1
66.7
6.4
12.0
28.5
32.1
7.5
3.1
10.4
30.0
28.6
29.7
31.5
45.0
39.5
41.6
70.0
71.4
70.3
68.5
55.0
60.5
58.4
3.3
8.9
22.5
30.7
16.6
3.4
14.6
Average Experience (years)
Average Monthly Hours Worked
Average Monthly Income
14.9
18.3
15.7
17.9
41.8
47.1
17.2
45.5
43.2
50.6
17.1
48.0
Constant Year 2000 Peso*
(US $)
4,016
(425)
4,927
(521)
4,647
(491)
3,280
(347)
4,026
(426)
3,767
(398)
Capital
Centre
Border
North
Yucatan
South
32.0
29.4
30.3
30.0
29.2
31.8
68.0
70.6
69.7
70.0
70.8
68.2
41.9
26.6
20.5
5.0
3.2
2.7
36.3
33.6
33.6
35.6
32.8
33.6
63.7
66.4
66.4
64.4
67.2
66.4
41.4
28.6
18.6
4.5
4.0
3.0
Average Percentage of Domestic
Migrant (% - State or Higher Level)
7.39
7.59
7.47
6.13
6.63
6.41
24.7
34.5
22.0
60.5
4.2
30.3
12.4
63.8
41.7
85.9
5.2
39.8
15.3
12.5
41.2
75.3
65.5
78.0
39.5
95.8
69.7
87.6
36.2
58.3
14.1
94.8
60.2
84.7
87.5
58.8
30.7
12.5
9.5
7.5
7.4
6.2
6.0
4.4
3.9
3.9
3.3
2.5
1.0
1.0
0.2
29.1
36.0
19.3
62.2
4.6
35.1
14.9
66.9
43.1
89.2
7.3
44.1
12.3
15.4
37.3
70.9
64.0
80.7
37.8
95.4
64.9
85.1
33.1
56.9
10.8
92.7
55.9
87.0
84.6
62.7
26.4
13.6
5.5
7.6
7.8
6.1
7.2
4.6
4.5
5.1
3.0
1.6
0.5
0.7
5.8
Size of Sample
Average Age (years)
Average Years of Schooling (%):
Regional** Population (%):
Employment Share (%):
Manufacturing
Wholesale and Retail Trade
Other Community and Personal Services
Education
Construction
Public Administration and Defence
Transportation and Communication
Health and Social Work
Hotels and Restaurants
Private Household Services
Agriculture, Fishing, and Forestry
Financial Services and Insurance
Mining
Electricity, Gas, and Water
Real Estate and Business Services
*
**
$1 US = 9.46 Peso (monthly average in year 2000 - source: IFS)
Border States: Baja California, Chihuahua, Coahuila, Nuevo Leon, Sonora, Tamaulipas
Capital States: Federal District, Mexico
Center States: Colima, Guanajuato, Hidalgo, Jalisco, Michoacan, Morelos, Puebla, Queretaro, Tlaxcala, Veracruz
North States: Aguascalientes, Baja California Sur, Durango, Nayarit, San Luis Potosi, Sinaloa, Zacatecas
South States: Chiapas, Guerrero, Oaxaca
Yucatan States: Campeche, Tabasco, Quintana Roo, Yucatan
36
Table (4.2) – Mexican States and Associated
Numerical Codes Matching the
Map of the Metropolitan Areas
Name of State
Numerical Code
Aguascalientes
Baja California
Baja California Sur
Campeche
Coahuila
Colima
Chiapas
Chihuahua
Distrito Federal
Durango
Guanajuato
Guerrero
Hidalgo
Jalisco
México
Michoacán
Morelos
Nayarit
Nuevo León
Oaxaca
Puebla
Querétaro
Quintana Roo
San Luis Potosí
Sinaloa
Sonora
Tabasco
Tamaulipas
Tlaxcala
Veracruz
Yucatán
Zacatecas
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
37
Graph (4.1) – Mexican Metropolitan Areas
Source: Secretaría de Dessarollo Social, Consejo Nacional de Pabloción, Instituto Nacional de Estadística, Geografía, e Informática (2004)
38
Table (5.1) – OLS and IV Estimation Results of Equation (28)
OLS
(1)
a
∆𝑹𝒄
3.80
(0.60)
∆𝑬𝑹𝒄
1.93
(1.64)
No
Industry Fixed
Effects (𝒅𝒊)
OLS
(2)
a
IV2
5
IV4
IV2 & IV4
a
a
IV5
○
a
∆
IV2 & IV5
a
IV4 & IV5
□
a
3.78
(0.59)
2.95
(1.24)
3.89
(1.20)
4.11
(1.18)
3.99
(0.65)
3.98
(0.64)
3.99
(0.65)
–
–
–
–
–
–
–
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Instrumented for
∆𝑬𝑹𝒄 × 𝒅𝒊
No
No
Yes
Yes
Yes
Yes
Yes
Yes
Obs.
820
820
820
820
820
765
765
765
0.24
0.28
–
–
–
–
–
–
First Stage
♣
Partiasl 𝑅2
–
–
0.33
0.35
0.35
0.75
0.72
0.75
First Stage
♥
Weak IV F-test
–
–
F(1,54) = 27.5
P-value = 0.00
F(1,54) = 25.1
P-value = 0.00
F(2,54) = 13.1
P-value = 0.00
F(14,50) = 19.7
P-value = 0.00
F(15,50) = 31.5
P-value = 0.00
F(13,50) = 14.2
P-value = 0.00
First Stage
♠
Joint F-test
–
–
F(16,54) =22.6
P-value = 0.00
F(16,54) = 24.7
P-value = 0.00
F(17,54) = 38.0
P-value = 0.00
F(28,50) = 79.5
P-value = 0.00
F(29,50) = 161
P-value = 0.00
F(28,50) = 79.5
P-value = 0.00
∆𝑬𝑹𝒄 × 𝒅𝒊
𝑅
2
n
(.): Robust, city-clustered standard deviation. a, n, 5: Respectively, significance at all of the conventional, none of the conventional, and 5% levels of significance. ○∆ : Due to
detected collinearity, the instrument in IV5 associated with initial employment share of Private Household Services sector is dropped out and IV5 here is a set of 13 instruments.
Result is not sensitive to choice of dropped out sector. ∆ : On this column, First Stage Weak IV F-test is reported for joint significance test of IV2 and IV5 in regression of
∆𝑅𝑐 on all exogenous variables plus all excluded instruments. □: Due to detected collinearity, instruments in IV5 associated with initial employment share of Private Household
Services and Other Community and Personal Services sectors are dropped out and IV5 here is a set of 12 initial sectoral employment shares. The result is not sensitive to choice
of dropped out sectors. On this column, First Stage Weak IV F-test is reported for joint significance test of IV4 and IV5‟s in regression of ∆𝑅𝑐 on all exogenous variables plus all
excluded instruments. ♣: Squared-partial correlation between excluded instruments and ∆𝑅𝑐 . ♥: Of the instrument on ∆𝑅𝑐 in first-stage regression of ∆𝑅𝑐 on associated IV and
instruments for ∆𝐸𝑅𝑐 × 𝑑𝑖 and rest of the exogenous variables (𝑑𝑖 ‟s). The test is robust to clustering and heteroskedasticity. ♠: The joint F-test of all excluded instruments in ∆𝑅𝑐
first-stage regression. The test is robust to clustering and heteroskedasticity.
39
Graph (5.1) – Controlled Variation of Changes in City-Sector Wages vs. Controlled
Variation of Changes in City Rents
4
3
2
1
0
-1
-.06
-.04
-.02
0
Controled Variation in City Rent Changes
coef = 3.78, (robust) se = .59, t = 6.42
40
.02
.04
Graph (5.2) – Controlled Variation of Changes in City-Sector Wages vs. Controlled
Variation of Changes in City Rents after Removing the Likely Outlier
1.5
1
.5
0
-.5
-1
-.06
-.04
-.02
0
Controled Variation in City Rent Changes
coef = 3.65, (robust) se = .60, t = 6.05
41
.02
.04
Graph (5.3) – Controlled Variation of Changes in City-Sector Wages vs. Controlled
Variation of Changes in City Rents after Removing Both Likely
Outliers
1
.5
0
-.5
-1
-.06
-.04
-.02
0
Controled Variation in City Rent Changes
coef = 3.65, (robust) se = .60, t = 6.05
42
.02
.04
Table (5.2) – Decomposition of ∆𝑹𝒄
OLS (1)
∆𝑹𝟏𝒄 =
∆𝑹𝟐𝒄 =
∆𝑹𝟑𝒄 =
∆𝑹𝟒𝒄 =
∆𝜼𝒄𝒊𝒕+𝟏 𝝊𝒊𝒕
3.40
a
𝒊
(0.85)
𝜼𝒄𝒊𝒕+𝟏 ∆𝝊𝒊𝒕+𝟏
4.72
𝒊
∆𝜼𝒄𝒊𝒕+𝟏 𝝊𝒊𝒕+𝟏
𝒊
𝜼𝒄𝒊𝒕 ∆𝝊𝒊𝒕+𝟏
a
IV5 & IV2
IV5
5
2.40
5
(0.99)
a
5.44
IV5 & IV4
5
2.39
2.39
(0.99)
(0.99)
a
5.46
5.46
a
(1.25)
(1.39)
(1.38)
(1.38)
–
–
–
–
OLS (2)
IV5
–
–
–
–
3.77
a
(0.71)
4.23
a
3.00
a
(0.74)
4.90
a
–
–
–
–
∆𝑬𝑹𝒄 × 𝒅𝒊
Yes
Yes
Yes
Yes
(0.96)
Yes
(1.01)
Yes
Industry Fixed Effects (di)
Yes
Yes
Yes
Yes
Yes
Yes
Instrumented for ∆𝑬𝑹𝒄 × 𝒅𝒊
No
Yes
Yes
Yes
No
Yes
Obs.
𝑅2
820
0.29
765
–
765
–
765
–
820
0.29
765
–
Test if:
coef. on ∆𝑅1𝑐 = coef. on ∆𝑅2𝑐
P-value = 0.47
P-value = 0.12
P-value = 0.12
P-value = 0.12
–
–
Test if:
coef. on ∆𝑅3𝑐 = coef. on ∆𝑅4𝑐
–
–
–
–
P-value = 0.72
P-value = 0.09
𝒊
∆𝑅1𝑐 :
First Stage
♣
Partial 𝑅2
–
∆𝑅2𝑐 :
∆𝑅1𝑐 :
R2 = 0.61
∆𝑅2𝑐 :
2
∆𝑅1𝑐 :
R2 = 0.61
2
R = 0.93
R = 0.92
∆𝑅2𝑐 :
R2 = 0.53
–
∆𝑅3𝑐 :
R2 = 0.63
2
R = 0.89
∆𝑅1𝑐 :
∆𝑅1𝑐 :
∆𝑅1𝑐 :
F(29,50) = 4.17
F(28,50) = 4.21
F(28,50) = 4.21
∆𝑅3𝑐 :
First Stage
P-value = 0.00
P-value = 0.00
P-value = 0.00
–
–
F(27,50) = 5.10
♠
Joint F-test
∆𝑅2𝑐 :
∆𝑅2𝑐 :
∆𝑅2𝑐 :
P-value = 0.00
F(29,50) = 230
F(28,50) = 167
F(29,50) = 167
P-value = 0.00
P-value = 0.00
P-value = 0.00
(.): Robust City-clustered standard deviation. a, 5: Respectively, significance at all conventional and 5% levels of significance. ♣: Squared-partial correlation between excluded
instruments and ∆𝑅𝑐 . ♥: Of the instrument on ∆𝑅𝑐 ‟s in first-stage regression of ∆𝑅𝑐 on associated IV, instruments for ∆𝐸𝑅𝑐 × 𝑑𝑖 , and rest of the exogenous variables (𝑑𝑖 ‟s). The
test is robust to clustering and heteroskedasticity. ♠: Joint F-test of all excluded instruments in ∆𝑅𝑐 ‟s first-stage regression. The test is robust to clustering and heteroskedasticity.
: Hansen‟s J Statistic; the test is robust to clustering and heteroskedastcity.
43
Table (5.3) – OLS and IV Estimation Results of Equation (28) after Correcting for Selection Bias
OLS
(1)
a
OLS
(2)
a
IV2
10
IV4
IV2 & IV4
a
a
IV5
○
a
∆
IV2 & IV5
a
IV4 & IV5
□
a
3.23
(0.59)
2.36
(1.21)
3.01
(1.18)
3.19
(1.16)
3.30
(0.58)
3.31
(0.57)
3.30
(0.58)
–
–
–
–
–
–
–
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Instrumented for
∆𝑬𝑹𝒄 × 𝒅𝒊
No
No
Yes
Yes
Yes
Yes
Yes
Yes
Obs.
820
820
820
820
820
765
765
765
0.22
0.26
–
–
–
–
–
–
First Stage
♣
Partial 𝑅2
–
–
0.32
0.34
0.34
0.76
0.76
0.76
First Stage
♥
Weak IV F-test
–
–
F(1,54) = 26.4
P-value = 0.00
F(1,54) = 24.3
P-value = 0.00
F(2,54) = 12.9
P-value = 0.00
F(13,50) = 13.7
P-value = 0.00
F(14,50) = 13.7
P-value = 0.00
F(13,50) = 13.7
P-value = 0.00
First Stage
♠
Joint F-test
–
–
F(16,54) = 22.7
P-value = 0.00
F(16,54) = 26.7
P-value = 0.00
F(17,54) = 40.3
P-value = 0.00
F(28,50) = 81.3
P-value = 0.00
F(29,50) = 184
P-value = 0.00
F(28,50) = 81.3
P-value = 0.00
∆𝑹𝒄
3.26
(0.60)
∆𝑬𝑹𝒄
2.63
(1.57)
No
Industry Fixed
Effects (𝒅𝒊)
∆𝑬𝑹𝒄 × 𝒅𝒊
𝑅
2
n
(.): Robust, city-clustered standard deviation. a, n, 5: Respectively, significance at all of the conventional, none of the conventional, and 5% levels of significance. ○∆ : Due to
detected collinearity, the instrument in IV5 associated with initial employment share of Private Household Services sector is dropped out and IV5 here is a set of 13 instruments.
Result is not sensitive to choice of dropped out sector. ∆ : On this column, First Stage Weak IV F-test is reported for joint significance test of IV2 and IV5 in regression of
∆𝑅𝑐 on all exogenous variables plus all excluded instruments. □: Due to detected collinearity, instruments in IV5 associated with initial employment share of Private Household
Services and Other Community and Personal Services sectors are dropped out and IV5 here is a set of 12 initial sectoral employment shares. The result is not sensitive to choice
of dropped out sectors. On this column, First Stage Weak IV F-test is reported for joint significance test of IV4 and IV5‟s in regression of ∆𝑅𝑐 on all exogenous variables plus all
excluded instruments. ♣: Squared-partial correlation between excluded instruments and ∆𝑅𝑐 . ♥: Of the instrument on ∆𝑅𝑐 in first-stage regression of ∆𝑅𝑐 on associated IV and
instruments for ∆𝐸𝑅𝑐 × 𝑑𝑖 and rest of the exogenous variables (𝑑𝑖 ‟s). The test is robust to clustering and heteroskedasticity. ♠: The joint F-test of all excluded instruments in ∆𝑅𝑐
first-stage regression. The test is robust to clustering and heteroskedasticity.
44
Table (5.4) – Robustness
OLS
∆𝑹𝒄
3.38
a
IV4
OLS
a
3.72
(0.63)
(1.18)
a
a
IV4
a
3.77
(0.58)
1 – Herfindahl
-0.57
(0.20)
-0.63
(0.23)
–
∆𝐁𝐀 +
–
–
9.1e-08
OLS
a
3.75
(1.16)
(0.57)
–
n
(8.4e-08)
n
4.4e-08
a
3.88
n
(8.5e-08)
n
IV4
3.30
OLS
a
(1.16)
a
IV4
a
4.06
3.98
(0.58)
(1.23)
OLS
3.68
a
–
–
-0.58
(0.20)
–
–
–
–
-3.6e-08
–
0.02
(0.05)
0.00
(0.05)
–
∆𝒍𝒐𝒈(𝑳𝒂𝒃𝒐𝒖𝒓 𝑭𝒓𝒐𝒄𝒆)
–
–
–
–
-0.09
(0.07)
∆𝑴𝒊𝒏𝒘𝒂𝒈𝒆
–
–
–
–
(1.31)
a
–
–
a
3.91
(0.60)
–
∆𝑨𝒗𝒆. 𝒚𝒓𝒔. 𝒔𝒄𝒉𝒍.
IV4
n
(1.1e-07)
n
5
-0.64
(0.31)
-3.4e-07
(2.6e-07)
n
–
–
0.07
(0.04)
-0.16
(0.11)
–
–
-0.07
(0.07)
–
–
-1.39
(0.75)
-1.48
(0.77)
-1.31
(0.85)
-1.91
(1.31)
–
n
n
10
n
10
n
0.01
(0.06)
10
-0.27
(0.16)
n
∆𝑬𝑹𝒄 × 𝒅𝒊
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Industry Fixed
Effects (𝒅𝒊 )
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Instrumented for
∆𝑬𝑹𝒄 × 𝒅𝒊
No
Yes
No
Yes
No
Yes
No
Yes
No
Yes
Obs.
820
820
820
820
820
820
820
820
820
820
0.31
–
0.28
–
0.29
–
0.30
–
0.32
–
𝑅
2
n
(.): Robust city-clustered standard deviation. a: Significant at all conventional levels of significance. n: Not significant at any conventional level of
significance. 5: Significant at 5%. 10: Significant at 10%.
45
Appendix
Appendix 2.H. Deriving the Identification Conditions
As described in the text, we are interested in the condition: 1
11
plim
𝐶,𝐼→∞ 𝐼 𝐶
which, using 𝑅 =
𝑗
𝐼
𝐶
∆𝑅𝑐 ∆𝜉𝑖𝑐 = 0 ,
(𝐻1)
𝑖=1 𝑐=1
𝜂𝑗𝑐 𝑤𝑗 − 𝑤1 , can be written as:
11
plim
𝐶,𝐼→∞ 𝐼 𝐶
𝐼
𝐶
1
𝐼
∆𝜂𝑗𝑐 𝑤𝑗 − 𝑤1 +
𝑖=1 𝑐=1 𝑗 =1
𝜂𝑗𝑐 ∆ 𝑤𝑗 − 𝑤1
∆𝜉𝑖𝑐
𝑗 =1
or:
11
plim
𝐶,𝐼→∞ 𝐼 𝐶
1
𝐼
𝐶
𝑤𝑗 − 𝑤1
𝑗 =1
1
∆𝜂𝑗𝑐
𝑐=1
𝐼
∆𝜉𝑖𝑐 +
𝑖=1
𝐶
∆ 𝑤𝑗 − 𝑤1
𝑗 =1
Throughout this appendix we omit the t subscript for simplicity.
46
1
𝜂𝑗𝑐
𝑐=1
∆𝜉𝑖𝑐 .
𝑖=1
(𝐻2)
We will handle the limiting arguments sequentially, allowing 𝐶 → ∞ first. Then, we are concerned with two components in (H2), which
we will handle in turn. The first is:
1
plim
𝐶→∞ 𝐶
Given the decomposition 𝜖𝑖𝑐 = 𝜖𝑐 + 𝜐𝑖𝑐𝜖 , where
𝜂𝑖𝑐 ≈
𝜖
𝑖 𝜐𝑖𝑐
1
1
+ 𝜋1 𝜖𝑖𝑐 −
𝐼
𝐼
𝐶
1
∆𝜂𝑗𝑐
𝑐=1
∆𝜉𝑖𝑐 .
(𝐻3)
𝑖=1
= 0, using equation (18) in the text we get:
𝑗
𝜖𝑗𝑐 + 𝜋2 𝑝𝑖 𝛺𝑖𝑐 −
47
1
𝐼
𝑗
𝑝𝑗 𝛺𝑗𝑐 ,
(18)
1
𝛥𝜂𝑗𝑐 = 𝜋1 𝛥𝜖𝑗𝑐 − 𝛥
𝐼
1
𝜖𝑗𝑐 + 𝜋2 𝛥𝑝𝑗 𝛺𝑗𝑐 − 𝛥
𝐼
𝑗
1
𝜖
= 𝜋1 𝛥 𝜖𝑐 + 𝜐𝑐𝑗
− 𝛥
𝐼
𝑗
𝜖
𝜖𝑐 + 𝜐𝑐𝑗
Also, since ∆𝜉𝑖𝑐 = 𝛾1 ∆𝜖𝑖𝑐 + 𝛾1
𝛾2
1−𝛾2
1
𝑗 𝐼 ∆𝜖𝑗𝑐 ,
we have:
48
𝑝𝑗 𝛺𝑗𝑐
1
+ 𝜋2 𝛥𝑝𝑗 𝛺𝑗𝑐 − 𝛥
𝐼
1
𝜖
= 𝜋1 𝛥𝜖𝑐 + 𝛥𝜐𝑐𝑗
− 𝛥𝜖𝑐 + 𝜋2 𝛥𝑝𝑗 𝛺𝑗𝑐 − 𝛥
𝐼
𝜖
= 𝜋1 𝛥𝜐𝑐𝑗
+ 𝜋2 𝛥𝑝𝑗 𝛺𝑗𝑐 − 𝛥𝑝𝛺𝑐 .
𝑗
(𝐻4)
𝑗
𝑝𝑗 𝛺𝑗𝑐
𝑗
𝑝𝑗 𝛺𝑗𝑐
∆𝜉𝑖𝑐 =
𝑖
𝜖
𝛾1 ∆ 𝜖𝑐 + 𝜐𝑐𝑖
+ 𝛾1
𝛾2 1
1 − 𝛾2 𝐼
𝜖
𝛾1 ∆ 𝜖𝑐 + 𝜐𝑐𝑖
+ 𝛾1
𝛾2
∆𝜖
1 − 𝛾2 𝑐
𝑖
=
𝑖
= 𝐼𝛾1 ∆𝜖𝑐 + 𝐼𝛾1
= 𝛾1 + 𝛾1
𝑗
𝜖
∆ 𝜖𝑐 + 𝜐𝑐𝑗
𝛾2
∆𝜖
1 − 𝛾2 𝑐
𝛾2
𝐼∆𝜖𝑐 .
1 − 𝛾2
(𝐻5)
Then, given that 𝐸 ∆𝜖𝑐 = 0 (again, recalling that we have removed economy-wide trends) and if ∆𝜖𝑐 is independent of 𝛥𝜐𝑖𝑐𝜖 and
𝛥𝑝𝑗 𝛺𝑗𝑐 − 𝛥𝑝𝛺𝑐 , it is straight forward to show that (H3) equals zero.
The second component is:
1
𝑝 lim
𝐶→∞ 𝐶
where
1
𝑖=1 ∆𝜉𝑖𝑐
𝐶
1
𝜂𝑗𝑐
𝑐=1
∆𝜉𝑖𝑐 ,
(𝐻6)
𝑖=1
is again given by (H5), while 𝜂𝑗𝑐 is given by equation (18) in the text. For (H6) to be zero we require in addition that ∆𝜖𝑐 be
independent of past values of 𝜐𝑖𝑐𝜖 and of 𝑝𝑗 𝛺𝑗𝑐 − 𝑝𝛺𝑐 . Thus, if ∆𝝐𝒄 is independent of the past and is independent of 𝜟𝝊𝝐𝒊𝒄 and of 𝜟𝒑𝒋 𝜴𝒋𝒄 −
𝜟𝒑𝜴𝒄, then (H1) equals zero and OLS is consistent.
49
We are also interested in the conditions under which our instruments can provide consistent estimates. Apart from the instruments being
correlated with Δ𝑅𝑐 , the condition we require for a given instrument, 𝑍𝑐 , is:
11
plim
𝐶,𝐼→∞ 𝐼 𝐶
𝐼
𝐶
𝑍𝑐 ∆𝜉𝑖𝑐 = 0,
(𝐻7)
𝑖=1 𝑐=1
for what we call IV1:
𝑍𝑐 =
𝑗
𝜂𝑗𝑐 𝑔𝑗 + 1
𝑖 𝜂𝑖𝑐 𝑔𝑖 + 1
𝑤𝑗 − 𝑤1 ,
where 𝑔𝑖 is the growth rate in employment in industry i at the national level. Given this, first allowing for 𝐶 → ∞, the left hand side in (H7)
becomes:
1
plim
𝐶→∞ 𝐶
𝐶
𝑤𝑗 − 𝑤1
𝑗
𝑐=1
𝜂𝑗𝑐 𝑔𝑗 + 1
𝑖 𝜂𝑖𝑐 𝑔𝑖 + 1
50
𝐼
∆𝜉𝑖𝑐 .
𝑖=1
(𝐻8)
Thus, (H8) equals zero under the same conditions under which (H6) equals zero. In other words, the conditions that 𝐸 ∆𝜖𝑐 = 0 and ∆𝜖𝑐 is
independent of the past values of 𝜐𝑖𝑐𝜖 and 𝑝𝑗 𝛺𝑗𝑐 − 𝑝𝛺𝑐 . Obviously this condition will be satisfied if 𝜖𝑐 behaves as a random walk with
increments independent of the past.
Similarly, the relevant condition when using IV2 is given by:
1
plim
𝐶→∞ 𝐶
𝐶
𝐼
Δ 𝑤𝑗 − 𝑤1
𝑗
𝜂𝑗𝑐
𝑐=1
∆𝜉𝑖𝑐 = 0,
(𝐻9)
𝑖=1
which is satisfied under the same conditions (that 𝐸 ∆𝜖𝑐 = 0 and ∆𝜖𝑐 is independent of the past values of 𝜐𝑖𝑐𝜖 and 𝑝𝑗 𝛺𝑗𝑐 − 𝑝𝛺𝑐 ) required for
(H8) to equal to zero.
OLS can provide consistent estimates and for it to do so requires the assumptions needed for the IV‟s to provide consistent estimates (that
changes in the absolute advantages for a city are independent of the initial set of comparative advantage factors for that city) plus the stronger
assumption that changes in absolute advantage and changes in comparative advantage are independent. Thus, if OLS and IV estimates are
equal then this is a test of the stronger assumption about independence in changes assuming the instruments are valid.
51