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Essays on retail price movements Kano, Kazuko 2007

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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{<Ti(x,£i)  = a}g(£i)d£i.  (3.6) Manufacturer  i forms  expectation about the action of  its rival according to the conditional choice probability, P5j(a_j = a\x). Define  the expected one-period profit  for  manufacturer  i conditional on a, ~ a and x as X^/f^lft'^'^E/V  I  d)U l(p' t,cud\al,a_t). (3.7) P'i  d' The ex ante one-period profit  depends on the transition probability of  prices and man-ufacturers'  actions. When making its decision, manufacturers  i forms  expectation with respect to d' and p[,  given the choice probabilities of  manufacturer  —i. Given state x, private information  and strategy a, let Vf(x,£i)  be the value function  of  manufacturer  i associated with an optimal choice a. Then, the Bellman equation of  manufacturer  % is Vf(x,£i)  = max m°(a,x)  + e? + 0 V f{x'|x,a t) f  {x^e'^g^d^},  (3.8) where f(x'\x,a,i)  = x)/(x'|x, a^ , a_,). Integrating out private information £i, I can rewrite the above Bellman equation (3.8) in terms of  commonly observable state variables x. Let V?{x)  be the integrated value function  of  manufacturer  i facing  state x given strategy a, V i<T(x)  — f  V?(x,£i)g(£i)d£i.  With the integrated value function,  the Bellman equation (3.8) is rewritten as V°(x)=  f  m^{nUa,x)+£<l  + pTf(x'\x,a l)V°(x')}g l(£ z)d£l. (3.9) The right hand side of  equation (3.9) defines  a contraction mapping in the space of  the integrated value functions.  For each manufacturer,  there exists a unique value function Vf  that solves the functional  equation (3.9), given an arbitrary strategy cr. For i = {1,2} and any (x,  £*), the best response function  for  manufacturer  i is defined as strategy o% such that ai(x,e i) = arg max {Ilf  (a, x) + £• + (3  V  f(x'\x,  a)V?{x')}.  (3.10) ae{0,i} x' The pair of  the best response functions,  {er*(xj, £j), cr*(x_i, £_»)}, which defines  the best responses of  the manufacturers  to their rival's best response functions,  characterizes a Markov perfect  equilibrium in this game. Following Milgrom and Weber (1985), a Markov perfect  equilibrium can be repre-sented in a probability space. Note that the functions,  II^(a, x), f(x'\x,a),  and Vf(x'), depend on the strategies of  the manufacturers  through the conditional choice probabilities P  associated with an arbitrary strategy a. The equilibrium best response probabilities, which is integrated smoothed best response function,  associated with a set of  Markov perfect  equilibrium strategy a*, is the fixed  point of  the following  mapping: Let P*  be the best response probabilities in matrix form.  The right hand side of  equation (3.11) can be represented with a mapping operator from probability space to probability space, A(P).  Since this mapping is continuous in choice probabilities P,  by Brower's fixed point theorem, there exist the best response probabilities P*  that satisfy  P* = A(P*). 17 A Markov perfect  equilibrium is, then, characterized by a solution to the coupled fixed  point problem consisting of  equations (3.9) and (3.11) due to the interdependence between the value functions  and the conditional choice probabilities. Given the condi-tional choice probabilities, I solve the dynamic programming problems of  equation (3.9) for  manufacturers  % = {1,2}. Given these value functions,  the best response probabilities are obtained from equation (3.11). 1 7In general, the uniqueness of  equilibrium is not guaranteed. 3.3 The estimation procedure In this chapter, estimating menu costs 7 includes the following  two steps. First, I con-struct demand conditions by estimating a demand equation, and then estimate the tran-sition probabilities of  the state variables after  discretizing these variables. Second, using the results of  the first  step, I estimate menu costs 7 with the NPL estimator developed by Aguirregabiria and Mira (2002). The NPL estimator includes the Conditional Choice Probability (CCP) estimator by Hotz and Miller (1993) and the Nested Fixed Point (NFXP) estimator by Rust (1987) as extreme cases. The NPL estimator gains efficiency  compared to the CCP estimator while it saves computational costs compared to the NFXP estimator.18 Recent develop-ments for  estimating dynamic discrete choice games provide several alternative estima-tors such as in Jofre-Bonet  and Pesendorfer  (2003), Bajari, Benkard and Levin (2005), Pakes, Ostrovsky and Berry (2005), and Pesendorfer  and Schmidt-Dengler(2003,2004). Although several studies examine small sample properties in their proposed estimators, 1 8 The applications of  the NPL estimator to dynamic discrete choice games include Aguirregabiria and Mira (2006) for  an entry and exit game in a local retail market in Chile, Zhang-Foutz and Kadiyali (2003) for  a game of  release date preannouncements of  movies, and Collard-Wexler (2005) for  an entry-exit game in the ready-mix concrete industry. The code of  NPL for  a single agent model by Victor Aguirregabiria is available at http://people.bu.edu/vaguirre/. The code I use in this paper both for  a single agent model and a multiple agent model is based on his code. I thank him for  making his program available. so far  no consensus has been reached about which estimator performs  better than oth-ers, in particular, for  my problem. Aguirregabiria and Mira (2006) conduct the Monte Carlo experiments based on an entry and exit game. Comparing the NPL estimator and the two-stage pseudo maximum likelihood estimator, they show NPL performs  better especially when strategic interactions are strong.19 3.3.1 Estimating the demand equation and transition probabil-ities To estimate the demand function  (3.1), I specify  the following  empirical demand equa-tion: M&t)  = dt + b0 In(p it) + h In(p_ it) + eit = {aio + Dta} + b0 ln(pji) + bx ln(p_ it) + eit, (3-12) where «o is a constant term, Dit is a vector of  demand shifting  variables, a is a vector of coefficients  on the demand shifting  variables, and e is a demand error. I estimate equation (3.12) by two-stage least squares (2SLS). With the estimated coefficients  {d?0, a} , demand conditions dit are constructed as {d t = a0 + Dta}. Estimated demand coefficients  b0 and b\ are used to construct the one-period profits  of  the manufacturers. I then discretize the demand conditions, wholesale prices, and shelf  prices. Using 1 9 The two-stage pseudo maximum likelihood estimator corresponds to the NPL estimator at K = l , where K is the number of  iteration in the nested algorithm. the discretized variables, I estimate transition probability density functions  f d , and /f ,  using the method by Tauchen (1986). The construction of  ff  needs some additional procedure since / f  is conditional on not only own past value and the current choice but on the values of  demand conditions and costs.20 Theoretically, the initial values of  conditional choice probabilities should not matter as the consistency of  NPL estimator does not require the consistency of  initial choice proa-bilities. In this chapter, I construct the initial values of  conditional choice probabilities the results of  a probit estimation in the following  manner. First, I estimate a reduced-form probit model, in which the dependent variable is an index function  I  {Ap  ^ 0} in the case of  binary choice problem, and the explanatory variables include constant, dt-x, pit-1, i ) t - i , p_ i (2)t-i, cit, c_i(i)t, c_i(2)t, where the subscripts - z ( l ) and -i(2) present rival firms.  The estimation is conducted using the discretised variables. Then, I obtain the conditional choice probabilities for  each value of  state space by evaluating the predicted probabilities at each bin of  state space. 3.3.2 Estimating menu costs This section derives the pseudo-likelihood function  to estimate menu costs. For conve-nience, define  the following  notations: the rival's expected price under its conditional choice probability P,  p?it = Ea_ie{o,i} P( a-i I xt) J2 P_uP-itf-i(P-it  I xt,a-i) for  a given 2 0For the details of  estimations of  transition probabilities and initial choice probabilities, see Appendix A.l. xt ; manufacturer  i's expected price associated with action a, p" t = ^2p.tPufi(Pit  I Pit-i,(kt,dt,ai t — a) for  a given pu-i- Given the estimated coefficients  of  the demand equation (3.12), b0 and b\, and the constructed demand conditions dt, I set up the ex-pected one-period profit  associated with action a as n f  (a, Xt)  = (p£ - clt) exp(dt  - b0 ln(p£) + £ In(p pl t)) - 7 / { a = 1}. For exposition, denote I lp (a,x t ) = zp9, where z®p = {(p"t — cit) exp(d t — b0\n(p°; t) + bi ln(p^it)), — I{a  = 1}} and 9 = {1,7}. Let F p be the transition probability matrix representing all the transition processes of  the state variables x under the conditional choice probabilities P, and ef*(a)  be the vector of  the expectation of  el conditional on x.2 1 The empirical counterparts of  the value functions  (3.9) and the best response prob-abilities (3.11) are derived according to the mapping expression by Aguirregabiria and Mira (2006). Let Tj(P) denote the mapping operator of  the value functions  (3.9) in vector form given conditional choice probabilities P,  and ^(a | x) be the operator representing the best response probabilities given ri(P).  22 rj(P) can be written as T^P) = Zp 9+Af, where Z f  = (I  - (5F P)~^  £ a 6 { 0 , i } /?(a)n , (a) and Af  = ( / - (3F P)~^  £ a G { 0 , i } e f ( a ) , 21 F p = Y.a-i p( ai) * p( a-i) * ( Fi  ® F-i  ®F d®F tc® Fti),  where * represents the element-by-element product, (g) represents the Kronecker product, and Ff  represents the matrix of  the transition probability ff,  respectively. 2 2For the details of  the derivation, see Appendix A.2. where the value of  discount factor  is assumed to be known a priori and fixed  at 0.99.23 Assume that private information  follows  an i.i.d. Type I Extreme Value distribution.24 Then, ef  (a)  = Euler's constant — ln(Pja), where Euler's constant is about 0.577. The mapping of  the best response probabilities ^ given P  is exp{z? te + PF a(Zfd  + \?)} I construct a pseudo-likelihood function  to estimate 9 treating the conditional choice probability as nuisance parameters. Let P°  and 9°  denote the true conditional choice probabilities and menu costs. Given the true conditional choice probabilities P°, the corresponding pseudo-log-likelihood function  is 2 oo E E E I{ ait — a} I n I xu P°i  9°),  (3.14) i=l t= 1 ae{0,l} where ^ ( a | x; P°,  9°)  shows the dependence of  ^ on conditional choice probabilities P° and menu costs 9°.  The NPL estimator is obtained by the following  procedure. First, I conduct the pseudo-maximum likelihood estimation of  9 given P0 (initial values of  P), and then obtain the updated P\ using 9i according to the mapping Second, I iterate this procedure for  K  > 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 = <T;(x,£;), where aj : M N  x —> 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, <f>  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  <f>  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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Cornell University, manuscript Appendix A Appendix for  chapter 3 A.l Constructing transition probability matrices Using the discretized variables, I construct the matrices for  transition probability: ff(d t\dt-i) , /f(cit|ci t_i), and f¥(pit\dit,Pit-i,CiuOit)-  To do that, I first  specify  stochastic processes of  pa, dt, and cit, as follows: r Pit = 5Pio + SpiiPit-i  + SPi2dt + SpisCu + epit pit ± pit-1, yPit = Pit-1  otherwise, where if  cpit follows  some distribution / cp. 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. 

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