Structural Econometric Modeling: Rationales and Examples from


Download Structural Econometric Modeling: Rationales and Examples from


Preview text

Structural Econometric Modeling: Rationales and Examples
from Industrial Organization

by
Peter C. Reiss Graduate School of Business Stanford University Stanford, CA 94305-5015 [email protected]

Frank A. Wolak Department of Economics
Stanford University Stanford, CA 94305-6072
[email protected]

Abstract
This chapter explains the logic of structural econometric models and compares them to other types of econometric models. We provide a framework researchers can use to develop and evaluate structural econometric models. This framework pays particular attention to describing different sources of unobservables in structural models. We use our framework to evaluate several literatures in industrial organization economics, including the literatures dealing with market power, product differentiation, auctions, regulation and entry.
Keywords: structural econometric model; market power; auctions; regulation; and entry.
JEL: C50, C51, C52, D10, D20, D40.

Prepared for the Handbook of Econometrics, Volume 5.

Contents

1 Introduction

1

2 Descriptive and Structural Models in Econometrics

2

3 Putting the ‘Econ’ Back into Econometrics

6

3.1 Sources of Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Why Use Structural Models? . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Regressions and Structural Modeling . . . . . . . . . . . . . . . . . . 13

3.4 Structural Models, Simultaneous Equations and Reduced Forms . . . 17

3.4.1 ‘Magic’ Instruments in Simultaneous Equations Models . . . . 21

3.4.2 The Role of Non-Experimental Data in Structural Modeling . 26

4 A Framework for Structural Econometric Models in IO

28

4.1 The Economic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 The Stochastic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2.1 Unobserved Heterogeneity and Agent Uncertainty . . . . . . . 31

4.2.2 Optimization Errors . . . . . . . . . . . . . . . . . . . . . . . 35

4.2.3 Measurement Error . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 Steps to Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.4 Structural Model Epilogue . . . . . . . . . . . . . . . . . . . . . . . . 42

5 Demand and Cost Function Estimation Under Imperfect Competi-

tion

42

5.1 Using Price and Quantity Data to Diagnose Collusion . . . . . . . . . 43

5.2 The Economic Model . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2.1 Environment and Primitives . . . . . . . . . . . . . . . . . . . 45

5.2.2 Behavior and Optimization . . . . . . . . . . . . . . . . . . . 46

5.2.3 The Stochastic Model . . . . . . . . . . . . . . . . . . . . . . 48

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6 Market Power Models More Generally

53

6.1 Estimating Price-Cost Margins . . . . . . . . . . . . . . . . . . . . . 54

6.2 Identifying and Interpreting Price-Cost Margins . . . . . . . . . . . . 57

6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

7 Models Differentiated Product Competition

62

7.1 Neoclassical Demand Models . . . . . . . . . . . . . . . . . . . . . . . 63

7.2 Micro-Data Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7.2.1 A Household-Level Demand Model . . . . . . . . . . . . . . . 71

7.2.2 Goldberg’s Economic Model . . . . . . . . . . . . . . . . . . . 72

7.2.3 The Stochastic Model . . . . . . . . . . . . . . . . . . . . . . 74

7.2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.3 A Product-Level Demand Model . . . . . . . . . . . . . . . . . . . . . 78

7.3.1 The Economic Model in BLP . . . . . . . . . . . . . . . . . . 79

7.3.2 The Stochastic Model . . . . . . . . . . . . . . . . . . . . . . 80

7.4 More on the Econometric Assumptions . . . . . . . . . . . . . . . . . 84

7.4.1 Functional Form Assumptions for Price . . . . . . . . . . . . . 84

7.4.2 Distribution of Consumer Heterogeneity . . . . . . . . . . . . 85

7.4.3 Unobserved “Product Quality” . . . . . . . . . . . . . . . . . 88

7.4.4 The Cost Specifications . . . . . . . . . . . . . . . . . . . . . . 91

7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

2

8 Games with Incomplete Information: Auctions

92

8.1 Auctions Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

8.1.1 Descriptive Models . . . . . . . . . . . . . . . . . . . . . . . . 94

8.1.2 Structural Models . . . . . . . . . . . . . . . . . . . . . . . . . 96

8.1.3 Nonparametric Identification and Estimation . . . . . . . . . . 100

8.2 Further Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

8.3 Parametric Specifications for Auction Market Equilibria . . . . . . . . 107

8.4 Why Estimate a Structural Auction Model? . . . . . . . . . . . . . . 112

8.5 Extensions of Basic Auctions Models . . . . . . . . . . . . . . . . . . 114

9 Games with Incomplete Information: Principal-Agent Contracting

Models

115

9.1 Observables and Unobservables . . . . . . . . . . . . . . . . . . . . . 117

9.2 Economic Models of Regulator-Utility Interactions . . . . . . . . . . . 118

9.3 Estimating Productions Functions Accounting for the Private Infor-

mation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 9.3.1 Symmetric Information Model . . . . . . . . . . . . . . . . . 125 9.3.2 The Asymmetric Information Model . . . . . . . . . . . . . . 125 9.4 Econometric Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 9.5 Estimation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 9.6 Further Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

10 Market Structure and Firm Turnover

133

10.1 Overview of the Issues . . . . . . . . . . . . . . . . . . . . . . . . . . 134

10.1.1 Airline Competition and Entry . . . . . . . . . . . . . . . . . 135

10.2 An Economic Model and Data . . . . . . . . . . . . . . . . . . . . . . 137

10.3 Modeling Profits and Competition . . . . . . . . . . . . . . . . . . . . 139

10.4 The Econometric Model . . . . . . . . . . . . . . . . . . . . . . . . . 141

10.5 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

10.6 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

11 Ending Remarks

146

3

1 Introduction
The founding members of the Cowles Commission defined econometrics as: “a branch of economics in which economic theory and statistical method are fused in the analysis of numerical and institutional data” Hood and Koopmans (1953, p. xv). Many economists today, however, view econometrics as a field primarily concerned with statistical issues rather than economic questions. This has led some to draw a distinction between econometric modeling and structural econometric modeling, the latter phrase being used to emphasize the original Cowles Commission vision for econometrics.
This chapter has three main goals. The first is to explain the logic of structural econometric modeling and to describe the basic elements of a structural econometric model. While it might seem obvious how to combine economic theories with statistical models, nothing could be further from the truth. Structural econometric models must simultaneously: (1) be flexible statistical descriptions of data; (2) respect the details of economic institutions under consideration; and, (3) be sensitive to the nonexperimental nature of economic data. Moreover, just because an empirical researcher includes errors in an economic model does not guarantee that the resulting statistical model will be coherent or realistic. In this chapter, we illustrate challenges that arise in constructing structural econometric models.
A second goal of this chapter is to propose a framework for developing and evaluating structural econometric models. Although some elements of this framework originated with the founders of the Econometric Society, we add elements that are specific to a field of interest to us – industrial organization (IO). Our discussion emphasizes that the process of building a structural model entails several interrelated steps. These steps require the modeler to trade off both economic and statistical assumptions. In our opinion, much more attention needs to be devoted to appreciating and understanding these trade-offs. Many econometric textbooks, for example, focus on teaching statistical techniques and not on how to build structural econometric models. This emphasis perhaps has only reinforced many economists’ impression that econometricians care more about statistics than economics.
Our third goal is to illustrate how structural modeling trade-offs are made in practice. Specifically, we examine different types of structural econometric models developed by IO researchers. These models are used to examine such issues as: the extent of market power possessed by firms; the efficiency of alternative market allocation mechanisms (e.g., different rules for running single and multi-unit auctions); and the empirical implications of information and game-theoretic models. We should emphasize that this chapter is NOT a comprehensive survey of the IO literature or the above topics. Rather, we seek to illustrate how IO researchers have used economic structure and statistical assumptions to identify and estimate magnitudes of economic interest. Our hope is that in doing so we can provide a sense of the benefits and limitations of
1

structural econometric models generally.
This chapter is organized as follows. We begin with several examples of what we mean by a structural econometric model. We go on to illustrate the strength and weaknesses of structural econometric models through an extended series of examples. These examples provide a context for our structural modeling framework. Following a description of this framework, we use the framework to evaluate select structural models drawn from the industrial organization literature. First, we consider homogeneousproduct models of competition where researchers estimate “conduct” or competitive “conjectural variation” parameters. We then discuss structural approaches to modeling competition in: differentiated-product markets; private or asymmetric information in auctions and principal-agent relations; and models of discrete strategic actions, such as entry and exit decisions.
2 Descriptive and Structural Models in Econometrics
Empirical work in economics can be divided into two general categories: descriptive and structural. Descriptive work has a long and respected tradition in economics. It focuses on constructing and summarizing economic data. Descriptive work often can proceed without any reference to an economic model. For example, economists measure such things as the size of the workforce or an unemployment rate without relying on particular models of employment or unemployment. The primary goal of most descriptive work in economics is to uncover trends, patterns or associations that might stimulate further analyses. An excellent early example is Engel’s (1857) work relating commodity budget shares to total income. Engel’s finding that expenditure shares for food were negatively related to the logarithm of total household expenditures has shaped subsequent theoretical and empirical work on household consumption behavior (see Deaton and Muelbauer (1980) and Pollak and Wales (1992)). A somewhat more recent example of descriptive work is the Phillips curve. Phillips (1958) documented an inverse relationship between United Kingdom unemployment rates and changes in wage rates. This work inspired others to document relationships between unemployment rates and changes in prices. In the ensuing years, many economic theories have been advanced to explain why Phillips curves are or are not stable economic relations.
When it goes beyond measurement and tabulation issues, descriptive work is concerned with estimating the joint population density of two sets of variables x and y, f (x, y), or objects that can be derived from it such as:
f (y | x), the conditional density of y given x;
2

E(y | x), the conditional expectation of y given x;
Cov(y | x), the conditional covariances (or correlations) of y given x;
Qα(y | x) the α conditional quantile of y given x; or
BLP (y | x), the best linear predictor (BLP) of y given x.
In practice, descriptive researchers can choose from a vast array of parametric statistical distributions when they want to describe f (x, y) (or related objects). A common issue that arises in making this choice is whether the chosen distribution is sufficiently flexible.
Recently, statisticians and econometricians have devoted substantial energy to devising flexible methods for estimating joint densities. For example, statisticians have proposed kernel density techniques and other data smoothing methods for estimating f (x, y). Silverman (1986), Hardle (1990) and Hardle and Linton (1994) provide introductions to these procedures. Matzkin (1994), among others, discusses how economic restrictions can be incorporated into non-parametric methods. Although these nonparametric estimation techniques allow the researcher to estimate the joint density of x and y flexibly, they have their drawbacks. Most important, smoothing techniques often require enormous amounts of data to yield much precision. Silverman (1986) also argues that researchers using these techniques face a “curse of dimensionality,” wherein the amount of data required to obtain precise estimates grows rapidly with the dimensions of x and y. His calculations suggest that in typical applications, economists will need hundreds of thousands of observations before they can place great faith in these flexible techniques.1
Even in those rare circumstances when a researcher has sufficient data to estimate f (x, y) flexibly, there are still compeling rationales for preferring to estimate a structural econometric model. A structural econometric model shows how economic activity places restrictions on the population joint distribution of x and y. These restrictions can be used to recover underlying economic primitives. Thus, as in descriptive work, structural econometric modeling is about characterizing the joint distribution of economic data. Unlike descriptive models, however, structural models seek to recover economic parameters or primitives from the joint distribution. The essential components of a structural model are the economic and statistical assumptions that allow a researcher to recover these economic primitives. These assumptions minimally must be economically realistic and statistically sound. For the structure to be realistic, it must reasonably describe the economic and institutional environments generating the data. For the model to be coherent, it must be possible to recover structural parameter estimates from all plausible realizations of x and y.
1Silverman (1986, Table 4.2) shows that more than ten times as much data is required to attain the same level of precision for a four-dimensional as a two-dimensional joint density. More than 200 times as much data is required for an eight-dimensional as a four-dimensional density.
3

To understand the process of building a structural model, consider the example of a researcher who wishes to use household consumption, price and demographic data to estimate household demand for a particular good. The first step in building a structural model is to show that economic theory places restrictions on the joint distribution of household consumption, prices and income. The structural modeler might start by assuming the existence of a household-level utility function U(x, z, θ) that is a function of the vector of quantities consumed, x, taste parameters, θ, and a vector of household characteristics, z. The modeler might then use consumer theory to derive a mathematical relationship between: the demand for each product; total expenditures, y; prices, p; and household characteristics, z: x = h(p, y, z, θ). Of course this theory will not perfectly explain the household’s purchases. The researcher therefore must either enrich the economic model or introduce error terms that represent variables outside the economic theory. These error terms might represent unobserved differences among agents, agent optimization errors, or errors introduced during data collection. For example, a structural modeler might assume that he does not observe all of the household characteristics that determine differences in preferences. He could model this incompleteness explicitly by introducing a vector of unobserved household characteristics, , directly into the household utility functions: U = U(x, z, θ, ). By maximizing household utility subject to the household’s budget constraint, we obtain demand functions that depend on these unobserved characteristics: x = h(p, y, z, θ, ).
To estimate the unknown utility (“structural”) parameters θ, the structural modeler will usually add assumptions about the joint population distribution for the unobserved tastes, , and z, p and y. For example, he might assume a specific joint parametric distribution for all these variables. From this joint distribution, he could apply a change of variables to derive the joint distribution of the observed data f (x, y, z, p) or other objects such as f (x | z, p, y). The critical question at this point is: Can he now find a method for estimating θ from the structure of f (x, y, z, p)? Ideally, the researcher must demonstrate that his econometric model is consistent with the observed joint density of x, y, z and p, and that he can consistently estimate θ using the available data.
To summarize, structural econometric modeling uses economic and statistical assumptions to derive a joint density for the observed data. Examples of economic assumptions are: What utility function should be used? What is the budget constraint faced by the consumer? Examples of stochastic assumptions are: What types of errors should be introduced and where should they be introduced? Do these stochastic assumptions characterize the complete distribution, or might estimation be based on a statistical object that can be derived from the complete distribution? In what follows, we discuss these and other choices that structural modelers make. We loosely group these choices into three main groups: economic, statistical, and tractability assumptions.
In closing this section, we should emphasize a fundamental difference between struc-
4

tural and descriptive econometric models. A general goal of descriptive work is to estimate the joint density of x and y. Most descriptive work, however, settles for estimating best linear predictors or the conditional density of predetermined, x, and endogenous variables, y. The distribution of data alone, however, cannot justify causal or behavioral statements. On the other hand, a structural modeler can recover estimates of economic magnitudes and determine the extent of causation, but only because he is willing to make the economic and statistical assumptions necessary to infer these magnitudes from his econometric model for the joint density of x and y. This is a major strength of a structural econometric model – by making clear what economic assumptions are required to draw specific economic inferences from the data, the structural modeler makes it possible for others to assess the plausibility and sensitivity of findings to these assumptions.
Some researchers believe that there is an intermediate style of empirical research, somewhere between descriptive modeling and structural modeling. Sometimes this style is referred to as “reduced form analysis.” Like in structural modeling, economics plays a role in these reduced form models, but only to the extent that it classifies variables as dependent or independent. Thus, the economics does not place structure on the form of f (x, y) other than what is x and what is y. Despite this, the term “reduced form analysis” has come to signal an econometric model where the endogenous variables, y, are on the left hand side and the exogenous variables, x are on the right. In addition, it is assumed the errors in these models are mean independent of the exogenous variables.
In a typical “reduced form analysis,” the researcher then uses linear regression analysis to estimate coefficients for the right hand side variables. These coefficients are interpreted as capturing by how much the dependent variable will change if the independent variable changes by one unit – holding everything else constant. One of the main goals of this chapter is to argue that this use of the term reduced form is invalid and not what members of the Cowles Commission intended. These regressions are in fact descriptive, and not an intermediate category of empirical model. The term reduced form should instead be reserved for cases in which economics and statistics delivers a set of equations where the endogenous variables are on the left hand side and the exogenous variables and disturbances are on the right. Thus, a goal of this chapter is to eliminate the term “reduced form analysis” from applied economists’ vocabularies. Note that it is not the term “reduced form” that we seek to eliminate, but rather the idea that there is a reduced form model that is independent of a correctly specified structural model.
5

3 Putting the ‘Econ’ Back into Econometrics
3.1 Sources of Structure
There are two sources of “structure” in structural models. First, economic theories deliver mathematical statements about the relationship between x and y. These mathematical statements often are deterministic, and as such do not speak directly to the distribution of noisy economic data. It is the applied researcher who adds the second source of structure, which are statistical and other stochastic assumptions that specify how data on x and y were generated. This second source is necessary to transform deterministic models of economic behavior into stochastic econometric models. Thus, the “structure” in structural models typically comes from both economics and statistics.
Varying degrees of economic and stochastic structure can be imposed. Purists believe that structural models must come from fully-specified stochastic economic models. Others believe that it is acceptable to add structure if that structure facilitates estimation or allows the researcher to recover economically meaningful parameters. For example, economic theory may make predictions about the conditional density of y given x, f (y | x), but may be silent about the marginal density of x, f (x). In this case, a researcher might assume that the marginal density of x does not contain parameters that appear in the conditional density. Of course, there is nothing to guarantee that assumptions made to facilitate estimation are in fact reasonable or true. Put another way, the “structure” in a structural model is there because the researcher chose explicitly or implicitly to put it there. One of the advantages of structural econometric models is that researchers can examine the sensitivity of structural models and estimators to alternative economic and statistical assumptions. This is, however, often easier said than done.
To illustrate how economists can introduce economic structure into a statistical model, we begin by examining two stylized econometric models. The purpose of the first model is to illustrate the difference between a descriptive and a structural model. This example shows that the same linear regression model can be a descriptive or a structural model depending on what economic and statistical assumptions the researcher is willing to make.
Example 1
We imagine an economist with a cross-section of firm-level data on output, Qt, labor inputs, Lt, and capital inputs, Kt, for each firm t. To describe the relationship between output and inputs, the researcher might estimate the following linear regression by
6

ordinary least squares (OLS):

ln Qt = θ0 + θ1 ln Lt + θ2 ln Kt + t,

(1)

where the θ’s are unknown coefficients and the t is an error term that accounts for the fact that the right hand side variables do not perfectly predict log output.
What do we learn by estimating this regression? Absent more information we have estimated a descriptive regression. More precisely, we have estimated the parameters of the best linear predictor of yt = ln(Qt) given xt = (1, ln(Lt), ln(Kt)) . Goldberger (1991, Ch. 5) provides an excellent discussion of best linear predictors. The best linear predictor of y given a univariate x is BLP (y | x) = a + bx, where a = E(y) − bE(x) and b = Cov(y, x)/V ar(x). Notice that the coefficients, a and b, of the best linear predictor function are statistical (and not economic) functions of the population moments of f (x, y).
If we add to our descriptive model the assumption that the sample second moments converge to their population counterparts

1T

1T

Tl→im∞ T t=1 xt xt = Mxx, and Tl→ im∞ T t=1 xtyt = Mxy,

and that Mxx is a matrix of full rank, then OLS will deliver consistent estimates of the parameters of the best linear predictor function. Thus, if we are interested in predicting the logarithm of output, we do not need to impose any economic structure and very little statistical structure to estimate consistently the linear function of the logarithm of labor and logarithm of capital that best predicts the logarithm of output.

Many economists, however, see regression (1) as being more than a descriptive re-
gression. They would base their reasoning on the observation that (1) essentially
looks like a logarithmic restatement of a Cobb-Douglas production function: Qt = A Lαt Ktβ exp( t). Because of the close resemblance, they would argue that (1) is in fact a “structural” and not a descriptive econometric model.

A critical missing step in this logic is that a Cobb-Douglas production function is deterministic relationship, whereas the regression model (1) includes an error term. Where did the error term in the empirical model come from? The answer to this question is critical because it affects whether OLS will deliver consistent estimates of the parameters of the Cobb-Douglas production function, as opposed to consistent estimates of the parameters of the best linear predictor of the logarithm of output given the logarithms of the two inputs. In other words, it is the combination of an economic assumption (production is truly Cobb-Douglas) and statistical assumptions ( t satisfies certain moment conditions) that distinguishes a descriptive model from a structural model.

7

Preparing to load PDF file. please wait...

0 of 0
100%
Structural Econometric Modeling: Rationales and Examples from