ESSAYS ON RETAIL PRICE MOVEMENTS by KAZUKO KANO B.A., Keio University, 1995 M.A., Hitotsubashi University, 1997 M.A., University of British Columbia, 1999 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Economics) THE UNIVERSITY OF BRITISH COLUMBIA January 2007 ©Kazuko Kano, 2007 Abstract The central questions asked in this thesis are (i) whether retail prices are sticky or not, and (ii) what economic factors are crucial for explaining retail price movements. In the second chapter, I first examine the predictions of two representative economic models to explain price movements - a time-dependent pricing model and a state-dependent pricing model. This chapter is different from past studies with respect to its micro data set including cost information. The results of this chapter based on de-scriptive statistics empirically support the state-dependent pricing model in explaining the observed retail price movements. Moreover, using reduced-form probit estimations to explain price changes, I show that the frequency of price changes is significantly affected by the degree of competitions among brands. The third chapter examines a state-dependent pricing model in the presence of fixed adjustment costs of prices - menu costs. A model with menu costs has the potential to explain an important characteristic of retail price movements: prices discretely jump. This chapter shows that the assumption about market structure is crucial in identifying menu costs. Specifically, prices in a tight oligopolistic market can be more rigid than those in more competitive market such as monopolistically competitive one. If so, the estimates of menu costs under the assumption of monopolistic competition in past studies are potentially biased upwards due to the rigidity from strategic interactions among brands. In addition, I show the estimate could be biased downwards without controlling for unobserved promotional activities. Developing and estimating a dynamic discrete-choice model with multiple agents to correct these potential biases, this chapter provides empirical evidence that menu costs as well as strategic interactions are important in explaining the observed degree of price rigidity in weekly price movements of a typical retail product, graham crackers. The fourth chapter provides a survey of the recently proposed estimators for struc-tural estimations in dynamic discrete choice games. This survey focuses on two-step estimators, which overcome the computational costs that used to be unavoidable in the course of structural estimations of dynamic discrete-choice models. Table of Contents Abstract ii Table of Contents iv List of Tables vii List of Figures ix Acknowledgements x Dedication xi 1. Overview and Summary 1 2. Rigidity of Retail Prices: Empirical Regularities in Scanner Data . . . 6 2.1 Introduction 6 2.2 Data description 12 2.3 State-dependent or time-dependent pricing? 19 2.3.1 Hazard rate of price changes 22 2.3.2 Correlation between frequency of price changes and variation in costs 24 2.4 Probit estimation 25 2.4.1 Effects of costs 33 2.4.2 Effects of price dispersion across stores 35 2.4.3 Effects of brand competition and market structure 36 2.4.4 Effects of duration 42 2.4.5 Probit estimations with all the variables 43 2.5 Concluding remarks 43 3. Menu Costs, Strategic Interactions, and Retail Price Movements . . 59 3.1 Introduction 59 3.2 The model 66 3.2.1 The environment 67 3.2.2 The problems of manufacturers 69 3.2.3 Markov strategy, Bellman equation, and equilibria 74 3.3 The estimation procedure 78 3.3.1 Estimating the demand equation and transition probabilities . . . 79 3.3.2 Estimating menu costs 80 3.3.3 Estimation with potential multiplicity of equilibria 83 3.4 Data, demand estimation, and transition probabilities 84 3.4.1 The data 85 3.4.2 Demand estimation and state variables 87 3.5 Results 90 3.5.1 Estimated size of menu costs 90 3.6 Conclusion 98 4. Estimation of Dynamic Discrete Choice Games: a Survey 106 4.1 Introduction 106 4.2 A single-agent discrete choice model 108 4.2.1 The basic model 109 4.2.2 Estimation methods 110 4.3 A multiple-agent discrete choice model 118 4.3.1 The basic model 119 4.3.2 Multiple equilibria and the identification problem 128 4.3.3 Estimation methods 130 4.4 Monte Carlo studies 139 4.5 Concluding remarks and future research 144 Bibliography 146 Appendix for chapter 3 157 A.l Constructing transition probability matrices 157 A.2 Alternative presentation of value functions and best response probabilities 158 A.3 The data 160 List of Tables 2.1 Predictions of Time-dependent Models and State-dependent Models . . . 46 2.2 Correlation between Frequency of Price Changes and Variations in Costs 46 2.3 Descriptive Statistics Related to Price Changes and State Variables . . . 47 2.4 Results of Probit Estimation - Costs 48 2.5 Results of Probit Estimation - Price Dispersion Across Stores 48 2.6 Frequency and Average Duration of Price Changes Across Package Sizes 49 2.7 Frequency of Price Changes by Brands, 64 oz 49 2.8 Results of Probit Estimation - Brand Competition 50 2.9 Results of Probit Estimation - Share of Package Sizes and Share of Brands 50 2.10 Results of Probit Estimation - Market Structure, HHI 51 2.11 Results of Probit Estimation - Market Structure, CR4 52 2.12 Results of Probit Estimation - Time-related Variables 53 2.13 Results of Probit Estimation - All Variables except PDEV B 54 2.14 Results of Probit Estimation - All Variables including PDEV B 55 3.1 Market Shares of Graham Crackers 100 3.2 Summary Statistics of Variables 100 3.3 Summary Statistics of Price Changes 100 3.4 Estimated Demand Equation 101 3.5 State Variables (Discretized Values) 101 3.6 Estimated Menu Costs 102 3.7 Estimated Menu Costs and Fixed Costs of Downward Price-Changes . . 103 3.8 Menu Costs in Previous Studies 104 List of Figures 2.1 Hazard Rates of Prices 56 2.2 Prices and AACs of Two Items in Refrigerated Orange Juice 57 2.3 Frequency of Price Changes and Standard Deviation of AACs 58 3.1 Shelf Prices of Three National Brands 105 Acknowledgements I am greatly indebted to my thesis supervisor, Professor Margaret Slade for her research directions and suggestions. I am also grateful to Professor Gorkem Celik, Pro-fessor Micheal Devereux, Professor Susumu Imai, Dr. Takashi Kano, Professor Thomas Lemieux, Professor Kevin Milligan, Professor Art Shneyerov for their support. I would also like to thank Professor Daniel Putner, Professor Atsushi Inoue, Professor Thomas Ross, and Professor Nicolas Schmitt, the examining committee and the external exam-iner at the final oral examination. Earlier versions of the third chapter of this thesis were presented at the University of British Columbia, the University of Warwick, Concordia University, Hitotsubashi University, the 2004 Canadian Economic Association Meetings at Ryerson University, and the poster session in the 2006 Numerically Intensive Economic Policy Analysis at Queen's University. I appreciate helpful comments from seminar par-ticipants, and, especially, those by the discussants, Professor Avi Goldfarb and Professor Victor Aguirregabiria. I have also benefited from the encouraging and thoughtful com-ments of Professor Daniel Levy and an anonymous referee in Managerial Decision and Economics. Finally, I would like to thank the James M. Kilts Center, Graduate School of Business, University of Chicago for the use of the data in this paper. I am responsible for all errors in this thesis. To Takashi, and Our Parents Chapter 1 Overview and Summary This thesis consists of two essays that contribute to an understanding of retail price move-ments, and an essay that provides a detailed survey of empirical methods for examining dynamic brand competition affecting retail price movements. The question of whether prices in retail markets are flexible or rigid has been fre-quently asked in economics. This question is fundamental for understanding not only how retailers determine their prices when they face changes in economic conditions, but also how monetary policy affects the real economy through monetary non-neutrality due to nominal price rigidity. Seeking a rigorous answer to the question, therefore, is one of the central tasks of economics. Empirical studies to answer this question have been conducted with respect to differ-ent products. Empirical evidence for price rigidity is, however, mixed so far. Nonethe-less, many papers have been written to find a rigorous explanation of the degree of price rigidity observed in data. In the literature of macroeconomics, there are two dom-inant competing models; a time-dependent pricing model and a state-dependent pricing model. The source of the observed price rigidity is also asked in the literature of the em-pirical industrial organization. The literature has claimed fixed adjustment costs, stock of goodwill, inventory behavior, market concentration, firm size, and "market thickness" as important factors in determining the degree of price rigidity. In the first essay, I empirically investigate how retail prices in a micro data set behave by asking the following two questions: (i) which of time-dependent or state-dependent pricing mechanisms does a better job in explaining retail price movements in the data set, and (ii) what economic factors are crucial for the behavior of the retail prices. I exploit the Dominick's Finer Food (DFF) data set, which is a scanner data set collected from a supermarket chain in the United States. The crucial feature of the data set, which is not common in other data sets used in the past studies, is the availability of unit costs retailers face - wholesale prices. While the questions asked in the first essay are not new in the literature, using DFF data provides important insight with respect to the dependence of price changes on the costs the retailers face. This feature of the DFF data is quite important, for example, because the state-dependent pricing models and time-dependent pricing models have quite different predictions about the relationships between price changes and cost movements. Using a descriptive approach, I first show empirical evidence for state-dependent pricing models over time-dependent pricing models. To answer the second question, I examine several factors that could explain the observed price changes. They include variations in costs, the degree of price dispersion across stores, the degree of brand competition, the size of the market and the market share of brand, and market structure reflected in the measures of market concentration within a package size. I estimate several probit models of a binary indicator of price changes with these state variables as explanatory variables. The most important inference obtained from these exercises with probit estimation is that the competition across brands significantly affects the retailers' decisions of price changes. Given the empirical evidence that brand competition is important in state-dependent pricing models, I investigate the source of price rigidity in more detail. The third chapter examines a state-dependent pricing model in the presence of fixed adjustment costs of price changes — menu costs. The model with menu costs has potential to explain an important characteristic of retail price movements: prices discretely jump. In this essay, I develop an economic model in which, faced with menu costs of changing their prices, manufacturers play a dynamic game of price competition. Estimating the structural model, this essay draws inferences on the observed degree of rigidity of retail prices. In particular, I estimate menu costs by taking into account a factor that potentially makes the estimates of menu costs under the assumption of monopolistic competition model in previous studies biased. Specifically, prices in a tight oligopolistic market could be more rigid than those in more competitive markets such as monopolistically competitive ones. If so, the estimate of menu costs under the assumption of monopolistic competition model can be potentially biased upwards due to the rigidity from strategic interactions among brands. In addition, I show the estimate could be biased downwards without controlling for unobserved profit-enhancing promotional activities of manufacturers accompanied with price reduction. Using prices of a narrowly defined product category from the DFF data set, after correcting these potential biases, I provide empirical evidence that menu costs are statistically significant as well as economically important in explaining the observed degree of price rigidity in the movements of weekly retail prices. Firstly, the size of the estimated menu costs is close to those estimated in past studies using data from different markets. This result, hence, supports the conclusions of past studies that menu costs play an economically important role in the weekly movements of retail prices. Secondly, I provide evidence that unobserved profit-enhancing promotional activities in fact result in statistically significant downward bias of the estimate of menu costs. Finally, comparing the results of the oligopolistic competitive market model with those of a monopolistic competitive market model statistically supports the hypothesis that strategic interactions among manufacturers cause the estimator based on the latter model to be biased upwards. In summary, the results of the third chapter not only confirms the findings of the past studies using the data of another product — fixed adjustment costs of price changes are statistically significant as well as economically important —, but also empirically reveal another potentially crucial source of price rigidity — strategic interactions among firms in oligopolistic competitive markets. The fourth chapter provides a detailed survey of the empirical methods to estimate structural parameters in dynamic discrete choice games. While the importance of these models in explaining many economic problems had been recognized, technical difficulties prevented this class of models from being investigated empirically. Recently, however, several important methodological breakthroughs have happened in empirical methods that enable structural estimation of dynamic discrete choice games. The building blocks of these empirical methods are seminal studies in single-agent dynamic discrete choice models. This chapter first surveys two estimators in the single-agent discrete choice models, which are known as the nested fixed point estimator and the conditional choice probability estimator. Subsequently, I review the pseudo-likelihood estimator, the nested pseudo-likelihood estimator, the minimum-x2 estimator, the method of moment esti-mator, and the asymptotic weighted least square estimator, for estimating a class of multiple-agent discrete choice games with Markov-perfect equilibria. After reporting the results of recent Monte Carlo exercises for the statistical performances of these newly developed estimators, I conclude. Chapter 2 Rigidity of Retail Prices: Empirical Regularities in Scanner Data 2.1 Introduction The question of whether prices in retail markets are flexible or rigid has been frequently asked in economics. This question is fundamental for understanding not only how retail-ers determine their prices when they face changes in economic conditions, but also how monetary policy affects the real economy through monetary non-neutrality due to price rigidity. Seeking a rigorous answer to the question, therefore, is one of the central tasks of economics. Empirical studies to answer this question have been conducted with respect to dif-ferent products. For example, using micro data of retail prices in mail-order catalogues, Kashyap (1995) reports empirical evidence supporting price rigidity: the retail prices in his data change in a significantly infrequent manner. The study by Dutta, Bergen and Levy (2002), on the other hand, shows empirical evidence against the hypothesis of price rigidity. Applying a vector autoregression (VAR) to data of three major brands of orange juice, which are collected in a supermarket chain in the United States, they examine how significantly the retail prices of these brands respond to changes in the wholesale prices as well as the spot market prices in the corresponding commodity market in the short run.1 Looking at the cumulative impulse responses of the retail prices, they observe that most of the retail prices are adjusted to cost changes in the short run; therefore, they conclude that the retail prices are flexible. Although evidence for price rigidity is mixed so far, many papers have been writ-ten to seek rigorous models for explaining the degree of price rigidity observed in the data. In the literature of macroeconomics, there are two dominant competing models, time-dependent models (e.g., Taylor (1979, 1980) and Calvo (1983)) and state-dependent models (e.g., Sheshinski and Weiss (1977), Caplin and Spulber (1987), Dotsey, King and Wolman (1999), Golosov and Lucas (2006), and Devereux and Siu (2005)). An essential difference between these two models is in the specification of how firms change their prices. In time-dependent models, price changes are completely exogenous: firms could change prices in a fixed interval (Taylor-type) or with fixed probability each period 1They define eight weeks as a short-run period. (Calvo-type). In state-dependent models, the decisions of price changes are endoge-nous: firms adjust their prices to exogenous changes in state variables according to their profit maximization problems. Recently, a number of studies test the predictions of time-dependent models as well as state-dependent models using various micro-data. On one hand, Klenow and Kryvtsov (2005) support time-dependent models showing the time-dependent models explain 95 percent of the variance of the inflation in the micro data collected by the United States Bureau of Labor Statistics. In addition, time-dependent rule can be relevant even at the product level since many firms review their prices periodically (e.g., every 12, 24, and 36 months)(Alvarez and Hernando (2005)). On the other hand, Bils and Klenow (2004) and Baharad and Eden (2004) find evidence against time-dependent models using micro-data from the United States and Israel, re-spectively.2 Moreover, Baumgartner, Glatzer, Rumler and Stiglbauer (2005) and Camp-bell and Eden (2005) find the importance of state-dependent pricing mechanisms over the time-dependent models for explaining the consumer prices in Austria and prices of individual product in scanner data in the United States, respectively.3 The source of 2 Bils and Klenow (2004) show that time-dependent models fail to produce observed persistency and volatility of inflation using the price data of 123 goods from the United States Bureau of Economic Analysis. Baharad and Eden (2004) use unpublished data of the retail prices of 381 products in Israel, and find evidence against a time-dependent model. They show that the observed relationship between price dispersion and the frequency of price changes is not consistent with the prediction of the time-dependent model. 3In addition, a number of studies including Alvarez and Hernando (2005) and Baumgartner et al. price rigidity is also analyzed in the literature of empirical industrial organization. In particular, Slade (1998, 1999) assumes that firms face fixed adjustment costs of price changes. The decision of price changes depends on costs and demand conditions, espe-cially a stock of goodwill. Aguirregabiria (1999) studies both inventory behavior and price changes using data from a supermarket in Spain. In his model, facing fixed costs of ordering as well as price adjustments, the retailer makes the decision of price changes by looking at her stock of inventory. Moreover, investigating retail prices of lettuce, Powers and Powers (2001) observe that the degree of price rigidity significantly depends on market concentration, firm size, and market thickness. In this chapter, I also empirically investigate how retail prices in a micro data set move by asking the following two questions: (i) which pricing mechanisms of time-dependent or state-dependent does a better job in explaining retail price movements, and (ii) what economic factors are crucial in determining the behavior of the retail prices. Of course, these questions are not new in the above literature. Rather, what makes this chapter (2005) test the predictions of these two pricing models using the individual consumer and producer prices in euro countries. They were conducted under a research project, the Eurosystem Inflation Persistence Network. The results of this project are presented in a number of working papers from the European Central Bank. For details of these studies, see the survey by Dhyne, Alvarez, Bihan, Veronese, Dias, Hoffmann, Jonker, Lunnemann, Rumler and Vilmunen (2005), Alvarez, Dhyne, Hoeberichts, Kwapil, Bihan, Lunnemann, Martins, Sabbatini, Stahl, Vermeulen and Vilmunen (2005), and the papers cited therein. different from those in the existing papers is the data set I exploit — the Dominick's Finer Food (DFF) data set. As described in section 2.2 in details, the DFF data set is a scanner data set collected from a supermarket chain in the United States. When approaching the two specific questions, the advantage of using this data set over other data sets analyzed in the past literature is that this data set contains information about costs retailers face. This feature of the DFF data is quite important because, on the one hand, state-dependent pricing models generally predict that retailers' decisions of price changes crucially depend on changes in costs. On the other hand, in time-dependent pricing models, price changes occur independently of changes in costs. Therefore, the information of costs helps identify the two pricing mechanisms from the retail price data more precisely. To answer the two questions in this chapter, I focus on a product category in the DFF, refrigerated orange juice, construct detailed descriptive statistics of product items within this category, and estimate several simple reduced-form econometric models of price changes. The main results of this chapter are summarized as follows. In section 3, I specifically approach to the first question estimating a hazard function of price changes, which provides a simple test for time-dependent pricing models, as in Alvarez and Hernando (2004), Baumgartner et al. (2005), and Campbell and Edens (2005) for their retail price data. Consistent with the observation of these studies, the estimated hazard function is downward sloping for short durations of the current price level. This result is, however, inconsistent with the predictions of the time-dependent pricing models. Furthermore, I observe that the frequency of price changes is correlated positively with the standard deviation of costs in a statistically significant manner. Therefore, the results of section 2.3 support a state-dependent pricing model as a more accurate pricing mechanism for explaining retail price movements in the DFF data than time-dependent pricing models. Given the evidence for state-dependent pricing models shown in section 2.3, I move to the second question in section 2.4. In particular, I examine the roles of the following state variables in the retailers' decision to change prices: variations in costs, the degree of price dispersion across stores, the degree of brand competition, the market size re-flected in the sales share of a package size, the market power reflected in the sales share of a brand, the market structure reflected in the measures of market concentration -Herfindahl-Hirschman Index (HHI) and Four-Firm Concentration Ratio(CR4) within a packaged size-, and the duration of no-price change. I estimate several probit models of a binary indicator of price changes with these state variables as explanatory variables. In order to define variables as those pre-determined, I use lagged variables by one period when appropriate. The most important finding obtained from these exercises with probit estimation is that the competition among brands significantly affects the retailers' deci-sions to change prices. More specifically, I observe that (i) the greater the deviation of the price of a product item of a brand from those of other brands, the higher the probability of a price change of the product item of the brand is, (ii) the greater the share of a brand in a market which is segmented by packaged sizes, the higher the probability of price changes is, (iii) the more concentrated a market, the lower the probability of changing prices of product items included in the markets is. Therefore, these observations in this chapter lead to an important inference on price changes: the frequency of price changes of a product item is crucially affected by the degree of the competition among brands within markets segmented by package sizes. The structure of this chapter is as follows. The next section describes the data I use in the empirical exercises. Section 2.3 shows the results of simple empirical exercises to test which of time-dependent and state-dependent models fits the data better. Section 2.4 presents the results of estimation of the probit models. The final section concludes. 2.2 Data description This chapter investigates a scanner data set from a supermarket chain in the United States, Dominick's Finer Foods (DFF). The data set is publicly available at James M. Kilts Center, the Graduate School of Business, the University of Chicago for academic use.4 DFF is the second largest supermarket chain in Chicago metropolitan area, covering 4The data set and its description are downloadable at the web site of James M. Kilts Center, http://www.gsb.uchicago.edu/kilts/. In addition, Hoch, Kim, Montgomery and Rossi (1995), Dhar and Hoch (1996), Peltzman (2000), Chintagunta, Dube and Singh (2003), and Chevalier, Kashyap and about 20 percent of grocery sales in the corresponding area. The data set contains weekly store-level transaction records collected from more than 80 stores operated by DFF. It includes unit sales, retail prices, retail margins, and deal codes indicating promotional activities at the universal product code (UPC) level.56 The original weekly data spans the 399 weeks from September 1989 to May 1997. The most important feature of this data set is that I can recover average acquisition costs (AACs) - a measure of wholesale prices - from retail margins and retail prices in the data set. This availability of the variations in average costs in the DFF data set clearly distinguishes my study of pricing behavior with micro data from those of Campbell and Eden (2005), Baharad and Eden (2004), Bils and Klenow (2004), Klenow and Kryvtsov Rossi (2003) are helpful for acquiring the information about DFF's practice in pricing, promotional activity, and inventory management. 5 A deal code indicates whether any promotional activity takes place for an item in a store in a week. DFF records three promotional activities of "bonus-buy" which is typically price reduction with shelf-tags for discount announcements, "simple" which is simple price reduction, and in-store "coupon". The examples of the shelf tags used for bonus-buy and in-store coupons are shown by Dhar and Hoch (1996). The promotional activity may or may not be associated with advertisement or in-store display, whose information, unfortunately, is not contained in the data set. 6The universal product code (UPC) is the number that identifies an individual item such as Tropicana Premium Choice 64 oz (a brand-size combination). A UPC in the DFF data set is typically a 10 digit number which identifies a manufacturer and a product-item. UPC could be used as a Stock Keeping Unit (SKU), which is, in general, assigned by manufactures/retailers for their internal use in inventory management. (2005), Alvarez and Hernando (2004), and Baumgartner et al. (2005): the data sets used by these authors do not contain cost data. To identify whether state-dependent pricing mechanisms are more suitable for explaining retail price movements than time-dependent pricing mechanisms, cost data is quite informative as discussed in the introduction. The AAC is determined as follows by DFF on the weekly basis: DFF purchases their products from manufacturers and determine prices and AACs by the night on Wednesday for the week starting on Thursday. The DFF defines AAC for the week t as follows: AACt = [(newly-bought stock at the end of t-1) * (wholesale price paid at t-1) + (quantity in stock at the end of t-2) * AACt-\}/ (quantity in stock at the end of t-1) A potential drawback of using AACs as cost data is that this measure of wholesale prices might be a contaminated proxy for the true wholesale prices. This is because AACs contain not only information of wholesale prices but also that of the inventory level of a store. As argued by Peltzman (2000), this construction of AACs potentially causes an endogeneity problem when estimating a retailer's pricing decision through the responses of the inventory level to the changes in retail prices. Nevertheless, as noted by Chevalier et al. (2003), due to the inventory policy of DFF, wholesale prices are quickly reflected in AACs. Particularly, Dutta et al. (2002) mention that, in the case of refrigerated orange juice, the turnover of the product is less than a week. Therefore, I use a AAC as a good proxy for wholesale prices. In this chapter, I study a product category — refrigerated orange juice.7 This cate-gory contains 58 product items which are differentiated by brands, tastes, and package sizes.8 I use the data of 37 product items out of 58 product items, omitting the product items which are sold less than 50 weeks on average across stores or which have too many missing values (more than one third of the available periods). The share in sales of the 37 items accounts for 93.50 percent of the total sales of refrigerated orange juice. The refrigerated orange juice is suitable to this study. First, the numbers of products, pack-age size, and brands are large. Second, as described in detail later, the products have great variation in the frequency of price change. These properties allow me to investigate what factors can explain the heterogeneity in price changes of products whose contents of products are essentially same. An important caveat for using the whole sample is that DFF and the University of Chicago conducted several experiments for the retailer's pricing, promotional activities, 7Montgomery (1997), Dutta et al. (2002), and Chintagunta et al. (2003) scrutinize the data of refrigerated orange juice products. This product category is constructed by extracting the sample of orange juice from the DFF product category, refrigerated juice products. This definition of the product category of refrigerated orange juice follows that of Chintagunta et al. (2003). 8 I define a "product item" by a combination of brand, size, and taste. For example, Tropicana Premium Choice 64 oz and Tropicana SB 64 oz are different product items with the same package size of the same brand. and shelf management.9 The potential problem of these experiments for my study is that, during the experiment, retail prices are manipulated so that pricing rules in most of stores deviate from regular pricing behavior of retailers. Unfortunately, the DFF data do not contain the information about the experiments for a particular product item in a particular week. The experiments are not observed and are different across stores and product categories. This unobservability of the experiments makes it quite difficult to identify the effects of the experiments on the retail prices, especially, of refrigerated orange juice products, whose prices are widely dispersed across stores. Even with the difficulty in the identification of the experiments, I try to infer the periods conducted the experiments for refrigerated orange juice by looking for a particular pattern of shifts in the regular prices. I identify 68 weeks from 1993 to 1994 as the periods of the experiments. In the following analysis, I investigate (i) the whole sample with 399 weeks and (ii) the subsample constructed by dropping the 68 weeks I identify as the experiments, in order to check the robustness of my inferences in this chapter. In general, however, I could not find any significant difference in my constructing statistics between the two samples. Therefore, throughout this chapter, I will report only the results with the whole sample. An unfortunate but inevitable characteristic of the DFF data is that there are missing data. This is simply because retail prices and profit margins in the DFF data are recorded only when product items are purchased: there is no record of transaction of a product 9For the details of the pricing experiments, see Dhar and Hoch (1996). item when there is no purchase or when the product item is stocked out. The problem of missing data is quite common among researches using scanner data.10 As a common practice in studies using scanner data, the missing retail prices could be imputed or simply omitted (list-wise deletion). The imputation of the missing retail prices could potentially create false pricing patterns. In addition, it is difficult to impute retail prices for some product items with frequent price changes. Therefore, in this chapter, I employ the list-wise deletion: I omit the data points unless prices both in current and previous periods are available. An important characteristic of retail price movements in the DFF data set is that they can be decomposed into flat and stable movements ("regular prices") and sharp reductions represented by downward spikes (temporary "deal" prices). A typical obser-vation in time-series plots of retail prices is that reductions of retail prices are followed by immediate returns to the price levels before the reductions, i.e., the regular prices.11 This pattern of temporary discounts or deals is prevalent in retail price movements. How-10For example, Erderm, Keane and Sun (1999) point out that about 80 percent of daily scanner data by Nielsen is also imputed using a complex ad-hoc procedure. Erderm et al. (1999) discuss a potential selection bias due to these missing data. Since the original transaction records and the pricing cycle in DFF are weekly, the percentage of missing data in the entire sample of the DFF data used in this chapter is smaller than those discussed by Erderm et al. (1999). Therefore, the effect of a selection bias due to the missing data on the inferences of this chapter would be also small. 11Rotemberg (2005) points out that the pattern that prices go back to the regular prices is one form of price rigidity. ever, the literature has not reached at a consensus about whether or not to include the temporary sales into data set. On the one hand, many studies including Levy, Bergen, Dutta and Venable (1997) and Slade (1998) use actual transaction prices to study price rigidity. There are several rationales for this empirical treatment. First, the original prices are those customers actually pay. Second, a large part of sales in a product item with frequent discounts might occur during the weeks in which the product item is on deal. Finally, most of retail price adjustments across periods are associated with tem-porary reductions of wholesale prices. These all mean that if I eliminate temporary price changes in retail price data by an ad-hoc way, I could potentially draw incorrect inferences on retail price adjustments. Thus, it is important to incorporate the tempo-rary price discounts into my analysis. On the other hand, there is no rigorous economic model that can explain both frequent price discounts represented by downward spikes and relatively stable long-run regular price movements at the same time. For this reason, Peltzman (2000) and Midrigan (2005) smooth out weekly price series into monthly series eliminating the effect of temporary sales in order to study price adjustment behavior of retailers. In this chapter, following the former empirical treatment, I use original retail prices provided by the DFF data set. 2.3 State-dependent or time-dependent pricing? As stated in the introduction, the essential difference between time-dependent and state-dependent models is in the specification of price changes. A typical time-dependent model has either of the following two specifications. In the first time-dependent model in Calvo (1983), a firm can change its price with a constant probability, which is given as an exogenous parameter. In the other time-dependent model by Taylor (1979, 1980), a timing of price change depends only on calender time. For example, a list price of a product is allowed to be renewed only with fixed timing such as once in a year. These time-dependent pricing mechanisms are widely employed in macroeconomics.12 On the other hand, in state-dependent models, price changes are endogenous so that the deci-sions of price changes are derived as outcomes of profit-maximization problems of firms. Firms determine whether or not to change their prices looking at state variables, which could include demand conditions, marginal costs of products, the duration of current price level, inflation rate, and so on. State-dependent and time-dependent pricing mechanisms can be distinguished in the following two ways. First, time-dependent and state-dependent models predict differ-ent shapes of unconditional hazard functions. Specifically, in a typical time-dependent pricing model of Calvo (1983), it is assumed that a firm changes its price with constant 12Representative studies include Yun (1996) and Christiano, Eichenbaum and Evans (2005b). For more details and recent developments, see, for example, Eichenbaum and Fisher (2003). probability A > 0 in any period. The surviving rate of the current price, S(d), is given by S(d) - exp(-Ad), (2.1) where d is the duration, of the current price level.The hazard rate, h(d), is derived by h(d) = -d In (S(d))/dd = A. (2.2) Therefore, the hazard rate in the Calvo pricing model is, in general, constant and flat against the duration of the current price level.13 Also, in a Taylor-pricing model, a firm revises its price every fixed period. Therefore, the predicted hazard functions have spikes with fixed periods of price changes, and the hazard rate is zero in the remaining periods. On the other hand, state dependent models, for example by Dotsey et al. (1999), generally predict upward unconditional hazard functions. This is because a firm 1 3 The hazard rate is, in general, defined as follows. Let T denote the duration of the current price level which is assumed to be a random variable with density / . Given duration d and a small number the hazard rate h{d) is defined as the probability at which the current price level is changed right after duration d, given the current price level lasts at least until d, i.e., = l i m P r o b e r ^ + ( 2 3 ) C-»oo Q = ft Prob(T £ d) C ( } 1 Prob(T s; d + C) - Prob(T ^ d) ,„ c 1 } = m s(dy where S(d) = 1 — Prob(T ^ d) is the surviving rate. (2.6) changes its price only when benefits from price changes accumulates large enough to cover adjustment costs.14 Second, I can also distinguish state-dependent and time-dependent pricing mech-anisms in terms of the predictions with respect to the price changes reacting to cost changes. In the state-dependent pricing mechanism, the frequency of price changes is correlated positively with the variation in costs. This is because, for example, price set-ters facing fixed adjustment costs when changing their prices do not revise their prices if the changes in costs are relatively small to fixed adjustment costs: only when the costs vary large enough to overcome fixed adjustment costs, they revise their prices. On the other hand, the time-dependent pricing mechanism in general implies no correlation be-tween the frequency of price changes and the variation in costs. In this pricing scheme, the probability for a firm to change its price is constant over time regardless of state variables. This prediction is not tested by the past studies mentioned above since the data used by the above authors do not contain cost information. Table 2.1 summarizes the theoretical predictions of the two models with respect to the unconditional hazard rates and the correlation between frequency of price changes and the volatility of costs. In the next two subsections, I investigate the two theoretical implications of state-dependent and time-dependent pricing mechanisms in detail. 14There is no closed form of hazard functions from Dotsey et al. (1999). 2.3.1 Hazard rate of price changes As mentioned above, an important implication of time-dependent pricing models is that the hazard rate of price changes is independent of the duration of the current price level. To estimate the hazard function h, I use the following discrete approximation. Let djjSit denote the number of weeks in which the price of product item i in store s at period t has not been changed until period t. Then, let D(d) denote the probability that the price is changed in the dth week. This probability is approximately obtained by D(d) = (NiNgNt) -1 Y^ Y1Y1 Wf = i s t where iVj, N s, and N t are the numbers of products in a stores, stores, and weeks, re-spectively, and /(.) is the index function which takes value one if the inside argument is true and zero otherwise. Using the definition of D(d) yields the discrete approximation of the surviving rate S(d), 11(d), as dmax n(d) = £ D(m) (2.8) m=d where dmax is the maximum duration in the entire sample. The discrete approximation of the hazard function h(d), 9(d), is then given by 9(d) = D(d)/U(d). Figure 2.1 plots the hazard function 9(d). Notice that the hazard function is steeply decreasing until around the duration of 20 weeks.15 This observation is consistent with 1 5 The steep downward hazard function could reflect the effect of deals, which typically last for 1-4 weeks. However, the range of weeks with the downward hazard function is longer than the possible number of weeks of deals. the finding reported by Alvarez and Hernando(2004) Baumgartner et al. (2005), and Campbell and Edens (2005) who argue that their hazard function does not match the implication of time-dependent pricing mechanisms. In addition, Alvarez et al. (2005) report the downward sloping unconditional hazard function is as one of stylized facts observed in various consumer and producer price data. 1 6 Although the hazard func-tion in Figure 2.1 is almost flat after the duration of 20 weeks, their argument against time-dependent pricing is also applicable to my DFF data set: the hypothesis of time-dependent pricing is not supported by the shape of the estimated hazard function. The downward hazard function might, however, result from the heterogeneity of the timing of price changes across product items. Baumgartner et al. (2005) observe the downward unconditional hazard function for their consumer price data. After control-ling for fixed-effects of products in their probit estimation, however, they find that the correlation between the duration of the current price level and the probability of price changes is positive. To control for the heterogeneity across products, I conduct the similar exercise in the section 2.4. The observation of downward unconditional hazard function is inconsistent with the prediction of state-dependent models, too. In the following section, I investigate whether another state-dependent variable, cost, explains the price movements. 16Alvarez et al. (2005) also report some evidence for Taylor-type time-dependent models. This is because, in some industries, prices are changed every twelve months. This is, however, not the case with the retail prices examined in this chapter. 2.3.2 Correlation between frequency of price changes and vari-ation in costs This subsection empirically examines the hypothesis of state-dependent pricing mecha-nisms — a positive correlation between the frequency of price changes and the variation in costs. As the first approach to this hypothesis, Figure 2.2 plots the retail prices of two particular product items, Minute Maid 32oz and Toropicana Premium 64oz, and the corresponding AACs in a particular store for the entire 399 weeks. The reason I choose these two product items is that the variation in the AAC of the latter product item is visually greater than that of the former product item. The figure clearly shows that the frequency of price changes in the latter product item is also greater than that of the former item. This observation leads to my conjecture that a product item with a higher variation in its cost tends to change its retail price more frequently. Figure 2.3 is the scatter plot of the frequencies of price changes against the standard deviations of the corresponding AACs. The frequency of price changes of a product item in a store is measured by the percentage of the weeks in which price changes occurred over the entire 399 weeks. The standard deviations of AACs are also computed for each product-item and store combination for the available periods in each store. The number of the scattered points is 2553, which is the number of stores times the number of product items. The most striking fact Figure 2.3 uncovers is that the frequencies of price changes are associated positively with the variations in the corresponding AACs. The scatter plot, however, shows significant heteroscedasticity in the joint distribution of the two variables of concern. To take into account this large degree of heteroscedasticity, I regress the frequencies of price changes on the standard deviations of AACs by OLS, including store as well as product-item specific dummies into the regression to control for the store and product-item specific fixed effects. Table 2.2 reports the results of the regression.17 Notice that the coefficient on the standard deviation of AACs is positive and statistically different from zero at any conventional significance levels: if the variation in costs is large, so is the frequency of price changes. This is evidence for state-dependent pricing mechanism to have explanatory power for price changes. 2.4 Probit estimation The preliminary evidence in the previous section suggests that state-dependent pricing models with cost variables have better explanatory power of price changes in the data than time-dependent models. In this section, using reduced-form probit models, I exam-ine if the factors emphasized in the literature of state-dependent pricing models could indeed explain price changes in the data of this chapter. Specifically, I estimate pro-1 7 The F statistic for the null hypothesis that all the coefficients of store dummy variables are jointly zero is 38.58 with the degrees of freedom 68 and 2447. The null is rejected at 1 percent significance level. The F statistic for the null hypothesis that all the coefficients for the product-item dummy variables is 1158.02 with degrees of freedom 35 and 2447. The null is also rejected at 1 percent significance level. bit models of a binary indicator of price changes of product items with the following potentially important factors: variation in costs, price dispersion across stores, brand competition and market structure within a package size, and time elapsed since price changes. The reasons why I focus on these state variables are the following. First, as men-tioned above, it is the most important claim of state-dependent pricing models that retail prices are adjusted to large changes in costs. In fact, as shown in Figure 2.2, the frequency of price changes is positively correlated with the standard deviation of AACs. In this section, I mainly use the absolute value of a percentage change in AAC, ABSAAC t = \[{AAC t - AACt-i)/AACt-i] * 100|, as an explanatory variable for price changes in order to examine the reaction of prices to cost factors. In ad-dition, I also construct the absolute value of a cumulative percentage change in cost, CUMAAC t = | i,t][( AACd ~ AACd-i)/AACd-i] * 100|, where d is the index initial-ized to one when a is changed. CUM AAC is a relevant state variable, for example, when a firm facing a fixed adjustment cost of price changes follows an (s, S) policy rule. Suppose that AAC is monotonically increasing for a while. According to an (s, S) rule, a firm revises its price to its optimal level when AAC reaches the threshold s, at which the benefit of changing prices becomes higher than the fixed adjustment cost. In this case, the firm might change its price even when current incremental change in AAC is very small. In addition, when retail prices periodically change, and react to changes in costs with lags, the variable CUM AAC could be important. In these cases, the expected sign of coefficient on CUM AAC is positive. On the other hand, if the retailers change their prices looking at only contemporaneous changes in costs, this variable might not be important for price changes. Second, in state-dependent pricing models, demand conditions are also important for retailers to determine their prices. The dispersion of prices across stores could be an important factor in demand conditions. For example, if the price of a retailer is far high away from those of the rivals, the demand for the retailer's product item could be extremely small. In this case, the retailer has a strong incentive to adjust its price to those of the rivals. As discussed in Campbell and Eden (2005), standard state-dependent pricing models with menu costs predict that the probability of price changes increases as a firm's price deviates from the average price over other stores. To capture this effect of the price dispersion across stores, I use the absolute value of the deviation of the price of a product item from the average price over the other stores as an explanatory variable in a specification of probit estimation.1819 More precisely, the price dispersion across stores 1 8 A caveat is that in this data, the price dispersion across stores does not reflect the price dispersion across competing retailers. Nevertheless, the price dispersion of a product item across stores could reflect the DFF's perception about competition among retailers. This is because, in DFF, the dispersion of prices across stores might reflect the degree of competitions with other retailers. 1 9 DFF adopts zone pricing that causes the price dispersion of an item in the chain. The DFF zone pricing strategy assigns the stores of DFF to several zones generally consisting of three zones with high, middle, and low prices, respectively. The number of zones and the distribution of stores over the zones is measured by the following variable PDEV store: for the price of product item i sold in store s at period t — 1, Pi tS,t-7-1 PDEV Z%ORE = Ns' 52s'Tts(Pi,s',t-l-Pi,s,t-l) N s> T.s'&Pi,*'*-1 where N s> is the number of the other stores. Pi,S,t-1 N/Es'^SPIIS't-l (2.9) Third, the deviation of the price of a product-item from the average price of product-items in competing brands in an identical package size measures the degree of brand competition in a store. This variable reflects the demand conditions for brands in a store. Slade (1995, 1998, 1999) discusses that brand competition in oligopoly markets is important for price changes in the weekly store-level data. To study the effect of competition among brands within the same package size on price changes, I use the absolute value of the deviation of the price of a product-item in a brand from the average price over the other brand as an explanatory variable. I take into account only the competition among brands within the same package size because there are significant variations in market characteristics across package sizes such as the pattern of price changes, the quantity sold, and the number of product items. The detail of the market segmentation is discussed in the section 2.4.3. Similarly to PDEV store, the price dispersion across brands is measured by the following variable PDEV B: for the price of are, however, varying depending on items and periods. Stores with low price zones are labeled as "Cub Fighters," which are supposed to take price strategies to compete with nearby warehouse-type retailers. For the details of zone pricing in DFF, see Hoch et al. (1995), Montgomery (1997), and Chintagunta et al. (2003). product-item i in brand b sold in store s at period t — 1, Pi tb,s,t-i, Pi,b,s,t-1 PDEV Bbst Yljeb' Pj,b',s,t-1 (2.10) where Ny is the number of the other brands and Nj is the number of items in bland b'. Fourth, the size of a market or a brand might influence the probability of price changes. For example, Powers and Powers (2001) empirically show that the market thickness, which reflects the importance of a product for retailers arid measured by relative shelf spaces in a product category, is positively correlated with the probability of price changes. Using the DFF data set, Besanko, Dube and Gupta (2005) show that major brands in terms of sales share receive higher pass-through elasticities from AAC to retail prices than minor brands do. In this chapter, to measure the size of a market which is segmented by package sizes, I use the share of the sales generated by the items in a package size in all the refrigerated orange juice products in a store. Also, to measure the size of a brand in a market, I use the sales share of a brand in a market defined by package sizes. Fifth, the frequency of price changes might be related to the intensity of brand competition reflected in market structure. There are theoretical predictions as well as empirical findings that prices tend to be rigid in concentrated markets. For example, Rotemberg and Saloner (1987) predict that, facing fixed adjustment costs, a monopolist has less incentive to change prices than duopoly firms. Empirically, Carlton (1986) shows that prices are rigid in concentrated industries. Therefore, by looking at variables presenting market structure, HHI and CR4,1 also examine the effect of market structure on price changes. These variables of market structure are of the previous week since I assume firms make their decisions looking at pre-determined variables. Finally, the effect of the duration of current price level on the probability of price changes is also considered. I estimate a probit model including a variable representing the number of weeks since last price change, TIME. The preliminary evidence in Figure 2.1 indicate that the probability of price changes could be negatively correlated with the duration of current price level. As shown in Baumgartner et al. (2005), however, the direction of this correlation could be changed once the heterogeneity of the timing in price changes across product-items are controlled for by product-item dummy variables. Previous studies using retail prices report the positive correlation between price changes and the duration of current price level. Using the prices of saltine crackers, Slade (1995) reports the positive correlation between the probability of price changes and the duration. Pesendorfer (2002) shows that the probability of having a sale, which is defined as down-ward price change below a certain price level, is positively correlated with time elapsed since the last sale on the products of Ketchup in several supermarket chains. In the model of Pesendorfer (2002) the accumulation of demand from low-valuation customers plays an important role to explain this positive correlation. In addition to the state variables mentioned above, the following variables are also included in the probit estimation: constant, a deal dummy which takes one when the product-item is on the deal DEAL, one-period lag of deal dummy DEALT_i, year dummy, and the dummy variables to control fixed effects specific to stores and product-items.20 In particular, I assume that the deal is known at the time of a price change since Chevalier et al. (2003) states that DFF generally knows the promotional activity beforehand. In summary, the variables and their definitions other than dummy variables are listed as follows. Subscripts i, b, k, s, t, represent product-item, brand, package size, store, and time, respectively. In the rest of this chapter, the subscripts are, however, omitted. Again, note that I use the values in the previous period for PDEV STOR E, PDEV B, SHARE SIZ E, SHARE*, HHI, a n d CRA. PINDX itb,k,s,t '• a binary variables that takes one when a price is changed from the previous period and zero otherwise. DEALi^,k,s,t a binary variable that takes one when the item is on deal and zero otherwise. The one-period lag of this variable is denoted by DEALT_I ABSAACifi,k,s,t absolute value of week-to-week percentage change in AAC CUMAACi :b,k,s,t '• absolute value of cumulative percent change in AAC since last price 20Year dummy variables are included to capture possible changes over time in pricing policy of DFF. For example, using the data from 1989 to 1992, Chintagunta et al. (2003) discuss that the pattern of zone pricing in 1992 differs from that before 1992. The difference in the zone pricing, which mainly determines the dispersion of price levels across stores, could influence the probability of price changes. change PDEVfTORE : the deviation of price of an item i sold in a store s from the average price of the item sold in the other stores PDEV® h k s t : the deviation of price of an item i of brand b from the average price of the items of the other brands in the same package size k in the same store s SHARE^ sztE : the sales share of a package size k in the product category in a store s SHARE® k s t : the sales share of a brand b in a package size A; in a store s HHIkjS,t • Hershman-Herfindahl index, the sum of squared sales shares of all brands in the package size k in the store s, divided by 10000 CR4k:s,t four brand concentration ratio, the sum of sales shares of top 4 brands in the package size k sold in the store s TIMEi^k,s,t the duration of current price level, the number of weeks since last price change21 Table 2.4 presents the descriptive statistics of variables shown above. The first row shows that on average the prices are changed for 44 percent of the whole sample periods. The second row shows that the items are on deal for 26 percent of the whole sample periods. The mean of ABSAAC is small, but its large standard deviation implies that 21Unfortunately, TIME and CUM AAC are censored when there is missing data points. ABSAAC is quite volatile. The descriptive statistics for PDEV STOR E and PDEV B reveal that PDEV B is more volatile than PDEV STOR E. HHI has the mean of 0.55, and ranges from a relatively competitive size 0.178 to a monopoly 1. HHI has a greater variation compared to CR4, whose mean is as high as 97 percent. The data I use are of weekly-store-UPC levels from 37 items in 69 stores. The maximum number of observation is 709075.22 The number of observations in the estimation, however, depends on the specifications of the probit models discussed below. 2.4.1 Effects of costs Table 2.4 presents the results of the probit estimations with two cost variables, ABSAAC and CUM AAC. The first column shows the result of the estimation with ABSAAC23; the second column the result using CUM AAC instead of ABSAAC. The coefficient on ABSAAC is significantly positive. As expected from the results in section 2.3.2, the probability of price changes increases as the size of the change in AAC increases in the absolute value. On the other hand, the coefficient on CUM AAC is significantly 2 2 The number of the maximum data points is smaller than 1018647(37 times 69 times 399) mainly because many items are not sold in the whole periods or in all the stores. The other reasons are because lags are taken and because there are missing data. The data points are omitted from the analysis when either or both of p,kbst and Pikbst-1 are missing as noted in the section 2.2. 23Alternatively, I also estimated the model using absolute value of percentage change in AAC instead of ABSAAC. Since the results are similar, I report only the results from ABSAAC. negative, which means that the accumulation of AAC does not contribute to an increase in the probability of price changes. However, the size of coefficient of CUMAAC is much smaller than that of ABSAAC. Therefore, the effect of CUM AAC would be marginal.24 These results suggest that, for the product-items in refrigerated orange juice, the size of a current change in AAC affects the retailers' decisions to change their prices more than the cumulative AAC does.25 Since controlling for the variations of costs is quite important to draw a precise inference on price changes, in the following estimations, I include ABSAAC as well as the benchmark variables DEAL, DEALt-i, and dummy variables for year, stores, and product-items, unless otherwise noted.26 2 4 I include either ABSAAC or CUM AAC because the correlation between ABSAAC and CUM AAC is high for items with frequent price changes. On average, the correlation between the two variables is 0.478. The correlations by product-items range from 0.190 to 0.852. The correlations by product-items are more than 0.5 in the most of products sold in 64 oz package size, whose prices are changed very frequently. 2 5This result, however, might reflect the consequences of fast turnover and perishability of refrigerated orange juice products. 2 6 I also tried an alternative specification including a dummy variable for premium or concentrated orange juice, a dummy variable for private or national brands, and package-size dummy variables, instead of the product-item dummy variables. The estimated coefficients of these dummy variables are statistically significant at 1 percent level. The results show that(l) the items of premium orange juice have lower probability of price changes compared to concentrated orange juice, that (2) the items of private brand has higher probability of price changes compared to those of national brands, and that 2.4.2 Effects of price dispersion across stores Table 2.5 presents the result of the probit estimation to examine the effects of the price dispersion across stores on retail price changes. More specifically, I estimate the probit model with the measure for the price dispersion PDEV STOR E. The first column shows the result without DEALT_i. The coefficient on PDEV STOR E is negative and statis-tically different from zero at any conventional significance levels. This result implies that the price is changed more frequently in the store around the mean of the price distribution. However, as shown in the third column of Table 2.3, when the estimation is conducted with DEALT_i, the sign of the coefficient on PDEV STOR E becomes posi-tive with statistical significance. Examining the correlation between PDEV STOR E and DEALT_I, I find that they are significantly and negatively correlated with each other with the corresponding correlation coefficient -0.18. This might reflect the fact that, in DFF, a deal is often accompanied with price reductions either to a price level identical across stores or by a similar percentage across many stores. Therefore, a deal tends to decrease PDEV STOR E. In this case, if DEALT_I is not included in the regression, the estimated coefficient on PDEV STOR E could be biased downward. Thus, including the lag of DEAL, which is important for deriving a proper inference on the effect of (3) compared to the items of package size of 64oz, the items sold in the other package size have lower probability of price changes. The estimated coefficients of the variables of interest are quite similar, but the log-likelihood is higher in the specification with product-item dummies. Therefore, I report only the results using the product-item dummy variables. PDEV STOR E on price changes, I find evidence for the hypothesis of state-dependent price models that the probability of price changes increases as a retailer's price deviates from the average price over other stores. 2.4.3 Effects of brand competition and market structure In this subsection, I examine the effects of brand competition and market structure on re-tail price changes. I estimate the probit model of retail price changes using variables cap-turing the degree of brand competition and market structure: PDEV 3, SHARE SIZ E, SHARE B, HHI, and CRA. In this subsection, I focus on the data variations over different package sizes be-cause the data reveal significant heterogeneity across package sizes with respect to many characteristics of product-items, in spite of the fact that the contents of items are ho-mogeneous. Indeed, the degree of heterogeneity is greater among different package sizes than among brands. Tables 2.6 and 2.7 present the statistics to highlight this obser-vation. The numbers presented in these tables are from a representative store of DFF (store 112). As shown in the first column of Table 2.6, there are six package sizes which are sold for the entire sample period. Table 2.6 reports the sales share of items included in the corresponding package size, the frequency of price changes in percentage terms, the frequency of changes in AACs, the number of items, and the number of brands. The numbers of items and brands can be different because several brands sell multiple items that are slightly differentiated over tastes.27 Note the striking heterogeneity in the fre-quency of price changes as well as in the mean duration of price changes across different package sizes. While items included in a package size are not in actual separate mar-kets, the observed great degree of heterogeneity in pricing pattern suggests that items included in different package sizes have different properties of pricing from each other, product-items included in different package sizes are also clearly distinguished from each other in terms of the sales share and the market structure (i.e., the numbers of items and brands). On the other hand, as shown in Table 2.7, the heterogeneity in pricing pattern across brands within the same package size is less obvious. Table 2.7 reports the sales share of items sold by each brand in the size of 64oz, the frequency of price changes in percentage terms, and the frequency of changes in AACs. In spite of the huge difference in sales shares across brands, there is no clear diversity in the frequency of price changes across brands. While the leading brand, Tropicana, has the largest share and experi-ences the more frequent price changes than other national brands, the difference in the frequency among brands are not as large as those among package size. This observation suggests that brands are competing with each other taking similar strategies within the same package size. The heterogeneity in the degree of price variability across package sizes might reflect the different degree of the variability of AACs across package sizes. However, the reason 2 7For example, Tropicana sells Premium Choice, Premium Choice Homestyle, etc. why movements of AACs as well as prices vary across different package sizes is not obvious because the inputs of refrigerated orange juice should be almost identical across package sizes. This suggests that the observed heterogeneity in price variability cannot be explained only by cost factors that manufacturers face. Therefore, other factors related to demand conditions and market structure might explain an important part of the observed heterogeneity in price variability. To see these factors, I investigate the effects of the number of brands and the size of the market in terms of sales share on price changes in Table 2.6. As shown in Table 2.6, price changes are the most frequent in the package size of 64oz, which has the largest market size in terms of sales share. The 64oz package size also has the largest number of brands and differentiated product-items. Notice that size category 128oz changes its price the second most frequently. There is only one product-item included in this size category, which is Heritage House 128oz, a store brand of DFF. This item shows typical two properties of the store brand of DFF: (i) the AACs of the store brands are much smoother than those of national brands and (ii) the store brands of DFF are subject to frequent price changes. The regular price of this product-item is, however, highly correlated with that of Minute Maid 96oz. This might imply that Heritage House 128oz targets Minute Maid 96 oz as a major competitor. Size category 16oz changes its price the least frequently. This size category contains only one item from Tropicana. The sales share of this item is also the smallest. Table 2.8 reports the result of the probit estimation to examine the role of brand competition in price changes. The explanatory variables include the measure of price dispersion across brands (PDEV B ) . The brands are assumed to compete with each other only within each package size. The sample includes product-items of three package sizes, 32oz, 64oz, and 96oz. The other package sizes are not included into the sample since there is only one product-item, i.e., no competitor in these package sizes. The table shows that the coefficient on PDEV B is positive and statistically different from zero. This observation means that the probability that the price of a product-item will change increases as the price of the item deviates far from the average price over the competitors' prices. Therefore, the competition across brands is an important factor to explain price changes in the DFF data. Recall that the statistics reported in Table 2.6 imply that in addition to changes in AACs, several other factors related to market size and and market structure— the sales share of a package size, the sales share of brands, and the measures of market concentration in a package size — might have some explanatory power for price variability in the data. The sales share of a package size could indicate the market size of the corresponding package size; the share of brands the market power of brands; and the measure of market concentration the degree of competition. To examine the effects of these factors, I estimate probit models including these factors as explanatory variables. Table 2.9 presents the results of the probit estimation of the specification including the share of package size and the share of brands in a package size. First, the coefficient on the sales share of package sizes is positive with statistical significance. Although the magnitude is small, this result implies that the greater the market size of a package size category is, the more frequently the prices of the items within the size category change. Second, the share of brands also has a statistically significant positive coefficient, which implies that the larger the share of brands entering the market is, the higher the probability of changing prices is. The positive association between the sales share of package sizes and the probability of price changes is consistent with the observation in Table 2.6. Size category 64oz has the largest sales share and the highest frequency of price changes; size category 96oz the second; size category 128 oz, whose characteristics is discussed above, has the third highest sales share; and size category 32oz the fourth. Notice that the market structure in terms of the number of brands is identical between size categories 96oz and 32oz. This means that the difference in the probability of price changes between these two size categories might reflect their difference in the size of markets.28 The relationship between the frequency of price changes and the sales share of brands in a package size is less clear from Table 2.7. However, after controlling for costs, deals, and fixed effects, I can observe that the coefficient is significantly positive. 2 8 In addition, the size of market appears to be positively related to degree of product differentiation, i.e, the number of product-items. Within size categories 64oz and 96oz, the brands compete not only by pricing but also by introducing differentiated products. Tables 2.10 and 2.11 present the results of estimating probit models including the measures of market structure, HHI and CRA. Table 2.10 shows the results with HHI, and Table 2.11 with CRA. The most important result observed in Table 2.10 is that, as shown in the second equation, the coefficient on HHI is significantly negative. This result is consistent with conventional observations in the literature of industrial organization: an increase in the degree of the concentration in a market leads to lower probabilities of price changes. It should be noted, however, that the above inference depends on controlling for the size effect presented by the sales share of package sizes as well as the sales share of brands. The result in the first equation in Table 2.10 shows that estimating the equation along only with the benchmark variables results in the positive sign of HHI. In this specification, while cross-sectional variations across product-items are controlled, the cross-sectional variations across package sizes and brands are not. As shown in the third equation of Table 2.10, controlling for the size effect presented by the sales share of package sizes as well as the sales share of brands leads to a statistically significant negative coefficient on HHI. Table 2.11 shows the results using CRA instead of HHI. The results of estimation are similar to those with HHI. 2 9 Therefore, these results lead to an important inference of this chapter: the frequency of price changes of a product-item is crucially affected by 2 9 The inference on the effect of week-to-week deviation of CRA from its means is obtained only through the variation of the items of 64 oz since CRA takes 100 for the items in the other package sizes. the degree of concentration of markets. 2.4.4 Effects of duration Table 2.12 shows the results using the time-related variables, TIME. The estimated coefficient on TIME is negative with statistical significance. This implies that the prob-ability of changing a current price is higher as the current price is younger. This result is inconsistent with the empirical results by Slade (1995a) and Baumgartner et al. (2005). Especially, Baumgartner et al. (2005) find the downward unconditional hazard function in their sample, but, once controlling for heterogeneity in their probit estimations, they find that the correlation between the duration of the current price level and the prob-ability of price changes positive. In my sample, however, I do not find evidence for positive correlations between the duration of current price level and the probability of price changes even after controlling for fixed-effects across product-items. This result is rather consistent with the results reported by Campbell and Eden(2005), who find the negative correlation using a reduced-form linea-probability model.30 3 0 One could argue that this is because I use actual transaction prices without omitting deal prices. Conducting estimations only with the observations from the package sizes with few number of deals such as Size 16oz and Size 32oz, however, does not alter the results. 2.4.5 Probit estimations with all the variables Finally, Table 2.13 and 2.14 reports the results of the estimations with all the variables individually examined in the above subsections. The results are robust to the inclusion of the other variables. These results show that the state variables, especially the variation of current cost changes, the demand conditions due to brand competition, and the market structure are important in explaining price changes. 2.5 Concluding remarks Investigating a micro scanner data set in a descriptive fashion, this chapter asks which of two competing pricing mechanisms, state-dependent pricing models and time-dependent pricing models, is better to explain retail price movements observed in the data set. Moreover, conditional on the findings for the above question, this chapter also tries to reveal what economic factors are crucial for the observed degree of rigidity of retail prices. What distinguishes the analysis of this chapter from those in the past papers is that the data set includes the information of retail prices as well as the corresponding costs, i.e., wholesale prices. This is important because controlling for the information of costs leads to more precise inferences on the sources for price rigidity in state-dependent pricing models. The results of this chapter first show evidence for state-dependent pricing models as a better mechanism to describe the data than time-dependent models. Second, I observe significant heterogeneity in the frequency of price changes across markets segmented by the package sizes, even though the ingredients of the product-items are homogeneous. Using the reduced form probit estimation, I examine what factors could explain the observed heterogeneity in the frequency of price changes. The results of the probit es-timation imply that the probability of price changes are positively correlated with the price differentials and the degree of competition among brands as well as the standard deviation of the wholesale prices, the size of markets and brands, and the price differen-tials across stores. Thus, I conclude that the brand competition is a crucial factor for the price changes of individual product-items in the grocery retail stores: given the effect of wholesale price movements, less competition among brands leads to less frequent price changes. The weakness of the above reduced-form approach is that we cannot identify how crucial a potentially important but unobserved source of price rigidity — fixed adjust-ment costs of price changes, which are known as menu costs — is for the observed degree of price rigidity. One approach to investigate the importance of adjustment costs is to estimate them as structural parameters and examine their statistical significance. This approach is promising since I can distinguish the effect of menu costs from those of the other factors in this chapters i.e., the demand conditions, the unit costs, the market structure, the degree of competition, and so on. In the next chapter, constructing a fully-structural dynamic discrete-choice model with multiple-agents, which explicitly in-corporates the profit-maximization behavior and the economic factors mentioned above, I provide identification of fixed adjustment costs associated with retail price changes. After estimating the model by using a recently developed estimator of the structural pa-rameters, I draw statistical as well as economic inferences on the role of fixed adjustment costs in retail price movements. Table 2.1: Predictions of Time-dependent Models and State-dependent Models Time-Dependent Models State-Dependent Models Slope of unconditional flat (Calvo-type) upward hazard functions spike for every certain interval (Taylor-type) Correlation between frequency of price changes and the volatility of cots none positive Table 2.2: Correlation between Frequency of Price Changes and Variations in Costs Variables Estimates (S.E.) S.D. of AAC 0.583 (0.098) Constant 0.366 ' (0.028) adj. R2 0.960 No. of obs 2553 Notel: Dependent variable is the frequency of price changes of a product-item in a store. Note 2: The OLS regression includes store and product-item dummies. Table 2.3: Descriptive Statistics Related to Price Changes and State Variables Variables Unit Mean S.D. Max Min PINDX binary 0.444 0.497 1 0 DEAL binary 0.260 0.439 1 0 ABSAAC % 3.58 23.96 10972.58 0 CUMAAC % 8.094 26.823 10972.58 0 PDEV store 0.046 0.057 0.939 0 PDEV b 0.163 0.139 3.38 0 SHARE B % 43.034 32.561 100 0.029 SHARE SIZ E % 44.909 30.847 100 0.016 HHI 0.551 0.213 1 0.178 CRA % 97.53 3.152 100 74.92 TIME weeks 4.319 7.087 99 1 Note 1: The numbers of observations are 646352 for PDEV B, and 709075 for the other variables. Note2: The value of the variables, PDEV STOR E, PDEV B, SHARE 8, SHARE SIZ E, HHI, CRA are lagged by one period. Table 2.4: Results of Probit Estimation - Costs Dependent Variable: PINDX Variables Estimates (S.E.) Estimates (S.E.) ABSAAC 0.030 (0.0001) CUMAAC -0.0003 (0.0001) DEAL 0.697 (0.004) 0.705 (0.004) DEAL(t-l) 0.710 (0.004) 0.707 (0.004) Constant -0.770 (0.019) -0.661 (0.019) Log likelihood Pseudo R2 -386618 0.206 -392810 0.194 Note 1: The estimation includes the dummy variables to control the fixed effects specific to years, product-items, and stores. Note 2: Pseudo R2 is McFadden's R-Square. Note3: The number of observations is 709075. Price changes are observed 314833 times. Table 2.5: Results of Probit Estimation - Price Dispersion Across Stores Dependent Variable: PINDX Variables Estimates (S.E.) Estimates (S.E.) PDEV store -0.210 (0.031) 1.143 (0.033) ABSAAC 0.029 (0.0003) 0.030 (0.0003) DEAL 0.869 (0.004) 0.700 (0.004) DEAL(t-l) 0.740 (0.004) Constant -0.588 (0.019) -0.849 (0.019) Log likelihood -403492 -386016 Pseudo R2 0.172 0.207 Note 1: The estimation includes the dummy variables to control the fixed effects specific to years, product-items, and stores. Note 2: Pseudo R2 is McFadden's R-Square. 48 Note3: The number of observations is 709075. Price changes are observed 314833 times. Table 2.6: Frequency and Average Duration of Price Changes Across Package Sizes Size Share(Std.dev.) Frequency Frequency No No of price(%) of AAC (%) of items of Brands 3.8 oz 1.30 (0.39) 26.72 49.31 1 1 16 oz 1.12 (0.42) 13.49 23.28 1 1 32 oz 2.41 (1.24) 16.45 30.94 4 3 64 oz 61.76 (7.20) 59.59 70.04 20 6 96 oz 27.79 (6.02) 31.65 60.06 10 3 128 oz 6.15 (2.46) 48.60 76.59 1 1 Table 2.7: Frequency of Price Changes by Brands, 64 oz Bland Name Share (Std.dev) Frequency of price (%) Frequency of AAC(%) Tropicana 55.75 (12.29) 61.89 70.11 Minute Maid 21.47 (10.97) 59.76 68.71 Heritage House 14.19 (7.50) 70.74 65.39 Florida National 4.88 (4.29) 51.20 81.10 Tree Fresh 2.69 (3.06) 57.75 75.94 Florida Gold 2.30 (4.08) 51.08 55.42 Note 1: The data is from store 112. Note 2: Share in the second column of Table 2.6 is the weekly mean of percentage of sales from the package size in the total sales of 37 items. Share in the second column of Table 2.7 is the weekly mean of percentage of sales from the brand in the total sales in 64oz. Note 3: Frequency of price is the percentage of the number of weeks with price changes in available number of weeks. Note 4: Frequency of AAC is the percentage of the number of weeks with AAC changes in available number of weeks. To compute the frequency of AAC, AAC is rounded at 1 cent. Note 5: No. of items and No. of brand are the maximum number of items and brands in a size, respectively. Table 2.8: Results of Probit Estimation - Brand Competition Dependent Variable: PINDX Variables Estimates (S.E.) PDEV b 0.537 (0.014) ABSAAC 0.028 (0.0003) DEAL 0.671 (0.004) DEAL{ t-1) 0.684 (0.004) Constant -0.795 (0.020) Log likelihood -353616 Pseudo R2 0.274 Note 1: The estimation includes the dummy variables to control the fixed effects specific to years, product-items, and stores. Note 2: The estimation uses the items with the package sizes of 32oz, 64oz, and 96oz, in which multiple brands sell the items. No. of'observations is 632652. Prices are changed 293180 times. Note 3: Pseudo R2 is McFadden's R-Square. Table 2.9: Results of Probit Estimation - Share of Package Sizes and Share of Brands Dependent Variable: PINDX Variables Estimates (S.E.) SHARE s 0.004 (0.0002) SHARE B 0.007 (0.0001) ABSAAC 0.030 (0.0003) DEAL 0.707 (0.004) DEAL(t-l) 0.635 (0.004) Constant -1.063 (0.023 ) Log likelihood -383994 Pseudo R2 0.212 Note 1: The estimation includes the dummy variables to control the fixed effects specific to years, product-items, and stores. Note 2: Pseudo R2 is McFadden's R-Square. ^Q Note3: The number of observations is 709075. Price changes are observed 314833 times. Table 2.10: Results of Probit Estimation - Market Structure, HHI Dependent Variable: PINDX Variables Estimates (S.E.) Estimates (S.E.) HHI 0.148 (0.013) -0.204 (0.015) Share of Size 0.005 (0.0002) Share of Brands 0.008 (0.0001) ABSAAC 0.030 (0.0003) 0.030 (0.0003) DEAL 0.698 (0.004) 0.707 (0.004) DEAL{t-l) 0.707 (0.004) 0.632 (0.004) Constant -0.841 (0.020) -1.058 (0.023) Log likelihood Pseudo R2 -386550 0.206 -383896 0.212 Note 1: The estimation includes the dummy variables to control the fixed effects specific to years, product-items, and stores. Note 2: Pseudo R2 is McFadden's R-Square. Note 3: The number of observations is 709075. Price changes are observed 314833 times. Table 2.11: Results of Probit Estimation - Market Structure, CR4 Dependent Variable: PINDX Variables Estimates (S.E.) Estimates (S.E.) . CR4 0.004 (0.001) -0.006 (0.001) Share of Size 0.004 (0.0002) Share of Brands 0.007 (0.0001) ABSAAC 0.030 (0.0003) 0.030 (0.0003) DEAL 0.698 (0.004) 0.707 (0.004) DEAL{t-l) 0.710 (0.004) 0.634 (0.008) Constant -1.174 (0.074) -0.504 (0.075) Log likelihood Pseudo R2 -386602 0.206 -383963 0.212 Note 1: The estimation includes the dummy variables to control the fixed effects specific to years, product-items, and stores. Note 2: Pseudo R? is McFadden's R-Square. Note3: The number of observations is 709075. Price changes are observed 314833 times. Table 2.12: Results of Probit Estimation - Time-related Variables Dependent Variable: PINDX Variables Estimates (S.E.) TIME -0.05 (0.0005) ABSAAC 0.029 (0.0003) DEAL 0.677 (0.004) DEAL(t-l) 0.625 (0.004) Constant -0.628 (0.019 ) Log likelihood Pseudo R2 -378117 0.224 Note 1: The estimation includes the dummy variables to control the fixed effects specific to years, product-items, and stores. Note 2: Pseudo R 2 is McFadden's R-Square. Note3: The number of observations is 709075. Price changes are observed 314833 times. Table 2.13: Results of Probit Estimation - All Variables except PDEV B Dependent Variable: PINDX Variables Estimates (S.E.) Estimates (S.E.) ABSAAC 0.030 (0.0003) 0.030 (0.0003) PDEV store 1.476 ( 0.034) 1.467 ( 0.0339) PDEV b SHARE size 0.008 (0.0002) 0.008 (0.0002) SHARE B 0.008 (0.0001) 0.007 (0.0001) HHI -0.375 ( 0.013) CRA -0.0139 (0.008) TIME -0.051 (0.0005) -0.052 (0.0004) DEAL 0.694 ( 0.004) 0.694 ( 0.004) DEAL(t-l) 0.587 ( 0.004) 0.593 ( 0.004) Constant -1.193 ( 0.020) -0.068 (0.075) Log likelihood Pseudo R? -375689 0.228 -375952 0.228 Note 1: The estimation includes the dummy variables to control the fixed effects specific to years, product-items, and stores. Note 2: Pseudo R 2 is McFadden's R-Square. Note3: The number of observations is 709075. Price changes are observed 314833 times. Table 2.14: Results of Probit Estimation - All Variables including PDEV B Dependent Variable: PINDX Variables Estimates (S.E.) Estimates (S.E.) ABSAAC 0.028 ( 0.0003 ) 0.028 ( 0.0003) PDEV store 1.361 ( 0.036 ) 1.351 ( 0.036) PDEV B 0.368 ( 0.014 ) 0.388 ( 0.014) SHARE size 0.005 ( 0.0002 ) 0.004 ( 0.0002) SHARE B 0.008 ( 0.0001 ) 0.007 ( 0.0001) HHI -0.260 ( 0.015 ) CRA -0.010 (0.001) TIME -0.056 ( 0.0005) -0.056 (0.001) DEAL 0.665 ( 0.004) 0.665 ( 0.004) DEAL( t-1) 0.554 ( 0.004) 0.556 ( 0.004) Constant -0.992 ( 0.024) -0.124 ( 0.075) Log likelihood Pseudo R? -342176 0.297 -342256 0.297 Note 1: The estimation includes the dummy variables to control the fixed effects specific to years, product-items, and stores. Note 2: Pseudo R 2 is McFadden's R-Square. Note3: The number of observations is 632652. Price changes are observed 293810 times. Figure 2.1: Hazard Rates of Prices Figure 2.2: Prices and AACs of Two Items in Refrigerated Orange Juice week Figure 2.3: Frequency of Price Changes and Standard Deviation of AACs p_freq = 0.2379 • ! . 0388 c_std 0.00 0.05 0.10 0.1S 0.20 0.25 0.30 0.3S 0.40 c_std Note 1: The vertical line shows the frequencies of price changes and the horizontal line shows the standard deviations of AACs. The data is for 37 items from 69 stores in refrigerated orange juice. The number of scattered points is 2553. Note 2: Frequency is measured in terms of percentages of the number of price changes in the total weeks in which an item is sold. The standard deviations of AACs are constructed for each item-store combination for the all weeks, in which an item is sold in a store. Chapter 3 Menu Costs, Strategic Interactions, and Retail Price Movements 3.1 Introduction In this chapter, I develop an economic model in which, faced with fixed adjustment costs of changing their prices, manufacturers play a dynamic game of price competition. Estimating the structural model, this paper draws inferences on a potential source of the discrete movements commonly observed in data of retail prices — menu costs ac-companied with firms' price changes. In particular, I estimate menu costs by taking into account a factor that potentially make the estimates of menu costs under the as-sumption of monopolistic competitions in the past studies biased upwards due to the rigidity from strategic interactions among brands in an oligopolistic market. In addition, I show that the estimate could be biased downwards without controlling for unobserved profit-enhancing promotional activities of manufacturers accompanied with price reduc-tion. In particular, the bias due to strategic interactions on the estimate of menu costs has not been investigated before. Using a scanner data set collected from a large super-market chain, after correcting these potential biases, I provide empirical evidence that menu costs are statistically significant as well as economically important in explaining the high-frequency, weekly movements of the retail prices in my data set. This chapter defines menu costs as any fixed adjustment costs a price setter has to pay whenever changing its price within a period, regardless of the magnitude and direction of the price change.1 These fixed adjustment costs may include not only the costs of relabeling price tags but also managerial costs and information-gathering costs, which might occur when firms changing their prices. Importantly, several recent papers provide evidence that these menu costs are empirically crucial. On the one hand, constructing direct measures of physical and labor costs in large supermarket chains in the United States, Levy et al. (1997) claim that menu costs play a crucial role in the price setting behavior of retail supermarkets. On the other hand, estimating menu costs as structural parameters of single-agent dynamic discrete-choice models in monopolistic competitive markets, Slade (1998) and Aguirregabiria (1999) find that menu costs are statistically significant. This chapter also adopts dynamic discrete choice models to estimate menu 1 Menu costs can be asymmetric: the fixed adjustment costs can differ across directions of price changes. In this chapter, however, I examine only symmetric menu costs. costs.2 As frequently observed in the recent macroeconomic literature, monopolistic com-petition is the most common market structure maintained by theoretical and empirical studies of price rigidity.3 This assumption of market structure, however, is problematic if the following two facts are taken into account. First, it is obvious that not all product markets in an economy are monopolistically competitive. If the market of a product is dominated by a small number of firms, the assumption of oligopolistic competition is appropriate for studying the pricing behavior of firms. Second, under oligopolistic competition, if we employ the estimates of menu costs in past studies under the main-tained assumption of monopolistic competition, the estimate might be potentially biased upwards. This is due to possible strategic interactions among firms in an oligopolistic market. For exposition, suppose that there are a few firms in an oligopolistic market, which compete with respect to their prices. While monopolistic competition models cre-ate strategic complementarity between each firm's price and the average price of all firms, each firm perceives its own market power so small that the average price is regarded as 2The definition of menu costs in this chapter follows those by Slade (1998) and Aguirregabiria (1999). 3For example, Blanchard and Kiyotaki (1987) show that menu costs combined with monopolistic competition may generate large effect of monetary shocks on output. To explain the persistent effects of monetary policy shocks on real aggregate variables observed in aggregate time series data, Yun (1996), Smets and Wouters (2003), and Christiano, Eichenbaum and Evans (2005a) introduce the staggered multi-period price setting mechanism of Calvo (1983) into dynamic stochastic general equilibrium models with monopolistically competitive firms. being exogenous. In contrast, in a tight oligopoly market, each firm takes into account strategic interactions among firms more explicitly. This would lead to stronger strategic complementarity, and firms may prefer less aggressive price competition. Because of their strategic interactions, the equilibrium price of the market might be rigid to some extent, regardless of the existence of menu costs. In the literature of empirical industrial organization, for example, Neumark and Sharpe (1992) and Carlton (1989) provide em-pirical evidence of positive correlation between price rigidity and market concentration.4 In this case, ignoring the effect of the strategic interactions on price rigidity makes an estimate of menu costs biased upwards. This means that, to derive an inference on menu costs, it is important to take into account the market structure of a product and the strategic interactions among the firms in the market. Although a slew of recent papers study price rigidity using micro data, almost none of them investigates the relationship between the price rigidity of a product and its market structure taking into account the effect of strategic interactions.5 There are, however, a few exceptions. Dutta and Rustichini (1995) and Lipman and Wang (2000) develop theoretical models in which, being faced with menu costs, firms in a duopoly market play 4In the second chapter of this thesis, I also show that the positive correlation between price rigidity and market concentration is found in the refrigerated orange juice products from the Dominick's Finer Food data set. 5Sheshinski and Weiss (1977), Carlton (1986), Cecchetti (1986), Kashyap (1995), and Lach and Tsiddon (1996) are among the earlier studies on price rigidity with micro data. a dynamic game under perfect information. 6 Unfortunately, it is not a straightforward exercise to construct econometric models from their theoretical implications. One alter-native approach used by Slade (1999) consists of estimating thresholds of price changes as functions of strategic variables within a reduced-form statistical model. Assuming that firms follow a variant of (s, S) policy, Slade (1999) observes that strategic inter-actions among firms engaging oligopolistic competition exacerbate price rigidity. This observation suggests possible upward bias of the estimates of menu costs, as discussed above. This chapter goes beyond the reduced-form model of Slade (1999) by developing a fully-structural dynamic discrete-choice model with menu costs and strategic interac-tions. I model oligopolistic competition and incorporate it directly into an econometric model. Since the effect of oligopolistic interactions on prices is captured by strategies in the model, the rigidity due to menu costs is separately inferred from that caused by strategic interactions.7 This approach leads to more precise estimates of the magnitude of menu costs if oligopolistic interactions are important in my sample. Estimated menu costs may also be biased downwards because of unobserved profit-6One source of price rigidity in an oligopolistic market would be collusion. Modelling collusion is, however, beyond the scope of this paper. For a theoretical model, see Athey, Bagwell and Sanchirico (2004). 7my econometric model does not impose the assumption that strategic interactions lead to price rigidity. Thus, I may find less or more price rigidity in my oligopoly model than in a monopolistic competition model. enhancing promotional activities of firms accompanied with a price reduction. To explain this potential downward bias, suppose that, given menu costs, promotional activities of firms reduce the prices of their products but, at the same time, increase the firms' profits. The problem is that when researchers cannot observe these promotional activities perfectly, it is not possible to control for the profit-increasing effects of downward price changes. As a result, the estimate of menu costs might be biased downwards because the estimates capture not only menu costs of price changes but also these profit-increasing effects as fixed adjustment costs of price changes.8 To deal with this possible downward bias of the estimates of menu costs due to unobserved promotional activities, I introduce a dummy variable specific to price reductions under the hypothesis that my estimate of menu costs increases when the dummy variable is included into my econometric model. With the weekly retail price data of graham crackers collected in Dominick's Finer Food, I identify menu costs based on a dynamic discrete-choice model with multiple agents. Since my price data are well characterized by frequent discrete jumps, I ex-ploit fixed adjustment costs to explain these observed discrete price changes, as in the dynamic discrete-choice models with a single agent under monopolistic competition by Slade (1998) and Aguirregabiria (1999). To take into account'the effect of strategic in-8 A promotional activity might be a demand shifter specific to price reduction and has a positive effect on a manufacturer's profit in this case. If a researcher cannot identify this demand shifter from data, the researcher captures the effect of the promotional activity as negative fixed adjustment costs of price changes. teractions among manufacturers on price rigidity, I develop a dynamic discrete-choice model with multiple agents in an oligopolistic market. I estimate my fully-structural dynamic discrete-choice model exploiting the nested pseudo likelihood algorithm (NPL) developed by Aguirregabiria and Mira (2002, 2004). The NPL includes the conditional choice probability (CCP) estimator of Hotz and Miller (1993) as well as the nested fixed point (NFXP) estimator of Rust (1987) as extreme cases. The major advantage of the NPL over the other two estimators is that the NPL gains efficiency compared to the CCP, while the NPL saves computational costs compared to the NFXP. Aguirregabiria and Mira (2002) develop the NPL for estimating dynamic discrete-choice models with a single agent. Aguirregabiria and Mira (2006) extend their NPL to a multiple agent setting that allows strategic interactions among players. I adopt their estimator to analyze the price-change game in an oligopolistic market.9 Firstly, I find that my estimates of menu costs are statistically significant. The size of the estimated menu costs is close to those estimated in the past studies using the data from different markets. Therefore, I conclude that menu costs explain the observed degree of price rigidity, and play an economically important role in the weekly movements of my price data. Secondly, estimating the augmented models with the dummy variable specific to price reductions, I provide evidence that unobserved profit-enhancing promotional 9In Chapter 4,1 review the recent developments in the estimation methods of dynamic discrete-choice games in detail. activities in fact leads to statistically significant downward bias of the estimate of menu costs. Finally, the comparison between the results of my oligopolistic market model with those of a monopolistic competitive market model statistically supports the empirical hypothesis that strategic interactions among manufacturers results in upward bias of the estimator based on the latter model. In summary, the results of this chapter not only confirm the inferences drawn by the past studies using the data of another product — fixed adjustment costs of price changes are statistically significant as well as economically important —, but also empirically reveal another potentially crucial source of price rigidity — strategic interactions among firms in oligopolistic markets. Section 3.2 introduces a dynamic discrete-choice model with multiple agents under an oligopolistic market. Section 3.3 describes the empirical model to identify and estimate menu costs. Section 3.4 discusses the data set and estimates the demand function for gra-ham crackers and transitory probabilities of descritized state variables. After reporting the main results in section 3.5, I conclude in section 3.6. 3.2 The model This section introduces a structural model in this chapter, which leads to identification of menu costs. This model describes a dynamic duopoly game between two manufacturers, who decide whether to change the retail prices of their products in the presence of menu costs. 3.2.1 The environment The purpose of the analysis in this chapter is to focus on dynamic brand competition with respect to price changes. To do so, I assume a specific structure of decision making of manufacturers. First, I assume that manufacturers - or brands - are competing with respect to prices. The assumptions on the strategic instruments manufacturers are competing with have potentially critical effect on the hypothesis to test. In this chapter, I investigate price competition as I am concerned with how firms adjust their prices facing fixed adjustment costs of prices. Second, I assume that the competition is among manufactures rather than among retailers. Manufactures sell their branded products through a retailer. They act to max-imize the sum of discounted profits and extract all the profits obtained in the retail store. This vertically-integrated structure is a strong assumption, but could be reasonable when a retailer is neutral and acts passively regarding the competition among manufacturers. There is evidence that this assumption could be justified for the data I use in this chap-ter. In this chapter, I analyze brand competition of a narrowly defined single product category - graham crackers. When we look at prices in a narrowly defined product with small sales such as graham crackers, the price differentials across brands might reflect the competition among brands rather than that among retailers. For example, conduct-ing an interview with a manager in a supermarket, Slade(1995, 1998, 1999) states that retailers are competing with their overall offering rather than through a single product. Chintagunta et al. (2003) confirm the claim by Slade through an interview with a store manager in Dominick's Finer Food (DFF), the supermarket whose data set I use. In addition, using the DFF data set, Montgomery (1997) state that the price movements across time reflects manufacturers' decision making rather than the retailer's. Besanko et al. (2005) also show that the pass-through elasticity between retail prices and wholesale prices is as high as 80 percent in the product category of crackers, which is the product analyzed in this chapter, in DFF.10 These description and evidence show that retailers generally act passively in the pricing of a single minor product. Third, I assume that the manufacturers maximize the profits gained within a store. I define a price-cost margin as the difference between a retail price and a wholesale price, but not that between a retail price and a marginal production cost. This assumption is imposed since I would like to extract the competitive aspect among manufacturers reflected in retail prices. Finally, the manufacturers decide only whether or not to change their current prices but not exact price levels. A price level is determined by a retailer, who follows a certain 1 0In the analysis of Besanko et al. (2005), the pass-through elasticity of 100 percent means that the retailer acts completely passive in pricing. On one hand, they shows that the pass-through elasticity of beer exceeds 500 percent. This indicates that the retailer tend to discount the product more heavily than manufacturers intend. They confirm that the retailer uses a package of beer as a loss-leader. On the other hand, the pass-through elasticity of toothpaste is as low as 20 percent. pricing rule. This assumption is made to capture the fact that the retailers act passively but it is hard to consider that the manufacturers can control the retail prices perfectly. These assumptions are strong and abstract actual vertical structure to a great extent. Ideally, the model would include an explicit vertical structure such that the manufactur-ers set their wholesale prices constructing expectations with respect to retailers' pricing and other manufacturers' pricing given their own production costs. This straightfor-ward structure of vertical integration is, however, difficult to incorporate into a dynamic oligopoly model with fixed adjustment costs, which is structurally estimated later.11 Also, even under the vertically-integrated structure, it might be ideal to assume that manufacturers decide both price changes and price levels. The extension to this direc-tion requires a model to have both discrete and continuous control variables. Although the extension to this direction would be fruitful, I focus on the most simple structure in this chapter. In the following, I formalize the model under the assumptions stated above. 3.2.2 The problems of manufacturers Consider a market in which two manufacturers compete with respect to their prices pit and p_jt for periods t = 1,2,..., oo. Let i = {1, 2} and — i = {2,1} denote the indices of a manufacturer and its rival, respectively. To sell their products, the manufacturers have 1 1For a static model with an explicit vertical structure, see Villas-Boas and Zhao (2005). to put their products on the shelf in a retail store with no cost. For simplicity, I con-sider a vertically integrated manufacturer-retailer relationship, in which the retailer acts passively: the manufacturers' decisions to change their prices are always implemented by the retailer. At the beginning of each period, the manufacturers know the wholesale prices of the two products, {q t , c_ j t } , and the values of demand conditions in the past period, {di t-i, d^u-i}. The current demand conditions, {d it, d-u}, and the consumption, {q it, q~u}, realize during the period t. The demand function for the product of manufacturer i = {1,2} is Qit = dit ~ bopu + hp-it, (3.1) where b0 > 0, b\ > 0, b\ < bo, and — i = {2,1}. qit and pit stand for the quantity sold and the price of the product of manufacturer i, respectively. Let ait denote the discrete action taken by manufacturer i at period t: alt = 0 means no price change and ait = 1 a price change, respectively. Changing prices incurs fixed price adjustment costs, 7 > 0, i.e., menu costs. In addition, manufacturer i receives private information sit that affects its profitability. Private information e i t is a vector including and e\t as its elements, where e\t is the private information of manufacturer i when taking action a = {0,1}. Subsequently, the manufacturers simultaneously decide whether to change their prices or not. Once manufacturer i decides to change its prices incurring menu costs, shelf price Pit is determined at the optimal level without menu costs. The actual shelf prices are set by the retailer. It is worth noting that this paper does not model the decision making with respect to shelf price levels, but this decision is modelled as a stochastic process that is of common knowledge across the manufacturers. For any price of manufacturer —i, p-u, the one-period profit of manufacturer i at period t is defined as nit(p it, da, Cit, aiu a-it) = (1 - ait)^t + aitirlt, (3.2) where 7if4 = (p it-1 - cit)(d it - b0pit^i + bip-a) + 4t (3.3) and ^It = (pa - cit){du ~ boPit + hp-it) -I- e}t - 7. (3.4) The one-period profit of manufacturer i depends on the action its rival takes, a_ t t, through the rival's retail price p_it. In particular, equation (3.3) shows the one-period profit for manufacturer i at period t when the manufacturer takes action ait = 0, while equation (3.4) is the profit when manufacturer i decides to change its price. The demand conditions and wholesale prices, which evolve independently from the actions taken by the manufacturers, follow stationary first-order Markov processes with the density functions ff(d it\dit-1) and ff(ci t\cit-i), respectively.12 The shelf prices of the 1 2 The assumption that the demand conditions are independently distributed implies that there is no interaction between manufacturers and consumers. The processes of the wholesale prices are also assumed to be exogenous because I focus on a price competition in a retail store. products depend on the actions of the manufacturers. Let ff (pit\dit-i,pit-i, cu, an) be the transition density function of the retail price of manufacturer i. Denote the transition density function of the retail price of manufacture i when it takes action alt = 1 by fi{Pit\dit-i,Pit-i,Ci t). Then, the transition density function is described as I ff (Pit I dit-iiPu-uCit) if ait — 1, degenerated at pn — pu-i if a-it — 0. The state variables in this model consist of commonly and privately observable com-ponents. The commonly observable component is denoted by a vector xt such that xt = {p it-i,p-it-i,du-i,d-it-i,Cit,C-it}. Private information eit is observable only for manufacture i, and is independently and identically distributed with a known density function g(eu) across actions, manufacturers, and time. Manufacturer i observes {x t, £u}, while a researcher observes only xt. Throughout this paper, I assume that the state space of xt, X, has a finite discrete support of dimension M. The assumption of i.i.d. private information is admittedly strong. This assumption would be, however, acceptable in a well-defined model. My model defines the observable components of the profit gained by brands in a store as precisely as possible based on the economic theory. In addition, my empirical model controls for dynamics and strate-gic interactions, which could be very important in the actual decision making of price changes, by directly incorporating theoretical counterparts into the empirical model. In addition, this assumption is necessary to implement an empirical method for a dynamic discrete choice model with multiple agents. For example, if the private information is correlated across manufacturers, each player can infer the private information of the other manufacturers based on its own private information. This requires a researcher to take additional integration with respect to private information. 1 3 Moreover, without the assumption of serially uncorrelated private information, manufacturers infer the current private information of the others based on the past state variables. Then, the size of the state space expands exponentially in the number of the size of state space, and therefore too large to be dealt with even for the problem with the small number of grids per state variable. Given the vector of the state variables and the expected sequence of its rival's action, manufacturer i maximizes the following objective function where (3 £ (0 1) is the discount factor, and E{• \ xt, e^} is the mathematical expectation operator conditional on the payoff relevant state variables at period t. The action of manufacturer — i, a^ls. affects the current profit of manufacturer i through P- ls. Since the time horizon is infinite and the problem has Markov structure, I assume Markov stationary environment in the following. 13Moreover, relaxing this assumption makes some part of the empirical method employed in this paper infeasible. oo (3.5) s-t 3.2.3 Markov strategy, Bellman equation, and equilibria The manufacturers solve the stationary Markov problem and play Markov strategies.14 Since the problem is stationary with infinite time horizon, I drop time-subscript t from the rest of the analysis. Instead, I use x' = d-, c-, and e' = (e^e^J to denote the state variables at the next period.15 The realization of one-period profit depends on the demand conditions and shelf prices at the end of a period, which are the state variables in the next period.16 When manufacturer i = {1,2} decides whether to change its price at the beginning of period t, the profit is random because the manufacturers do not determine the levels of their shelf prices, which are stochastic with the density function ff. Therefore, the manufacturers have to form expectation with respect to the levels of the shelf prices at the time of decision making. In addition, I assume that demand conditions realize after the man-ufacturers made their decision abut price changes. Therefore, the manufacturers form their expectation with respect to demand conditions as well. Let da = d-u = dt so that 1 4I f {xu,£it} = {xi s ,£is}, then manufacturer i's decision at period t and s are the same (an = djS). 1 5 The following description about strategies, the Bellman equation, and the equilibria is based on Aguirregabiria and Mira (2006), who characterize a dynamic game structure with discrete choices and space. Aguirregabiria and Mira (2006) analyze an entry-and-exit game as an example of the application of their basic model structure. 16Puterman (1994) shows that I can set up a problem with a one-period payoff depending on the state in the next period. demand conditions are symmetric across manufacturers. Let a = {<7j, cr_j} be a set of arbitrary strategies of the manufacturers, where a* defines a mapping from the state space of (x,£i) into the action space; that is, Oi : M x R2 {0,1}. Given a, the conditional choice probability for manufacturer i to choose action a, P°(ai = a\x), is defined as P?(ai = a\x) = Prob[ai(x,£i) = a\x] = J I{ 1 stages. In the estimation, the K-stage pseudo-log-likelihood is constructed as: 2 T E E E I{a lt = a}\nyi{a\x t-PK-\9). (3.15) i=1 t=1 ae{0,l} 2 3 The value of 0.99 is chosen so that the discount factor is close to one. As stated later in the section of the data, the decision modeled in this paper is weekly and very short-run. 2 4 The cumulative distribution function of the type I extreme distribution is G{E) = exp( — Letting 0 denote the structural parameter that maximizes equation (3.15), I can obtain the K-stage estimator of conditional choice probabilities: PK = ^(P K~ 1]eK). (3.16) This estimator is also known as a quasi-generalized M-estimator. Under standard reg-ularity conditions, it is consistent and asymptotically normal. Moreover, the estimator gains efficiency by repeating for K > 1 stages compared to the estimator without the it-eration in terms of K. In practice, I conduct the estimation for K stage until PK = PK-I or, equivalently, Ok = &K-i is obtained.25 3.3.3 Estimation with potential multiplicity of equilibria The parameter of menu costs might not be point-identified because the model potentially could have multiple equilibria. Without knowing the selection mechanism of the game, it is not possible to construct the likelihood functions. As in Aguirregabiria and Mira (2006) and the other studies who propose alternative estimators such as Pesendorfer and Schmidt-Dengler (2003), and Pakes et al. (2005), I assume that the observed data are generated from one equilibrium. This assumption implies that, given a vector of observable state variables, a certain strategy is chosen with probability one in the data. Therefore, this assumption avoids the problem of unknown selection mechanism. This 2 5See Gourieroux and Monfort (1995) for the asymptotic characterizations of a quasi-generalized M-estimator. For the asymptotic characterization of the NPL estimator, see Aguirregabiria and Mira (2002, 2004). assumption is not, unfortunately, testable, and whether this assumption is satisfied or not is the problem and data specific. In the context of entry-exit game with cross-sectional data with short time periods, Aguirregabiria and Mira emphasize the importance of the condition where players are the same across markets. Similarly, Pesendorfer and Schmidt-Dengler(2003, 2005) argue that this assumption is more likely to be satisfied in a single market with the same players than multiple markets with different players. In my model, the players are fixed for the entire periods, and they play in the same market. Therefore, it is not likely that the players switch the equilibrium they play. Moreover, as I will discuss the property of my data below, the markets are similar to each other as they are in the same city. 3.4 Data, demand estimation, and transition proba-bilities I analyze the empirical model stated above choosing one product, graham crackers, sold in a large supermarket chain in the United States. This section first describes the property of the data. Second, I report the empirical results to prepare for the estimation of menu costs: the estimated demand equation and transition probabilities of the state variables. 3.4.1 The data The data in this chapter are from the weekly scanner data set collected in the branch stores of Dominick's Finer Food, the second largest supermarket chain in metropolitan Chicago during my sample period from September 1989 to May 1997.26 The data set con-tains information such as shelf prices, quantities sold, and importantly a proxy variable of wholesale prices (average acquisition costs) by stores as well as by Universal Product Codes, which distinguish products. The products in the data set are priced on weekly basis, which matches my sample frequency. My sample contains 21 stores out of the total 84 stores in the supermarket chain. These stores are chosen based on the availability of transaction records.27 I choose graham crackers as the product to be analyzed because (i) a small number of manufacturers dominate the market; (ii) there is only one similar size of package; (iii) Graham crackers are minor products so that I can avoid the possibility that pricing is affected by loss-leader motivation of the retailer. There are three national brands (Sarelno, Keebler, and Nabisco), and one private brand of Dominick's. The sizes of packages are 15 oz or 16 oz. 2 8 2 6 The data set is available at http://gsbwww.uchicago.edu/kilts/research/db/dominicks/. 2 7 I omit the stores with too many missing data from the sample. For the details of choice of the stores, see Appendix A.3. 2 8 In addition, the data set provides a code that show whether DFF buys a product is directly from manufacturers or through wholesalers. According the code, DFF buys graham crackers directly from Table 3.1 shows the market shares of the manufacturers in the total sales of graham crackers in these 21 stores. The market share of the four brands is about 97 percent of the total sales of graham crackers. Among them, the three national brands have the market share of 72.24 percent. I analyze the competition among these national brands.29 Figure 3.1 plots the shelf prices of three national brands in a representative store. The figure shows two important aspects of the data. First, the shelf prices discretely jump both upwards and downwards. Second, most of downward price changes are followed by upward price changes almost the same magnitude within a quite short period. In particular, I can interpret the second aspect as promotional activities with "temporary discounts". In total, I have an unbalanced panel data with 21978 observations (7326 observations for three brands). The number of weeks available ranges from 328 to 362 depending on the numbers of missing data. Tables 3.2 and 3.3 report the summary statistics of prices, quantity sold, and costs in the sample, and the descriptive statistics associated with price changes.30 Prices and costs are nominal. Prices are changed for 32.6 percent of times in the sample. The manufacturers. assume that the private brand, Dominick's, does not join the game among the national brands, and treat the price of Dominick's as being exogenous. This is because (1) the correlation between the price of private brand and those of national brands are weak, and (2) the prices and AACs of private brand behaves in a different way. 3 0 I use average acquisition costs (AACs) as the measure of wholesale prices. For the differences of average acquisition cost from the wholesale price, see Appendix A.3. magnitudes of downward price changes and upward price changes are similar to each other with 0.28 dollars and 0.27 dollars on average, respectively. In the estimation of dynamic discrete choice game, using the data set with a long sample period has an advantage over short panel data sets often used for an entry-exit game. In my sample, I can observe the actions of each player and transitions of state variables repeatedly. This feature leads to more precise estimate of conditional choice probabilities and transition probability matrices. 3.4.2 Demand estimation and state variables I estimate the demand equation (3.12) by 2SLS. The dependent variable is the log of quantity sold. The explanatory variables include the following variables. First, price variables are p (the log of prices), rp(the log of simple average of rivals' prices), dp (the log of prices of Dominick's, store brand). The effect of promotional activity is controlled by a dummy variable bonus, which takes one when deal code indicates that "bonus" takes place. Bonus is a promotion activity, which is typically price reduction associated with a shelf-tag announcing promotion.31 Once in a while, DFF bundles multiple units into one package. To capture this effect, bundle is created which shows additional units bundled. In addition, variables that capture persistent effects of promotional activity and pricing 3 1 Unfortunately, the variable of bonus does not capture all the promotional activities. According to the description of the DFF data set, there could be promotional activities even when there is no record in the data set. are created: durd (the duration since the last discount more than five percent) and durb (the number of weeks in the duration of bonus). 3 2 Also, cc (the log of customer count) controls for the effect of store traffic on the demand of graham crackers. Customer count is the number of customers who purchased at least one item in the store. A unit of customer count is 100. 3 3 Holiday dummy variables are also created to capture seasonality in demand if there is any. Chevalier et al. (2003) report that the demand of several goods exhibits some degree of seasonality using the data of various goods from Dominck's. According to the week coding in the original data set, I create dummy variables that takes one in the week, which include any holiday and its previous week. 3 4 In addition to these variables, the estimated demand equation includes constant, brand dummy variables, and store dummy variables. To take into account possible endogeneity causing correlations among the current prices of three national brands, pu and p~u, and the demand error term, I use instru-mental variables. The instruments include wholesale price, cit, and the average of rivals' wholesale prices, c_,:i, and a variable constructed by multiplying the wholesale prices by a variable income.35 Since the variation of wholesale prices cit and c_jt across stores is 3 2For the construction of these variables when the data point is missing, see the Appendix A.3. 3 3 Customer count in the original data set is recorded daily. The daily average of customer count in each week is used in the analysis. 34Alternatively, including monthly dummy variables is also considered. However, the results were similar and the results with holiday dummy variables are reported. 3 5 The variable income is the log of the median of incomes from U.S. Census-data in 1990. Income small, then the variable created by multiplying cit by income is used to control the vari-ation of prices across stores as well as weeks. In this estimation, the price of Dominick's and promotional variable are regarded as exogenous since, as stated above, the price of Dominick's has weak correlation with the prices of three national brands. Also, bonus is assumed to be exogenous. In the estimation of demand system using the data from DFF, Chintagunta et al. (2003) assumes that variables related to promotional activity, which include bonus, are exogenous. They justify this assumption since the schedule of promotional activity is generally determined in advance. Thus, according to their claim, bonus is a pre-determined variable. Table 3.4 reports the results of the demand estimation. Most of the coefficients appearing in the table are statistically different form zero at the 5 % significance level while dp, bonus, and bundle are insignificant. To construct demand conditions, I use the estimates of a constant, the shelf prices of Dominick's, and customer count. The constructed demand conditions are the same across manufacturers: dt = dit = d_it. Since the customer count and Dominick's prices are not known by manufacturers at the time of decision making, I assume that manufacturers form expectation with respect to these variables. The ex ante one-period profit includes expected demand conditions dt+if d(d t+i | dt) for a given dt, where dt consists of customer count and Dominick's price in the previous period as well as a constant, differs by ZIP codes, and so by stores. The NPL estimator needs to discretize the state variables. Each state variable is divided into two regions according to its empirical distribution so that each cell of a variable has probability 0.5 to be visited. Variables consisting of state variables are evaluated at the lower bound of each grid of each state variables — the values at one percentile and 51 percentile. Table 3.5 presents the descriptive statistics of the discritized variables. The total number of grids in the discretized state space is 128. I construct the transition probability matrices and the construction of the initial values of conditional choice probabilities as described in the previous section. 3.5 Results 3.5.1 Estimated size of menu costs Table 3.6 reports the estimate of menu costs 7. The estimate is 1.009 and statistically different from zero at any conventional significance levels based on the standard error 0.025.36 This point estimate implies that the marginal cost of a price change is 1.009 U.S. dollars. This amount of menu costs is 3.54 percent of average weekly graham cracker sales per store in a week. The estimate of menu costs in the above benchmark specification, however, might be biased downwards by unobservable promotional activities. To explain this potential 3 6 The standard error is calculated by 10000 non-parametric bootstrapping resamples. downward bias, suppose that, given menu costs, there is an unobservable factor that increases the profit of the manufacturer only when the manufacturer reduces its price. If the econometric model does not control this profit-enhancing factor specific to downward price changes, the estimate of menu costs is biased downwards because the coefficient 7 captures not only fixed adjustment costs of price changes but the profit-increasing effect in this case.37 The most likely interpretation of this profit-increasing factor is promotional activities due to the following two reasons. First, a promotional activity for a product usually takes place with not only reducing the price of the product temporarily but also conducting demand-enhancing activities such as advertisements and in-store displays. Second, a promotional activity might decrease the marginal costs the retailers have to pay when the prices are temporarily discounted.38 Through these two possible effects, the unobserved promotional activities of the manufactures might lead to downward bias of the estimate of menu costs. In fact, I find that the shelf prices in the sample are characterized by frequent downward price changes followed by immediate increases in 3 7For exposition, consider the following simple one-period profit of a manufacture (p — c)q + XI[Ap < 0] — 7 where A > 0 is the profit-increasing factor specific to downward price changes . If I do not observe and control A, I estimate menu costs as XI[Ap < 0] — 7, which leads to a downward bias of the estimate of fixed adjustment costs of price changes 7. 3 8For example, with a single agent dynamic discrete choice model with a sample different from ours, Aguirregabiria (1999) observes that menu costs of downward price changes are significantly much lower than those of upwards price changes. He argues that this is because the retailer does not pay for the costs associated with the profit-increasing promotional discounts. the prices back to the "regular" price levels. This important characteristic of my price data suggests that promotional activities accompanied with temporary price discounts frequently occur in the sample. To correct this potential downward bias of the estimate of menu costs 7 in the benchmark specification, I create a dummy variable specific to downward price changes, A. More specifically, I consider the following specification: zn = {71"^ (a), — I{Ap it 0},I{Ap it < 0} } and 0 = {1,7, A}. If the profit-increasing factor specific to downward price changes is important in the sample, I should observe that (i) the sign of A is positive and (ii) the estimate of coefficient 7 is greater than that in the benchmark specification. 39 Table 3.7(a) shows the results of the estimation with the augmented specification. As expected, the sign of point estimate of the coefficient on the dummy variable specific to downward price changes, A, is positive with the value 2.840 and statistically differ-ent from zero at any conventional significance levels with nonparametric bootstrapping standard error 0.028. This implies that the unobservable profit-increasing factor specific to downward price changes is crucial in explaining the behavior of the price data in this chapter. The identified downward-price-change specific factor A would include both the demand-shifting factors and the effect reducing in-store-cost. After controlling the 39While I emphasize the importance of controlling possible promotional activities, I assume that these activities are exogenously given. These activities can, however, be endogenous and strategic. Incorporating multi-strategic instruments into the model is left to be a future research. profit-enhancing effect, the point estimate of menu costs 7 turns out to be 2.578, which is greater than that in the benchmark model. The standard t-statistic rejects the null that the point estimate 2.578 is equal to that of the benchmark model, 1.009, at any conventional significance level (t-statistic = 60.35). Therefore, the downward bias of the benchmark specification is significant statistically as well as economically. The most important advantage of the dynamic discrete-choice model with an oligopolis-tic market in this chapter over the standard monopolistic competition model, which is employed by previous studies to estimate menu costs, is that this model takes into ac-count the potential effect of strategic interactions among manufacturers on price rigidity. For example, Slade (1999) shows that prices of oligopolistic firms, which follow a variant of (s,S) strategy, are stickier than those with monopolistically competitive firms because the thresholds of price changes widen as price level goes up. If the effect of strategic interactions among firms on price rigidity is crucial in the observed price behavior, the estimate of menu costs 7 with a monopolistic competitive market might be biased up-wards. This is because, with a monopolistic competition model that does not identify any strategic interaction, menu costs 7 capture not only fixed adjustment costs of price changes but also the price rigidity due to strategic interactions among firms. To examine the above conjecture, I next estimate menu costs 7 under a monopolistic competition model with the dummy variable specific to downward price changes. In the monopolistic competition model, the decision rule of price changes does not depend on the conditional probabilities of the other firms. Therefore, a manufacturer constructs her expectation over only the evolution of exogenous state variables, the future values of the unobservable state variable, and her own future actions. The model contains the three state variables of the demand condition, the wholesale price, and the past price. Since a firm regards the prices of her rival brands as being exogenous, I include the rivals' price as a part of the demand conditions, which is exogenously evolved. Then I compare the estimate of menu costs under the second specification with that under my monopolistic competition model. The oligopolistic model corrects the upward bias in the monopolistic competition model if the strategic interactions among the manufacturers are important in the sample. Table 3.7(b) reports the results of the monopolistic competition model.40 First, the coefficient of the dummy variable specific to downward price changes has a positive sign with the magnitude of 3.870, and is statistically different from zero at 1 percent signifi-cance level with its standard error 0.04. Thus, the profit-enhancing factor is important regardless of the assumption about the competition among the manufacturers in the sample. Second, the estimated coefficient on menu costs 7 under the assumption of monopolistic competition model is 3.443 and significantly different from zero with its 4 0 The estimation of a monopolistic competition model is conducted using the framework of a single-agent dynamic discrete choice model in Aguirregabiria and Mira (2002). Rivals' prices are assumed to evolve exogenously, and included in the demand condition. In addition, the manufacturers do not take into account the conditional choice probabilities of other manufacturers. standard error of 0.037. This estimated size of menu cost, 3.443, is greater than the counterpart in my oligopolistic model, 2.578. The standard t-statistic rejects the null that these two point estimates are equal at any conventional significance level (t-statistic = -23.38). The difference between the estimated sizes of menu costs reveals the up-ward bias associated with the identification under the assumption of the monopolistic competition model. Moreover, this result implies that oligopolistic strategic interactions explain some part of price rigidity, which would be captured by menu costs if a researcher estimates the model under monopolistic competition model. Table 3.8 compares the results of this chapter with those of the previous studies. While products and empirical strategies are different, these estimates are comparable since they are measured in the same unit: the U.S. dollars. In addition, all the estimates measure adjustment costs in grocery stores. The first row of the table shows the result of my benchmark specification with binary choice and oligopolistic competitions, but with-out controlling for downward bias due to unobserved promotional activities. My point estimate of menu costs, 1.009, is close to the result under the assumption of symmetric menu costs by Aguirregabiria (1999), 1.117, but much smaller than that of Slade (1998), 2.55.41 It is not surprising that the estimate of this chapter is greater than the direct measure of menu costs calculated by Levy et al. (1997), 0.52, because the estimate could 4 1 The result of Aguirregabiria (1999) 1.117 is calculated from the estimated value of menu costs, 72.62, in the specification (2) in Table 5 and the number of stores in a supermarket chain, 62, in Aguirregabiria (1999). capture anything associated with price changes, whereas that reported by Levy et al. (1997) includes only physical and labor costs of price changes. The second row shows the estimate of menu cost based on the oligopoly model in this chapter, which is close to the estimated size by Slade (1998). The estimated size is also close to the one obtained by Aguirregabiria (1999) with asymmetric menu costs. As before, the estimated menu cost is much greater than that of Levy et al. (1997). This comparison suggests that dynamic discrete choice models yield similar results in identification of the size of menu costs. My result, 7.56 percent, is, however, much greater than that by Aguirregabiria (1999) in terms of the percentage of menu costs in revenues while it is closer to that by Slade (1998). Note that Aguirregabiria (1999) estimates menu costs using 534 brands in various products while Slade (1998) examines a single product as in this chapter. This comparison suggests that the menu costs might be relatively uniform across products, and that the large percentage of my estimate in terms of revenues simply reflects the small revenues generated by graham crackers. Therefore, across different products, menu costs might be more comparable in magnitude under the same currency unit rather than in terms of percentage in revenues as long as retail-grocery products are concerned. The estimated size of menu costs is also significant in comparison with previous theoretical studies. Although I do not emphasize the implication of the estimated size of menu costs from the single product, the comparison between my estimate and the size of menu costs appearing in the past theoretical studies in macroeconomics would help highlight the importance of the estimated size of menu costs. Under a general equilibrium model with monopolistic competitions and menu costs, Blanchard and Kiyotaki(1987) calculate the size of menu costs that suffices to prevent firms from adjusting their prices. The calculated size of menu costs is 0.08 percent of revenue. The subsequent studies in macroeconomics consider the size of 0.5 - 0.7 percent of revenue to be reasonable, and to have significant impact on price adjustments. For example, using a monopolistically competitive model, Ball and Romer (1990) show that the cost needed to prevent price adjustment to a monetary shock is 0.7 percent of revenue under the reasonable values of mark-up and labor supply elasticity.42 Golosov and Lucas (2006) use the value of menu costs of 0.5 percent of revenues in their calibration showing their state-dependent model explains the observed correlation between inflation rates and frequency of price changes in past studies well. My estimate, more than 7 percent in revenue, is considerably greater than the values appearing in these theoretical studies. It is, however, worth noting that the estimate in Aguirregabiria (1999), which is similar to my estimate in terms of a nominal value, is just 0.7 percent in revenue. As mentioned before, the large menu costs in my estimate in revenue could result from small sales in graham crackers. Therefore, if we conduct the analysis using various products, the estimated size in terms of revenue could be smaller than the result using only graham crackers. In order to verify this statement, it is, however, necessary to conduct the analysis with various products. At 4 2 The mark-up is 15 percent of revenue, and the labor elasticity is 0.15. this stage, I left the empirical exercise using other products as a future study. The results from graham crackers show that the size of estimated size of menu costs is great enough to have significant effects on price adjustments. Therefore, I conclude that menu costs have significant implication for price adjustment behavior economically as well as statistically. In addition, this chapter has shown that strategic interactions could induce rigidity in a tight oligopolistic market. This result implies an important conclusion in this chapter: not only menu costs but strategic interactions among manufacturers are important for explaining the observed degree of price rigidity. 3.6 Conclusion This chapter studies weekly price movements of a typical product sold in retail stores, graham crackers. As observed commonly in retail price data, the price movements of the product are well characterized by frequent discrete jumps. To explain the discreteness of price changes, I employ a dynamic discrete-choice model with menu costs as the hy-pothesized data-generating process. Since the market of graham crackers are dominated by a few manufactures, I further assume oligopolistic competition to reflect this market structure, and examine possible effects of oligopolistic strategic interactions among man-ufacturers on the discrete behavior of my price. I estimate this dynamic discrete-choice model with oligopolistic competition by exploiting a recent development in the estima-tion of dynamic discrete choice games, the NPL estimator. The results show that menu costs are important statistically and economically. However, I claim that adopting the conventional estimators in explaining my price data could lead to two possible biases in the estimate of menu costs. The first bias is downward and due to unobserved pro-motional activities. If a promotional activity is profit-enhancing, the estimates without controlling this factor result in a downward bias. The results of this chapter show that correcting this bias is important for a precise inference on menu costs. The second source of a bias in conventional estimators is their assumption of monopolistic competition. If strategic interactions among manufacturers affect the pricing behavior in the sample, the estimated menu costs with a monopolistic competition model is biased upwards because strategic interactions in an oligopolistic competition potentially create price rigidity, the estimate in the conventional estimator is biased upwards. The results show that the es-timate of menu costs under oligopolistic market is smaller than and statistically different from that under monopolistic competition. This means that oligopolistic competitions explain some part of price rigidity, which is captured by menu costs unless a researcher incorporates oligopolistic strategic interactions. Thus, at least in the sample of this chap-ter, I conclude that oligopolistic strategic interactions could be an important source of price rigidity. Table 3.1: Market Shares of Graham Crackers Manufacturer size of a box share in four brands (%) share in three brands (%) Sarelno 16 oz 16.78 23.06 Keebler 15 oz 20.24 27.83 Nabisco 16 oz 35.72 49.11 Dominick's 16 oz 27.26 — -* Shares in total sales in 21 stores. Table 3.2: Summary Statistics of Variables Mean Std.Dev. Min Max Quantity (box) 11.29 11.45 1 370 Price ($ U.S.) 2.52 0.28 1.35 3.09 Cost ($ U.S.) 1.8 0.20 1.17 2.21 Table 3.3: Summary Statistics of Price Changes NOB Mean of |Ap| Mean of price Mean of Aq Ap = 0 14800 (67.3 %) 0 2.54 0 Ap ± 0 7178 (32.6 %) 0.28 2.47 Ap < 0 3390 (15.4 %) 0.28 2.33 7.65 Ap > 0 3788 (17.2 %) 0.27 2.61 -6.29 Table 3.4: Estimated Demand Equation Variable* Coefficient Standard error** constant 1.462 0.036 P -3.782 0.116 rp 0.819 0.106 dp 0.006 0.028 bonus 0.013 0.009 bundle 0.003 0.069 durb -0.016 0.003 durd 8:52e-6 0.0002 cc 0.836 0.036 brand2 (Keebler) 0.19 0.008 brandS (Nabisco) 0.471 0.028 * The store-level fixed effects and holiday-dummy variables are also included but not reported. ** White's heteroscedasticity-robust standard errors Table 3.5: State Variables (Discretized Values) Variable State 1 State 2 dt 3.015 4.048 ln(pu) 0.2601 0.3784 ln(p21) 0.2765 0.4133 ln(p3t) 0.2765 0.4298 ln(cit) 0.1086 0.1893 In (c 2t) 0.1475 0.2653 ln(c3t) 0.2085 0.2824 Table 3.6: Estimated Menu Costs Variable Estimate S.E.* 7 1.009 0.025 Results at K = 8 Log-likelihood = -18780 *The standard errors are based on 10000 non-parametric bootstrapping re-samples. Table 3.7: Estimated Menu Costs and Fixed Costs of Downward Price-Changes (a) Oligopoly model Variable Estimate S.E. 7 2.578 0.026 A 2.840 0.028 Results at K = 30 Log-likelihood = -21910 (b) Monopolistic competition model Variable Estimate S.E. 7 3.443 0.037 A 3.870 0.040 Results at K = 10 Log-likelihood = -24521 *The standard errors are based on 10000 non-parametric bootstrapping re-samples. Table 3.8: Menu Costs in Previous Studies Menu costs % in revenues this study (1) Ap^O 1.009 2.96 % (2) Ap^O 2.578 7.57 % Levy et al. (1997) 0.52 0.70 % 1.33 0.72 % Slade(1998) Ap^O 2.55 (5.11 %) f Aguirregabiria(1999) (1) Ap^O (1.117) | Ap ^ 0 (3-06)11 0.7 % Ap > 0 2.23 0.31 % Ap < 0 0.83 0.39 % | The value is calculated from Table IA and VB as the share-weighted average, j The value is calculated from the result of specification (2) in Table 5 and the number of stores, jf The value is calculated by the author from Table 6 according to / { A P > 0} + I{AP < 0} = / { A P ± 0}. Figure 3.1: Shelf Prices of Three National Brands 3 5 1 15 29 tt 57 71 65 S9 113 127 141 155 1 ® 183 197 211 225 239 253 267 281 29S 309 323 337 351 365 379 393 WM* Sareino mmm* KflftWflf Nabtsco Chapter 4 Estimation of Dynamic Discrete Choice Games: a Survey 4.1 Introduction The importance of estimating structural parameters in empirical dynamic games has been widely recognized in the literature of industrial organization. Especially, struc-tural estimation is indispensable to recover many important but usually unobservable variables such as marginal costs, sunk costs of entry, and fixed costs of changing prices. Major obstacle for the estimation of empirical games was the indeterminacy problem due to potential existence of multiple equilibriums and the computational costs in solving dynamic programming problems. Recently, several estimators have been developed for estimating a class of dynamic discrete choice games. The object of this chapter is to survey these recently developed methods: (i) the Pseudo-Likelihood (PML) estimator, (ii) the Nested Pseudo-Likelihood (NPL) estimator by Aguirregabiria and Mira (2006), and (iii) the minimum-^2 estimator and the method of moments estimator proposed by Pakes et al. (2005), and (iv) the asymptotic weighted least square estimator by Pesendorfer and Schmidt-Dengler (2004). Importantly, these estimators could overcome the problems of multiple equilibriums and computational costs. The computational costs of these four estimators are significantly smaller compared to conventional structural estimation methods since they avoid solving dynamic pro-gramming problems. The estimation algorithms consist of the following "two-steps": 1. The transition probabilities and choice probabilities are estimated from the data. The continuation values are also recovered from these estimates. 2. The parameters in the payoff functions are estimated. This two-step estimation procedure is originally employed by conditional choice proba-bility (CCP) estimator developed by Hotz and Miller (1993). The CCP estimator is the seminal two-step approach that first appeared in the literature to estimate the structural parameters in a single-agent dynamic discrete choice model. The CCP estimator greatly reduces computational costs inevitable in a conventional estimation procedure that re-quires solving a dynamic programming problem numerically for each step of iterations in the estimation process of structural parameters. Given the continuation values recovered from the data, the computational burden of the CCP is no more than that of the static discrete choice model. The CCP estimator is a building block of the two-step estimators for dynamic discrete choice games. Also, the Nested Fixed Point (NFXP) algorithm by Rust (1987) is another important estimator in the development of the estimators stated above. This chapter reviews NFXP and CCP estimator as well as the four estimators in detail. In the following, I first describe the framework of a single agent dynamic discrete choice model and introduce two important estimators, the Nested Fixed Point (NFXP) algorithm by Rust (1987) and the conditional choice probability (CCP) estimator by Hotz and Miller (1993). Then, I extend the basic framework to a multiple agent model and review the four different estimators mentioned above. After discussing the recent results of Monte Carlo studies on the statistical performances of the estimators, I conclude. 4.2 A single-agent discrete choice model As mentioned in the introduction, all the two-step estimators in dynamic discrete games surveyed in this chapter are based on the development of two-step estimators to reduce computational burden in single agent dynamic discrete choice models.1 In the following, I provide a simple framework of a single agent dynamic discrete choice model, and describe 1For the conventional estimators of a dynamic discrete choice models, see Eckstein and Wolpin (1989) and Miller (1997). two important contributions, Rust (1987) and Hotz and Miller (1993), which are building blocks of the two-step estimators in dynamic discrete games. 4.2.1 The basic model Consider a problem of an agent who sequentially chooses an action at £ A to maximize his/her payoff for periods t = 1,..., oo. The space of actions at, A, is discrete with the J actions that are mutually exclusive. The economic condition the agent faces at period t is defined by the vectors of state variables, xt £ X and et £ The state space of xt, X, is discrete with a finite dimension M. While xt is observable to the agent and a researcher, £t is observable only to the agent but not to the researcher. The vector of the unobservable state variables, et consists of as many elements as J. State variables (x t,£t) are assumed to follow a first-order controlled Markov process, whose transition probability is described by f(x t+i, £t+i\x t, et, at). The rational expectation is employed as an underlying identification assumption: the transition probability is not only the beliefs of the agent but also the true law of motions of the state variables. Conditional on the state variables, the agent chooses his/her actions sequentially to maximize the discounted expected sum of payoffs where ir(a s, xs, £s) is the current period payoff associated with action as conditional on the state (xs, e s) and (3 £ (0,1) is a constant subjective discount factor, respectively. The oo (4.1) s=t agent forms his/her expectation with respect to the future values of the state variables. The Bellman equation of this problem is V(x t,et) = max{7f(a t,xt,£ t) + pV(x t+1,£t+1)f(x t+1,£t+1\xt)£t,at)}. (4.2) ateA The optimal choice of the agent at period t, a*t , is to choose the action at such that al = argmaxVr(xt,et). (4-3) ateA The optimal decision rule al = a*(x t, et) is a deterministic function from the view point of the agent. It is, however, a stochastic rule from the viewpoint of the researcher because of the unobservable component of the state variables. 4.2.2 Estimation methods The objective of the researcher is to draw inferences on the primitives in the model — the structural parameters in the current profit function, policy function, and the transition probabilities - from the distributions of data, which are considered to be generated from the model and the equilibrium conditions. The value of the discount factor is assumed to be a certain value a priori. In the following, I review two important estimators for recent developments in the two-step estimation approach in dynamic discrete choice games: the nested fixed point es-timator (NFXP) by Rust (1987) and the conditional choice probability estimator (CCP) by Hotz and Miller (1993). They impose the following assumptions. Assumption 1: The current period payoff is additively separable with respect to the observable term and unobservable term: Assumption 2: The transition probability function satisfies the conditional indepen-dence (Rust 1987): where f(x t+i\xt, at) is the transition probability of the observable state variables xt, and g(e t+i\xt) is the continuous distribution function of the vector of the unobservable, e. The conditional independence assumption implies that the unobservable state variable et is serially uncorrelated and that the next-period observable state variable is conditional on only the current-period observable state variables and control variables. Under this assumption, an agent bases her decision on the current observable state variables and the current unobservable state variables. However, since et is assumed to be conditional on the observable state variables, a researcher can predict the decision rule of an agent based only on the information of the observable state variables: the unobservable state variable is basically noise. This assumption is restrictive but crucial in deriving an empirically tractable estimators and facilitating empirical analysis. First, it avoids the need of multiple integrations over the serially correlated unobservable state variables. Second, the conditional independence assumption enables a researcher to estimate the transition jr(a t,xt,£t) = n(a t,xt) + et. (4.4) f(x t+1,£t+1\xt,£t,at) = f{x t+1\xt, at)g(£ t+1\xt) (4.5) probabilities of observable state variables separately from estimating parameters in the current payoff function. 2 In addition, these two assumptions are crucial for a researcher to integrate out the unobservable state variable from the decision rule of an agent, and to construct a system of conditional choice probabilities P(a\x t). Given the above two assumptions, the Bellman equation in terms of observable state variables is defined as V{x t)= / max{7r(a t,xi) + et + (3V(x t+l)f(x t+l\xt,at)}dg(£ t\xt)) (4.6) J ateA where V(x t) = f V(x t, £t)dg(e t\xt). In order to construct maximum likelihood functions, it is necessary to impose an distributional assumption on the unobservable. Assumption 3: e is identically independently distributed across alternatives and times. For example, a researcher can assume that e follows an i.i.d. extreme value distribu-tion 3 : g(e\x) = JJ exp{—£a + v) exp { - exp{ -e a + u}}. (4.7) ateA 2Rust (1994b) discusses that the conditional independence assumption is testable: under the null hypothesis that the conditional independence is valid, the past actions of agents have no effect on the current-period actions. Therefore, one can test the conditional independence by additively including a function of past actions in the payoff functions and examining the statistical significance of the coefficient associated with the function. 3This distribution is also called Type I extreme value distribution and Gumbel distribution where v is a Euler's constant, 0.557. The assumption of i.i.d. extreme-value distribution is useful since the probability that an alternative a is chosen conditional on xt, P(a\x t), has the following closed form: P( a\xt) = „ (4.8) The i.i.d. assumption is important in facilitating empirical analysis. Regarding the question of whether the assumption of the i.i.d. extreme value distribution is appropriate or not, the standard argument such as the independence from irrelevant alternatives in static discrete choice models applies. This assumption is also crucial in development of the two-step estimators, and will be discussed later. The NFXP The NFXP consists of two algorithms: the outer algorithm that numerically calculates continuation values using standard value-function iterations and the inner algorithm that updates the estimates of the structural parameters by the Maximum Likelihood estimation given the continuation values calculated in the first step. The NFXP is important for the estimation of dynamic discrete choice games since all estimators developed recently employ the idea of the algorithms in the NEXP and the assumptions stated above, some of which are originally imposed for NFXP. In addition, a result from the NFXP is efficient providing a benchmark to other estimators. Ackerberg, Benkard, Berry and Pakes (2005) emphasize that the efficiency of the NFXP comes from the fact that the value functions do not contain sampling errors since they are computed numerically as in a standard dynamic programming problem. This contrasts to the two-step approaches following the NEXP, which use the information from data to derive the continuation values in single-agent models as well as multiple-agent models. Yet, the computational cost in the NEXP is still large because the vector of the continuation values must be obtained by value-function iterations for each trial value of the vector of the structural parameters. This means that the researcher needs to solve the dynamic programming problems by value function iterations for each step of iterations to get a convergence in the Maximum Likelihood estimation stage. Furthermore, the extension of the NFXP to a multi-agent model exacerbates the curse of dimensionality since the number of states at which value functions are evaluated increases exponentially as the number of players increases.4 The CCP A major breakthrough affecting the recent developments in estimators for dynamic dis-crete choice games is the CCP by Hotz and Miller (1993). The CCP estimator avoids the computation of value functions for each step of iterations in the estimation of the vector of parameters. Instead of numerically solving the fixed points for each trial set 4There are few papers that actually use the NFXP in multiple agent settings. The exceptions include Seim (2002) who studies a two-stage entry-exit game and Gowrisankaran and Town (1997) who combine the NEXP with the algorithm by Pakes and McGuire (1994). of structural parameters, the continuation values consistent with transition probabilities and choice probabilities implied by the data are calculated directly from the data. When the continuation value can be expressed as a linear function of the vector of parameters, the calculated continuation values are fixed during the estimation of the structural pa-rameters. This procedure greatly reduces the computational burden caused by the fixed point problems in the NEXP. The main difficulty in the CCP is that the estimator could be inefficient compared to that of the NFXP since the calculated continuation values are potentially subject to large sampling errors. The CCP estimator exploits the one-to-one mapping between the continuation values and the choice probabilities. The choice probabilities can be first estimated nonparamet-rically. The continuation values are then backed out from the estimated choice probabil-ities by solving a linear system of equations. Under the assumptions of the conditional independence and the additive separability, let the choice specific conditional value be defined by va(x t,£t) = n(a,x t) + pE[V(x t+1,£t+1)f(x t+1\xt,at)}. (4.9) The conditional choice probability for the agent to choose action a is the probability of the following condition to hold: Prob(a\x t) = Prob(v a(x t) + £at > va'(x t) + e?) (4.10) for all the alternatives a' ^ a, similarly to static discrete choice models. The inversion theorem developed by Hotz and Miller (1993) shows that, for any continuous and strictly monotonic distribution of the unobservable, there is a mapping from the conditional choice probabilities to the difference between the choice-specific value functions. In the cases with commonly used distributions such as logit and normal, the mappings can be expressed in simple closed-form presentations as follows: va(x t) — va'(x t) = hi(Prob(a\x)) — ln(Prob(a'\x)) for logit distribution with binary c^Scfe) va(x t) — va'(x t) = Q~1(Prob(a\x)) for normal distribution with binary choice (4-12) where $ is a cumulative distribution function of the standard normal distribution. These presentations are used to represent the expected value of the disturbance in terms of conditional choice probabilities conditional on that the choice a is optimal. E[e a\xt,a*t = a] = E[ £a\£a-£a' >va'{x t)-va(x t)} (4.13) = E[£ a\£a > £a' +va\xt)-V a{x t)) (4.14) = E[£ a\£a > £a' - \n(Prob(a\x)) + ln(Prob(a'\x))} (4.15) = v - ln(Prob(a\x)) (4.16) where u is Euler's constant. The above is for a binary choice model with logit distribu-tion when the location parameter of the distribution is zero. The equivalent expression for the standard normal is A($_1(Pro6(a|x))) where A is the inverse Mill's ratio. The continuation value is obtained by inserting these presentations and conditional choice probabilities into the Bellman equation.5 There could be an alternative way to compute the continuation values. Hotz, Miller, Sanders and The potential difficulties in the CCP estimator are the following. First, the estimator could be inefficient relative to the NFXP estimator. Aguirregabiria and Mira (2002), however, show that the efficiency of the CCP estimator could be improved by iterating the CCP estimation. Second, since the assumptions of the conditional independence and the additive separability of the unobservable state variable are essential in the CCP estimator, these maintained assumptions make the CCP inappropriate to be applied to many economic models that require the serially correlated unobservable. If the model of interest is Markov stationary, the computational costs could be further reduced. Let V, p(a), 7r(a), ea and F denote the vectors of continuation values, condi-tional choice probabilities, payoffs associated with choice a, conditional expectation of private information with choice a expressed in terms of conditional choice probabilities (equation 4.13) and the matrix of conditional transition probabilities, respectively.6 The vector of continuation values V{x) could be then calculated from the following mapping, V = [I- m~l £ > ( < 0 * br(a) + ea] (4-17) a where * is the element-by-element multiplication operator. The above presentation is shown by Aguirregabiria (1993), Rust (1994b), and Miller (1997). Showing that the value function can be expressed by a linear function with respect to current profit and Smith (1994) exploit the estimated choice probabilities to simulate synthetic data and use the average of the discounted expected payoffs calculated with those simulated data as the continuation values. 6In the case where its parameters in Markov stationary environment, this expression is also the key to the estimators of dynamic discrete choice models. All the estimators surveyed in this chapter assume that the game environment is Markov stationary. Applying the above representation saves computational time to a great extent. Applications and Practical Issues It is worth noting that a part of parameters could be estimated before one conducts NFXP or CCP estimation. For example, analyzing a price-change problem, Slade (1998) first estimates the demand function, and then adjustment costs of price changes using the CCP estimator. Also, Aguirregabiria (1999) estimate the demand function first, and estimate fixed ordering costs and menu costs using CCP estimator. Estimating parameters outside of the NFXP/CCP estimation greatly reduces the computational costs. Whether or not a certain parameter could be estimated depends on the model. The parameters could be estimated outside the NFXP/CCP as long as the identification of the parameter depends on the value function. 4.3 A multiple-agent discrete choice model In this section, I review a multiple agent model and estimation methods. 4.3.1 The basic model Consider an oligopolistic market where firms i = 1,...,N competing for periods t = 1,..., oo. At the beginning of period t, the state of a firm i is characterized by a pair of state variables (xu, £u)- The state space of xt — (xi t, ...,xN t), X, is a discrete space with a finite dimension M N. eit consists of the elements contingent on the choice set of firm % but not on those of the other firms. The space of private information is spanned in the real line, Let at = (an, ...,ajvt) be the vector of the actions the firms could take at period t. The space of possible actions ait, A, is also a discrete space and contingent on xt. Let J denote the number of available actions in each period. In each period, firms simultaneously choose their actions. Conditional on the state variables, firm i chooses her actions sequentially to maximize the discounted expected sum of profits: where 7Tj(a;s, a_ i s , xs, eis) is the current period's payoff of firm i, and (3 £ (0,1) is the constant discount factor. Firm % forms her expectation with respect to the future values of the state variables and the actions taken by the other firms in the current and future periods. Note that the current period's profit depends on the actions of the other firms as well as her own action.7 7Pakes et al. (2005) consider a special case of the games discussed in this section. In their entry-exit oo (4.18) s=t All the estimators discussed in this chapter relies on the following assumptions. Assumption 1: Additive separability in the profit function: a current period's profit function is additively separable with respect to commonly observable variables and unob-servable variables: TTi(a it, a_it, xu eit) = 1fj(a it, a_it, xt) + eit{a it). Assumption 2: Incomplete information: The vector of state variables xt is observable to all the players and the researcher. eit is, however, observable only by firm i. The other firms, jointly indexed by —i, and a researcher do not observe £u but only know its distribution function, g{£u\xit). Under the assumption of incomplete information, the other firms and a researcher have the same information set with respect to the state variables of firm i. This means that the choice probability of a firm to take an action is perceived to be the same by the researcher and the other firms. The strategy taken by the other firms are expressed in terms of choice probabilities conditional on only the commonly observable state variables. In addition, the assumption of incomplete information helps a Markov perfect equilibrium in a dynamic discrete choice exist as it enables the equilibrium to be represented either in the probability space or in the value-function space. For example, Doraszelski and Satterthwaite (2005) show that the existence of equilibrium is guaranteed in the game with pure-strategy examined by Ericson and Pakes (1995) if the incomplete information model, the current period's payoff of a firm does not depend on the actions of the other firms in the current period. is allowed. 8 Assumption 3: The transition probability function satisfies the conditional indepen-dence: Assume that the vector of the state variables (x t, £u) follows the follwoing first-order Markov process, whose transition probability is described by f(x t+1,£it+1\xt,£it,at) = f(x t+i\xt,at)g(£ it+1\xt+1), (4.19) where f(x t+i\xt,at) is the transition probabilities of the commonly observable state vari-ables xt, and g(£ it+i\xt+i) is the continuous distribution function of the vector of private information £u+\-This is the conditional independence assumption for multiple-agent models. This as-sumption implies that the actions taken by the firms could influence the evolution of the commonly observable state variables but not on the unobservable state variable £u+i-Moreover, the commonly observable state variables are sufficient to predict the decision rule of players. Without this assumption, the past decision of players could affect the decisions of players. This causes computational difficulty making state space too large for even a simple problem. Assumption 4: i.i.d. private information In the multiple-agent model, the private information is assumed to be i.i.d. across firms as well as times and alternatives. The i.i.d. private information implies that the choice 8Analyzing the game of complete information is more difficult. See Cilberto and Tamer(2003) for example. of a firm is not correlated with those of the other firms through unobservable. Under this assumption, an agent and a researcher can construct the expectation with respect to actions taken by the other agents simply by the product of choice probabilities of the other firms. If this assumption is not imposed, the private information of a firm gives some information of the private information of the other firms. In this case, the computation of the choice probabilities becomes complex. In addition, it is assumed that The environment of the game is Markov-stationary so that the payoffs, the transition probabilities, and the decisions depend only on the state variables not on period. Hence, I omit the time subscript hereafter, and use (x, e) and (x',e') to denote the state variables in the current and next periods, respectively. Let us define the value function of firm i facing state (x, £*) by Vi(x, £*), which is the value of firm i when she behaves optimally now and in future. Then, given the actions of the other firms, a_j, firm €s problem is to choose action such that ai = argmax£{7fj(aj,a_j,x,£j) + f3 / f(x'\x,ai,a-i)V i(x',£' i)dg(e' i\x')}. (4.20) me A J Assume that the firms follow a Markov strategy. Let an arbitrary strategy of firm i be denoted by <7j = The vector of the strategies of all the firms is cr = cr(x,£i, ...,£N). Let / { • } be an indicator function which takes one when the statement inside the bracket is true. Conditional on strategies cr, I define the conditional choice probability of action a;, Pa(aj|x), by Pficiilx) = Prob(ai(x,£i) = a^x) = J /(cr^x,^) = ai)dg{£i\x). Suppose that the other firms —i follow strategies cr. Let Vf(x,£i) be the value function of firm % when it behaves optimally now and in future given strategy cr; that is to say, V?(x,ei) = vajaxE{nf(a i,x,£i)+13 / /a(x'|x, a^V^x', ^dg^x')}, (4.21) aiEA I where 7T? a-i / 7Ti(ai,a-i,x,£i), (4.22) and r(x'|x,a,) = > ] ( \ \Pna j\x) )f(x'\x,a l,ai), (4.23) i) e (n pj( ai x)) /(xix' by the assumption of i.i.d. private information. Let Vi(x) denote the integrated value function under strategies cr V i°{x) = J V^(x,£i)dg(ei\x). (4.24) With the additive separability of the payoff function, the integrated Bellman equation is then given as Vf{x) = [ max[n°(a ux) + £l(a i) + pyjr(x'\x,a i)V ia(x')]dg(£ i\x). (4.25) A stationary Markov-perfect equilibrium in this problem is defined by a set of strate-gies such that each strategy taken by each firm consists of the best response to the strategies of the other firms. In other words, the vector of strategies, a*, is a Markov-perfect equilibrium strategy set if a*(x, £i) = arg max{^f (a u x) + el(a i) [x'\x, ajVf (x')}, (4-26) CF ; (= A * * Let Pa* denote the conditional choice probability set in a Markov-perfect equilibrium, which is given by Recall that the current profit function nf (a*, x) and the transition of the commonly ob-servable state variables f a(x'\x,a,i) depend on the strategies of the firms only through the conditional choice probabilities Pa. The same is the case for the integrated value function Vf(x). This means that the above equation of the conditional choice probabil-ity Pf*(ai\x) for all % constructs a mapping Pa* = A(P a*). Hence, the Markov-perfect equilibrium conditional choice probability Pa* is a fixed point of the mapping A which is called the best response probability. Since the best response probability is continuous, 9The existence of the equilibrium in this class of the problem is first suggested by Rust (1994a) under the assumptions of additive separability and conditional independence. Rust (1994a) argues that the equilibrium can be obtained as a solution of a coupled-fixed point problem that is solved for the fixed points of equilibrium choice probabilities and value functions. for any firm i and any state (x, £i). 9 (4.27) there exists at least a fixed point by Brower's fixed point theorem. Notice that the Bell-man equation (4.25) is a function of Pa*. The Markov-perfect equilibrium is, therefore, given by the solution of a coupled-fixed point problem consisting of the Bellman equation (4.25) and the best response probability PCT* = A(P<7*). In order to obtain the equilibrium value functions and the equilibrium choice proba-bilities, a researcher needs to solve N Bellman equations. To avoid solving N dynamic programming problems, all the two-step approaches discussed in this survey provide al-ternative presentations of equilibrium conditions either in the space of the conditional choice probabilities or the expected discounted sum of payoffs. For example, Aguirre-gabiria and Mira (2006) and Pesendorfer and Schmidt-Dengler (2004) present equilibrium conditions in the space of choice probabilities, whereas Pesendorfer and Schmidt-Dengler (2003) exploit a representation of equilibrium conditions in the space of value functions. 10 1 0 Presenting the equilibrium condition in the space of value functions, Doraszelski and Satterthwaite (2005) show that there exists a pure-strategy Markov equilibrium in the class of game discussed in Ericson and Pakes (1995) by adding the private information. The original game discussed in Ericson and Pakes (1995) and Doraszelski and Satterthwaite (2005) permits both continuous and discrete choices. Ackerberg et al. (2005) state that typically it is possible to obtain equilibrium strategies and their associated values even when all the conditions are not satisfied using the algorithm to compute the Markov perfect equilibrium. Thus, if one obtains estimated parameters, the existence of the equilibrium could be confirmed using the algorithm to compute the equilibrium. If the convergence is obtained, the existence is confirmed. However, in general, the equilibrium for a given parameter need not be unique. Exploiting the Markov stationarity of the problem, Aguirregabiria and Mira (2006) obtain the value functions in probability space as follows. First, given the conditional choice probability in a Markov perfect equilibrium, Pa*, the Bellman equation (4.25) can be rewritten as (*) = E PF ^ to' + ( o i . + E f a * ^ ' i * ' ( 4 - 2 8 ) at x' where ea* [a u x) is defined by ea* (a*, x) = E[£i(a,i)\x, a, = a*(x,£;)] (4-29) Let vectors V°*, P?*(a,i), e f (a^), and 7rf (a,), and matrix f a*(ai) be defined by i f E ^ ^ - Y M ' (4-30) Pf io i ) = [i?*(a i|x1) . . - i?*(a i|xM)] , > (4.31) e f (a i ) = [ e f (a»> • • • e f xm)]', (4.32) (a*) = [<*(ai,xi) • • • irf (a u xM)}', (4.33) r \ o i ) = {f'ix'jlxuOi)} for i,j = 1 • •. Af. (4.34) Then the Bellman equation (4.28) results in the following vector-value equation [/ - {5F°*\Vf = Y> a^APf(a x) * [<* (aO + e f (a,)] (4.35) where F a* = J2 a ( ai)f a* ( ai) a n d * the element-by-element multiplication operator, respectively. The above vector-value equation solves each continuation value V?* (xj) for firm i and state Xj. This fact leads to significant reduction of the computational costs of the problem. Let Tj be the solution of this linear equation system such that V?*(x) = T i(x;P a"). For arbitrary Pa, the operator Ti(x; Pa) gives the continuation value for firm i when all the firms behave according to Pa.11 Then, the best response mapping Pa) = {^(a^x; Pa)} is tyi(a i\x;P a) = / I(a.i ^argma,x{nf(a i,x)+£ i(a i)+py^f a{x'\x,a i)T i(x-,P' 7)})dg(£ i\x). } meA *—* X 1 (4.36) The Markov-perfect equilibrium of this problem is also characterized as the fixed point of the mapping Pa* = 3>{P a*). Given these presentation, one can estimate the parameters as follows: 1. First, obtain the estimates of the transition probability and choice probabilities. The distribution function of the private information is also imposed. 2. Obtain the continuation values using the above presentation. 3. Given the continuation values, the choice probabilities implied by the model is derived. Then, the parameter can be estimated. As stated in the end of the last section, some parameters could be either estimated 1 1 See Aguirregabiria and Mira (2006). beforehand or jointly estimated in the step 3 in the above process. For example, in an entry/exit model, the parameters that are independent of sunk costs/scrap values could be estimated beforehand. 4.3.2 Multiple equilibria and the identification problem An important caveat to the above solution of the multiple-agent model is that the equi-librium of the dynamic discrete choice problem needs not be unique. Especially when the best response functions are non-linear with respect to the actions taken by other players, the multiple equilibriums could emerge. This property of equilibrium leads to identifi-cation problems when estimating the parameters of the payoff function. As discussed in an earlier version of Aguirregabira and Mira (2006), the uniqueness of the equilibrium is neither a necessary nor a sufficient condition for the identification of the parameters. However, in order to estimate the parameters using a conventional estimator such as the Maximum Likelihood estimator, it is necessary to know which equilibrium is played in each data points; i.e., a researcher has to observe an equilibrium-selection mechanism in a game. Without the knowledge of an equilibrium-selection mechanism, which is generally unobservable, the conventional Maximum Likelihood estimation is infeasible. The two-step approaches surveyed in this chapter presume that the data is generated from only one path of Markov perfect equilibriums, as in Pesendorfer and Schmidt-Dengler (2003, 2004) and Aguirregabiria and Mira (2006). Pakes et al. (2005), on the other hand, do not directly assume that the data is generated from one path of multiple equilibriums. Instead, they show that the assumption that the firms condition their beliefs only on the commonly observable state variables implies that there is only one equilibrium policy consistent with the data generating process. The assumption that the data are generated from one equilibrium path is, however, not testable in an empirical practice. Whether or not this assumption is satisfied depends on the nature of the data. Pakes et al. (2005) do not discuss the identification issue explicitly in their paper. They state that the identification could be problem-specific since "many of parameters determining behavior in dynamic games can be estimated without ever computing an equilibrium, and those parameters that remain depend on the nature of the problem and data availability". Given the above assumption that the data is generated from one equilibrium path is satisfied, Aguirregabiria and Mira (2006), and Pesendorfer and Schmidt-Dengler (2003, 2004) discuss the identification issue of the structural parameters is explicitly discussed. Pesendorfer and Schmidt-Dengler (2003, 2004) show that the necessary condition for identification is that the model includes at most as many parameters as the number of the best response mappings that characterize equilibrium conditions, N x M N x J. Then, the parameters are identified up to a scale factor as in static discrete'choice models. The assumption that the data is generated from only one path of MPE is likely to be satisfied when all the data are considered to be generated from the same MPE. Therefore, this assumption is more difficult to be satisfied with the cross-sectional data from many different markets than with those from a single market or markets with similar characteristics. This is because, as discussed in Pakes et al. (2005), the assumption implies that all the data in different markets should be generated from the same MPE without explicitly allowing the heterogeneity across markets. Aguirregabiria and Mira (2006) allow the heterogeneity across firms only when players' decisions are observed in all or most of the markets. For example, when an entry/exit game is examined using the data from many independent markets, the assumption is likely to be satisfied if the players are same across markets (e.g., observe the entry/exit decisions of MacDonald and Burger King in all markets in data). However, if the players are different in every markets (e.g., players consist of local restaurants and pubs), it is unlikely that every markets play according to a single strategy among possible multiple strategies. In this case, it would be better to estimate the game for each market separately. 4.3.3 Estimation methods This subsection describes the estimators of the two-step approaches, which are recently proposed by Pesendorfer and Schmidt-Dengler (2003), Aguirregabiria and Mira (2006), and Pakes et al. (2005), and Pesendorfer and Schmidt-Dengler (2004). The results of Monte Carlo studies conducted in these papers are also reported. Pseudo-maximum likelihood estimator Pesendorfer and Schmidt-Dengler (2003), Aguirregabiria and Mira (2006), and Pakes et al. (2005) develop pseudo-maximum likelihood estimators. Commonly in their estima-tors, the pseudo likelihood function is defined as T 1V QM(0, P) = E E l n P, E) (4.37) t=1 1=1 where P is an arbitrary vector of conditional choice probabilities that need not to be generated from equilibrium, and <3/ is an equilibrium mapping in the space of choice prob-ability. A pseudo-likelihood estimator is obtained by maximizing the above likelihood function with the consistent estimator of the true conditional choice probabilities, P°: OPML = argmaxQM{6, P°). (4.38) 0 Aguirregabiria and Mira (2006) show that the above pseudo-likelihood estimator is root-M consistent and asymptotically normal under regularity conditions. The estimator proposed by Pesendorfer and Schmidt-Dengler (2003) is also a pseudo-maximum likeli-hood estimator. In their model, they assume that the private information follows i.i.d. normal distribution. Aguirregabiria and Mira (2006), however, discuss the weakness of the above pseudo-likelihood estimator. There are two difficulties in this estimator. First, the asymptotic variances of the pseudo-likelihood estimates depend on the asymptotic variances of the estimates of conditional choice probabilities. Therefore, if the sampling errors in the first-stage estimation of conditional choice probabilities are large, the pseudo-likelihood estimator is inefficient. Second, if the first-stage nonparametric estimates of conditional choice probabilities are biased with finite sample, so is the second-stage estimate. Nested pseudo likelihood estimator To circumvent the problems of the pseudo-likelihood estimator, Aguirregabiria and Mira (2002) propose the nested pseudo-likelihood (NPL) estimator that iterates the procedure of the pseudo-likelihood estimation recursively in the following steps 1. First, obtain an initial guess of the conditional choice probability, P0, which need, not to be an consistent estimator. 2. Then, obtain the pseudo-likelihood estimate 0\ = argmax^ QM{Q, PO) 3. Then, update the conditional choice probabilities P\ = ^(#1, Po). 4. Repeat steps 2 and 3 until the convergence criterion in the choice probability, | PK — PK_i| < e, is satisfied at the Kth iteration where e is the tolerance. The fixed point of the NPL estimator is obtained as the limit of the above procedure. Under regularity conditions on the Jacobian matrix, the NPL is asymptotically more efficient than any two-step estimator regardless of the initial guess of the conditional choice probabilities, Po- The NPL estimator does not require that the initial guess of the choice probabilities should be consistent estimates since the choice probabilities are updated by the equilibrium conditions. Methods of moments and minimum x2 estimators To estimate the model of an entry-exit game, Pakes et al. (2005) propose several alterna-tive ways including a method of moments. They analyze a single-market model as well as a multiple market model. In their single market model, the same number of potential entrants enters to the market through time. The state variables consist of the number of firms active at the beginning of each period, nt, and the vector of profit shifters, zt. which follow a first order Markov process. Both the sunk costs for entry and the scrap values for exit follow i.i.d. distributions over time and firms. The sunk costs and the scrap values are private information. Observing the private information, the firms decide whether to enter, exit, or stay in the market. The firms condition their beliefs about the entry and exit of the other firms only on the commonly observable state variables, (n t,zt). As the environment is assumed to be Markov stationary, I drop time subscripts hereafter. Let if(n, z) be the profit of the incumbent, the scrap value, and k the entry costs. The Bellman equation of the incumbent firm is V(n, z) = 7T(n, z) + PE^[max{4>, VC(n, 2)}] (4.39) where VC is the continuation value defined below. The first component of the second term in the Bellman equation is the option value of exit, while the second component is that of staying in the market. Let e, x, f(n'\n, z, stay), g(z'\z) be the number of entrants, the number of exitors, the belief of the incumbent about n' = n + e — x conditional on the incumbent's staying in the market, and the belief of the incumbent about the evolution of z, respectively. Then, the continuation value of an incumbent conditional on the incumbent's staying in the market is VC(n,z) = ^V{n\z')f{n'\n,z,stay)g(z'\z) (4.40) e,x,z' In their setting, the current actions of the other firms do not affect the current period payoff of the firm i. 1 2 The decision rule of the incumbent is to stay in if < VC(n, z). Note that, to determine the continuation value, only the number of firms in the market and its evolution as well as the transition of the demand shifter matter but the identity of the firms (i.e., which firms exit, enter, and stay in) does not. Let M l be the matrix whose elements is f(n'\n, z, stay)g(z'\z). In the vector form, the continuation value can be written as VC = M l7T + / ? M % [ m a x { V C } } . (4.41) Given the distribution of and the current profit function which is known up to param-eter values, the unknowns are M l and VC in the above vector equation. Letting px(n, z) be the probability of exit given state (n, z) and assuming that the distribution of the 12Although this setting helps the computation simplify, one can extend the model so that the current actions of firms affect the payoffs of the other firms as done by Aguirregabiria and Mira (2006) and Pesendorfer and Schmidt-Dengler (2004). scrap value is exponential, it is the case that E^maxicj), VC(n, z)}] = [1 - px(n, z)]VC{n, z) + px(n, z)£4 VC(n, z)$A2) = [1 - px(n, z)]VC(n, z) + px(n, z)£ + px(n, z)VC{n, $.43) = VC{n,z) + £px(n,z) (4.44) where £ is the parameter of the exponential distribution. Then, VC = ATtt + pM^VC + Zpx] (4.45) oo T = 1 This leads to the closed-form expression of the continuation values of incumbent firm, VC(8) = [I- pM^M^ir + f3Zp x\ (4.47) Mi and px can be estimated from the data. Define T(n, z) be the set of periods when (nt, zt) = (n, z). Then, the empirical counterpart of px is px(n, z) = V - — ^ (4.48) ^ nl{(n t,zt) = (n,z)} and the empirical counterpart of the element of M l is M'{n', An,,) = y ' " " X > ) I { { n r 1 ' 7 f ) } f { ( y } = (4.49) t (n-x t)I{(n t,zt) = (n,z)} These estimates, which are called the empirical Markov matrices, are consistent when the number of observations such as (n t, zt) = (n, z) approaches infinity. M l is the weighted average of the actual transition with the number of the incumbents in each period in the corresponding state as the wight. Substituting these empirical probabilities into (4.47) yields the consistent estimate of VC: VC{6) = [I- PM^M^N + PTF]. (4.50) As in the mapping shown in the section of the basic model, the mapping of the con-tinuation value is linear in the payoff function. The problem of an entrant is formulated in the analogous manner. The estimator of Pakes et al. (2005) does not require to impute the probabilities of the states that are not visited in the data. Thus, the resulting con-tinuation values are the average of the discounted values of the returns of the incumbent firms. The term involving the probabilities is fixed during the estimation. In the estimation of the parameters of this model, they use the pseudo log-likelihood estimator, a pseudo minimum x 2 estimator, and a method of moments estimator. In particular, the pseudo minimum x 2 estimator minimizes the sum of the squared differ-ences in the entry and exit rates between the observed data and the prediction of the model for each state. On the other hand, the method of moments estimator minimizes the across-state average of the sum of the squared differences in the entry and exit rates between the observed data and the prediction of the model.13 1 3For the estimation of transition probabilities, they use the empirical Markov matrix, which is stated above, and the "structural" transition matrix, which generate the Markov matrix using the binomial or multinomial formula from the observed state-specific entry and exit probabilities. For the computation of the continuation values, they use the discounted sum of future profits discussed above, and those The variances of the estimated parameters depend on the estimates of the transi-tion matrices and choice probabilities. They propose to compute the variances of the parameters using a parametric bootstrap. Asymptotic least square estimator Pesendorfer and Schmidt-Dengler (2004) propose an estimator unifying all the estimators stated above by reinterpreting them as an asymptotic weighted least square estimator. They show that the differences in the existing estimators can be explained with the different weights. They then propose an asymptotically optimal weight. Their model is the same as the basic model of the multiple agent model described above. The equilibrium conditions are expressed in the space of conditional choice prob-abilities, PA = 9). For the first stage estimation of conditional choice proba-bilities and transition probabilities, either the sample frequency or the nonparametric Kernel estimation is used. Let PT and FT be auxiliary consistent estimators of the con-ditional choice probabilities and the transition probabilities. With a symmetric positive definite matrix WT, an asymptotic least square estimator 9 is in general given as the solution for the following minimizing problem: min[PT - V(P T, F T, D)]'W T[P T - V(P T, F T, 0)]. (4.51) 9 obtained as a solution of a single-agent NFXP, which is obtained solving the Bellman equaiton, 4.39. Extending the algorithm to multi-stage is always possible. They reinterpret the estimators discussed in this section as variants of the least square estimators with different weighting matrix. Assuming that the conditional choice prob-abilities are estimated as the sample frequency P, they show that the pseudo-likelihood estimator is equivalent to the least square estimator with the weighting matrix being the inverse of the covariance matrix of the conditional choice probability vector evaluated at (P T,F T,9). The K-stage nested pseudo likelihood estimator replaces P — ^(PT, FT, 9) with the following functions in each stage: P — , F T, 9) for k stage pseudo likelihood estimation. They also show that the method of moment estimators with two different empirical Markov transition matrix in Pakes et al. (2005) are equivalent to (1) the asymptotic least square estimator with diagonal elements in the weight matrix equal to one on the fixed finite states that are visited in the sample and zero otherwise, and (2) that with diagonal elements in the weight matrix equal to the inverse of the number of observations per state, again, on the fixed finite states. They show that the asymptotic efficiency of the weighted least square estimator depends on the derivatives of $ with respect to the choice probabilities and the transition probabilities, and the variance-covariance matrix of these auxiliary probabilities. They argue that most of the two-step estimators reviewed in this chapter are not in the class of the best asymptotic weighted least square estimator.14 1 4For example, the pseudo likelihood estimator and the nested pseudo likelihood estimator are not 4.4 Monte Carlo studies Aguirregabiria and Mira (2006), Pakes et al. (2005), and Pesendorfer and Schmidt-Dengler (2004) conduct Monte Carlo experiments to compare the statistical performances of the estimators. Aguirregabiria and Mira (2004) Taking an entry-exit game as an example, Aguirregabiria and Mira (2004) compare the performance of the two-stage pseudo-likelihood (PML) estimators (K = 1) with that of the NPL estimator. Six Monte Carlo experiments are conducted with different sets of parameters including the parameter reflecting the degree of the intensity of strategic interaction across firms. The state variable is a demand shifter such as market sizes. They consider 400 separate markets, each of which contains the state space with size 5. The results are from 1000 sample draws based on a set of equilibrium policy functions. The true transition probability is assumed to be known and fixed during the experiments. The first stage estimation of choice probabilities are calculated using sample frequency or evaluating the results of logit estimations. The benchmark case is the infeasible pseudo-likelihood estimation with true policy function. This is the case where a researcher knows efficient. The moment estimator of Pakes et al. (2005) is equivalent to the efficient asymptotic least square estimator when the parameters are exactly identified. The method of moments estimator of Hotz and Miller (1993) is an efficient, asymptotic least square estimator if the instrument is chosen appropriately. the true equilibrium-selection mechanism and the true conditional choice probabilities. They evaluate PML and NPL in terms of the mean value of the estimates, the standard deviation, and the mean squared error. They obtain the following resulsts: (i) the results of the pseudo-likelihood estimation depend on how the first stage estimation is conducted; (ii) the NPL always converges to the same values of the estimates regardless of which first stage estimates are used; (iii) there is a significant gain in using the NPL compared to the results of the estimates with K — 1 in terms of mean square error; (iV) the larger the degree of strategic interaction, the greater the gain of the NPL. Thus, they claim that the NPL estimator performs better than the two-step PML . Pakes, Ostrovsky, and Berry (2005) Pakes et al. (2005) also conduct Monte Carlo experiments to analyze the statistical properties of several alternative estimators based on two entry-exit games with Cournot competition. They construct seven artificial panel data sets with different sample sizes. The size of panel data varies with 5 and 15 periods and 50, 250, 500, and 1000 cross-sectional dimensions. One equilibrium is computed using the algorithm of Pakes and McGuire(1994) for each game and used to simulate synthetic data sets. A Cournot model with a linear demand is used in their single location example. The state variables are the number of active firms at the beginning of each period, the demand shifter such as the market size (population in a market), and the growth rate of market size where the growth rate follows a first-order Markov process. The entry fees follow a unimodal distribution function, and the scrap value is distributed with an exponential function. Their state space is with about 2700 points. They examine 12 different estimates from the combination of the estimation methods, i.e., the estimates of the first stage choice probabilities and the computational methods of continuation vales. The estimation methods they examine include the "simple" method of moments15, the pseudo-likelihood, and the pseudo minimim-x2 estimator. 1 6 For the estimates of the first stage probabilities, "structural" transition probabilities17 or empirical transition probabilities (px and M l shown above) are used. The first stage estimate of continuation values are computed either by a nested fixed point algorithm (the value function iteration) or a single matrix inversion (4.39). They examine each combination of these methods in terms of the performance and computational costs. Their Monte carlo experiments show the following results: (i) pseudo-likelihood esti-mator does not work well; (ii) the minimim x 2 estimator is also biased with a large vari-ance; (iii) the estimator with a simple moments of method works well. They claim that 1 5 The "simple" method of moments is meant to be the fitting between the mean over all observations of the choice probabilities predicted by the model with the data. 1 6 The pseudo minimim-x2 is the minimization of the sum of the squares of the difference between the empirical and the estimates of the state specific entry probabilities weighted by the inverse of the number of times that the data visited the state 1 7 The "structural" transition probabilities is constructed using the binomial formula from the esti-mates of entry and exit problem the fail of the pseudo-likelihood estimator is due to the fact that the pseudo-likelihood estimator assigns zero probability to the event that could happen in the data but the con-tinuation value is lower than the other option. They conclude that the "simple" method of moments does the best job in terms of parameter estimates and the corresponding standard errors. When the "simple" method of moments is used, the statistical performances are identical among the estimation procedures with different first-stage estimates of prob-abilities and continuation values. Using the nested fixed point calculation to calculate the continuation value improves the statistical performance quite marginally. Because the computational burden of the nested fixed point calculation is large, they conclude that the estimation with matrix inversion and empirical probabilities works best both in large and small sample sizes. The kernel smoothing of the continuation value improves the performance with small samples. They also examine the performance of a nested algorithm using the simple method of moments, structural transition, and nested fixed point estimates. They observe that the estimates of the parameters are oscillating and moving away from the equilibrium. They also point out the possibility that, during iterations, the multiple equilibria may not be ruled out due to the update of choice probabilities for each iteration. Pesendorfer and Schmit-Denger (2004) Pesendorfer and Schmidt-Dengler (2004) conduct Monte Carlo experiments for the least square estimators with four different weighting matrices: their efficient weighting matrix, an identity matrix, the inverse of the variance-covariance matrix of choice prob-abilities (the pseudo-likelihood), and its k-step iteration (k — 20). The model which generating the data is an entry-exit game with two firms. There are four states pre-sented by the firms' actions of whether to stay in market or not. Each firms receive the monopoly profit or duopoly profit at each stage. Unlike the other two studies which work with one equilibrium, they conduct experiments for three equilibria in the model. The choice probabilities are calculated using sample frequency. The sample sizes are 100, 1000, 10000, and 100000. According to their results, the efficient least square estimator does not work well with the small sample size 100. However, as the sample size rises, the performance of the efficient least square estimator is improved. The pseudo-likelihood (PML) estimator performs the second best. With the small sample size, the PML works better than the efficient least square does. The estimate with the identity matrix as weight is ranked the third. The estimate with the identity matrix has the largest standard error and the severest small sample bias. The performance of the k-PML (k=20) is the worst. Although the k-PML performs best in one of the equilibria, in the other equilibria, the values of estimated parameters do not converge to the true parameters. Their results show that the k-PML is unstable depending on the equilibria. They conclude that the efficient least square estimator works best with moderate and large sample sizes. 4.5 Concluding remarks and future research This chapter provides a detailed survey of recently proposed two-step estimators of struc-tural parameters of dynamic discrete-choice games. The survey makes it clear that the following three points are important as future researches in this literature. First, the literature has reached no consensus about the way of dealing with possible multiple equilibria and their identification. All the estimators reviewed in this chapter rely on the presumption that the data is generated from one equilibrium path. There has been developed no test to see whether or not this maintained assumption is truly satisfied. 18 Developing an empirical method to detect multiple equilibria and map the data to them is a promising future research topic. Second, the results of the existing Monte Carlo studies this chapter reviews are not conclusive about the finite sample properties of the recently developed two-step estima-tors. The reason for this lack of conclusion might be the absence of a common, canonical model in the Monte Carlo experiments the recent papers conducted. For example, the model used by Aguirregabiria and Mira (2006) and Pesendorfer and Schmidt-Dengler (2004) allow the action of a firm to affect the current period profit of other firms, while the model exploited by Pakes et al. (2005) does not. We need to compare different estimators using the same canonical model to make the comparison fair. 1 8 As discussed in Pesendorfer and Schmidt-Dengler (2003), this assumption is likely to be satisfied when a panel data set consists of time series from a single market. Third, although the curse of dimensionality in the conventional meaning associated with dynamic programming problems is avoided, the state space could be too large to handle. This is because, in general, the transition probability matrix exponentially expands in the number of players and choices. Suppose that, in a single-agent problem with j choices and one state variable, the size of state-space is M. Then the size of transition matrix is M x M. If we extend the problem to a multiple-agent model with n players keeping the size of state space the same, we need jn sets of transition matrices, each of which is conditional on each combination of choices. In addition, if the number of state variables increases linearly with the number of players, say to n, the size of state variables increases to M n. Because of these problems, the application is limited to relatively simple problems with small state space. 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The demand conditions and costs are dis-tributed independently from other state variables and the decisions of manufacturers: dt = 5dio + fidiidt-i + <4 and Cit = $ciO + &cilcit-l + e1ti where ef and ecit follow some distribution function f ed and /ec (. To construct the transition probability matrices, I use the method by Tauchen (1986). Based on the estimations of the density of residuals and the transition processes, the transition probability matrices are constructed. This is done by matching the values of residuals of the process evaluated at discretised space to those of evaluated points used for Kernel density estimation. A.2 Alternative presentation of value functions and best response probabilities According to Aguirregabiria and Mira (2006), I derive an alternative presentation of value functions and conditional choice probabilities, which are used in the pseudo-likelihood estimation of a menu-cost parameter. Let P* be a matrix of equilibrium probabilities, which are best response probabilities, and Vf* be the corresponding value functions of manufacturer i. Using P* and Vf*, I can rewrite the Bellman equation (3.9) as ae{o,i} x'ex where f p*(x'\x) is the transition probability induced by P*, and ef* (a) is the expectation of el conditional on x. 1 In vector form, equation (A2-1) is vr= E p;(amr(a)+er(a)}+pY,FP^ ^ a€{0,l} x'£X where V p*, P*(a), Uf*, and ef * (a) are the vectors of the corresponding elements in equa-tion (A2-1) with dimension M. F p* is a matrix of transition probabilities of f p*(x'\x). Under the condition (3 < 1, the value function given P* can be obtained as the solution of the following linear equation: cI-(3F p*)Vr= E p;(a)[ur(a) + er(a)}, (A2-2) ae{0,l} where I is an identity matrix with dimension M. Denote the mapping for the solution of equation (A2-2) as r,(x;P*). For an arbitrary set of probabilities P, the mapping operator r^rr; P) gives the values for manufacturer i when all the manufacturers behave according to P. Note that this mapping is constructed given the conditional choice probabilities of manufacturer i as well as those of its rival manufacturer. Using this mapping T, instead of V p in equation (3.9), I define a mapping ^ to calculate the expected value for manufacturer i to choose action a, for P: Vi(a\x) = f I{a = arg max [nf {a, x) + e» + f{x'\x, a)T p(x')]}^)^, (A2-3) J ae{0,l} x' I use the two mappings Tj(x; P) and ^ ( a | x) to estimate menu costs, 7. xThat is, f p'(x' | z) = Ea_ ; Pit* I I *)/(*' I x.Oi.a-i). A.3 The data This section describes the details about the construction of the two variables, a measure of wholesale prices and promotional code, the choice of stores, and the problem of missing data in the third chapter. The variables used in the analysis ' The analysis in the third chapter uses the following variables in the original data set: UPC, store code, week code, price, move(quantity sold), profit rate, the code for promo-tion, bundling (the number of units bundled together), OK (a code to show whether data points are reliable or not), customer count (the number of customers who purchased at least a single good), income (the median of income), each in the original name in the original data set. Recovering a measure of wholesale prices As stated in the main chapter, one of the advantages to use the DFF data set is that I could observe a good measure of wholesale prices. The data set contains a variable "profit rate", which presents gross margin of the retailer in terms of percentage in the revenue: Pit~AACCu * Using profit rates and prices, I can recover average acquisition costs (AAC)". AAC is a moving average of wholesale prices of existing inventory AACt = [(Quantity bought at the end of t-1) * (wholesale price paid at t-1) + (Quantity in stock at the end of t-2) * AACt~ 1]/ (Quantity in stock at the end of t-1). AAC are different from the current period wholesale price if the store carries any inven-tory bought at different prices in any previous periods. As mentioned in other studies for example, Chevalier et al. (2003), the policy of DFF is such that wholesale prices are reflected to acquisition cost fast. The code of promotional activities In the DFF data, the code for promotion records three activities of bonus, simple, and in-store coupons. In the analysis, I use only bonus as a measure of promotional activity creating a dummy variable when bonus is recorded. I do not use simple and coupons because of the following reasons. First, bonus is the most frequently used. As stated in the main chapter, bonus is typically price reduction with an announcement tag on shelves. Bonus is recorded about 20 percent of the entire period while simple and coupons are recorded only 2 percent and 0 percent, respectively. Second, according to the file description of the DFF data set, simple is described as "simple price reduction". However, the effects of mere price reductions are captured by prices in the demand estimation. In addition, the record for simple is somewhat suspicious in the record of graham crackers. There are several data points that simple is associated with price increases, not price reductions. Finally, coupons are never used in graham crackers. It should be noted that bonus does not perfectly record the promotional activities in the DFF. First, the data set does not contain the information of whether or not bonus is associated with advertisement or in-store display. Second, as stated in the description of the DFF data set, the record of this variable is somewhat inconsistent with the actual implementation of promotional activities. That is, the promotional activities may take place even when bonus and simple are not recorded. Missing data, the choice of stores, and the data sample In the DFF data set, some data points are missing because the scanner data is the transaction records at the checking-out counters in the supermarket. If the prices of a product in a store are missing just for a couple of weeks, the reason for missing data points would be because even a single unit of the product is not bought in a week in the store or because the product is temporarily stocked out. If the data points are missing for relatively long time, the reason could be because the store does not carry the product on regular basis or because the store is not operating. For the analysis, I use the data from the stores which are likely to carry the product on regular basis. This is because I need consecutive data series as long as possible. In addition, the stores who do not carry the product regularly might adopt different retail strategies toward the product from those who do. Newly opened stores or closed stores may also adopt different strategies, too. Accordingly, I first omit the stores whose records do not cover the entire sample period. Second, I also omit the stores with too many missing data: I take into account the pricing policy of Dominick's that is represented by pricing zone (low, middle, and high) to reflect the variation in prices across stores, and omit the stores if more than 2 percent of data are missing for the stores in middle and low pricing zones and 2.5 percent for the stores in high pricing zone.2 3 As a result, I select 21 stores.4 For my analysis, I need the data points with the following information: a data of a brand in a certain week in a certain store should contain current prices of all the brands including the private brand, and past prices of its own brand. As the imputation of prices could potentially lead to false information, I simply omit the data points unless all the information is not available. In addition, there is a code called OK, which is attached by University of Chicago to show whether each data point is suspicious or not.5 These data points are also excluded from the analysis. In total, I have an unbalanced 2Dominick's assigns 16 pricing zones to each stores based on the competitiveness of outlets with other retailers. University of Chicago arranges these 16 zones into three zones. 3There is no stores, of which missing data are less than 2 percent, in the high-price zone. 4The store numbers of selected outlets are as follows: 8, 14, 44, 56, 62, 71, 73, 74, 78, 80, 81, 83, 84, 101, 102, 109, 114, 121, 122, 126, 132. 5In the total data points of 129516 in the raw data for four brands in all the stores (more than 80 stores) in a chain, two percents of data are labeled as suspicious. data with 21978 observations (7326 observations for three brands). The discontinuity of records due to missing data affects the construction of the two variables used in the demand estimation, durd and durb. durd is the number of weeks since last price reduction by more than five percent, durb represents the number of weeks since bonus has started. I construct these two variables as follows when encountering missing data. Suppose that the price of a brand is changed more than five percent in the third week but the price in the fourth week is missing. Then, durd takes two in the fifth week. In the case durb, the value of fifth week simply takes zero unless bonus is held in the fifth week. These duration variables are created before list-wise deletion due to missing information of other brands.