Incorporating Price and Inventory Endogeneity in Firm Level Sales Forecasting


 Martin Henderson
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1 Incorporating Price and Inventory Endogeney in Firm Level 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, capal investment per store, selling expendure, 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 timeseries models that ignore inventory and gross margin as well as forecasts from equy 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. * KenanFlagler Business School, Universy of North Carolina at Chapel Hill, Chapel Hill, NC Johnson Graduate School of Management, Cornell Universy, Ithaca, NY Harvard Business School, Boston, MA
2 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 probabily 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 s 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 lerature 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 tradional 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 Smh and Achabal (1998). An increase in gross margin (or price) increases the optimal inventory as shown in the joint pricing and inventory lerature, see for example, Chen and SimchiLevi (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 lerature, see Gallego and van Ryzin (1994) and Smh and Achabal 1
3 (1998). The interrelationship between gross margin and sales is wellknown from the familiar supply and demand curves in microeconomics. Practioners 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 practioners find difficult to discern them quantatively 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 practioners. Consider for example Joseph A. Bank Clothiers, Inc (NYSE: JOSB), where sales, inventory, and gross margin increased substantially during By 2004, JOSB s inventory turns were half that of comparable retailers (e.g., JOSB had 353 days of inventory while competor 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 s logic in the absence of a quantative relationship between inventory and sales. These skeptical investors felt that JOSB s inventory was unnecessarily high. At Home Depot in , 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 quantative 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 quantative relationship that quantified the impact of addional 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 firmlevel annual and quarterly data for a large crosssection 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
4 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 causaly represented in the triangular model at a firmlevel. 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 oneyearahead forecasts of sales, inventory, and gross margin. We evaluate sales forecasts from our method against those from tradional timeseries techniques as well as against forecasts of sales provided by equy analysts. Next, we use our model to benchmark performance of Home Depot during in sales, inventory, and gross margin simultaneously. Our paper yields the following results which contribute to the lerature. 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 lerature and extend the lerature 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 supplydemand model wh 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 elasticy, inventory elasticy, stocking propensy to sales, stocking propensy to margin, markup propensy to sales, and markup propensy 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 expendure has a direct effect on retailers sales. Further, due to the triangular model, selling and advertising expendure 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, capal investment per store, index of consumer sentiment, and lagged values of sales and margin. 3
5 Third, we employ our model for forecasting sales, inventory, and gross margin as functions of predetermined variables. Tradional 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 tradional forecasting methods because accounts for sales being managed wh inventory and gross margin and forecasts for sales, inventory, and gross margin simultaneously taking into account their mutual interdependence and their codetermination by predetermined variables. In our evaluations, our model produces sales forecasts that are more accurate than those from two base models based on timeseries historical data as well as forecasts from equy analysts. For the test datasets from , 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 equy analysts are available, the MAPE of forecast errors from our model and from equy analysts are 3.43% and 8.62%, respectively. Our results build on the previous empirical research on firmlevel 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, albe 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 s component variables, inventory and sales. Our data set is richer than in the previous lerature since we expand the set of explanatory variables in the model to include the proportion of new inventory, selling expendure, store growth, index of consumer sentiment, and lagged timeseries variables. We also control for the number of stores, and redefine the metrics for gross margin and capal investment to obtain better 4
6 statistical properties for our model. Finally, we show sales forecasting as a new application of empirical models of firmlevel inventory. The results of our paper are useful to equy analysts as well as retail managers. Equy analysts typically forecast sales and then employ this forecast to predict earnings and determine the equy 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 elasticy and inventory elasticy, as well as to examine their own past actions by measuring stocking propensy to sales and margin and markup propensy to sales and inventory. The rest of this paper is organized as follows: 2 presents our hypotheses; 3 describes the dataset and definions 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 limations 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 causaly. Though we motivate our hypotheses in the context of the retailing industry, they may apply to other industries as well. Our un of analysis is a firmyear. 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 definions of all variables are provided in Hypotheses on Sales and Inventory Increase in inventory can be attributed to an increase in stocking quanties 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 vis a store when 5
7 they expect a high service level. Hall and Porteus (2000) and Gaur and Park (2006) study models in which customers swch to competors 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 visibily and could also signal populary 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 s 10K reports for the years , the company names inventory instock as one of s 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 causalies (and not correlations) so we distinguish between the directions of causaly 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 quanty 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 s merchandise or liquidate the merchandise through clearance sales. Hence, we expect gross margin to decrease wh inventory. This hypothesis is consistent wh the results obtained by Petruzzi and Dada (1999) who show that the optimal price is decreasing in stocking quanty when demand is linear in price wh an addive error term. This hypothesis is also consistent wh Gallego and van Ryzin (1994) who consider dynamic pricing for a seasonal em whose demand rate is a function of price and show that the optimal price trajectory is decreasing in the stocking quanty. Smh and Achabal (1998) obtain a similar result for a model wh deterministic demand rate as a multiplicative function of price and stocking quanty. 6
8 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 quanties 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 s pricing strategy, competive posion, demand for s products, cost of products, etc. For a given cost, margin increases wh 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 supplydemand model states that price increases wh 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 causalies between variables. Curve shifters are variables that affect one endogenous variable but not the others. The heterogeney in these curve shifters is then exploed to estimate bidirectional causalies 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 supplydemand of married 7
9 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 supplydemand 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 Definion of Variables We collect financial data for 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 We also collect data on the number of stores and total selling space 1 (in square feet) of each retailer in each year from 10K 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 information 2. We consider only retailers that have at least five years of data on number of stores to enable us to perform longudinal 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 dig 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 squarefootage information. Further, many retailers provide total space inclusive of warehouse and DCs whout separating the selling space. Thus, we use squarefootage data only to validate the results in the paper. 2 Of these 479 retailers, only 122 retailers do not possess stores. While a majory 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
10 (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 wh retailers familiar wh 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 wh 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 whtled down from 527 retailers to 302 retailers wh 2006 firmyear 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 Universy 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 equy analysts from Instutional Brokers Estimate System (I/B/E/S). We use the following notation. From the Compustat annual data, for firm i in year t, let SR be the total sales revenue (Compustat field DATA12), COGS be the cost of goods sold (DATA41), SGA be the selling, general and administrative expenses (DATA189), LIFO be the LIFO reserve (DATA240), and RENT,1, RENT,2,, RENT,5 be the rental commments 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 PPE q be the net property, plant and equipment (DATA42), AP q be the accounts payable (DATA46), and I q be the ending inventory (DATA38). Let N be the total number of stores open for firm i at the end of year t. 9
11 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 compatibily across observations. The value of PPE could vary depending on the values of capalized leases and operating leases held by a retailer. We compute the present value of rental commments for the next five years using RENT,1,,RENT,5, and add 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 wh d = 10% as well. From these data, we define the following variables: Average sales per store, CS = COGS LIFO + LIFO, 1 N i t Average inventory per store, 4 1 IS = I + LIFO N q 4 q = 1 Gross Margin, GM = SR COGS LIFO + LIFOi, t 1 Average SGA per store, SGAS = SGA N Average capal investment per store, CAPS PPE N RENTτ = q 4 + τ q= 1 τ = 1 (1 + d) Store growth, G = N N 1 Proportion of new inventory, 4 4 PI = AP q I + 4 = 1 q LIFO q 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, PI, mers 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 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
12 inventory purchased by the retailer whin the cred 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 uns of inventory purchased one year ago, and in the second case, a retailer carries one un of inventory purchased two years ago and a second un purchased today. Assume that each un costs a dollar and the cred 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 un of analysis because most retailers report store level data only at the annual level. Moreover, annual data are auded, and hence, of better qualy 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 definions, we compute the logarhm of each variable in order to construct a multiplicative model. The variables obtained after taking logarhm are denoted by lowercase letters, i.e., cs, is, gm, sgas, caps, g, pi, and ics t. 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 lerature; (b) multiplicative models of supply equations are commonly used in economics and (c) a multiplicative model has been found to f aggregate inventory levels in previous research (Gaur et al. (2005) and Roumianetsev and Netessine (2007)). We consider only whinfirm variations in the variables of study because acrossfirm variations can be caused by variables omted from our study such as differences across firms in accounting policies, 11
13 management abily, firm strategy, store appearance, location, competive environment in the industry, etc. We control for differences across firms by using timeinvariant firm fixed effects in each equation. Based on Hypotheses 16, several control variables, and firm fixed effects, we specify the three equations as: cs = F +α is +α gm +α sgas +α pi +α g +α ics +ε (1) i i, t t 1 is = J +α cs +α gm +α cs +α pi +α g +α caps +η (2) i i, t 1 24 i, t i, t 1 gm = H +α cs +α is +α gm +υ (3) i i, t 1 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 (F i, J i, and H i ), coefficients of endogenous variables, coefficients of predetermined variables, and error terms (ε, η, and υ ). The estimates of α 11, α 12, α 21, α 22, α 31, and α 32 enable us to test our six hypotheses. We call these coefficients the inventory elasticy, the price elasticy, the stocking propensy to sales, the stocking propensy to margin, the markup propensy to sales, and the markup propensy 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 capal 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 wh 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 elasticy and price elasticy follow common terminology. In the remaining equations, our terminology is motivated by Haavelmo (1943) who called the coefficients in a simultaneous equations model as propensies. 12
14 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 activies that help to implement a retailer s competive strategy (Palepu et al. 2004). We expect sales per store to increase wh SGA per store since prior work has shown that improvement in customer service and increase in advertising expenses 4 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, pi, would be endogenous wh average inventory per store, is. We control for store growth because the composion 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 condions. Carroll et al. (1994) find that the index of consumer sentiment is a leading indicator of change in personal consumption expendures, 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 condions for the following year. Aggregate inventory equation: We control for lagged proportion of new inventory because we expect average inventory per store to decrease wh this variable. If a retailer had a larger fraction of new inventory in one year, then would sell more and carry over less inventory in the following year. In this 4 We repeated our analysis wh 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 wh 1351 firmyear observations. 13
15 reasoning, we equate proportion of new inventory wh qualy or freshness of products sold at the retailer. We control for capal investment per store since can have substution and/or complementary effects on inventory. First consider the substution 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 s inventory, thus resulting in a lower average inventory level throughout the chain. Cachon and Fisher (2000) ce several benefs 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 benefs of information technology. Next consider the complementary effect of capal investment per store on inventory. When a retailer increases s capal investment per store, is able to use inventory more productively and hence may find profable to carry more inventory. For example, the retailer may remodel s store, buy new store fixtures or set up a warehouse that enables to support s stores wh a larger assortment of products. Thus, we expect that average inventory per store could increase or decrease wh increase in capal 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 s 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 profabily at a retailer. Note that the set of predetermined 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 curveshifters in order to identify the coefficients of endogenous variables in the model. Their abily to serve as curveshifters depends on our reasoning that each of these variables directly affects only those endogenous variables in 14
16 whose equations appears. For example, SGA per store acts as a curve shifter since affects sales per store directly and inventory per store and gross margin through s 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 firstdifferencing or meancentering the observations for each firm. We use firstdifferencing due to s usefulness for sales forecasting and superior statistical properties. Firstdifferencing expresses yeartoyear 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 model 5 cannot be used for sales forecasting whout assuming that mean of each variable remains unchanged when a new year is added  an assumption that is violated by the nonstationary of our variables. Meancentering is also problematic because produces inconsistent estimates in the presence of lagged endogenous variables (Wooldridge 2002). Firstdifferencing equations (1)(3) gives us the following structural equations: Δ cs =α +α Δ is +α Δ gm +α Δ sgas +α Δ pi +α Δ g +α Δ ics +Δε (4) i, t t 1 Δ is =α +α Δ cs +α Δ gm +α Δ cs +α Δ pi +α Δ g +α Δ caps +Δη (5) i, t 1 24 i, t i, t 1 Δ gm =α +α Δ cs +α Δ is +α Δ g +α Δ gm +Δυ (6) i, t 1 Here, Δ prefix for each variable denotes firstdifference. 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), wh coefficients denoted as β 10,, β 37 which are functions of the structural parameters, α 10,, α 34. Δ cs = β + β Δ cs +β Δ gm + β Δ sgas +β Δ pi + β Δ g +β Δ caps + β Δ ics + ν (7) i, t 1 12 i, t i, t i, t 1 17 t 1 Δ is = β + β Δ cs +β Δ gm +β Δ sgas +β Δ pi +β Δ g +β Δ caps +β Δ ics + θ (8) i, t 1 22 i, t i, t i, t 1 27 t 1 Δ gm =β + β Δ cs +β Δ gm +β Δ sgas +β Δ pi +β Δ g +β Δ caps +β Δ ics +ω (9) 30 31, 1 32, , , 1 37 t 1 5 For completeness, we estimated the meancentered model as well. We found all hypotheses to be supported (p< 0.05). 15
17 We first conduct a test of endogeney to determine if sales per store, inventory per store, and gross margin are endogenous in each of the equations. The test results support endogeney in all three equations at p< Our test of endogeney 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 condion required to recover the structural parameters, α 10,, α 34, 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 overidentified, 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 heteroscedasticy and (AR1) autocorrelation among observations whin 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<
18 Our results support findings from operations management theory. They show that sales, inventory, and gross margin aggregated at the firm level follow causaly relationships that are consistent wh theory developed at the em level. First, consider the aggregate sales equation and the aggregate inventory equation. The inventory elasticy 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 s 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 propensy to sales is 0.828, showing that increase in sales also causes an increase in inventory. Since this estimate is less than 1, 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 propensy to margin is and markup propensy to inventory is Thus, an increase in gross margin causes an increase in inventory consistent wh results from stochastic inventory theory. And an increase in inventory causes a decrease in margin, which supports findings from dynamic pricing lerature (Gallego and van Ryzin 1994; Smh and Achabal 1998) that show that optimal pricing trajectory decreases wh inventory. Finally, in the aggregate sales equation and the aggregate gross margin equation, price elasticy is and retailers markup propensy to sales is 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 demandsupply model in microeconomics. Besides being consistent wh theory, these results provide new evidence of the impact of inventory on sales and the interrelationship between margin and inventory. Further the large magnudes 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 oppose 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 elasticy, i.e., by reducing lost sales, stimulating 17
19 demand, and/or by increasing choice to customers. Addionally, 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 (0.245)(3.934) = 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 and (= (3.934)(0.828)) showing that first and higher order effects can be of oppose 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 = (0.242)(0.627) = Effect of inventory on margin = (0.224)(0.242) = , Effect of margin on sales = (0.627)(0.224) = , Effect of sales on margin = (0.828)(0.245) = We note from these estimates that sales and inventory are very sensive to changes in each other and in margin. However, margin is less sensive 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 wh 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 propensy to margin is negative and significant. Further, we find that even when the coefficients are significant, their magnudes differ considerably across segments. For example, inventory 18
20 elasticy is found to vary between for Home Furnishing Stores to 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< 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 activies, then on average, 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, has higher order effects on inventory per store and margin. The total effects are captured by the reduced form equation. The magnudes of the coefficients of SGA per store can be used to determine whether is profable 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 on average across all retailers in our dataset. Using the definions 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* = , an increase in cost of goods sold per store to $2.452M * = $2.4689M, 19
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