Matthew J Higgins
email: mhiggins@mgt.gatech.edu

Andrew T Young
email: atyoung@olemiss.edu

Daniel Levy
email: levyda@mail.biu.ac.il

Robust Correlates of County-Level Growth in the U.S

Keywords: Economic Growth, Conditional Convergence, Extreme Bounds Analysis, County-Level Data

 demonstrate that such correlations are hardly ever robust to changing the combination of conditioning variables included. We ask whether the same is true for the variables identified as important by Higgins et al. Using the extreme bounds analysis of Levine and Renelt, we find that the majority of the partial correlations can be accepted as robust. The variables associated with those partial correlations stand solidly as variables of interest for future studies of U.S. growth.

1.

Introduction [START_REF] Higgins | Growth and convergence across the U.S: evidence from county-level data[END_REF] Evans (1997aEvans ( , 1997b) ) and include all 41 conditioning variables in the cross-sectional regressions.

However, [START_REF] Levine | A sensitivity analysis of cross-country growth regressions[END_REF], employing a version of [START_REF] Leamer | Let's take the con out of econometrics[END_REF][START_REF] Leamer | Sensitivity analysis would help[END_REF] extreme bound analysis (EBA), show that growth regression estimates can be very sensitive to small changes in the set of conditioning variables. 1 In order to determine whether findings (1)-( 4)

from [START_REF] Higgins | Growth and convergence across the U.S: evidence from county-level data[END_REF] are model dependent, we replicate Levine and Renelt's EBA using the same data set as Higgins et al. We find that 7 out of 11 variables of interest are robust partial correlates with U.S. county-level growth.

Section 2 outlines the EBA methodology and describes the data. Section 3 reports our results. We conclude in Section 4.

Extreme Bounds Analysis (EBA)

In response to sensitivity issues, [START_REF] Leamer | Let's take the con out of econometrics[END_REF][START_REF] Leamer | Sensitivity analysis would help[END_REF] proposes an EBA to identify "robust" empirical relations. For a specific variable of interest, the extreme bounds of the Since it is well-established that initial income be included in growth regressions, the EBA for "income" is constituted by the results of C(40,3) = 9,880 OLS regressions. The EBAs for the remaining 10 conditioning variables of interest are constituted by the results of C(39,3) = 9,139 OLS regressions.

Results of Extreme Bounds Analysis

The extreme bounds for coefficients are reported in Table 2 along with their 95 percent confidence intervals. A full 7 of 11 variables, found to be significant correlates with economic growth by [START_REF] Higgins | Growth and convergence across the U.S: evidence from county-level data[END_REF], are robust as defined by the EBA. Furthermore, the robust correlates according to the EBA carry the same signs as reported in [START_REF] Higgins | Growth and convergence across the U.S: evidence from county-level data[END_REF].

First, the initial level of income is a robust, negative correlate with per capita income growth. This confirms that a conditional convergence effect exists across the U.S. at the countylevel. This can also be viewed as consistent with studies by [START_REF] Barro | Convergence[END_REF], Evans andKarras (1996a, 1996b), Sala-i-Martin (1996), and Evans (1997aEvans ( , 1997b) ) who document conditional convergence using state-level data; and with [START_REF] Young | Heterogeneous convergence[END_REF] who find conditional convergence within many individual U.S. states using county-level data. 3 The existence of conditional convergence is always encouraging in the limited sense that it implies that if a relatively poor economy can emulate the policies and institutions of its wealthier counterparts, then it can expect to grow faster and catch-up in terms of its per capita income.

The robust estimated effects of educational attainment variables appear reasonable. The larger a percent of a county's population not obtaining the remedial communication and analytical skills associated with completing high school, the lower is the county's growth rate.

Likewise, a larger percent of a population achieving at least 4 college-years-worth of human capital correlates with a higher rate of growth. 4 Of note, the effect associated with some college attainment, but less than a bachelor degree, is fragile. This can be viewed as consistent with Higgins et al.'s (2006) finding that no statistically significant effect is associated with that variable. One interpretation is that the opportunity costs of education at those levels of attainment are comparable to the social returns.

Turning to the government employment variables, the robust negative correlations associated with federal and local government are consistent with [START_REF] Higgins | Growth and convergence across the U.S: evidence from county-level data[END_REF]. However, Higgins et al. also found state government employment to be negatively correlated (significantly at the 1 percent level) with growth using 3SLS estimation. Here we find that, using an EBA, that negative correlation is fragile. Of note, [START_REF] Higgins | Growth and convergence across the U.S: evidence from county-level data[END_REF] also report, as a baseline, OLS coefficient estimates. For the state employment variable the Higgins et al. OLS estimate is negative but insignificant; the EBA here produces a stronger finding that, changing the set of conditioning variables, can produce both negative and positive statistically significant coefficient estimates.

Two of the industry employment variables are robust, positive correlates with countylevel growth. In both cases the positive sign is consistent with the findings of [START_REF] Higgins | Growth and convergence across the U.S: evidence from county-level data[END_REF]. While growth effects associated with entertainment and recreational services are not widely documented in the literature, the robust positive correlation of growth with the prevalence of finance, insurance, and real estate industry is in agreement with existing cross-country evidence.5

Conclusions

Higgins et al. (2006) report several statistically significant partial correlates with U.S. per capita income growth at the county-level. However, [START_REF] Levine | A sensitivity analysis of cross-country growth regressions[END_REF] demonstrate that, for cross-country data sets, such correlations are hardly ever robust to changing the combination of conditioning variables included. It is natural, then, to ask whether the same is true for the variables identified as important by Higgins et al.

We carry out an extreme bounds analysis of the [START_REF] Levine | A sensitivity analysis of cross-country growth regressions[END_REF] type using the U.S. county-level data of [START_REF] Higgins | Growth and convergence across the U.S: evidence from county-level data[END_REF]. We find that the majority of the partial correlations put to test (7 out of 11) can be accepted as robust correlates with U.S. county-level growth. The variables associated with those partial correlations stand solidly as variables of interest for other studies of U.S. growth.

Tables

  study U.S. county-level income growth from 1970 to 1998 controlling for 41 demographic conditioning variables. Their findings include: (1) conditional -

	convergence; (2) federal, state and local government employment negatively correlate with
	growth; (3) the relationship between educational attainment and growth is nonlinear; and (4)
	finance, insurance, and real estate industry employment and entertainment industry employment
	correlate positively with growth, whereas education employment correlates negatively with
	growth. Higgins, et al. use a consistent 3SLS estimation method of

  The dependent variable is per capita real income growth from 1970 to 1998. See Higgins et al (2006, Table 1 and Section III) for a description of the data set, including the list of the remaining 30 conditioning variables.

	distribution of the associated coefficient estimates are calculated as the smallest and largest
	values that are not rejected at the 0.05 significance level given all possible combinations of the
	remaining conditioning variables taken 3 at a time. If the two bounds have differing signs, then
	the variable is labeled as fragile; otherwise it is labeled robust. 2
	The 11 conditioning variables of interest are listed in Table 1. These variables are 1970
	values for 3,058 U.S. counties.

  U.S. Bureau of Economic Analysis; Census denotes the U.S. Census Bureau. All variables are 1970 values for 3,058 U.S. counties.

	Table 1
	Extreme Bounds of Coefficients on Variables of Interest
	Variable	Upper Bound	Source
	Income	Real per capita personal income (excluding transfer	U.S. BEA
		payments)
	Education:	Percentage of the population with 11 years of education or	Census
	9-11 years	less
	Education:	Percentage of the population with a high school diploma	Census
	H.S. diploma	but no more education
	Education:	Percentage of the population with college education but not	Census
	some college	having obtained a bachelor degree
	Education:	Percentage of the population holding a bachelor or higher	Census
	bachelor +	level degree
	Federal government	Percentage of the population employed by the federal	BEA
	employment	government
	State government	Percentage of the population employed by the state	BEA
	employment	government
	Local government	Percentage of the population employed by the local	BEA
	employment	government
	Entertainment and	Percentage of the population employed in entertainment or	Census
	recreational	recreational services
	services
	Finance, insurance,	Percentage of the population employed in finance,	Census
	and real estate	insurance or real estate
	Educational	Percentage of the population employed in educational	Census
	services	services
	BEA denotes the

[START_REF] Levine | A sensitivity analysis of cross-country growth regressions[END_REF] find that, using an international sample, very few variables are robust correlates with growth.[START_REF] Sala-I-Martin | Determinants of long-term growth: a Bayesian averaging of classical estimates (BACE) approach[END_REF] introduce an alternative Bayesian sensitivity analysis. Their analysis is motivated by the belief that Levine and Renelt's "test is too strong for any variable to pass: any one regression model (no matter how well or poorly fitting) carries a veto" (p. 814). In contrast, we conclude that the majority of variables identified as important by[START_REF] Higgins | Growth and convergence across the U.S: evidence from county-level data[END_REF] are not "vetoed" by the Levine and Renelt test.

Statistically insignificant coefficient estimates are discarded from the analysis; including them would make for an unreasonably demanding test. An insignificant coefficient estimate of a different sign than extreme bounds of like signs is, rather than a contradiction of those bounds, merely a tentative acceptance of the null of zero partial correlation.

This type of convergence is known as -convergence and is necessary but not sufficient for -convergence, i.e., for a narrowing of the income distribution over time.[START_REF] Young | Sigma convergence versus beta convergence: evidence from US county-level data[END_REF] find that, over the same 1970 to 1998 time period, statistically significant -divergence actually occurred.

Our conditioning variables include a dummy variable that takes the value of 1 if the county includes a college or university with enrollment of 10,000 or more and accounts for at least

percent of the total population. In[START_REF] Higgins | Growth and convergence across the U.S: evidence from county-level data[END_REF] the inclusion did not render the bachelor+ coefficient estimate insignificant.

[START_REF] Levine | Finance and growth: theory and evidence[END_REF] provides an overview of the empirical findings, as well as the theoretical literature motivating the studies.

Tables (cont.)


Daniel Levy
email: levyda@mail.biu.ac.il.

Frank Smets

Guest

Price Setting and Price Adjustment in Some European Union Countries Introduction to the Special Issue

Keywords: D21, D40, E12, E31, E50 Price Rigidity, Price Flexibility, Cost of Price Adjustment, Menu Cost, Managerial and Customer Cost of Price Adjustment, Pricing, Price System, Price Setting, New Keynesian Economics, Store-Level Data, Micro-Level Data, Product-Level Data

eleven empirical studies of price setting and price adjustment that are included in this special issue. The studies, which use data from several European countries, were conducted as part of the European Central Bank's Inflation Persistence Network.

INTRODUCTION

An age old question in economics is, to what extent do prices adjust to changes in market conditions? In other words, how rigid or flexible prices are. Up until the beginning of the 1990s, there were hardly any studies of price rigidity that use micro level (i.e., store-level and/or product-level) data on actual transaction prices. The last 15-20 years have witnessed a remarkable revival in the popularity of New Keynesian models which incorporate various forms of rigidities as the main source of friction needed to generate monetary non-neutrality. See, for example, Mankiw andRomer (1991a, 1991b) and Sheshinski and Weiss (1993), and the references cited therein.

The renewed attention to the theoretical New Keynesian research program has revived the economists' interest in empirical aspects of price rigidity. Consequently, the New Keynesian macroeconomics and to a lesser extent the industrial organization literatures, began offering empirical studies of price rigidity using various types of microlevel data from the US as well as from the European Union countries. 1 Two previous special issues of the Managerial and Decision Economics were devoted to this research topic. One previous special issue (Levy, 2007a) was devoted to some recent theoretical developments in this line of research. A sequel (Levy, 2007b), reported the findings of some of the most recent empirical studies of price rigidity. The goal of this special issue is to report the results of some recent additional empirical studies that use micro level retail and wholesale transaction price data as well as some new survey data from several European Union countries. These studies were undertaken by the European Central Bank's (ECB's) Inflation Persistence Network (IPN). The IPN, which was a research team consisting of economists from the ECB and the national central banks of the Eurosystem, was set up by these institutions in order to conduct a coordinated research project on the patterns, determinants and implications of inflation persistence in the euro area and in its member countries. The main issues the IPN studies addressed were (a) existence and characterization of nominal rigidities in the Euro area, (b) determinants of nominal rigidities, and (c) empirical testing of alternative price setting models.

1 See [START_REF] Willis | Implications of structural changes in the U.S. economy for pricing behavior and inflation dynamics[END_REF] and [START_REF] Wolman | The frequency and costs of individual price adjustment[END_REF] for recent surveys of some of these studies.

2

These studies addressed numerous particular questions of interest. For example: Are prices rigid in the Euro area? Are these rigidities symmetric? What are the sources of the price rigidities? Are these sources similar across countries? Are there systematic patterns in the differences found across countries, products and/or sectors in the degree of price stickiness? Which factors beyond nominal rigidities are needed to explain this inflation persistence? Etc.

For this purpose, the ECB and the member countries' central banks have provided the IPN teams with incredibly detailed data sets, containing numerous macroeconomic and sector-level variables, as well as data and information on price-setting and price adjustment behavior at the individual firm level. The individual price records used in the construction of both consumer price index (CPI) and producer price index (PPI) have been made available in a large number of EU countries. In addition, the IPN has conducted surveys on price setting behavior in nine countries of the euro area.

IN THIS ISSUE

This special issue of the Managerial and Decision Economics contains 11 contributions from various IPN teams. 2 These papers address empirically various aspects of price rigidity and flexibility from different angles and points of view, using different types of data from 10 different countries. These include Germany, Italy, Spain, Portugal, the Netherlands, Austria, Hungary, Luxembourg (two studies), Slovakia, and Romania.

In "Price Adjustment in German Manufacturing: Evidence from Two Merged Surveys," Harald Stahl presents evidence on formation of producer prices in German manufacturing, using two survey data sets on 1,200 German firms. He finds that twenty percent of the firms' price setting behavior resembles time-dependent price setting rules.

It turns out, however, that neither Taylor nor Calvo-type price setting rules seem to describe well the sampled firms' price setting behavior. Stahl concludes that fixed contracts and coordination failures are the main reason for delay in price adjustment decisions although the hazard rates for price changes do not support this.

2 Other published studies from the IPN network include [START_REF] Alvarez | Sticky prices in the Euro area: a summary of new microevidence[END_REF], [START_REF] Angeloni | New evidence on inflation persistence and price stickiness in the euro area: implications for macro modeling[END_REF], [START_REF] Dhyne | Price changes in the Euro Area and the United States: Some facts from individual consumer price data[END_REF][START_REF] Fabiani | Pricing Decisions in the Euro Area: How Firms Set Prices and Why[END_REF]. For a complete list of the studies that were conducted as part of the IPN network, see http://www.ecb.int/home/html/researcher_ipn_papers.en.html.

In "Price Adjustment in Italy: Evidence from Micro Producer and Consumer Prices," Silvia Fabiani, Angela Gattulli, Giovanni Veronese and Roberto Sabbatini report that Italian producer prices remain unchanged for about 6 months, while consumer prices exhibit a longer duration of 10 months. They report that prices are more flexible at the production stage. They find that a higher labour share in total costs is related to lower frequency of price adjustment. The authors hypothesise that the structure and functioning of the retail sector in Italy together with other specific factors (e.g., menu costs or psychological pricing policies), may slow price adjustment at the consumption stage.

In "Price Setting Behaviour in Spain: Evidence From Micro PPI Data," Luis Álvarez, Pablo Burriel and Ignacio Hernando study the price setting behaviour at Spain's manufacturing industry using a rich micro-level PPI dataset. They find that some important variables, including the cost structure, degree of market competition, demand conditions and inflationary pressures have substantial impact on the frequency of price adjustment and thus contribute to the heterogeneity of price stickiness across industries.

They compare the consumer and the producer price setting practices and find evidence that producer prices are more flexible than consumer prices.

In "Price Stickiness in Portugal: Evidence from Survey Data," Fernando Martins presents evidence indicating a considerable degree of price stickiness in Portugal. For example, he finds that most firms do not change prices more than once a year. He also finds that the time lag in price reaction to shocks is significant. Implicit contracts between firms and their customers which "promise" stable prices seem to prevent the firms from changing their prices more frequently.

In "Price Setting Behaviour in the Netherlands: Results of a Survey," Marco Hoeberichts and Ad Stokman use a survey of Dutch firms' price setting practices. Their primary finding is that Dutch firms' price setting behaviour depends critically on both a firm's size and the competitive environment. For example, small firms' prices are more rigid. The weaker the competition is, the stickier the firm's price. Their survey suggests that contracts (explicit and implicit) may be the key source of price rigidity.

In "How Are Prices Adjusted in Response to Shocks? Survey Evidence from Austrian Firms," Claudia Kwapil, Johann Scharler, and Josef Baumgartner, study price response to shocks using Austrian firm survey data. They find that firms are more likely to adjust prices after a cost shock than after a demand shock. Their analysis suggests customer loyalty plays a key role in explaining price rigidity in response to demand shocks. Furthermore, a lack of competition seems to play a substantial role in explaining price stickiness. Finally, they find asymmetric responses after cost and demand shocks. It seems that after cost shocks, downward price rigidity is more frequent than upward price rigidity, while the opposite is true after shifts in demand.

In "Price Setting in Hungary -A Store-Level Analysis," Péter Gábriel and Ádám Reiff study the Hungarian micro CPI data to characterize store-level pricing practices in Hungary. They report the frequency and average size of price changes, the duration distribution of price spells and the mean durations for different product categories. They decompose the observed variations in the inflation rate into variations in frequencies and variation in sizes. Finally, they estimate the inflation effects of three general VAT-rate changes during the sample period.

Patrick Lünnemann and Thomas Y. Mathä contribute two papers to the special issue. In "Consumer Price Behavior: Evidence from Luxembourg Micro Data," they report that the median duration of consumer prices is about 8 months. With the exception of services, prices are not rigid downwards. They find that price changes are determined both by state-and time-dependent factors. Accumulated price and wage inflation, and the cash changeover increase the price change probability, while pricing at psychological pricing points and price regulation have the opposite effect. Automatic wage indexation increases the probability of price increases, especially in consumer product categories where prices are closely linked to wage costs.

In "Regulated and Services Prices' Rigidities and Inflation Persistence: Some Observations," Lünnemann and Mathä study price level rigidity and inflation persistence using about 1500 price indices for the EU15 member states. Services and harmonized indexes of consumer prices (HICP) subject to price regulation exhibit a larger degree of nominal price rigidity, with less frequent but larger changes as well as stronger asymmetries between increases and decreases. For most of the EU15 countries as well as for the EU15 and the euro area aggregates, excluding services from the full HICP reduces the measured degree of inflation persistence.

In "Price Setting and Market Structure: An Empirical Analysis of Micro Data in Slovakia," Fabrizio Coricelli and Roman Horváth find that the market structure is a key determinant of price setting behaviour. Market structure has two opposing effects on inflation persistence. On the one hand, increased competition may reduce persistence by increasing the frequency of price changes. On the other hand, higher competition may increase persistence through inertial behaviour induced by strategic complementarities among price setters. They find that the latter effect dominates. Indeed, the dispersion of prices is higher while persistence is lower in the non-tradable sectors, suggesting that higher competition is not conducive to lower persistence. Furthermore, they report that the frequency of price changes depends negatively on the price dispersion and positively on the product-specific inflation.

In "Survey Evidence on Price Setting Patterns of Romanian Firms," Mihai Copaciu, Florian Neagu, and Horia Braun-Erdei present the results of the first micro survey on price setting patterns among the New Member States of the European Union (NMS).

Diverging from other IPN studies' findings, small firms perceive higher competitive pressures and set prices using a state dependent rule. Lower perceived competition seems to be more relevant for the medium and large firms which use mark-up pricing. Prices are reviewed and changed more often than for EMU firms and are found to be more flexible than wages. Similar to other IPN studies' evidence, contracts are the main sources of price stickiness. The survey also suggests full price transmission of large unanticipated financial shocks.

ACKNOWLEDGEMENTS

Much of the editorial work on this special issue was completed at the Monetary Policy and Research Division of the Research Department, at the European Central Bank (Frankfurt), where Daniel Levy was a Visiting Scholar. He is grateful to the Bank's Research Department Director at the time, Vitor Gaspar, to the Department's research economists, and to the Bank's management and the administrative staff for their kind hospitality. Daniel Levy gratefully acknowledges also the financial support from the Adar Foundation of the Economics Department at Bar-Ilan University. We thank our colleagues Bob Barsky, Mark Bergen, Allan Chen, Sourav Ray, Paul Rubin, Avichai

process. MDE's managing editors during the last few years, Soyong Chong, Mike Hammock, Shireen Meer, and Mary Schroeder provided efficient editorial assistance in the editing and processing of the special issue. We are grateful to the MDE's editor-inchief, Paul Rubin for the continuous support and encouragement. Finally, we thank Wiley's editorial personnel, Russel Graham and Ivry Tan, for their assistance in preparing and processing this special issue for publication.


Daniel Levy
email: levyda@mail.biu.ac.il

Dongwon Lee

Allan ) Chen
email: hchen@mays.tamu.edu

Robert J Kauffman
email: rkauffman@asu.edu

Mark Bergen
email: mbergen@csom.umn.edu

Marco Bonomo

Alan Kackmeister

Attila Ratfai

Julio Rotemberg

Harald Stahl

Jonathan Willis

Alex Gershon Wolman

Bob Alperovich

Alan Barsky

Leif Blinder

Mark Danziger

Carlos Gertler

Jacob Marques

Paroush

Péter Benczúr

Michael Ehrmann

David Genesove

Peter Gabriel

Zvi Hercowitz

Heinz Herrmann

Johannes Hoffmann

Péter Karádi

Ed Knotek

Saul Lach

Benoît Mojon

Ádám Reiff

Frank Smets

Manish Aggrawal

Ning Liu

Avichai Snir

Chris Forman

Hemant Bhargava

D J Wu

Barrie Nault

Fred Riggins

Sri Narasimhan

Rahul Telang

Sunil Milthas

Price Points and Price Rigidity

Keywords: Price Point, 9-Ending Price, Price Rigidity Forthcoming

We study the link between price points and price rigidity, using two datasets: weekly scanner data, and Internet data. We find that: "9" is the most frequent ending for the penny, dime, dollar and ten-dollar digits; the most common price changes are those that keep the price endings at "9"; 9-ending prices are less likely to change than non-9-ending prices; and the average size of price change is larger for 9-ending than non-9-ending prices. We conclude that 9-ending contributes to price rigidity from penny to dollar digits, and across a wide range of product categories, retail formats and retailers.

I. Introduction

With the increased popularity of new Keynesian models, understanding the sources of nominal price rigidity has become even more important. 1 One of the recent theories of price rigidity is price point theory, which [START_REF] Blinder | Asking About Prices: A New Approach to Understanding Price Stickiness[END_REF] list among the twelve leading theories of price rigidity. According to the authors (p. 26), practitioners' "… belief in pricing points is part of the folklore of pricing …" Consistent with this observation, they offer evidence from interviews on the importance of price points. In their study of 200 U.S. firms, they found that 88 percent of retailers assigned substantial importance to price points in their pricing decisions. [START_REF] Kashyap | Sticky Prices: New Evidence from Retail Catalogues[END_REF], the first to explore the link between price points and price rigidity, found that catalog prices tended to be "stuck" at certain ending prices. After concluding that the observation cannot be explained by existing theories, he offered price point theory as a possible explanation.

As [START_REF] Blinder | Asking About Prices: A New Approach to Understanding Price Stickiness[END_REF] note in the opening quote above, however, a major difficulty with price point theory is that not much is known about the actual importance of price points or about their relationship to price rigidity. Price points will be particularly important for macroeconomics if they can be shown to contribute to price rigidity across a wide range of products and retailers.

The literature offers growing evidence on the use of price points, but still there is a lack of direct evidence linking price points and price rigidity. The literature documenting a link between price points and price rigidity using U.S. data is limited to [START_REF] Kashyap | Sticky Prices: New Evidence from Retail Catalogues[END_REF] and [START_REF] Blinder | Asking About Prices: A New Approach to Understanding Price Stickiness[END_REF].

Kashyap has emphasized the need for more direct evidence, stating that a "study focusing on more goods … would have much more power to determine the significance of price points."

Our goal is to fill this gap in the literature by offering new evidence on the link between price points and price rigidity using two particularly appropriate but different datasets. One is a large weekly scanner price dataset from a major Midwestern U.S. retailer, covering 29 product categories over an eight-year period. The second comes from the Internet and includes daily prices over a two-year period for 474 consumer electronic goods, such as music CDs, digital cameras, notebook PCs, etc., from 293 different e-retailers, with a wide range of prices. Taken together, the two datasets cover a diverse set of products, a wide range of prices, different retail formats, and multiple retailers and time periods.

The following summarizes our findings. "9" is the most popular price point for the penny, dime, dollar and the ten-dollar digits across the two datasets. The most common price changes are those that keep the terminal digits at these "9"endings. When we estimated the probability of a price change, we found that the 9-ending prices are less likely to change in comparison to non 9-ending prices. For the Dominick's data 9-ending prices are at least 43-66 percent less likely to change than non-9-ending prices. For the Internet data, these probabilities are in the range of 25-64 percent. The average size of the 9-ending price changes are larger in comparison to non-9ending prices, which further underscore the extent of the 9-ending price rigidity.

The paper is organized as follows. We describe the data in section II. In section III, we study the distribution of price-endings. In section IV, we assess the distribution of price changes.

In section V, we estimate the effect of 9-endings and 99-endings on price rigidity. In section VI, we evaluate the link between price points and the size of price changes. In section VII, we discuss the robustness of the findings. Section VIII concludes. [START_REF] Kashyap | Sticky Prices: New Evidence from Retail Catalogues[END_REF] price point theory suggests that price points should be most important to retail firms [START_REF] Blinder | Asking About Prices: A New Approach to Understanding Price Stickiness[END_REF][START_REF] Stahl | Price Adjustment in German Manufacturing: Evidence from Two Merged Surveys[END_REF]. We examine retail prices from two large datasets.

II. Two Datasets

One is Dominick's weekly price data for 29 different supermarket product categories over an eight-year period. The other contains daily prices from the Internet on products that include music CDs, DVDs, hard disks, and notebook PCs, among others.

The two datasets cover a wide variety of products, a wide range of prices, and different retail formats. In addition, although Dominick's prices are set on a chain-wide basis, our Internet data come from many different retailers, which presumably employ different pricing decision models. Thus, the conclusions that we draw are not specific to a particular retail format, a retailer, a product, or a price range. Dominick's is a large supermarket chain in the Chicago metropolitan area. During the period of our study, it operated 93 stores with a market share of about 25 percent. The data consist of up to 400 weekly observations of retail prices in 29 different product categories, covering the period from September 14, 1989 to May 8, 1997. The prices are the actual transaction prices as recorded by the chain's checkout scanners. If an item was on sale, then the price data reflect the sale price of the item. Although Dominick's prices are set on a chain-wide basis at the company headquarters, there is some price variation across the stores depending on the price tiers to which the stores belong. Dominick's divides its stores into four price tiers. These are "Cub-fighter," "low," "medium," and "high." The stores designated as Cub-fighters are typically located in proximity to a Cub Foods store and thus compete directly with it. The other three price tier stores employ a pricing strategy that fits best given their local market structure and competition conditions.

We report results from analyzing the prices in four stores, one from each price tier. The stores were selected at random and include Store #8 ("low" price tier), #12 ("high" price tier), #122 ("Cub Fighter"), and #133 ("medium" price tier). To study the behavior of regular prices, we removed data points if they involved bonus buys, coupon-based sales, or simple price reductions. For this, we relied on Dominick's data identifiers which indicated the occurrences of such promotions. Dominick's did not use loyalty cards during the time period studied.

In total, the Dominick's data contain over 98 million weekly price observations on 18,037 different grocery products in 29 product categories. 2 The four-store sample contains 4,910,129 weekly price observations on 16,105 different products. [START_REF] Barsky | What Can the Price Gap between Branded and Private Label Products Tell Us about Markups?[END_REF], [START_REF] Chevalier | Why Don't Prices Rise during Periods of Peak Demand? Evidence from Scanner Data[END_REF], and [START_REF] Levy | Holiday Price Rigidity and Cost of Price Adjustment[END_REF] offer more details about the data. 3 Table 1 presents descriptive statistics for the Dominick's data for the four stores.

Our Internet data were obtained through the use of a price data-gathering software agent.

We programmed it to download price data from BizRate (www.bizrate.com), a popular price comparison site. It accessed the site for data collection from 3:00 a.m. to 5:00 a.m. over a period of more than two years from March 26, 2003to April 15, 2005. We generated a large sample of product IDs using stratified proportionate random sampling (Wooldridge 2002) from a list of products available at BizRate. The software agent automatically built a panel of sales prices given the product IDs. 4 The resulting dataset consists of 743 daily price observations for 474 personal electronic products in 10 product categories from 293 different Internet-based retailers.

The categories include Music CDs, Movie DVDs, Video Games, Notebook PCs, Personal Digital Assistants (PDAs), Software, Digital Cameras and Camcorders, DVD Players, PC Monitors, and Hard Drives. 5 In total, the Internet data contain over 2.5 million daily price observations. Table 2 presents descriptive statistics for the Internet data.

III. Evidence on the Popularity of 9-Ending and 99-Ending Prices

I asked the best economist I know, at least for such things-my wife, if she recalled a price not ending in a "9" at our local grocery store. "Not really," she said. "Maybe sometimes there are prices ending in a "5," but not really." Jurek Konieczny (2003, Discussant Comment) We begin by presenting the results on the frequency distribution of price-endings in the two datasets. In the analysis of Dominick's data, our focus was on 9¢ and 99¢ price-endings because the overwhelming majority of the prices in retail grocery stores were well below $10.00 during the study period. 6 In the Internet data, the price ranges were different: from a minimum of $3.99 to a maximum of $6,000.00, with the average prices in different categories spanning $13.46 to $1,666.68 in the study period. The wider price range in the Internet data enables us to study not only 9¢ and 99¢ price-endings, but also other 9-ending prices in both the cents and the dollars digits, including $9, $9.99, $99, and $99.99.

In Figure 1, we report the frequency distribution of the last digit of the prices in Dominick's data. If a digit's appearance as a price-ending were random, then we should have seen 10 percent of the prices ending with each digit. As the figure indicates, however, about 69 percent of the prices ended with a "9." The next most popular ending was "5," accounting for only 12 percent of all price endings. Only a small proportion of the prices ends with other digits.

Next, we consider the frequency distribution of the last two digits. With two digits, there are 100 possible endings, 00¢, 01¢, …, 98¢, and 99¢. Thus, with a random distribution, the probability of each ending should be only 1 percent. According to Figure 2, however, most prices end with either 09¢, 19¢, …, or 99¢. This is not surprising since "9" was the dominant singledigit ending. But of these, more than 15 percent of the prices ended with 99¢. In contrast, only about 4 percent to 6 percent of the prices ended with 09¢, 19¢, …, and 89¢.

Figure 3 displays the frequency distribution of the last digit in the Internet data. We can see that "9" was the most popular terminal digit (33.4 percent), followed by "0" (24.1 percent), and "5" (17.4 percent). The frequency distribution of the last two digits, which is shown on Figure 4, exhibits a similar pattern, with 99¢ as the most popular price-ending (26.7 percent), followed by 00¢ (20.3 percent), 95¢ (13.8 percent), and 98¢ (4.8 percent).

As mentioned above, the Internet dataset also includes some high-price product categories, which allowed us to examine price-endings in dollar digits as well. In Figure 5, therefore, we present the frequency distribution of the last dollar digit in the Internet data.

According to the figure, "9" was the most popular ending for the dollar digit, with $9 priceendings over-represented with 36.1 percent, followed by $4 price-endings with 9.9 percent, and $5 price-endings with 9.2 percent. The popularity of $4 and $5 ending prices stems from the fact that the actual prices in the low price product categories (Music CDs, Movie DVDs, and Video Games) often are in the $14-$15 range.

A similar pattern emerged for the last two dollar digits, as shown in Figure 6. Not surprisingly, the last two dollar digits of most prices contained "9" also, such as $99, $89, and $09. But more prices ended with $99 than any other two dollar digit endings. Moreover, almost 10 percent ended with $99 among the 100 possible dollar endings of $0 through $99.

We also examined the frequency distribution of the last three digits of prices in the Internet data. According to Table 3 (first column), among the 1,000 possible endings $9.99 was the most popular ending for the last three digits (13.2 percent), followed by $9.00 (10.0 percent), and $9.95 (4.9 percent). When we examined the last four digits of the prices (second column) among the 10,000 possible endings $99.99 was the most popular ending (3.47 percent), followed by $99.00 (3.46 percent), and $19.99 (2.16 percent).

To summarize, in both datasets, "9" was the most popular terminal digit overall. But the popularity of "9" was not limited to the penny digit. Rather, it was popular in the dime, dollar, and ten-dollar digits too. The fact that our data include a variety of products with wide-ranging prices and different retail formats further underscores the popularity of "9" and "99" as a terminal cent and dollar digits.

IV. Frequency Distribution of Price Changes

Having documented the dominance of "9" and "99" price endings as the terminal digits in both datasets, we next assessed the extent to which the specific price points "9" and "99" may be contributing to the retail price rigidity. To characterize the price change dynamics, we conducted a 10-state Markov chain analysis for price changes that affect one digit of a price (the penny digit and the dollar digit), and a 100-state Markov chain analysis for price changes that affect two digits of a price (the penny and the dime digits, and the dollar and the 10-dollar digits).

Table 4 displays the 10-state transition probability matrix for the penny digit for the Dominick's data at the four sampled stores. For ease of interpretation, the figures in the matrix (as well as in the remaining matrices) have been normalized, so that the probabilities in all rows and columns combined add up to 1. Considering all 100 possible transition probabilities, it is clear that 9¢-ending prices are the most persistent: 37.87 percent of the 9¢-ending prices preserve the 9¢-ending after the change. Moreover, when non 9¢-ending prices change, they most often end up with 9¢-ending than with any other ending. Considering the diagonal elements of the matrix, after 9¢-ending prices, 5¢-ending prices seem to be the second most persistent with a transition probability of 0.84 percent, followed by 0¢-ending prices, with a transition probability of 0.64 percent. Overall, however, it seems that most of the transition dynamics takes place in the movement to and from 9¢-ending prices. Proportionally, there is very little transition from any particular non-9¢-ending prices to another non-9¢-ending price.

Table 5 displays the 10-state transition probability matrix for the penny digit for the Internet data. Focusing on the diagonal terms, we find that on the Internet 0¢-ending prices are the most persistent, with a transition probability of 20.35 percent. 9¢-ending prices are the second most persistent with a transition probability of 17.68 percent, followed by 5¢-ending prices with a transition probability of 10.63 percent.

Table 6 displays the 10-state transition probability matrix for the dollar digit for the Internet data. Focusing on the diagonal terms, we find that $9-ending prices are significantly more persistent than any other dollar-ending prices, with a transition probability of 11.75 percent. $4-eding prices are the second most persistent with a transition probability of 2.73 percent, followed by $5-ending prices with a transition probability of 2.52 percent. The persistence of the $4 and $5 endings stems from the fact that many price changes in the low price product categories (Music CDs, Movie DVDs, and Video Games) take place in the penny and in the dime digits.

Comparing the figures presented in Tables 5 and6, it appears that the Internet retailers tend not to use 9¢-ending proportionally as often. Instead, they use $9-ending more often. Thus, the use of 9 as a terminal digit increases as we move from the penny and dime digits to the dollar and the 10-dollar digits. Below we offer more evidence consistent with this behavior.

We next report the results of 100-state Markov chain analysis for the terminal two-digits of the price, for the penny and the dime digits for both data sets, and for the dollar and the 10dollar digits for the Internet data. The resulting transition probability matrix, however, is 100  100. We, therefore, present only partial results of these analyses. The figures presented in these matrices are normalized as before, so that the probabilities in the entire table add up to 1.

Table 7 lists the top 25 transition probabilities for the penny and the dime digits at the four Dominick's stores. According to these figures, the most common transitions are from 89¢ending prices to 99¢-ending prices with the transition probabilities of 1.34 percent, 1.09 percent, 0.87 percent, and 0.82 percent, for Stores #8, #12, #122, and #133, respectively. These probabilities seem quite high considering the fact that in the 100-state Markov chain there are 10,000 possible transitions. The second most common movement is from a 99¢-ending to a 89¢ending with the transition probability of 1.03 percent, 0.86 percent, and 0.70 percent, at Stores #8, #12, and #122, respectively. In Store #122, the second most common movement is from a 39¢-ending to a 49¢-ending, with a transition probability of 0.65 percent. The third most common movement in Stores #8 and #122 is from a 99¢-ending to a 19¢-ending with the transition probability of 0.86 percent and 0.61 percent, respectively, in Store #12 from a 79¢ending to a 99¢-ending with a transition probability of 0.83 percent, and in Store #133 from a 79¢-ending to a 89¢-ending with a transition probability of 0.62 percent.

The transition from 99¢-ending prices to 99¢-ending prices come only in the 13 th , 12 th , 15 th and 18 th places for Stores #8, #12, #122, and #133, respectively, with the corresponding transition probabilities of 0.66 percent, 0.61 percent, 0.43 percent, and 0.43 percent. While these figures are quite high, it appears that other movements are more dominant than this particular transition. The reason for this, we believe, is the fact that the average price in the Dominick's data is $2.67. Moreover, in all but two product categories, Analgesics and Laundry Detergents (Beer and Cigarette categories are not discussed as mentioned in footnote 2), the average prices are $3.00 or less. A move from a 99¢-ending price to a 99¢-ending price, therefore, will result in a minimum price increase of 33-50 percent on average and a minimum price decrease of 25-33 percent, on average. Changes of this magnitude seem fairly large and, therefore, we suspect that they are not as frequent.

Table 8 lists the top 25 transition probabilities for the internet data, for the penny and dime digits on the left-hand side and for the dollar and the 10-dollar digits on the right-hand side.

The top three transitions for the penny and dime digits are from 00¢-ending prices to 00¢-ending prices with a transition probability of 18.36 percent, from 99¢-ending prices to 99¢-ending prices with a transition probability of 11.89 percent, and from 95¢-ending prices to 95¢-ending prices with a transition probability of 8.83 percent. The top three transitions for the dollar and the 10dollar digits are from $14-ending prices to $14-ending prices with a transition probability of 1.47 percent, from $11-ending prices to $11-ending prices with a transition probability of 1.36 percent, and from $15-ending prices to $15-ending prices with a transition probability of 1.28 percent. The transition from $99-ending price to $99-dollar ending price came in only the 6 th .

The frequent use of the $11-, $14-, and $15-ending prices stems from the fact that in the low-priced product categories which include Music CD's, Movie DVD's and Video Games' categories, these are not just price endings; these are actual prices. In these categories, therefore, the most common price changes are in the penny and the dime digits, which leave the dollar and the 10-dollar digits unchanged. This finding suggests that price change patterns likely differ between low-priced and high-priced product categories. To explore this possibility, we separated the Internet data into two groups: (1) low-priced product categories which include Music CDs, Movie DVDs, and Video Games, and (2) high-priced product categories which include Computer Monitors, Digital Cameras, DVD Players, Hard Drives, Laptop Computers, PDAs, and Software.

The results of the analyses are reported in Table 9. Beginning with the low-priced product categories, we find that for the penny and the dime digits, the most common transition is from 99¢-ending to 99¢-ending with a transition probability of 16.32 percent, followed by a movement from 98¢-ending to 98¢-ending with a transition probability of 1.80 percent, and a movement from 95¢-ending to 95¢-ending with a transition probability of 1.75 percent. For the dollar and the $10 digits, we find that $14-, $11-, and $15-ending prices are the most popular.

Next, moving to the high priced product categories, we find that for the penny and the dime digits, the most common transition is from 00¢-ending to 00¢-ending with a transition probability of 28.59 percent, followed by a movement from 95¢-ending to 95¢-ending with a transition probability of 12.77 percent, and a movement from 99¢-ending to 99¢-ending with a transition probability of 9.42 percent. For the dollar and the 10-dollar digits, we find that the top three transition probabilities are from $99-ending prices to $99-ending prices with a transition probability of 1.51 percent, from $99-ending prices to $49-ending prices with a transition probability of 0.65 percent, and from $49-ending prices to $99-ending prices with a transition probability of 0.60 percent.

In sum, we find that for the low-priced product categories, price changes that keep the terminal digits at "9" are the most popular in the penny digit, in the penny and dime digits, and in the dollar digit. For the high-priced product categories, price changes that keep the terminal digits at "9" are the most popular in the dollar digit, and in the dollar and 10-dollar digits. These results suggest that the persistent use of 9-ending prices is more likely to occur in the right-most digits for low-priced products, but shift to the left as the products became more expensive. This is consistent with the finding discussed above that "99¢"-to-"99¢" transitions were less common in the Dominick's dataset, which consists of mostly low-priced products.

V. The Effect of Price Points on Price Rigidity

To study the link between 9-ending prices and price rigidity more directly, we use a binomial logit model to estimate price change probabilities. Using the method of maximum likelihood, we estimated the parameters ,  and  of the following equation:

ln (q/(1 -q)) =  +  9_Ending jt +  Product jt + t (1)
where q is the probability of a price change and 9_Ending jt is a 9-ending dummy variable. For the Dominick's data, we estimate two versions of the regression. In the first, the 9_Ending jt dummy equals 1 if the price for product j at time t ends with "9¢" and 0 otherwise. In the second regression, the 9_Ending jt dummy equals 1 if the price for product j at time t ends with "99¢" and 0 otherwise. For the Internet data, we estimate six versions of the regression, corresponding to the six different values of the 9_Ending jt dummy variable for 9¢, 99¢, $9, $9.99, $99 and $99.99.

Product jt represents a set of product-specific dummy variables based on universal product codes (UPCs) in the Dominick's data and other unique product identifiers in the Internet data.

They permit us to account for product-specific effects. For example, products for which 9-ending prices are more common, may tend to be more rigid. 7 The estimation results for the Dominick's data are reported in Table 10. In the table, we present the estimated coefficients of each dummy along with the corresponding odds ratios. For all 27 product categories, the coefficient estimates for the 9¢-ending dummy are negative (all pvalues < 0.0001). The odds ratios, which equal e Coefficient , are all smaller than 1, indicating that 9¢-ending prices are less likely to change than prices that do not end with 9¢. On average, prices that ended with 9¢ were 66 percent less likely to change than prices that did not end with 9¢.

We obtained similar results for the 99¢-ending prices. The coefficient estimates for the 99¢-ending dummy are all negative. For 25 out of 27 categories, they are statistically significant, as shown on the right-hand panel in Table 10. The odds ratios indicate that prices that ended with 99¢ were on average 43 percent less likely to change than prices that did not end with 99¢.

Next, we estimated the same logit regression model for the Internet data, using dummies for 9¢, 99¢, $9, $9.99, $99, and $99.99, in turn, as the independent variables. As with the Dominick's dataset, we included product dummies to account for product-specific effects. The estimation results are reported in Table 11. Similar to what we found with the Dominick's dataset, 9-ending prices were less likely to change than other prices. Overall, 9¢-ending prices were 25 percent, 99¢-ending prices 36 percent, $9-ending prices 36 percent, $99-ending prices 55 percent, $9.99-ending prices 45 percent, and $99.99-ending prices 64 percent less likely to change than other prices. We obtained similar results for the individual product categories. In 96 percent (52 out of 54 categories) of all possible cases in the category-level analyses, the effect of 9 price-endings on the probability of price changes was negative and significant.

Thus, prices seem to be "stuck" at 9-and 99-endings, making them more rigid: 9¢-and 99¢-ending prices at Dominick's as well as on the Internet are less likely to change than other prices. On the Internet, the findings hold also for $9-, $9.99-, $99-, and $99.99-ending prices.

VI. The Effect of Price Points on the Size of Price Change

If pricing points inhibit price changes, then they might also be expected to affect the sizes of price increases. Specifically if prices that are at price points are fixed longer than other prices, then any subsequent price adjustments might be expected to be larger than average. Anil Kashyap (1995, p. 267) If 9-ending prices are less likely to change in comparison to non-9-ending prices, then the average size of change of 9-ending prices should be larger when they do change, in comparison to non-9-ending prices. This assumes that the cost of a price change is the same regardless of the price-ending, which we believe is indeed the case according to the menu cost estimates of [START_REF] Levy | The Magnitude of Menu Costs: Direct Evidence from Large U.S. Supermarket Chains[END_REF], 1998[START_REF] Knotek | has focused on other types of pricing practices, especially the common use of round prices, which he terms "convenient prices" because their use reduces the amount of the change used in a transaction[END_REF]), Zbaracki, et al. (2004, 2007), and [START_REF] Dutta | Menu Costs, Posted Prices, and Multiproduct Retailers[END_REF] for large U.S.

supermarket and drugstore chains.

In Table 12, we report the average size of price changes for 9-ending and non-9-ending prices for both datasets. In the table, we also report the corresponding results for the low quartile of the products in terms of the popularity of 9-ending prices. The goal of this analysis is to assess the possibility that the findings we are documenting in this section may be driven by the frequent use of 9-endings. By limiting the analysis to the low quartile of the products in terms of the use of 9-endings, we are offering the most conservative test for this hypothesis.

In the Dominick's dataset, the average price change was 75¢ if the price ended with 9¢, in contrast to a 40¢ change when it did not end with 9¢, an 88 percent difference. The findings for the 99¢-ending prices are also consistent: the average price change was 91¢ if the price ended with 99¢, in contrast to a 55¢ change when it did not end with 99¢. This amounts to a 65 percent difference.

Similarly, when we focused on the low quartile of products in terms of the popularity of 9-ending prices, the average price change was 38¢ if the price ended with 9¢, in contrast to a 33¢ change when it did not end with 9¢, a 15 percent difference. For the 99¢-ending prices, the average price change was 49¢ if the price ended with 99¢, in contrast to a 34¢ change when it did not end with 99¢. This is a 44 percent difference.

With the Internet data, we considered prices ending with 9¢, 99¢, $9, $9.99, $99, and $99.99, again for the entire dataset, as well as for the low quartile of products. When we considered the entire Internet dataset, for the 9-ending prices, the average price changes were $15.54, $22.40, $32.13, $33.97, $66.15, and $63.04 for 9¢-, 99¢-, $9-, $9.99-, $99-, and $99.99ending prices, respectively. The corresponding non-9-ending average price changes were $18.07, $16.78, $12.83, $16.30, $15.20, and $16.88, respectively. In other words, the 9-ending price changes were higher than non-9-ending price changes by about -14 percent, 33 percent, 150 percent, 108 percent, 335 percent, and 273 percent, respectively. Only in one case (Notebook PCs, 9¢-vs. non-9¢-endings), was the average 9-ending price change lower than the average non-9-ending price change. See Table R22 in supplementary appendix.

When we considered the low quartile data, for 9-ending prices, the average price changes were $24.02, $27. 78, $11.93, $22.47, $49.61, and $38.24 for the 9¢-, 99¢-, $9-, $9.99-, $99-, and $99.99-ending prices, respectively. The corresponding non-9-ending average price changes were $21.03, $20.76, $7.21, $7.38, $18.27, and $19.21, respectively. Thus, the 9-ending price changes for the low quartile products were higher than non-9-ending price changes by about 14 percent, 34 percent, 65 percent, 204 percent, 172 percent, and 99 percent, respectively. Thus, the average size of the 9¢-ending and 99¢-ending price changes systematically exceed the average size of the non-9¢-ending and non-99¢-ending price changes, respectively.

The fact that the results are similar for the overall data and the products in the low quartile suggests that in terms of the 9¢ use, the difference is unlikely to be driven by product-specific effects that could simultaneously increase the prevalence of 9-ending prices and the magnitude of the price changes. If that were the case, we should not have observed larger price changes for 9-ending and 99-ending prices in the low quartile of products for which 9-ending prices are less common. These findings are consistent with our predictions: since 9-ending and 99-ending prices are less likely to change, the average sizes of the changes of the 9-ending and 99-ending prices are systematically larger when they do change, in comparison to the non-9-ending and non-99ending prices, respectively.

VII. Robustness

To explore the robustness of the findings, we conducted several additional analyses, much of them following the referees' comments and suggestions. The findings we have reported for the Dominick's data were based on the analysis of the price data from the chain's four stores.

We, however, have also analyzed the data for each of the four sampled stores individually, as well as the chain's entire dataset which include the price information from all 93 stores. In each case, we have considered the data for all 27 categories combined, as well as for each individual product category. For the Internet data, we have primarily reported the results of the aggregate data analysis. However, most of the analyses were repeated for each product category. In general, the results of these additional analyses are similar to the results that we have reported.

Here we offer some details about these analyses and the findings. More detailed presentation of these analyses is included in the supplementary appendix.

A. Evidence on the Frequency Distribution of 9-and 99-Ending Prices

We found that 9¢-and 99¢-ending prices were more popular than other endings at the Dominick's data (for all 93 stores combined), and at each one of the four individual stores sampled. At the category level, we found that 9¢-ending prices were more popular than other endings at all 27 product categories, while 99¢-ending prices were more popular than other endings in 23 of the 27 product categories.

For the Internet data, we found that 9¢-ending and 99¢-ending prices were more popular than other endings for four product categories, while the 0¢-ending was the most popular for the remaining six categories. For the dollar digit, 9-endings were more popular than other endings in 8 of the 10 categories. For the last two dollar digits, $99-ending prices were more popular than the other price-endings in 6 of the 10 categories. 8 We have also considered the possibility that the use of 9-and 99-ending prices is related to the sales volume. The analysis of 9-and 99-ending prices by sales volume, however, suggests no such systematic relationship. The results suggest that 9-ending prices are popular for both products that have a large sales volume and products that have a small sales volume.

B. Evidence on the Frequency Distribution of Price Changes

Similar to the other results that we have reported in this paper, we found that for regular prices in each of the four Dominick's stores, as well as for all 93 stores combined and for all prices, "9"-to-"9" was the most popular price change. For example, 37.74 percent of the transition takes place from 9¢-ending to 9¢-ending prices. 5¢-to-5¢ and 0¢-to-0¢ ending transitions only occur with 0.90 percent and 0.66 percent probabilities. The 9¢-ending prices are the most persistent if we consider the entire Dominick's data as well. "99"-to-"99" is not the most popular price change for any of the four stores, similar to the results reported earlier in the paper, but it is the most popular when all prices from all stores are considered. For the Dominick's dataset, in all but one category (Front-End Candies), there were considerably more price changes that were multiples of dimes and dollars for 9-ending prices.

For the Internet data, in the low-priced product categories, we found considerably more price changes that were multiples of dimes and dollars for 9-ending prices. For high-priced product categories, we found more price changes that were multiples of $10 and $100 for 9ending prices.

C. Evidence on the Link between 9-and 99-Ending Prices and Price Rigidity

We find a strong positive link between price points and price rigidity at the level of the entire Dominick's chain, as well at each one of the four sampled stores examined. Beginning with Store #8, we find that the probability of a change of a 9¢-ending and a 99¢-ending prices are on average 60 percent and 28 percent lower than non-9¢-ending and non-99¢-ending prices, respectively. The result holds true for most product categories: overall, in 50 of the 54 cases ( 27coefficients for the 9¢-ending dummy and 27 coefficients for the 99¢-ending dummy) the coefficient of the 9-ending dummy was negative. In 48 of these 50 cases, they were statistically significant. We found similar results for the remaining 3 stores. For example, at Store #12, the estimated coefficient was negative in 51 of the 54 cases, with 48 of them being statistically significant. At Store #122, the estimated coefficient was negative in 53 of the 54 cases, with 50 of them being statistically significant. At Store #133, the estimated coefficient was negative in 53 of the 54 cases, with 51 of them being statistically significant. The findings for the entire Dominick's dataset are even stronger: all 54 estimated coefficients were negative and statistically significant.

D. Evidence on the Link between 9-and 99-Endings and the Size of Price Changes

In the Dominick's dataset, in 23 of the 27 categories the average price change was higher for 9¢-ending than for non-9¢-ending prices. The findings that we obtained for the 99¢-ending prices are even stronger. In 26 categories (the exception is Frozen Entries), the average change was higher for 99¢-ending than for non-99¢-ending prices. Similarly, when we focused on the low quartile of products in terms of the popularity of 9-ending prices, we found that in 21 categories the average change was higher for 9¢-ending than for non-9¢-ending prices. For the 99¢-ending prices, in 25 categories the average price change was higher for the 99¢-ending than for non-99¢-ending prices.

With the Internet data, we considered prices ending with 9¢, 99¢, $9, $9.99, $99, and $99.99, again for the entire dataset, as well as for the low quartile of products. For the entire dataset we find that the average price change was higher if the price ended with 9 in comparison to non-9 ending prices in 8, 9, 9, 9, 8, and 7 categories for 9¢, 99¢, $9, $9.99, $99, and $99.99 ending prices, respectively. 9 Thus, in 50 of the 56 cases, the average size of the price change was higher if the price ended with a 9-ending price point in comparison to non-9¢-ending prices.

The results for the low quartile of products are similar. Specifically, we find that the average price change was higher if the price ended with 9 in comparison to non-9 ending prices in 7, 10, 9, 9, 6, and 6 categories for 9¢, 99¢, $9, $9.99, $99, and $99.99 ending prices, respectively. 10 Overall, in 47 of the 54 cases the average size of the price change was higher if the price ended with a 9-ending price point than with a non-9-ending price.

VIII. Conclusion

To our knowledge, this is the first study that directly examines the effect of price points on price rigidity across a broad range of product categories, price levels, and retailers, in the traditional retailing and the Internet-based selling formats, using data from the U.S. We found that 9-ending prices were the most popular and were less likely to change compared to non-9ending prices. Further, the most common price changes preserve the terminal digits at "9" and the size of the price changes was larger for these 9-ending prices than for non-9-ending prices.

We also discovered that there is a shift in this preservation of 9-ending prices with the price level: for more expensive product categories we saw less frequent persistence of 9's in the penny and the dime digits, but more frequent persistence of 9's in the dollar, $10, and $100 digits.

Overall, we find that for the Dominick's data 9-ending prices are at least 43-66 percent less likely to change than non-9-ending prices. For the Internet data, these probabilities are in the range of 25-64 percent. These figures seem to us quite substantial. We conclude therefore, that 9-ending and 99-ending prices form a considerable barrier to price changes, offering direct evidence on the link between price points and price rigidity. Combining this with the robustness of the findings-occurring in both datasets, across a wide range of product categories with a wide range of prices, products, retail formats and retailers, suggests that price points might be substantial enough to have broader macro implications. This is reinforced by the finding that the use of 9s shifts leftwards as the products' average price increases, which suggest that the phenomenon of 9-ending prices rigidity may exist in markets for other goods and services in more expensive product categories where the use of 9-endings in $1, $10, $100 digits, etc. is quite common. These include prices of the goods sold at department stores such as clothes, shoes, fragrances, jewelry, and high tech equipment, as well as other high priced products and services such as musical instruments, furniture, cars, home appliances, hotels, air travel, car rentals, and even in pricing of homes and apartments. Taken together, these goods and services comprise a substantial proportion of the aggregate consumption and thus may have a considerable economic significance.

The use of 9-ending prices seems to be relevant in the context of public policy issues as well. For example, the use of 9-ending prices is often debated in countries where low-denomination coins have been abolished. When small denomination coins are no longer used, transactions involving small changes must rely on rounding, as is the case in Israel, Hungary, or Singapore. In Israel, for example, the 1-Agora coin was abolished in 1991, and the 5-Agora coin was eliminated in 2008. The law, therefore, requires that the final bills be rounded up (if it ends with 5-Agora-9-Agora) or down (if it ends with 1-Agora-4-Agora) to the nearest 10-Agora. It turns out, however, that the Israeli retailers use 9-ending prices extensively, which irritates consumers, who claim that 9-ending prices are unethical given the absence of 1-Agora coin. The Israeli Parliament has twice rejected a proposed law which would outlaw the use of 9-ending prices. 11 This may extend to other countries soon. For example, dropping the smallest currency unit has been a recent topic of debate in the U.S., Canada and Europe 12 . Australia has stopped issuing 1¢ and 2¢ coins in 1989. New Zealand ceased issuing the 1¢ and 2¢ coins in 1989. The common use of price points has also received considerable attention in some European Union countries in the context of the conversion of prices from local currencies to the euro. The concern has been about the possibility that retailers may have acted opportunistically by rounding their prices upward after conversion to the euro in their attempt to preserve the price points. This appears to be true, for example, in the case of products that are sold through automated devices, such as soda and candy bar vending machines, parking meters, coin-operated laundry machines, etc. [START_REF] Bils | Some Evidence on the Importance of Sticky Prices[END_REF]Klenow 2004, Levy and[START_REF] Levy | The Real Thing: Nominal Price Rigidity of the Nickel Coke, 1886-1959[END_REF][START_REF] Campbell | Rigid Prices: Evidence from U.S. Scanner Data[END_REF][START_REF] Campbell | Rigid Prices: Evidence from U.S. Scanner Data[END_REF] avenues for future research.

We end by noting that the Internet provides a unique context for micro-level studies of price setting behavior [START_REF] Bergen | Beyond the Hype of Frictionless Markets: Evidence of Heterogeneity in Price Rigidity on the Internet[END_REF]. The ability to access transaction price data using software agents has allowed us to explore pricing and price adjustment patterns at a low cost and with a previously unimaginable level of microeconomic detail. This approach also allows empirical research methods to take advantage of natural experiments in the real world (Kauffman andWood, 2007, 2009). With the expanding retail activities on the Internet, and new techniques and tools that have become available, we expect such opportunities to increase further in the future.

Note:

The data are weekly. The sampled stores belong to four price tiers as follows: Store #8 -"low" price tier, #12 -"high" price tier, #122 -"Cub Fighter," and #133 -"medium" price tier. See section II for details. 2.17 8 0.15 0.10 0.11 0.14 0.14 0.29 0.13 0.13 0.12 1.43 9 3.40 1.58 1.45 1.88 2.34 3.15 1.85 1.77 0.85 37.87 Note: Each cell contains the percentage of the price change compared to the total price change (i.e., [START_REF]For recent surveys[END_REF]374,142). The top three highest transition probabilities on the matrix diagonal are indicated in boldface.

  Denmark stopped issuing the 5-and 10-ores in 1989. The Dutch eliminated the 1¢ of the guilder in 1980 and ceased issuing the 1¢ and 2¢ of Dutch euro coins in 2006. In Finland, the 1¢ and 2¢ of Finnish euro coins are not in general use any longer. In 2008, Hungary eliminated the 1 and 2 forint coins. France, Norway, Britain and Singapore have also eliminated lowdenomination coins.

  Each cell contains a coefficient and odds ratio in parenthesis; significance levels: *** < 0.01, ** < 0.05, * < 0.10. The estimated coefficients in italics indicate unsupportive results.

Figure 1

 1 Figure 1. Frequency Distribution of the Last Digit

Table 1 . Descriptive Statistics for the Dominick's Price Data, Stores #8, #12, #122 and #133 Category Number of Observations Number of Products

 1

	Mean	Std.	Min.	Max.
	Price	Dev.	Price	Price

Table 2 . Descriptive Statistics for the Internet Price Data Category Number of Observations Number of Products Number of Retailers Mean Price

 2 The table covers 743 daily price observations fromMarch 26, 2003 to April 15, 2005, from 293 Internet retailers for 474 products. The retailers have many different product categories (e.g., Amazon.com sells books, CDs, DVDs, computer products and electronics, etc.). Consequently, the sum of the number of retailers in each product category will not necessarily be consistent with the total number of stores in all product categories. In addition, some retailers do not have all products (e.g., in our sample, Amazon has 15 Music CDs while Barnes & Noble has 20). Also, the length of individual product's price time series varies due to different life cycle of products. Thus, the number of observations in the Music CDs category, for example, 302,914, is less than total available combinations(i.e., 46  15  743 = 512,670.)

	Std.	Min.	Max.
	Dev.	Price	Price

Table 4 . Transition Probability Matrix Conditional on a Price Change for a 10-State Markov Chain, Dominick's Data, Stores #8, #12, #122, #133, Regular Prices Only, for the

 4

					Penny Digit
					Next Period Ending Digit (¢)
		0	1	2	3	4	5	6	7	8	9
	Current Ending Digit (¢)	0 0.64 0.25 0.29 0.31 0.33 0.79 0.26 0.23 0.16 1 0.26 0.14 0.18 0.21 0.14 0.44 0.14 0.13 0.09 2 0.25 0.13 0.15 0.18 0.19 0.36 0.18 0.15 0.09 3 0.28 0.20 0.16 0.33 0.22 0.47 0.20 0.24 0.15 4 0.30 0.12 0.17 0.18 0.29 0.40 0.23 0.17 0.11 5 0.72 0.30 0.32 0.42 0.33 0.84 0.43 0.49 0.26 6 0.26 0.15 0.18 0.21 0.26 0.37 0.20 0.29 0.14 7 0.23 0.14 0.15 0.28 0.21 0.41 0.25 0.24 0.13	3.68 2.98 1.81 2.47 2.93 3.81 2.15

Table 5 . Transition Probability Matrix Conditional on a Price Change for a 10-State Markov Chain, Internet Data, for the Penny Digit

 5 Each cell contains the percentage of the price changes compared to the total number of price changes (41,034). The top three highest transition probabilities on the matrix diagonal are indicated in boldface.

					Next Period Ending Digit (¢)
		0	1	2	3	4	5	6	7	8	9
	Current Ending Digit (¢)	0 20.35 0.35 0.35 0.34 0.33 1.40 0.39 0.38 0.52 1 0.32 0.39 0.33 0.32 0.34 0.29 0.30 0.28 0.30 2 0.40 0.33 0.47 0.34 0.34 0.27 0.24 0.31 0.34 3 0.34 0.29 0.32 0.47 0.33 0.35 0.32 0.30 0.41 4 0.37 0.34 0.37 0.31 0.66 0.52 0.40 0.38 0.37 5 1.45 0.33 0.30 0.34 0.48 10.63 0.45 0.34 0.53 6 0.34 0.29 0.31 0.34 0.43 0.48 0.86 0.41 0.30 7 0.39 0.27 0.27 0.37 0.36 0.32 0.33 0.66 0.49 8 0.54 0.33 0.30 0.37 0.44 0.58 0.41 0.48 2.95 9 1.54 0.42 0.42 0.48 0.87 2.19 0.54 0.56 1.47 17.68 1.69 0.40 0.32 0.43 0.87 2.04 0.66 0.58 1.21
	Note:

Table 6 . Transition Probability Matrix Conditional on a Price Change for a 10-State Markov Chain, Internet Data, for the Dollar Digit

 6 Each cell contains the percentage of the price changes compared to the total number of price changes (41,034). The top three highest transition probabilities on the matrix diagonal are indicated in boldface.

					Next Period Ending Digit ($)
		0	1	2	3	4	5	6	7	8	9
	Current Ending Digit ($)	0 1.58 0.85 0.45 0.40 0.42 0.43 0.35 0.41 0.68 1.38 1 0.98 2.18 1.06 0.49 0.40 0.35 0.33 0.40 0.43 0.97 2 0.58 1.19 1.72 1.01 0.76 0.56 0.34 0.32 0.48 1.12 3 0.46 0.67 1.23 1.99 1.12 0.65 0.50 0.42 0.51 1.00 4 0.55 0.49 0.87 1.30 2.73 1.32 0.69 0.65 0.62 1.98 5 0.49 0.44 0.61 0.90 1.50 2.52 1.01 0.67 0.54 1.45 6 0.36 0.37 0.42 0.52 0.88 1.15 1.47 0.86 0.64 1.04 7 0.33 0.30 0.41 0.48 0.79 0.79 1.14 1.27 0.88 1.22 8 0.49 0.39 0.38 0.57 0.56 0.72 0.71 1.11 1.73 1.79 9 1.08 0.83 0.81 0.91 1.98 1.56 1.25 1.47 2.09 11.75
	Note:

Table 8 . Top 25 Transition Probabilities Conditional on a Price Change for a 100-State Markov Chain, Internet Dataset, for the Penny and Dime Digits (LHS) and for the Dollar and $10 Digits (RHS)

 8

			Cents			Dollars
		Current	Next		Current	Next
	Rank	Ending	Ending	%	Ending	Ending	%
	1	00	00	18.36	14	14	1.47
	2	99	99	11.89	11	11	1.36
	3	95	95	8.83	15	15	1.28
	4	98	98	1.13	09	09	1.23
	5	00	99	0.89	13	13	1.16
	6	99	00	0.85	99	99	1.01
	7	99	95	0.72	12	12	0.80
	8	00	95	0.66	10	10	0.67
	9	99	98	0.64	08	08	0.63
	10	99	49	0.62	14	15	0.59
	11	49	99	0.62	16	16	0.58
	12	95	00	0.62	15	14	0.54
	13	95	99	0.57	14	13	0.49
	14	98	99	0.54	12	11	0.48
	15	49	49	0.28	13	14	0.48
	16	00	50	0.25	11	12	0.44
	17	88	88	0.24	22	22	0.43
	18	50	00	0.23	12	13	0.42
	19	85	85	0.20	13	12	0.42
	20	96	96	0.19	99	49	0.42
	21	89	99	0.19	19	19	0.41
	22	00	90	0.18	11	10	0.39
	23	96	99	0.18	21	21	0.39
	24	24	99	0.17	49	99	0.38
	25	97	97	0.16	10	11	0.35
	Note: Total number of price changes = 41,034

Table 9 . Top 25 Transition Probabilities Conditional on a Price Change for a 100-State Markov Chain, by Price Level, Internet Data, for the Penny and Dime Digits (LHS) and for the Dollar and $10 Digits (RHS)

 9

					Cents					Dollars
		Low-Priced	High-Priced	Low-Priced	High-Priced
		Categories	Categories	Categories		Categories
	Ran k	Curre nt Endin g	Next Endi ng	%	Curr ent Endi ng	Next Endin g	%	Cur rent Endi ng	Next Endi ng	%	Cur rent ng Endi	Next ng Endi	%
	1	99	99	16.32 00	00	28.59 14	14	4.03 99	99	1.51
	2	98	98	1.80 95	95	12.77 11	11	3.72 99	49	0.65
	3	95	95	1.75 99	99	9.42 15	15	3.53 49	99	0.60
	4	99	98	1.19 00	99	1.34 09	09	3.31 99	79	0.54
	5	49	99	1.04 99	00	1.29 13	13	3.21 79	99	0.40
	6	98	99	0.97 00	95	1.02 12	12	2.18 99	89	0.39
	7	99	49	0.95 95	00	0.96 10	10	1.84 49	39	0.33
	8	96	96	0.50 99	95	0.94 08	08	1.62 49	49	0.28
	9	24	99	0.45 98	98	0.76 14	15	1.59 89	79	0.28
	10	99	24	0.42 95	99	0.75 16	16	1.55 79	69	0.28
	11	96	99	0.40 99	49	0.44 15	14	1.40 39	29	0.27
	12	89	99	0.37 00	50	0.39 13	14	1.26 49	29	0.25
	13	88	88	0.37 49	99	0.39 14	13	1.25 29	99	0.25
	14	99	95	0.34 50	00	0.35 12	11	1.17 99	69	0.25
	15	99	19	0.33 99	98	0.33 11	12	1.16 99	94	0.24
	16	82	82	0.28 49	49	0.32 22	22	1.15 59	49	0.23
	17	99	89	0.27 98	99	0.30 12	13	1.12 99	98	0.23
	18	19	99	0.26 85	85	0.29 13	12	1.06 79	49	0.22
	19	95	99	0.26 00	90	0.27 19	19	1.06 19	99	0.22
	20	99	39	0.25 97	97	0.22 21	21	1.01 69	59	0.21
	21	99	29	0.25 90	00	0.20 11	10	0.94 89	99	0.21
	22	49	59	0.24 94	99	0.18 10	11	0.90 99	29	0.20
	23	49	49	0.22 90	90	0.17 23	23	0.84 29	19	0.20
	24	09	95	0.21 99	94	0.17 16	17	0.78 09	99	0.20
	25	59	69	0.21 88	88	0.17 17	16	0.74 19	09	0.18
	Note: Low-priced categories include CDs, DVDs, and Video Games. High-priced categories
	include Computer Monitors, Digital Cameras, DVD Players, Hard Drives, Laptop Computers,
	PDAs, and Software.

Table 10 . Results of the Logit Regression (Equation 1) Estimation for the Dominick's Data, Regular Prices, Stores #8, #12, #122 and #133

 10 Ending j are dummy variables, which equal 1 if the price ends with 9 or 99, and 0 otherwise. All p-values < 0.0001, except for the coefficients formatted in italic (Bathroom Tissues and Frozen Juices, for 99¢-ending dummy), for which p > .10. The average odds ratios reported in the last row of the table are the simple averages of the odds ratios for each product category.

		9¢-Ending	99¢-Ending
		(9-Ending 9 = 1)	(9-Ending 99 = 1)
	Category	Coefficient Odds Ratio	Coefficient	Odds Ratio
	Analgesics	 1.4820	0.23	 0.3599	0.70
	Bath Soap Bathroom Tissues	 1.6871  0.4763	0.19 0.62	 0.7683  0.0353	0.46 0.97
	Bottled Juices	 0.7232	0.49	 0.4984	0.61
	Canned Soup	 0.4553	0.63	 0.6055	0.55
	Canned Tuna	 0.7692	0.46	 0.5518	0.58
	Cereals	 0.5013	0.61	 0.3582	0.70
	Cheeses	 1.7457	0.17	 1.1008	0.33
	Cookies	 2.1156	0.12	 1.1052	0.33
	Crackers	 1.8639	0.16	 0.9784	0.38
	Dish Detergent	 1.0433	0.35	 0.7082	0.49
	Fabric Softeners	 0.6951	0.50	 0.3909	0.68
	Front-End Candies	 0.8917	0.41	 1.5532	0.21
	Frozen Dinners	 1.3773	0.25	 0.6168	0.54
	Frozen Entrees Frozen Juices	 1.1704  0.3795	0.31 0.68	 0.6649  0.0395	0.51 0.96
	Grooming Products	 2.2234	0.11	 0.6918	0.50
	Laundry Detergents	 1.5275	0.22	 0.5607	0.57
	Oatmeal	 1.0142	0.36	 0.2450	0.78
	Paper Towels	 0.6164	0.54	 0.7879	0.45
	Refrigerated Juices	 0.8902	0.41	 0.4119	0.66
	Shampoos	 2.1695	0.11	 0.3264	0.72
	Snack Crackers	 1.9320	0.14	 0.8181	0.44
	Soaps	 1.6669	0.19	 0.6347	0.53
	Soft Drinks	 3.1645	0.04	 0.6425	0.53
	Tooth Brushes	 0.9833	0.37	 0.5719	0.56
	Tooth Pastes	 0.6796	0.51	 0.6291	0.53
	Average		0.34		0.57
	Note: 9-

Table 11 . Results of Logit Regression (Equation 1) Estimation for the Internet Dataset

 11

		9¢-	99¢-	$9-	$99-	$9.99-	$99.99-
	Category	Endings	Endings	Endings	Endings	Endings	Endings
	Music CDs	-0.0727 *** (0.9299)	-0.5463 *** (0.5791)	-0.0125 *** (0.9876)		-0.4430 *** (0.6421)
	Movie DVDs	-0.4716 *** (0.6240)	-0.5827 *** (0.5584)	-0.3551 *** (0.7011)		-0.9068 *** (0.4038)
	Video Games	0.1630 *** (1.1770)	0.0729 *** (1.0756)	-0.3572 *** (0.6996)		-0.2807 *** (0.7553)
	Software	-0.3185 *** (0.7272)	-0.4998 *** (0.6067)	-0.5892 *** (0.5548)	-1.0831 *** (0.3385)	-0.8032 *** (0.4479)	-1.4014 *** (0.2463)
	PDAs	-0.1496 *** (0.8611)	-0.2253 *** (0.7983)	-0.4370 *** (0.6460)	-0.5944 *** (0.5519)	-0.4041 *** (0.6676)	-0.8986 *** (0.4071)
	Hard Drives	-0.2276 *** (0.7964)	-0.2777 *** (0.7575)	-0.3368 *** (0.7141)	-0.3242 *** (0.7231)	-0.5197 *** (0.5947)	-0. 6072 *** (0. 5449)
	DVD Players	-0.5161 *** (0.5968)	-0.5808 *** (0.5595)	-0.7455 *** (0.4745)	-0.5246 *** (0. 5918)	-0.6718 *** (0.5108)	-0.6074 *** (0.5448)
	PC Monitors	-0.1893 *** (0.8275)	-0.3734 *** (0.6884)	-0.5445 *** (0.5801)	-0.7598 *** (0.4678)	-0.7457 *** (0.4744)	-1.3102 *** (0.2698)
	Digital
	Cameras

Table 12 . Comparing Average Size of Price Change Between 9-and Non-9-Ending Prices: for Dominick's (Regular Prices; Stores #8, #12, #122 and #133) and for the Internet All Products Low Quartile of Products in Terms of Popularity of 9-Ending Prices

 12

			Non-9-			9-	Non-9-
	9-Endings	Endings t-Stat p-Value	Endings	Endings t-Stat p-Value
	Dominick's
	9¢	$0.75	$0.40	934.87	.000	$0.38	$0.33	27.61	.000
	99¢	$0.91	$0.55	721.24	.000	$0.49	$0.34	53.64	.000
	Internet
	9¢ $15.54	$18.07	-4.50	.000	$24.02	$21.03	2.75	.006
	99¢ $22.40	$16.78	5.55	.000	$27.78	$20.76	4.56	.000
	$9 $32.13	$12.83	33.65	.000	$11.93	$7.21	5.67	.000
	$9.99 $33.97	$16.30	17.34	.000	$22.47	$7.38	5.99	.000
	$99 $66.15	$15.20	42.89	.000	$49.61	$18.27	8.56	.000
	$99.99 $63.04	$16.88	19.93	.000	$38.24	$19.21	4.78	.000

The products in Beers and Cigarettes categories are highly regulated, which might skew the results[START_REF] Besley | Sales Taxes and Prices: an Empirical Analysis[END_REF]. We, therefore, do not discuss the results for these two categories.

Dominick's data are available at http://research.chicagobooth.edu/marketing/databases/dominicks/stores.aspx. The site contains detailed information about the location of the stores, as well as detailed description of the data files, product categories included, etc. The site also discusses various measurement issues.

When the sellers' websites were inaccessible or the price information was not available, instances of missing data occurred. The software agent used the following algorithm to address this issue. If 10% or more observations were missing for a product, then that series was excluded from the data altogether. If less than 10% of the data were missing, then the algorithm examined if the prices for the day before and the day after were the same. If they were the same, then the software agent automatically filled in the missing data with that price. Otherwise, it filled in the missing data with the price for the day after. Only 0.075% of the Internet dataset was interpolated this way because of missing observations, and thus missing data are unlikely to affect our results.

Research Division, at the Research Department of the European Central Bank, where Daniel Levy was a visiting scholar. He is grateful to the Bank's Research Department for the hospitality. Daniel Levy gratefully acknowledges also the financial support from the Adar Foundation of the Economics Department at Bar-Ilan University. Dongwon Lee's research is supported by an eBRC Doctoral Support Award from Pennsylvania State University and a Research Grant from Korea University. Rob Kauffman acknowledges partial support from the MIS Research Center, and the W.P. Carey Chair in Information Systems, Arizona State University.

 Wooldridge, J. (2002), Econometric Analysis of Cross Section and Panel Data (Cambridge, MA: MIT Press).

Note:

The figures in each column are ordered from the most frequent to the least frequent. Bold-marked prices in the first three rows indicate that they are in the top three most frequent in each category. The right-most column shows the top ten most frequent price changes after three product categories (Music CDs, Movie DVDs, and Video Games) are excluded from the analysis. 6 Indeed, according to [START_REF] Dutta | Menu Costs, Posted Prices, and Multiproduct Retailers[END_REF] and [START_REF] Levy | The Magnitude of Menu Costs: Direct Evidence from Large U.S. Supermarket Chains[END_REF]Levy et al. ( , 1998)), the average price of an item in large U.S. supermarket chains during 1991-1992 was about $1.70. [START_REF] Bergen | On the Inefficiency of Item Pricing Laws: Theory and Evidence[END_REF] have noted that the figure increased to $2.08 by 2001. In our four-store sample, the average price is $2.67. See Table 1.

50 7 In an earlier analysis, we ran the above regression without the product dummies and obtained similar results. When we correlated the proportion of 9-ending prices for each product category with the regression coefficient of the 9-dummy from this earlier analysis, we obtained a significantly negative correlation for the 9¢ ending prices, suggesting the presence of some product specific effects. For the 99¢-ending prices the correlation coefficient was positive but statistically insignificant. We chose to include the product dummies in the results we report here.

8 Three individual product categories with low average prices exhibited some variation in their price endings. For example, for the dollar digit, the $3, $4 and $5 price-endings were the most common for CDs and DVDs. That is because the prices in these categories usually range between $13 and $16. Also, the $99 and $99.99 endings were not common in those two categories or the category of Video Games, because the average prices in these categories are less than $100. We, therefore, did not see frequent 9-endings for the dollar and ten-dollar digits in these categories.

9 Two categories, Music CDs and Video Games, contained no prices with a $99-and $99.00 endings.

10 There were no Music CDs, Music DVDs or Video Games with $99-or with $99.99-ending prices.

11 See, for example, http://www.globes.co.il/news/article.aspx?did=1000403091 (in Hebrew).

12 In the July 19, 2001 issue of the USA Today, L. Copland reported that "France, Spain and Britain quit producing low-denomination coins in recent decades because production costs kept going up while the coins' purchasing power went down." More recently, it has been reported that in many European countries which have adopted the Euro, the public seems to be exhibiting resistance to the use of 1-cent and 2-cent denomination coins. This is due to the inconvenience to make the U.S. penny obsolete. The bill was defeated. Previous attempts made in 1990 and 1996 also died in Congress.

13 A recent New York Time report [START_REF] Toy | Sometimes, Lucky Numbers Add Up to Apartment Sales[END_REF] lists numbers that have particular significance in some cultures. Even the sounds of the numbers can suggest good or bad luck. For example, the number 8 represents luck to Cantonese Chinese because it sounds like multiply or get rich (fa in Cantonese). In Japan, 8 also has great symbolic significance because the writing of the number 8 looks like a mountain ("八"), and thus the number 8 signifies growth and prosperity. In the Jewish culture the number 18 has a special significance because numerically it is equivalent to chai which means life, and therefore, donations made by Jews are often in multiples of 18. In the Indian society, the person's birth date is used to determine the person's lucky numbers based on Vedic astrology.

14 The cultural importance of numbers is not limited to "happy endings." For example, according to [START_REF] Mirhadi | Hotels Reach Overseas to Fill Beds, Restaurants, Showrooms[END_REF], when the Masquerade Tower was added to Hotel Rio in Las Vegas in 1997, the architects decided to skip the 40th to the 49th floors because the Arabic numeral "4" in Chinese sounds similar to the word "death." The elevators in the building went directly from the 39th floor to the 50th floor. According to [START_REF] Toy | Sometimes, Lucky Numbers Add Up to Apartment Sales[END_REF], in many residential and commercial buildings the 13th floor is missing, skipping from the 12th floor to the 14th floor.