Incorporating Price and Inventory Endogeneity in FirmLevel Sales Forecasting
Saravanan Kesavan*, Vishal Gaur†, Ananth Raman‡
October 2007
Abstract
As numerous papers have argued, sales, inventory, and gross margin for a retailer are interrelated. We
construct a simultaneous equations model to establish these interrelationships at a firm level. Using
publicly available financial data, we estimate the six causal effects among sales, inventory, and gross
margin. Our results show that sales, inventory, and gross margin are mutually endogenous. In particular,
we provide new evidence of the impact of inventory on sales and the interrelationship between gross
margin and inventory. We also estimate the effects of exogenous explanatory variables such as store
growth, proportion of new inventory, capital investment per store, selling expenditure, and index of
consumer sentiment on sales, inventory, and gross margin. We show that our model can be used to
generate sales forecasts even when sales are managed using inventory and price. In numerical tests, sales
forecasts from our model are more accurate than forecasts from time-series models that ignore inventory
and gross margin as well as forecasts from equity analysts. Finally, we use an example to illustrate how
our model can be used to benchmark retailers’ performance in sales, inventory, and gross margin
simultaneously.
*
Kenan-Flagler Business School, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599. Email:
skesavan@unc.edu.
†
Johnson Graduate School of Management, Cornell University, Ithaca, NY 14853. Email: vg77@cornell.edu.
‡
Harvard Business School, Boston, MA 02163. Email: araman@hbs.edu.
1. Introduction
The sales, inventory, and gross margin for a retailer are interrelated due to operational reasons. Retailers
often use inventory and gross margin to increase sales. Conversely, sales provide input to retailers’
decisions on inventory and gross margins. Inventory and gross margin also influence each other since
procuring more inventory increases the probability of markdowns, whereas higher gross margin increases
the incentive for retailers to carry more inventory.
It is valuable to quantify these relationships among sales, inventory, and gross margin since they
can be used in multiple applications, including forecasting, benchmarking performance, and planning.
Standard time series forecasting methods can be improved by incorporating inventory and gross margin as
causal variables because historical sales of a retailer are influenced by the amount of inventory carried
and the gross margin realized during those periods. Furthermore, quantifying the bidirectional
relationships among sales, inventory and gross margin enables the retailer to simultaneously forecast all
three variables for the future and simultaneously benchmark its historical performance in the three
variables. It also enables the retailer to actively manage sales, inventory and gross margin because a
change to any of them affects the other two.
Theoretical literature in operations management has postulated several causes for the
interrelationships among sales, inventory, and gross margin. An increase in sales leads to an increase in
average inventory due to economies of scale as shown by the traditional EOQ model. Conversely, an
increase in inventory leads to an increase in expected sales by improving service levels, as is commonly
argued in stochastic inventory theory, as well as due to a demand stimulating effect studied by
Balakrishnan et al. (2004), Dana and Petruzzi (2001), and Smith and Achabal (1998). An increase in gross
margin (or price) increases the optimal inventory as shown in the joint pricing and inventory literature,
see for example, Chen and Simchi-Levi (2004), Federgruen and Heching (1999), and Petruzzi and Dada
(1999). On the flip side, an increase in inventory decreases the gross margin as shown in the markdown
management and clearance pricing literature, see Gallego and van Ryzin (1994) and Smith and Achabal
1
(1998). The interrelationship between gross margin and sales is well-known from the familiar supply and
demand curves in microeconomics.
Practitioners are often interested in understanding and quantifying the relationships among sales,
inventory, and gross margin. However, even though these relationships have been postulated in theory,
most practitioners find it difficult to discern them quantitatively because data observed in practice
represent the joint outcome of all the interrelationships. Raman et al (2005) provide examples where
quantifying these relationships would have provided useful insights to practitioners. Consider for example
Joseph A. Bank Clothiers, Inc (NYSE: JOSB), where sales, inventory, and gross margin increased
substantially during 2000-2004. By 2004, JOSB’s inventory turns were half that of comparable retailers
(e.g., JOSB had 353 days of inventory while competitor Men’s Wearhouse had 164 days of inventory on
hand in 2004). JOSB management claimed that the added inventory was needed to drive customer service
and sales but was unable to persuade some investors of its logic in the absence of a quantitative
relationship between inventory and sales. These skeptical investors felt that JOSB’s inventory was
unnecessarily high. At Home Depot in 2001-2002, sales and inventory declined while gross margins went
up. Some investors wondered if the lower sales were caused by the reduction in inventory levels but were
unable to establish conclusively –in the absence of a quantitative relationship -- that the reduced sales
were because of reduced inventory levels. Finally, at least some investors (e.g., David Berman, a hedge
fund manager) believe that sales growth caused by increased inventory levels are usually less sustainable
than other types of sales growth (e.g., those driven by enhanced consumer acceptance of the retail
concept). A quantitative relationship that quantified the impact of additional inventory on sales would
enable them to distinguish the two types of sales growth and value retailers differently for each of these
components.
Our paper studies the interrelationships among sales, inventory and gross margin, and examines
their implications stated above. Our analysis is conducted using firm-level annual and quarterly data for a
large cross-section of U.S. retailers listed on NYSE, AMEX or NASDAQ. Since sales, inventory and
gross margin are mutually endogenous, we represent them by a triangular model shown in Figure 1 and
2
set up a system of three simultaneous equations, namely, an aggregate sales equation, an aggregate
inventory equation, and an aggregate gross margin equation. By estimating these equations, we test
hypotheses in all six directions of causality represented in the triangular model at a firm-level. Thus, our
analysis decomposes changes in sales, inventory, and gross margin into their various causal components.
We then present two applications of our model. In the first application, we simultaneously generate oneyear-ahead forecasts of sales, inventory, and gross margin. We evaluate sales forecasts from our method
against those from traditional time-series techniques as well as against forecasts of sales provided by
equity analysts. Next, we use our model to benchmark performance of Home Depot during 2001-2002 in
sales, inventory, and gross margin simultaneously.
Our paper yields the following results which contribute to the literature. First, we determine the
effects of sales, inventory, and gross margin on each other. Our estimates of these effects support several
assumptions and analytical findings from the theoretical operations literature and extend the literature by
measuring these effects at firm level. We find that not only an increase in sales leads to an increase in
inventory, but also an increase in inventory leads to an increase in sales. We also find that an increase in
gross margin leads to an increase in inventory, whereas an increase in inventory leads to a drop in gross
margin. Finally, we find support for the supply-demand model with increase in sales leading to an
increase in gross margin and increase in gross margin resulting in lower sales. We term these six causal
effects as price elasticity, inventory elasticity, stocking propensity to sales, stocking propensity to margin,
markup propensity to sales, and markup propensity to inventory.
Second, we define curve shifters in each equation which enable us to identify each causal effect.
Moreover, the effects of curve shifters and other predetermined variables on sales, inventory, and margins
are of independent interest. For example, we find that selling and advertising expenditure has a direct
effect on retailers’ sales. Further, due to the triangular model, selling and advertising expenditure
indirectly affects inventory and margin, which in turn has ripple effects on all three endogenous variables.
The other predetermined variables considered are proportion of new inventory, store growth, capital
investment per store, index of consumer sentiment, and lagged values of sales and margin.
3
Third, we employ our model for forecasting sales, inventory, and gross margin as functions of
predetermined variables. Traditional forecasting models assume that sales are exogenous, and hence,
espouse forecasting for sales and then using the sales forecast to determine inventory. Our model
significantly improves upon traditional forecasting methods because it accounts for sales being managed
with inventory and gross margin and it forecasts for sales, inventory, and gross margin simultaneously
taking into account their mutual interdependence and their co-determination by predetermined variables.
In our evaluations, our model produces sales forecasts that are more accurate than those from two base
models based on time-series historical data as well as forecasts from equity analysts. For the test datasets
from 2004-2006, the mean absolute percent errors (MAPE) of forecast errors from our model are 4.05%
and from the two base models are 6.95% and 7.66%. During the same period, for firms for which
forecasts from equity analysts are available, the MAPE of forecast errors from our model and from equity
analysts are 3.43% and 8.62%, respectively.
Our results build on the previous empirical research on firm-level inventories both
methodologically and by offering new insights. Several authors have studied firm level inventories but
most have performed correlation studies between inventory turns and independent variables, e.g., Gaur et
al. (2005), Gaur and Kesavan (2006), and Roumianetsev and Netessine (2007). A notable exception is the
study by Kekre and Srinivasan (1990), which employs inventory as a variable in a simultaneous equations
model, albeit to study a different issue. Methodologically, ours is the first causal model to conduct joint
estimation of a system of equations to analyze simultaneous variations in sales, inventory, and gross
margin. Instead of inventory turnover, we use sales, inventory, and gross margin as three distinct
dependent variables. This enables us to decompose the variation in inventory turnover into its component
variables, inventory and sales. Our data set is richer than in the previous literature since we expand the set
of explanatory variables in the model to include the proportion of new inventory, selling expenditure,
store growth, index of consumer sentiment, and lagged time-series variables. We also control for the
number of stores, and redefine the metrics for gross margin and capital investment to obtain better
4
statistical properties for our model. Finally, we show sales forecasting as a new application of empirical
models of firm-level inventory.
The results of our paper are useful to equity analysts as well as retail managers. Equity analysts
typically forecast sales and then employ this forecast to predict earnings and determine the equity
valuation for a firm. Our model not only beats analysts’ forecasts of sales, but also results in simultaneous
forecasts for sales, inventory, and gross margin, which can be used in firm valuation. Retail managers can
use our model to measure consumers’ reactions such as price elasticity and inventory elasticity, as well as
to examine their own past actions by measuring stocking propensity to sales and margin and markup
propensity to sales and inventory.
The rest of this paper is organized as follows: §2 presents our hypotheses; §3 describes the dataset
and definitions of variables used in our study; §4 discusses the resulting model and the estimation
methodology; the estimation results are presented in §5; §6 shows the applications of our model; and §7
discusses limitations of our study and directions for future work.
2. Hypotheses on Sales, Inventory, and Margin
We discuss the hypotheses between pairs of variables to differentiate the directions of causality. Though
we motivate our hypotheses in the context of the retailing industry, they may apply to other industries as
well. Our unit of analysis is a firm-year. We use cost of sales as a measure of volume of sales. We define
inventory to be the average of total dollar inventory carried by a firm during the year. Finally, we measure
gross margin by the ratio of revenue to cost of sales and assume that any change in gross margin is due to
change in prices only. Complete definitions of all variables are provided in §3.
2.1 Hypotheses on Sales and Inventory
Increase in inventory can be attributed to an increase in stocking quantities and/or an increase in variety.
Hence, increase in inventory could affect sales by increasing service level, by stimulating demand, or by
increasing choice for consumers. The service level effect of inventory takes place by reducing the
incidence of lost sales when demand is stochastic. The demand stimulating effect can take place in several
ways. Dana and Petruzzi (2001) show a model in which customers are more willing to visit a store when
5
they expect a high service level. Hall and Porteus (2000) and Gaur and Park (2006) study models in
which customers switch to competitors after experiencing stockouts. Balakrishnan et al. (2004) show the
example of a retailer who follows a “Stack them high and let them fly” strategy in which presence of
inventory enhances visibility and could also signal popularity of a product. Raman et al. (2005) discuss
another retailer, Jos Bank, which has strategically increased inventory in order to drive an increase in
sales. In its 10-K reports for the years 2002-2005, the company names ‘inventory in-stock’ as one of its
four pillars of success. van Ryzin and Mahajan (1999) present a model of assortment planning in which
increase in variety results in an increases in sales. Since all of the effects described above are in the same
direction, we hypothesize that increase in inventory causes an increase in sales.
HYPOTHESIS 1. Increase in inventory causes increase in sales.
Note that all our hypotheses are on causalities (and not correlations) so we distinguish between
the directions of causality between variables. While Hypothesis 1 states that inventory causes sales to
increase, we would also expect sales to cause an increase in inventory. This hypothesis is easy to argue
from inventory theory. For example, the EOQ model implies that a retailer’s average stocking quantity is
increasing in mean demand. The newsvendor model also implies this relationship for commonly used
demand distributions such as normal or Poisson. Hence, we set up the following hypothesis.
HYPOTHESIS 2. Increase in sales causes increase in inventory.
2.2 Hypotheses on Inventory and Gross Margin
As inventory increases, the retailer may be forced to take larger markdowns on its merchandise or
liquidate the merchandise through clearance sales. Hence, we expect gross margin to decrease with
inventory. This hypothesis is consistent with the results obtained by Petruzzi and Dada (1999) who show
that the optimal price is decreasing in stocking quantity when demand is linear in price with an additive
error term. This hypothesis is also consistent with Gallego and van Ryzin (1994) who consider dynamic
pricing for a seasonal item whose demand rate is a function of price and show that the optimal price
trajectory is decreasing in the stocking quantity. Smith and Achabal (1998) obtain a similar result for a
model with deterministic demand rate as a multiplicative function of price and stocking quantity.
6
Pashigian (1988) argues that increase in variety leads to increase in markdowns relative to sales. Hence
irrespective of whether an increase in inventory occurs due to increase in stocking quantities or variety,
we expect margin to decline.
HYPOTHESIS 3. Increase in inventory causes decrease in gross margin.
Gross margin has a direct effect on inventory because higher margin induces retailers to carry
more inventory. In the classical newsvendor solution, as margin increases underage cost increases and
results in a higher optimal safety stock. Hence, an increase in margin implies a higher average inventory
level. Further, van Ryzin and Mahajan (1999) show that increase in margin increases the incentive to
stock higher levels of variety. Hence we expect increase in margin to result in higher inventory.
HYPOTHESIS 4. Increase in margin causes increase in inventory.
2.3 Hypotheses on Sales and Gross Margin
A retailer’s gross margin depends on several factors including its pricing strategy, competitive position,
demand for its products, cost of products, etc. For a given cost, margin increases with price. As margin
increases, sales would be expected to decline because demand is generally downward sloping in price.
This motivates Hypothesis 5.
HYPOTHESIS 5. Increase in gross margin causes decrease in sales.
The supply equation in supply-demand model states that price increases with demand. For a given
level of inventory, we expect that as sales increases, retailers would have fewer promotions and clearance
sales. Hence retailers would have higher gross margin.
HYPOTHESIS 6. Increase in sales causes increase in gross margin.
In order to measure these six causal effects, we need to determine curve shifters that would
enable us to decouple causalities between variables. Curve shifters are variables that affect one
endogenous variable but not the others. The heterogeneity in these curve shifters is then exploited to
estimate bidirectional causalities between endogenous variables. Based on Wooldridge (2002), we define
curve shifters as exogenous variables that follow certain exclusion restrictions necessary for the
identification of the simultaneous equations model. Using the example of supply-demand of married
7
women to labor market, Wooldridge (2002: p. 214) shows that presence of kids affects supply of married
women to labor market while the women’s past experience affects labor demand for them. Thus, the
presence of kids and past experience act as curve shifters in estimating supply-demand equations for
married women to labor market.
In the next section, we define the curve shifters and other variables used in our model and
describe the data used in our analysis.
3. Data Description and Definition of Variables
We collect financial data for 1993-2006 of the entire population of public retailers listed on the US stock
exchanges, NYSE, NASDAQ and AMEX, from Standard & Poor’s Compustat database using the
Wharton Research Data Services (WRDS). There were 1217 retailers that report at least one year of data
to the Stock Exchange Commission (SEC) during 1993-2006. We also collect data on the number of
stores and total selling space1 (in square feet) of each retailer in each year from 10-K statements accessed
through the Thomson Research Database. However, since Generally Accepted Accounting Principles
(GAAP) do not mandate retailers to reveal store related information, far fewer retailers report their
number of stores in their chain in their 10K statements. We find that 479 retailers do not report store
information2. We consider only retailers that have at least five years of data on number of stores to enable
us to perform longitudinal analysis. As Table 1a shows, 527 out of the 1217 retailers report store
information for at least five years during our study period.
The Compustat database provides Standard Industry Classification (SIC) codes for all firms,
assigned by the U.S. Department of Commerce based on their type of business. The U.S Department of
Commerce includes eight categories, identified by two digit SIC codes, under retail trade. The retail
categories are Lumber and Other Building Materials Dealers (SIC: 52); General Merchandise Stores
1
Only 349 of the 1217 retailers reported any square-footage information. Further, many retailers provide total space
inclusive of warehouse and DCs without separating the selling space. Thus, we use square-footage data only to
validate the results in the paper.
2
Of these 479 retailers, only 122 retailers do not possess stores. While a majority of the remaining retailers do not
report store information in the 10K statements, we also encountered other issues such as missing firms, missing 10K
statements or corrupt 10K files in the Thomson Research Database.
8
(SIC: 53); Food Stores (SIC: 54); Eating and Drinking Places (SIC: 55); Apparel and Accessory Stores
(SIC: 56); Home furnishing stores (SIC: 57); Automotive Dealers and Service Stations (SIC: 58) and
Miscellaneous Retail (SIC: 59). Table 1b reports the number of retailers in each of these categories. Since
retailers in categories Eating and Drinking Places and Automotive Dealers and Service Stations contain
significant service components to their business we remove them from our sample. Further, we exclude
jewelry firms, which are part of the Miscellaneous Retail sector, from our dataset because we found in
discussion with retailers familiar with this sector that many of the arguments used in our hypotheses (e.g.,
the EOQ model or the demand stimulating effect of inventory) do not apply to jewelry retail. For this
reason, jewelry retailers could not be combined with the rest of the retailers, and further, since there were
only twelve jewelry firms, we could not create a separate group to analyze them. After we remove outliers
in our dataset, our final dataset whittled down from 527 retailers to 302 retailers with 2006 firm-year
observations. All further analysis was performed on this dataset. Table 1c presents the summary statistics
for these firms.
Besides financial data, we obtain index of consumer sentiment (ICS) collected and compiled by
University of Michigan. The index of consumer sentiment represents consumers’ confidence and is
collected on a monthly basis. Finally, we obtain forecasts of annual sales made by equity analysts from
Institutional Brokers Estimate System (I/B/E/S).
We use the following notation. From the Compustat annual data, for firm i in year t, let SRit be
the total sales revenue (Compustat field DATA12), COGSit be the cost of goods sold (DATA41), SGAit
be the selling, general and administrative expenses (DATA189), LIFOit be the LIFO reserve (DATA240),
and RENTit,1, RENTit,2, …, RENTit,5 be the rental commitments for the next five years (DATA96, DATA
164, DATA165, DATA166, and DATA167, respectively). From the Compustat quarterly data, for firm i
in year t quarter q, let PPEitq be the net property, plant and equipment (DATA42), APitq be the accounts
payable (DATA46), and Iitq be the ending inventory (DATA38). Let Nit be the total number of stores open
for firm i at the end of year t.
9
We make the following adjustments to our data. The use of FIFO versus LIFO methods for
valuing inventory produces an artificial difference in the reported ending inventory and cost of goods
sold. Thus, we add back LIFO reserve to the ending inventory and subtract the annual change in LIFO
reserve from the cost of goods sold to ensure compatibility across observations. The value of PPE could
vary depending on the values of capitalized leases and operating leases held by a retailer. We compute the
present value of rental commitments for the next five years using RENTit,1,…,RENTit,5, and add it to PPE
to adjust uniformly for operating leases. We use a discount rate d = 8% per year for computing the present
value, and verify our results with d = 10% as well.
From these data, we define the following variables:
Average sales per store,
CSit = ⎡⎣COGSit − LIFOit + LIFOi ,t −1 ⎤⎦ N it
Average inventory per store,
⎡1 4
⎤
ISit = ⎢
I itq + LIFOit ⎥ N it
⎢⎣ 4 q =1
⎥⎦
Gross Margin,
GM it = SRit ⎡⎣COGSit − LIFOit + LIFOi ,t −1 ⎤⎦
Average SGA per store,
SGASit = SGAit N it
Average capital investment per store,
5
⎡1 4
RENTitτ ⎤
CAPSit = ⎢
PPEitq +
N it
τ ⎥
τ =1 (1 + d ) ⎥
⎣⎢ 4 q =1
⎦
Store growth,
Git = N it N it −1
Proportion of new inventory,
PI it =
∑
∑
4
∑
APitq
q =1
∑
⎡ 4
⎤
⎢ I itq + 4 LIFOit ⎥
⎣⎢ q =1
⎦⎥
∑
Here, average sales per store, average inventory per store, and gross margin are the three
endogenous variables in our study, and the rest are exogenous variables. Proportion of new inventory, PIit,
merits explanation. We define the proportion of new inventory in order to measure the fraction of
inventory that has been purchased recently (see Raman et al. 2005 for the application of this measure to
Jos Bank). Retailers typically pay their suppliers in a fixed number of days as defined in their contracts to
take advantage of favorable terms of payment. Hence, accounts payable represents the amount of
10
inventory purchased by the retailer within the credit period. Hence, larger this ratio more recent the
inventory. This measure differs from the average age of inventory which is defined as 365 divided by
inventory turns. To illustrate this difference, consider two cases. In the first case, a retailer carries two
units of inventory purchased one year ago, and in the second case, a retailer carries one unit of inventory
purchased two years ago and a second unit purchased today. Assume that each unit costs a dollar and the
credit period is less than one year. Then, the accounts payable is zero in the first case, and $1 in the
second case. Hence, proportion of new inventory is zero and 0.5 in the two cases whereas average age of
inventory is 365 in both cases.
We use the annual time period as the unit of analysis because most retailers report store level data
only at the annual level. Moreover, annual data are audited, and hence, of better quality than quarterly
data. We also normalize our variables by the number of retail stores in order to avoid correlations between
sales and inventory that could arise due to scale effects caused by increase or decrease in the size of a
firm.
Using the above definitions, we compute the logarithm of each variable in order to construct a
multiplicative model. The variables obtained after taking logarithm are denoted by lower-case letters, i.e.,
csit, isit, gmit, sgasit, capsit, git, piit, and icst.
4. Model
4.1 Structural Equations
We set up three simultaneous equations, one for each endogenous variable. We use a multiplicative model
for each equation because: (a) a multiplicative model of demand is used extensively in theoretical
operations management and marketing literature; (b) multiplicative models of supply equations are
commonly used in economics and (c) a multiplicative model has been found to fit aggregate inventory
levels in previous research (Gaur et al. (2005) and Roumianetsev and Netessine (2007)).
We consider only within-firm variations in the variables of study because across-firm variations
can be caused by variables omitted from our study such as differences across firms in accounting policies,
11
management ability, firm strategy, store appearance, location, competitive environment in the industry,
etc. We control for differences across firms by using time-invariant firm fixed effects in each equation.
Based on Hypotheses 1-6, several control variables, and firm fixed effects, we specify the three
equations as:
csit = Fi + α11isit + α12 gmit + α13 sgasit + α14 pii ,t −1 + α15 git + α16icst −1 + εit
(1)
isit = J i + α 21csit + α 22 gmit + α 23csi ,t −1 + α 24 pii ,t −1 + α 25 g it + α 26 capsi ,t −1 + ηit
(2)
gmit = H i + α31csit + α32isit + α33 gmi ,t −1 + υit
(3)
Equation (1) models average sales per store, (2) average inventory per store and (3) gross margin. We
name the equations as the aggregate sales equation, the aggregate inventory equation, and the aggregate
gross margin equation, respectively. Each equation consists of firm fixed effects (Fi, Ji, and Hi),
coefficients of endogenous variables, coefficients of predetermined variables, and error terms (εit, ηit, and
υit). The estimates of α11, α12, α21, α22, α31, and α32 enable us to test our six hypotheses. We call these
coefficients the inventory elasticity, the price elasticity, the stocking propensity to sales, the stocking
propensity to margin, the markup propensity to sales, and the markup propensity to inventory,
respectively.3 It is useful to interpret the aggregate sales equation as measuring the consumers’ response
to retailer’s actions and the aggregate inventory and aggregate gross margin equations as measuring the
retailer’s actions on inventory and gross margin.
Figure 2 depicts the endogenous and predetermined variables in equations (1)-(3). The set of
predetermined variables includes SGA per store, lagged proportion of new inventory, lagged capital
investment per store, store growth, lagged index of consumer sentiment, lagged sales per store and lagged
gross margin. We do not lag SGA per store and store growth with the assumption that a retailer can plan
SGA expenses and store growth accurately for the next year. We explain the use of these variables in each
equation as follows:
3
The terms inventory elasticity and price elasticity follow common terminology. In the remaining equations, our
terminology is motivated by Haavelmo (1943) who called the coefficients in a simultaneous equations model as
propensities.
12
Aggregate sales equation: We control for SGA per store, lagged proportion of new inventory,
store growth, and macroeconomic factors. Selling, general and administrative expense per store depends
on costs involved in building brand image, providing customer service and other operational activities
that help to implement a retailer’s competitive strategy (Palepu et al. 2004). We expect sales per store to
increase with SGA per store since prior work has shown that improvement in customer service and
increase in advertising expenses4 have both led to increase in sales (Bass and Clarke 1972).
We control for lagged proportion of new inventory because a mere increase in average inventory
would not increase sales if some of the inventory is stale or obsolete. We expect sales per store to increase
as proportion of new inventory increases. We do not use current proportion of new inventory as a separate
variable because current proportion of new inventory, piit, would be endogenous with average inventory
per store, isit.
We control for store growth because the composition of new and old stores would affect total
sales differently. Contribution of sales from new stores differs from old stores because they are opened
during the middle of the year, and hence, their sales contribution to total sales depends on the number of
days for which the stores were open. Since we do not have information on when the stores were opened
during a year, we use an aggregate measure of change of stores at a retailer as a control variable.
Finally, we use lagged index of consumer sentiment as a leading indicator of macroeconomic
conditions. Carroll et al. (1994) find that the index of consumer sentiment is a leading indicator of change
in personal consumption expenditures, a factor that would affect demand faced by a retailer. Since this
index is calculated every month, we use the index from December as a leading indicator of
macroeconomic conditions for the following year.
Aggregate inventory equation: We control for lagged proportion of new inventory because we
expect average inventory per store to decrease with this variable. If a retailer had a larger fraction of new
inventory in one year, then it would sell more and carry over less inventory in the following year. In this
4
We repeated our analysis with only advertisement expenses instead of SGA and found support for all hypotheses.
Since many firms do not report advertising expenses separately, this sample contained 224 firms with 1351 firmyear observations.
13
reasoning, we equate proportion of new inventory with quality or freshness of products sold at the retailer.
We control for capital investment per store since it can have substitution and/or complementary effects on
inventory. First consider the substitution effect where retailer investment in warehouses, information
technology, ERP or supply chain management systems etc. could lead to lower inventories. Presence of
warehouses enables a retailer to pool its inventory, thus resulting in a lower average inventory level
throughout the chain. Cachon and Fisher (2000) cite several benefits of information technology including
shorter lead times, smaller batch sizes and better allocation to stores that lower average inventory levels in
the retail chain. Several other studies (Clark and Hammond 1997; Kurt Salmon Associates 1993) have
suggested lower average inventory levels as one of the benefits of information technology. Next consider
the complementary effect of capital investment per store on inventory. When a retailer increases its
capital investment per store, it is able to use inventory more productively and hence may find it profitable
to carry more inventory. For example, the retailer may remodel its store, buy new store fixtures or set up a
warehouse that enables it to support its stores with a larger assortment of products. Thus, we expect that
average inventory per store could increase or decrease with increase in capital investment per store.
Since average inventory levels can be sticky we control for lagged sales per store to include the
effect of sales from previous period on the persistence of inventory levels. Finally, we control for store
growth since average inventory per store could vary across old and new stores. For example, the 2005
Annual Report of Jos Bank states that its new stores tend to carry less inventory than old stores.
Aggregate gross margin equation: We use lagged gross margin as a control variable. It is used
as a proxy for the drivers of profitability at a retailer.
Note that the set of pre-determined variables differs across equations. The variables that do not
appear in each equation are SGA expenses, proportion of new inventory, index of consumer sentiment,
lagged sales per store, and lagged gross margin. We use these variables as curve-shifters in order to
identify the coefficients of endogenous variables in the model. Their ability to serve as curve-shifters
depends on our reasoning that each of these variables directly affects only those endogenous variables in
14
whose equations it appears. For example, SGA per store acts as a curve shifter since it affects sales per
store directly and inventory per store and gross margin through its effect on sales per store.
4.2 Estimation of the Simultaneous System of Equations
Due to the presence of firm fixed effects, our model should be estimated by first-differencing or meancentering the observations for each firm. We use first-differencing due to its usefulness for sales
forecasting and superior statistical properties. First-differencing expresses year-to-year changes in sales,
inventory, and gross margin as functions of lagged and exogenous variables. Thus, changes in sales,
inventory, and gross margin can be forecasted using estimates from our model. In contrast, a meancentered model5 cannot be used for sales forecasting without assuming that mean of each variable remains
unchanged when a new year is added - an assumption that is violated by the non-stationarity of our
variables. Mean-centering is also problematic because it produces inconsistent estimates in the presence
of lagged endogenous variables (Wooldridge 2002).
First-differencing equations (1)-(3) gives us the following structural equations:
Δcsit = α10 + α11Δisit + α12 Δgmit + α13 Δsgasit + α14 Δpii ,t −1 + α15 Δgit + α16 Δicst −1 + Δεit
(4)
Δisit = α 20 + α 21Δcsit + α 22 Δgmit + α 23 Δcsi ,t −1 + α 24 Δpii ,t −1 + α 25 Δgit + α 26 Δcapsi ,t −1 + Δηit
(5)
Δgmit = α30 + α31Δcsit + α32 Δisit + α33 Δgit + α34 Δgmi ,t −1 + Δυit
(6)
Here, Δ prefix for each variable denotes first-difference. We solve this simultaneous system to obtain the
reduced form of the equations by expressing each endogenous variable as a function of predetermined
variables only. The three reduced form equations as shown in (7)-(9), with coefficients denoted as β10, …,
β37 which are functions of the structural parameters, α10, …, α34.
Δcsit = β10 + β11Δcsi ,t −1 + β12 Δgmi ,t −1 + β13 Δsgasit + β14 Δpii ,t −1 + β15 Δgit + β16 Δcapsi ,t −1 + β17 Δicst −1 + νit
(7)
Δisit = β20 + β21Δcsi ,t −1 + β22 Δgmi ,t −1 + β23 Δsgasit + β24 Δpii ,t −1 + β25 Δgit + β26 Δcapsi ,t −1 + β27 Δicst −1 + θit
(8)
Δgmit = β30 + β31Δcsi ,t −1 + β32 Δgmi ,t −1 + β33 Δsgasit + β34 Δpii ,t −1 + β35 Δgit + β36 Δcapsi ,t −1 + β37 Δicst −1 + ωit
(9)
5
For completeness, we estimated the mean-centered model as well. We found all hypotheses to be supported (p<
0.05).
15
We first conduct a test of endogeneity to determine if sales per store, inventory per store, and
gross margin are endogenous in each of the equations. The test results support endogeneity in all three
equations at p<0.001. Our test of endogeneity is based on Wooldridge (2002: p. 121).
We then perform the order test to determine if our equations meet exclusion restrictions necessary
for identification (Wooldridge 2002: p. 215). The order test is a necessary condition required to recover
the structural parameters, α10, …, α34, from the coefficients of the reduced form equations. One or more
of the structural parameters cannot be recovered if the coefficients of curve shifters in the reduced form
equations are not statistically significant. We find that all three equations are over-identified, and thus, all
structural parameters can be recovered from the coefficients of the reduced form equations.
We consider different estimation techniques such as three stage least squares (3SLS), 2SLS, and
IVGLS Instrument Variable Generalized Least Squares Method (IVGLS). Both 3SLS and 2SLS require
errors to be homoscedastic. However, the errors in our reduced form model can be both heteroscedastic
and autocorrelated because shocks to sales, inventory, and margin can be contemporaneously correlated
as well as correlated across time periods. For example, a shock to sales in a year could cause inventory to
be high or low in that year as well as in the next year resulting in correlations between residuals in the
aggregate sales equation and aggregate inventory equation. Hence, we choose IVGLS procedure that
takes into account both heteroscedasticity and (AR1) autocorrelation among observations within each
firm (Wooldridge 2002).
5. Estimation Results
5.1 Endogenous Variables
Table 2 presents estimation results for the three structural equations (4)-(6) using data for all 302 firms.
The structural equations capture the effects of the endogenous variable on each other, and thus, enable us
to test our hypotheses. We find that all six hypotheses are supported by the coefficients’ estimates at
p<0.001.
16
Our results support findings from operations management theory. They show that sales,
inventory, and gross margin aggregated at the firm level follow causality relationships that are consistent
with theory developed at the item level. First, consider the aggregate sales equation and the aggregate
inventory equation. The inventory elasticity is 0.224, showing that an increase in inventory causes an
increase in sales. It implies that if a retailer increases inventory in order to increase sales then its
inventory turns would decrease since a 1% increase in inventory per store causes an increase of 0.224%
increase in sales per store. The stocking propensity to sales is 0.828, showing that increase in sales also
causes an increase in inventory. Since this estimate is less than 1, it supports economies of scale in
inventory. It implies that an increase in consumers’ purchases increases the inventory turns of the retailer.
Next, consider the aggregate inventory equation and the aggregate gross margin equation. The stocking
propensity to margin is 0.627 and markup propensity to inventory is -0.245. Thus, an increase in gross
margin causes an increase in inventory consistent with results from stochastic inventory theory. And an
increase in inventory causes a decrease in margin, which supports findings from dynamic pricing
literature (Gallego and van Ryzin 1994; Smith and Achabal 1998) that show that optimal pricing
trajectory decreases with inventory. Finally, in the aggregate sales equation and the aggregate gross
margin equation, price elasticity is -3.934 and retailers’ markup propensity to sales is 0.242. Thus, an
increase in margin causes a decrease in sales, whereas an increase in sales causes an increase in margin as
expected according to the demand-supply model in microeconomics. Besides being consistent with
theory, these results provide new evidence of the impact of inventory on sales and the interrelationship
between margin and inventory. Further the large magnitudes of these estimates make these relationships
economically significant and managerially relevant.
Our model distinguishes between direct and indirect effects of sales, inventory, and gross margin
on each other. We find that the second order indirect effect can sometimes be larger or be of opposite sign
compared to the first order direct effect. This implies that both first and higher order effects are
manifested in the data collected on retailers. For example, consider the effects of inventory on sales.
Inventory directly affects sales through inventory elasticity, i.e., by reducing lost sales, stimulating
17
demand, and/or by increasing choice to customers. Additionally, inventory indirectly affects sales through
margin: an increase in inventory leads to a decrease in margin, which in turn leads to a further increase in
sales. The sum of first and second order effects of inventory on sales is equal to 0.224 + (-0.245)(-3.934)
= 1.187. Here, the second order effect of inventory on sales is much larger than the corresponding first
order effect. Next consider the effect of gross margin on inventory. We find that the first and second order
effects of margin on inventory are 0.627 and -3.257 (= (-3.934)(0.828)) showing that first and higher
order effects can be of opposite signs. In other words, the effect of an increase in margin on sales
outweighs the direct effect of margin on inventory in the observed data. Likewise, for the remaining pairs
of variables we get:
Effect of sales on inventory = 0.828 + (0.242)(0.627) = 0.979
Effect of inventory on margin = -0.245 + (0.224)(0.242) = -0.191,
Effect of margin on sales = -3.934 + (0.627)(0.224) = -3.793,
Effect of sales on margin = 0.242 + (0.828)(-0.245) = 0.039.
We note from these estimates that sales and inventory are very sensitive to changes in each other
and in margin. However, margin is less sensitive to changes in sales and inventory. Thus, we see that
small changes in margin in our dataset lead to large changes in sales and inventory. Changes in sales and
inventory also lead to large changes in each other, but do not affect margins by much.
To test our hypotheses at the segment level and to determine how the coefficients vary across
segments, we estimate the structural equations separately for different industry segments. Since the retail
category Lumber and other building materials contained only 13 firms, we combine these firms with
Home Furnishing Stores. Table 3 reports the coefficient estimates of the six causal effects. We have a
total of thirty hypotheses tests consisting of six hypotheses on five segments. Of these thirty tests, our
hypotheses are statistically supported in 27 cases (p<0.001). Two of the three remaining cases have
estimates in the direction hypothesized but not statistically significant. In the case of General Stores, we
find that the stocking propensity to margin is negative and significant. Further, we find that even when the
coefficients are significant, their magnitudes differ considerably across segments. For example, inventory
18
elasticity is found to vary between 0.188 for Home Furnishing Stores to 0.321 for Apparel. This implies
that and increase in inventory leads to a larger increase in sales for apparel retailers than for home
furnishing retailers. Future research could investigate the reasons for these differences.
5.2 Predetermined Variables
We use both structural and reduced form equations to interpret the coefficients of predetermined
variables. Estimates from reduced form equations are useful because they capture the overall effect of the
predetermined variables on each endogenous variable whereas the structural equations capture only first
order effects. They are also useful for forecasting sales per store, inventory per store, and gross margin.
Tables 2 and 4 show the estimates of coefficients of structural equations and reduced form equations,
respectively. We summarize some of the insights from these estimates as below. All coefficients are
statistically significant at p<0.001.
Consider the effect of selling and advertising expenses on the endogenous variables. Table 2
shows that a 1% increase in SGA per store increases sales per store by 0.702%. Further, Table 4 shows
that the overall effects of a 1% increase in SGA per store are an increase in sales per store by 0.688%, in
inventory per store by 0.570%, and in gross margin by 0.033%. All of these effects are statistically
significant at p<0.01. Thus, when a retailer increases spending in selling and advertising activities, then
on average, it increases sales and margin while carrying more inventory. While SGA per store appears
only in the equation for sales per store in the structural form, it has higher order effects on inventory per
store and margin. The total effects are captured by the reduced form equation.
The magnitudes of the coefficients of SGA per store can be used to determine whether it is
profitable for a retailer to increase SGA spending at the margin. As shown in Table 1c, SGA per store is
$0.968M, sales per store is $2.452M, inventory per store is $0.585M and gross margin is 1.542 on
average across all retailers in our dataset. Using the definitions of these variables given in §3, we see that
a 1% increase in SGA expense, i.e., a $9680 increase in SGA expense, leads to:
an increase in gross margin to 1.542*1.00033 = 1.5425,
an increase in cost of goods sold per store to $2.452M * 1.00688 = $2.4689M,
19
an increase in sales per store to $2.4689M * 1.5425 = $3.8082M,
an increase in inventory per store to $0.585M * 1.0057 = $0.5883M.
Thus, the gross profit of the retailer increases from $(2.452*1.542 – 2.452) = $1.3290M to $(3.8082 –
2.4689) = $1.3394M, which is an increase of $10,400 and is greater than the increase in SGA expenses of
$9,680. Since inventory per store has gone up by $3334.5, the increase in profit is justified if inventory
holding costs are sufficiently low (i.e., below 21.5% per year). Thus, our results show how additional
investment in SGA expenses may or may not be profitable for the retailer. Similar analysis can be done
for the other predetermined variables in our model as well.
As to the effect of lagged proportion of new inventory on the endogenous variables, Table 2
shows that a 1% increase in proportion of new inventory causes sales per store to increase by 0.036% and
inventory per store to decrease by 0.014%; these effects are in the directions expected. However, Table 4
shows that the overall effect of proportion of new inventory on the three endogenous variables is different
from the first order effects. A 1% increase in proportion of new inventory leads to 0.023% decrease in
sales per store. This change in sign between the first and overall effect is partly explained by the
simultaneous decrease in inventory per store (0.014%) caused by the increase in proportion of new
inventory. Finally, Table 4 shows that the increase in proportion of new inventory is associated with a
0.014% increase in gross margin. This is expected because more new inventory would imply a larger
proportion of full price sales and less old inventory to be marked down.
We find that a 1% increase in lagged capital investment per store increases both sales per store
(0.011%) and inventory per store (0.052%) while decreasing margin (-0.004%). In §4.1, we expected
capital investment per store to either cause an increase or decrease in average inventory per store
depending on whether the substitution or the complementary effect dominates. Our results support
presence of complementary effects in our sample.
The rest of the coefficients’ estimates for predetermined variables are as expected and support the
reasoning in §4.1. In particular, increase in store growth leads to a decrease in sales per store (-0.169%),
decrease in inventory per store (-0.247%), and increase in margin (0.025%). Increase in the index of
20
consumer sentiment leads to increases in sales per store (0.037%), inventory per store (0.088%), and
gross margin (0.023%).
6. Applications
In this section, we present two applications of our model. In the first application, we use the reduced form
equations to generate sales forecasts for our sample of retailers and test the predictive power of the model.
In the second application, we use residuals from the structural equations to estimate performance
differences in endogenous variables across firms and over time that are not accounted for by changes in
the predetermined variables. We illustrate the application by benchmarking the performance of Home
Depot during 2001-2002.
6.1 Sales Forecasting
Simultaneous equations models have been extensively used in forecasting (Pindyck and Rubenfeld 1997).
We forecast sales per store and gross margin simultaneously using our reduced form model. Since sales
per store are computed at cost, we combine forecasts of sales per store and margin in order to obtain a
forecast of sales revenue per store. Thus, from equations (7)-(9), after removing first-differencing, we get:
l 1it = cs + β + β Δcs + β Δgm + β Δsgas + β Δpi + β Δg + β Δcaps + β Δics + ρ (cs
l 1it −1 − cs )
cs
10
11
12
13
14
15
16
17
it −1
it −1
it −1
it −1
it
it −1
it
it −1
it −1
1i
(10)
m = gm + β + β Δcs + β Δgm + β Δsgas + β Δpi + β Δg + β Δcaps + β Δics + ρ ( gm
m − gm )
gm
30
31
32
33
34
35
36
37
it −1
it −1
it −1
it
it −1
it
it −1
it −1
it −1
1it
1it
2i
(11)
l it = cs
l it + gm
m
sr
it
(12)
Here, β 10 ,..., β 37 are the coefficients’ estimates shown in Table 4, ρ 1i and ρ 2i are estimates of panel-
l it denotes the forecast of logged sales revenue per store.
specific AR(1) coefficients and sr
We compare forecasts from our model against forecasts from two base models and forecasts
provided by equity analysts. In the first base model, we use Box-Jenkins method to model sales per store
and gross margin as independent ARIMA (1,1,1) processes. We then estimate the coefficients of these
models using generalized least squares to incorporate heteroscedasticity and AR(1) autocorrelation.
l 2it = cs + γ + γ Δcs + ρ (cs
l 2it − cs )
cs
it −1
10
11
it −1
3i
it −1
21
(13)
m = gm + γ + γ Δgm + ρ ( gm
m − gm )
gm
it −1
it −1
it −1
20
21
4i
2 it
2 it
(14)
The second base model is a linear exponential smoothing model which allows forecasting of data
with trends. The sales revenue forecast for period t is
l it = L + b
sr
i ,t −1
i ,t −1
(15)
where Lt and bt denote estimates of level and slope of series at time t respectively. They are given by
l it + (1 − α )( L + b ) ,
Lit = α sr
i ,t −1
i ,t −1
bit = β ( Lit − Li ,t −1 ) + (1 − β )bi ,t −1 ,
where α and β are smoothing constants with values α=0.75 and β=0.15.
We follow the methodology provided by Pindyck and Rubenfeld (1997) to compare forecasts
from our model against forecasts from base models. We evaluate forecasts from ex-post simulations, i.e.,
forecasts for the fit dataset, and from ex-post forecasts, i.e., forecasts for the test dataset. We use three fit
datasets and three test datasets. First, we use data for 1993 to 2003 to fit our model, leaving out
observations for 2004 as the test sample. Next we use data from 1993 to 2004 to fit our model and
observations from 2005 as the test sample. Then we repeat the same using observations from 2006 as the
test sample.
The results of the ex-post simulations are given in Table 5. The reported forecast errors are based
on dollar figures, not the logged values obtained from the models. As we see the MAPE, MAD, and
RMSE of errors from reduced form model are lower than those of errors from the time series and
exponential smoothing models in all three fit datasets. For example, the MAPE, MAD and RMSE of
errors from simulations of reduced form during 1993-2003 are 4.44%, 0.32, and 0.94 while those of the
errors from the Time Series model are 9.27%, 0.82, and 2.87 and those of the errors from the exponential
smoothing model are 9.72%, 0.84, and 2.86. Hence, we see that the forecasts from the reduced form
model are more accurate than those from the two base models.
Table 5 also presents the accuracy measures computed for the reduced form model and the base
models for the ex-post forecasts. We find that the MAPE, MAD, and RMSE of forecast errors from the
22
reduced form model are lower than those from the two base models for 2004-2006. For example, the
MAPE, MAD and RMSE values for 2004 of forecast errors from reduced form model, time series model,
and exponential smoothing model are (4.41%, 0.39, 1.14), (7.60%%, 0.60, 1.39), and (7.72%, 0.65, 1.46),
respectively.
Next we compare forecasts from the reduced form model against those from equity analysts for
years 2004-2006. We use analyst forecasts from I/B/E/S database made thirteen months before the fiscal
year end date. For example, if the end date of fiscal year 2004 was January 31, 2005, then we use
forecasts made on or before December 31, 2003 for this year. To be comparable, we generate one-year
ahead forecasts for 2004-2006 using the reduced form model. Since analysts do not cover all firms, fewer
firms have analysts’ forecasts available. The sample sizes for years 2004-2006 are 114, 114, and 84
respectively. Since analysts forecast directly for sales revenue, we could either divide their forecasts by
number of stores to obtain sales revenue per store or, equivalently, scale up forecasts of sales per store
from our model to obtain sales forecasts for entire retailer chain. We choose the former for our analysis.
Table 6 presents the results of this analysis for 2004-2006. We find that forecasts from the reduced form
model are more accurate than forecasts from equity analysts based on MAP, MAD, and RMSE values for
all three years. For the entire period 2004-2006, the MAPE, MAD, and RMSE for forecasts from our
model and for those from equity analysts are (3.43%, 0.25, 0.50) and (8.62%, 0.63, 1.19) respectively.
The above results demonstrate the effectiveness of our model to forecast sales. The model has
more potential use in forecasting earnings since it jointly forecasts sales, inventory and margin.
Commonly, equity analysts forecast earnings performance of a firm by forecasting sales and then using
the sales forecasts as input to forecast line items in income statement and balance sheet by making
assumptions about gross margin, inventory turns etc. based on past observations (Palepu et al. 2004). Our
methodology provides considerable improvements to such a sequential forecasting process by not only
producing sales forecasts that are more accurate, but also forecasting inventory and gross margin.
23
6.2 Performance Benchmarking
Since our model differentiates amongst changes in sales due to various causes, its predictions can be used
to benchmark retailers’ performance on sales, inventory levels, and gross margin. This section discusses
the computation of benchmarks and illustrates our methodology using data for Home Depot for the years
2000-2002. When Home Depot’s sales declined in 2001 and 2002, some investors claimed that the sales
decline was in line with their predictions based on the management’s decision to lower inventory.
However, Home Depot’s gross margin increased during this period. Our model enables us to separate the
impact of inventory and gross margin changes and measure if the sales decline was commensurate with
the lower inventory carried by Home Depot.
Our methodology is as follows. We use the estimates for the structural equations (4)-(6) obtained
for the Home Furnishing Stores segment, which includes Home Depot, to define the following residuals:
Abnormal sales per store = Δcsit − (0.001 + 0.188Δisit − 2.834Δgmit + 0.672Δsgasit + 0.011Δpii ,t −1
−0.023Δg it + 0.019Δicst −1 )
Abnormal inventory per store = Δisit − ( −0.013 + 0.919Δcsit + 0.22Δgmit + 0.051Δcsi ,t −1 − 0.007 Δpii ,t −1
−0.1001Δg it + 0.046 Δcapsi ,t −1 )
Abnormal margin per store = Δgmit − (0.001 + 0.022 Δcsit − 0.039 Δisit + 0.048Δgmi ,t −1 )
Abnormal sales per store is interpreted equivalently to distance from a demand curve. It indicates the
amount by which sales differ from the model prediction given inventory level, gross margin and other
explanatory variables. Abnormal inventory per store is equivalent to distance from a supply curve. It
indicates the amount by which the retailer has higher or lower inventory than predicted for given values
of price, sales and other explanatory variables. Abnormal margin per store is also equivalent to distance
from a supply curve. It indicates the amount by which the retailer is charging higher or lower price than
predicted for given inventory supply, sales, and remaining variables. Thus, the three residuals take into
account market conditions as well as the retailer’s actions on all explanatory variables, endogenous and
exogenous. We recall that sales per store is computed at cost as defined in §2.
Table 7 shows the raw data and residuals for Home Depot for 2001-2002. We observe that both
the sales per store and the inventory per store declined, but gross margin increased from 2001 to 2002.
24
The last three rows of Table 7 show the abnormal sales per store, abnormal inventory per store and
abnormal margin per store. In 2001 and 2002, based on the abnormal sales per store values, we find that
Home Depot’s sales per store were higher than predicted after controlling for retailer actions including
those that resulted in lower inventory per store. By attributing the entire sales decline to decrease in
inventory, some investors underestimated the sales performance of Home Depot during these years. Our
results show that once retailer actions have been controlled for, market conditions contributed to higher
sales than predicted for Home Depot in both years.
Next we investigate the abnormal inventory per store for Home Depot in 2001 and 2002. Here we
find that once we control for market condition, margin and other explanatory variables, Home Depot
carried lower inventory than predicted by our model in 2001 and more inventory than predicted in 2002.
Hence inventory related costs could have been lower than predicted in 2001 and higher than predicted in
2002. Finally we find the abnormal gross margin for Home Depot to be zero in 2001 and positive in 2002.
This indicates that Home Depot’s margin is improving after we take into account the market condition
and retailer actions.
In this section, we used the structural equations model to compute performance benchmarks. One
may also use the reduced form equations (7)-(9) to compute predicted values for benchmarking. The
predictions from the reduced form model can be constructed a priori in the same manner as we did in §6.1
for forecasting. We used the structural equations in this section because they were based on not only the
predetermined variables, but also the endogenous variables. Hence, they took into account the effect of
actual actions of Home Depot in evaluating its performance. Further comparisons between these two
types of predictions are left for future research.
7. Conclusions, Limitations, and Future Work
In this paper, we constructed a simultaneous equations model to examine the relationships among sales
per store, inventory per store, and gross margin. We defined curve shifters that permit us to determine the
directions of causality among these variables. We showed that sales per store, inventory per store, and
25
gross margin are mutually endogenous. We further showed that the reduced form model of the
simultaneous equations enables us to forecast sales, inventory, and gross margin jointly from historical
data after accounting for the interrelationships among them. Our tests showed the sales forecasts thus
generated to be more accurate than forecasts from two base models and from equity analysts. Finally, we
used the example of Home Depot to illustrate how one can benchmark a retailer’s performance in sales,
inventory, and gross margin simultaneously.
Our estimates of the six causal effects have some limitations due to the use of aggregate financial
data. Since we use quarterly and annual financial data to estimate these effects, their estimates could
suffer from aggregation bias due to aggregation across SKUs, stores, and time. Future research may refine
these estimates with disaggregated data. Also, since financial data can be noisy our estimates of the six
causal effects may be improved by using more accurate or detailed firm-level data. Examples of such data
include capital investment, distribution of the age of inventory, square-footage of stores etc.
We identify the study of inventory elasticity and various propensities measured in this paper as
one of the areas of future research. Our study shows that inventory elasticity exhibits considerable
heterogeneity among retail segments. This heterogeneity provides opportunities for future research to
examine the drivers of these differences. Also, our study is the first to measure the stocking and markup
propensities of retailers and the results indicate that these propensities are not only high in magnitude but
also exhibit variability across the retail segments. Some of the factors that can be studied to analyze the
differences in these estimates are extent of stock-outs, substitutability of products, lead times for
procurement, product characteristics, competition, etc. Further, with sufficient data, it may be possible to
repeat the analysis for each firm and determine the drivers of elasticities and propensities at firm level.
Another direction for future research may be availed by applying our model for predicting
earnings. We showed that our model is superior to traditional forecasting since it yields joint forecasts for
sales, inventory, and margin. We may use these forecasts to predict future earnings of firms and
determine whether the resulting method yields more accurate forecasts than those provided by equity
analysts. Finally, this methodology may be tested for manufacturers and wholesalers.
26
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van Ryzin, G., S. Mahajan. 1999. On the Relationship between Inventory Costs and Variety Benefits in
Retail Assortments. Management Science. 45 1496-1509.
Wooldridge, H. M. 2002. Econometric Analysis of Cross Section and Panel Data. The MIT Press,
Cambridge, Massachusetts.
28
Figure 1: Triangular model of endogeneity among sales, inventory, and margin
Figure 2: Effects of endogenous and exogenous variables on sales, inventory, and margin
SGA per store, sgait
Lagged index of consumer
sentiment, icsi,t-1
Sales per
store, csit
Store growth, git
Inventory per
store, isit
Lagged proportion of new
inventory, pii,t-1
Lagged sales per store, csi t-1
Gross margin,
gmit
Lagged cap. investment per
store, pii,t-1
Lagged gross margin, gmi,t-1
29
Table 1a. Frequency table of number of years in which store related information was reported by
retailers during 1993-2006
Number of years
in which store
information was Number of
reported
firms
0
479
1-2
92
3-4
119
5-6
136
7-8
110
9-10
66
11-12
67
13-14
148
Table 1b. Store reporting across different retail sectors during 1993-2006
Retail Sector
Two Digit
SIC code
Examples
Number of
Retailers
Number of Retailers that
reported store
information for atleast 5
years
Lumber and other
building materials
52
Home Depot, Lowe's,
National Home Centers
42
13
General
Merchandise Stores
53
Costco, Dollar General,
Wal-mart
104
48
Food Stores
54
Safeway, Dairy Mart
Convenience stores,
Shaws
127
57
Eating and Drinking
Places
55
Ruby Tuesday, Pizza Inn,
Planet Hollywood
65
139
Apparel and
Accessory Stores
56
Mens Wearhouse, Harolds,
Childrens Place
117
73
Home Furnishing
Stores
57
Williams-Sonoma, Jennifer
Convertibles, Circuit City
99
44
Automotive Dealers
and Service Stations
58
Pep Boys, Discount Auto
Parts, Carmax
286
37
Miscellaneous Retail
59
Toys R Us, Officemax,
Walgreen
377
116
30
Table 1c Variable Definitions and Summary Statistics
Variables
Mean
Standard
Deviation
Min
Max
Average Sales per store ($ M)
csit
2.452
4.229
0.005
121.875
Average Inventory per store ($ M)
isit
0.585
3.732
0.003
32.04
gmit
1.542
1.2
0.949
18.211
Average SGA per store ($ M)
sgasit
0.968
3.611
0.013
50.451
Average Capital Investment per store ($ M)
capsit
1.09
3.504
0.012
30.938
Store growth
git
1.067
1.189
0.285
4.874
Proportion of new inventory
piit
0.437
1.737
0.044
9.796
Index of consumer sentiment
icsit
95.679
1.061
86.661
105.32
Definitions
Margin
Note: Summary statistics based on 2006 firm-year observations.
Table 2: Estimation Results of Structural Equations
Aggregate Sales
Equation (Δcsit)
Sales per store (Δcsit)
Inventory per store (Δisit)
Gross Margin (Δgmit)
0.224**
(0.021)
-3.934**
(0.226)
Lagged proportion of new
inventory (Δpiit-1)
Store growth (Δgit)
Lagged index of consumer
sentiment (Δicsi-1)
Lagged capital investment per
store (Δcapsit-1)
Aggregate Gross
Margin Equation (Δmut)
0.242**
(0.002)
-0.245**
(0.002)
0.627**
(0.212)
0.045**
(0.006)
Lagged sales per store (Δcsit-1)
SGA per store (Δsgasit)
Aggregate Inventory
Equation (Δisit)
0.828**
(0.011)
0.702**
(0.021)
0.036**
(0.003)
-0.006*
(0.003)
0.105**
(0.008)
-0.014**
(0.004)
-0.098**
(0.009)
0.049**
(0.002)
Constant
0.001*
(0.0004)
-0.014**
(0.001)
0.067**
(0.003)
-0.001**
(0.000)
Wald χ2
2349336**
398186.96**
27214.01**
Lagged gross margin (Δgmit-1)
Notes: All variables have been first-differenced. *,** denote statistically significant at 0.05 and < 0.001, respectively.
The numbers in brackets below the parameter estimates are the respective standard errors. Wald test statistic
compares fit of model including explanatory variables to fit of model with only the intercept.
31
Table 3: Summary Results of Structural Equations for Different Retail Segments
Retail Industry
Segment
Number
of Firms
Number of
Observations
47
317
51
Apparel and
Accessory Stores
Inventory
Elasticity
Stocking
Propensity to
Sales
Stocking
Propensity to
Margin
Markup
Propensity to
Sales
Markup
Propensity to
Inventory
-2.592**
(0.102)
0.193**
(0.008)
0.930**
(0.021)
-2.718**
(0.178)
0.082**
(0.016)
-0.062**
(0.016)
325
-3.708**
(0.287)
0.270**
(0.0156)
0.737**
(0.013)
1.515**
(0.242)
0.029**
(0.007)
-0.027**
(0.008)
68
518
-1.338**
(0.111)
0.321**
(0.021)
0.752**
(0.017)
0.430**
(0.154)
0.163**
(0.024)
-0.118**
(0.025)
Home Furnishing
Stores
51
345
-2.834**
(0.334)
0.188**
(0.009)
0.919**
(0.019)
0.220
(0.376)
0.022
(0.015)
-0.039**
(0.013)
Miscellaneous
Retail
85
501
-3.685**
(0.155)
0.199**
(0.009)
0.832**
(0.011)
0.687**
(0.167)
0.057**
(0.006)
-0.053**
(0.006)
General
Merchandise
Stores
Food Stores
Price
Elasticity
Note: ** denotes statistically significant at p<0.001. The numbers in brackets below the parameter estimates are the respective standard errors.
32
Table 4: Estimation Results of Reduced Form Equations
Sales per store (Δcsit)
Inventory per store
(Δisit)
Gross Margin (Δgmit)
Lagged sales per store
(Δcsit-1)
0.143**
(0.001)
0.179**
(0.005)
-0.021**
(0.001)
SGA per store (Δsgasit)
0.688**
(0.004)
0.570**
(0.003)
0.033**
(0.001)
Lagged proportion of
new inventory (Δpiit-1)
-0.023**
(0.003)
-0.025**
(0.002)
0.014**
(0.000)
Store growth (Δgit)
-0.169**
(0.002)
-0.247**
(0.005)
0.025**
(0.001)
Lagged capital
investment per store
(Δcapsit-1)
0.011**
(0.001)
0.052**
(0.002)
-0.004**
(0.000)
Lagged gross margin
(Δgmit-1)
0.145**
(0.008)
0.383**
(0.013)
0.003**
(0.003)
Lagged index of
consumer sentiment
(Δicst-1)
0.037**
(0.002)
0.088**
(0.003)
0.023**
(0.001)
Constant
-0.007**
(0.000)
-0.019**
(0.000)
0.001**
(0.000)
Wald χ2
2122433**
1709768**
21314.76**
Note: All variables have been first-differenced. ** denote statistically significant at <0.001. The numbers in brackets
below the parameter estimates are the respective standard errors.
Table 5: Evaluation of Forecast Accuracy for Various Fit and Test Samples
Ex-post Simulations
Ex-post Forecasts
Linear
Reduced Time
Linear Exp.
Reduced Time
Exp.
Form
Series Smoothing
Form
Series
Smoothing
Model
Model
Model
Model
Model
Model
MAPE(%) 4.44%
9.27%
9.72%
4.41%
7.60%
7.72%
Fit = 1993-2003, Fit = 1548,
MAD
0.32
0.82
0.84
0.39
0.6
0.65
Test = 2004
Test = 179
RMSE
0.94
2.87
2.86
1.14
1.39
1.46
MAPE(%) 4.44%
9.06%
9.57%
3.38%
7.32%
8.14%
1993-2004, 2005 1727, 168
MAD
0.33
0.79
0.82
0.28
0.78
0.83
RMSE
0.92
2.74
2.75
0.6
2.8
2.64
MAPE(%) 4.35%
8.90%
9.39%
4.35%
5.94%
7.11%
1993-2005, 2006 1895, 111
MAD
0.32
0.79
0.82
0.43
0.71
0.79
RMSE
0.9
2.75
2.74
0.97
1.96
2.07
Time periods No. of obs.
for fit and test
for fit and
samples
test samples
33
Table 6: Comparison of Forecast Accuracy with Equity Analysts’ Sales Forecasts
Year
Reduced Form
Model
Analysts' Forecasts
3.61%
9.11%
MAD
0.26
0.58
RMSE
0.56
1.11
3.09%
8.59%
MAD
0.24
0.66
RMSE
0.46
1.25
n
MAPE(%)
2004
114
MAPE(%)
2005
114
MAPE(%)
2006
84
3.60%
8.16%
MAD
0.26
0.64
RMSE
0.49
1.2
3.43%
8.62%
MAD
0.25
0.63
RMSE
0.50
1.19
MAPE(%)
Overall (2004-2006)
312
Table 7: Benchmarking Home Depot’s performance in Sales, Inventory, and Gross Margin during
2001-2002
Values of variables for each year
Y-to-Y differences between log
values
2000
2001
2002
[2001] - [2000]
[2002] – [2001]
Sales per store, CSt
27.74
27.49
25.61
-0.01
-0.07
Inventory per store, ISt
5.67
5.33
5.10
-0.06
-0.04
Gross margin, GMt
1.45
1.46
1.48
0.01
0.02
SGA per store, SGASt
8.37
8.38
8.01
0.00
-0.04
Lagged proportion of new
inventory, PIt-1
0.46
0.43
0.48
-0.07
0.11
Store growth rate, Gt
1.22
1.18
1.15
-0.04
-0.02
12.16
12.33
11.98
0.01
-0.03
98.4
88.8
86.7
-0.10
-0.02
Abnormal sales per store
0.02
0.01
Abnormal inventory per
store
-0.04
0.03
Abnormal gross margin
0
0.02
Lagged capital investment
per store, CAPSt
Lagged index of consumer
sentiment, ICSt-1
Residuals from the Structural Equations Model
34