Structural Econometric Modeling in Industrial Organization Handout 1

Transcription

1 Structural Econometric Modeling in Industrial Organization Handout 1 Professor Matthijs Wildenbeest 16 May

2 Reading Peter C. Reiss and Frank A. Wolak A. Structural Econometric Modeling: Rationales and Examples from Industrial Organization. Handbook of Econometrics 6A, Chapter 64, Sections 1-4,

3 Background on Empirical IO Structural versus nonstructural econometrics Constructing structural models Framework for structural econometrics models in IO 3

4 Structural versus Nonstructural Econometrics Example: auctions Suppose we observe winning bids, y = {y 1,..., y T }, in a large number of T similar auctions, as well as the number of bidders in each market, x = {x 1,..., x T }. Goal exercise: understand equilibrium relationship between winning bids and the number of firms. Nonstructural approach: regress winning bids on the number of bidders. use nonparametric smoothing techniques to estimate the conditional density of winning bids given the observed number of bidders, i.e., f (y x). Does the regression coefficient tell us what happens when we add another bidder? Not without further knowledge about the auction under study. For instance, information paradigm matters. 4

5 Structural versus Nonstructural Econometrics 5

6 Structural versus Nonstructural Econometrics 6

7 Structural versus Nonstructural Econometrics Structural approach: Use the structure of an auction model to say something about winning bids and the number of firms. For example, Paarsch (1992, j econometrics) shows that for first-price sealed-bid auctions with Pareto-distributed private value bidders, the conditional density of winning bids given the number of firms f (y x) is f (y x, θ) = θ [ ] 2x θ1 θ 2 (x 1) θ2 x y θ, 2x+1 θ 2 (x 1) 1 so that the expected value of the winning bid given the number of bidder is [ ] θ1 θ 2 (x 1) θ 2 x E(y x, θ) = θ 2 (x 1) 1 θ 2 x 1. 7

8 Structural versus Nonstructural Econometrics Why use economic theory in this example? Helps us to clarify how institutional and economic conditions affect the relationship between x and y. Think of type of auction (sealed-bid versus open-outcry or first-price versus second-price), bidder behavior (risk neutral versus risk averse), and information paradigm (common versus private values). Three general reasons for specifying and estimating a structural econometric model: 1 Estimate unobserved parameters that could not otherwise be inferred from the data (costs, elasticities, valuations). 2 Perform counterfactuals or policy experiments. 3 Compare the predictive performance of two competing theories. 8

9 Structural versus Nonstructural Econometrics Although using structural econometrics has many advantages, this does not always mean structural models should be favored over nonstructural models. Think of a situation where there little or no useful economic theory to guide the empirical work. Levitt (1997, am econ rev): using electoral cycles in police hiring to estimate the effect of police on crime. Studies the effect of police on reducing crime. Previous studies found little evidence, likely due to simultaneity problems. Levitt proposes a new instrument: timing of elections. Effects the size of the police force, but does not belong directly to the crime production function. 9

10 Constructing Structural Models Sources of structure 1 economics 2 statistics Since economic models are often deterministic we have to add statistical structure to rationalize why economic theory does not perfectly explain the data. 10

11 Constructing Structural Models Example Cross-section data on output, Q i, labor inputs, L i, and capital inputs K i. Estimate the regression ln Q i = θ 0 + θ 1 ln L i + θ 2 ln K i + ɛ i, by ordinary least squares (OLS). Error term ɛ i necessary because right hand side variables do not perfectly explain log output. Interpretation? Best Linear Predictor (BLP) of ln Q i given a constant, ln L i and ln K i : only statistical structure needed (sample second moments converge to their population counterparts). Estimation of Cobb-Douglas production function: structure needed from both economics and statistics. 11

12 Constructing Structural Models Only structure from economics not enough to estimate (logarithmic transformation) of Cobb-Douglas production function Q i = AL α i K β i : we have to add an error term as well: Q i = AL α i K β i exp ɛ i. Where does the error term come from? If ɛ i is measurement error distributed independently of the right hand side variables the estimated OLS parameters can be interpreted as the coefficient of the Cobb-Douglas production function. Moreover, firms should produce on their production function. Note that if the error includes unobserved differences in productivity, OLS fails to deliver consistent estimates of the production function parameters. 12

13 Constructing Structural Models Linear regression model y = α + xβ + ɛ. From a statistical perspective we can always regress y on x (or the other way around): the coefficients have statistical interpretations (Best Linear Predictor). However, we need economic arguments to make a case about causation. Moreover, without an economic model the OLS regression only gives (under certain conditions) consistent estimates of a best linear predictor function. 13

14 Constructing Structural Models Usually not possible to test a deterministic economic model by running a regression. Many descriptive studies treat the linear regression coefficient estimates as as if they were estimates of the derivative of E(y x) with respect to x, although β = BLP(y x)/ x is usually not equal to E(y x)/ x. 14

15 Constructing Structural Models Nonexperimental data raises significant modeling issues. Estimating the demand curve q d t = γ 0 + γ 1 p t + γ 2 x 1t + ɛ 1t by OLS only gives consistent estimates of the demand curve parameters if price p t and a demand shifter like income x 1t are uncorrelated with the error ɛ 1t. If we perform experiments where we randomly select prices and observe the quantity demanded this will work. Same for the supply curve q s t = β 0 + β 1 p t + β 2 x 2t + ɛ 2t, where x 2t is now a supply shifter like input prices. 15

16 Constructing Structural Models In the experiments the quantity supplied will in general not be equal to the quantity demanded. However, no problem since we observe the quantity demand and supplied directly for each randomly generated price. Prices around us are nonexperimental. OLS no longer possible because of correlations between explanatory variables and error term. But if we use economics and impose the market-clearing equation q s t = q d t, we could apply instrumental variable techniques to get consistent estimates of the simultaneous equation model. 16

17 Constructing Structural Models Simultaneous equations models When dealing with endogeneity it is important to think about a complete simultaneous equations model. Example Researcher estimates: p i = POP i θ 1 + COMP i θ 2 + ɛ i, where p i is the price in market i, POP i is population size, and COMP i is a dummy for whether the firm faces competition. Has this equation a structural meaning? Could be: θ 2 measures effect of competition on prices. 17

18 Constructing Structural Models Simultaneous equations models Problem: COMP i is likely to depend on p i : COMP i = POP i γ 1 + p i γ 2 + η i. Therefore COMP i will be correlated with ɛ i, so OLS will give inconsistent estimates of θ 2. Possible solution: use average income Y i as instrument for COMP i, since one can argue Y i is correlated with COMP i but not with ɛ i. Statistical rationale. 18

19 Constructing Structural Models Simultaneous equations models To be completely convincing two things need to be done: 1 explain why Y i is not part of p i. 2 make the case that Y i is part of COMP i. Therefore, specify the complete system: p i = POP i θ 1 + COMP i θ 2 + ɛ i ; COMP i = POP i γ 1 + p i γ 2 + Y i γ 3 + η i. This requires the researcher to think carefully about the economic model underlying the simultaneous system of equations. 19

20 Framework for Structural Econometrics Models in IO A structural model has two main components: 1 economic model; 2 stochastic model. The economic model should have the following components: description of economic environment (market, actors, information available); list of primitives (technologies, preferences, endowments); exogenous variables (variables outside the model); decision variables and objective functions (utility/profit maximization); equilibrium concept (nash equilibrium) 20

21 Framework for Structural Econometrics Models in IO The stochastic model transforms the (usually) deterministic economic model into an econometric model. Main difference between the two is inclusion of unobservables. Major stochastic specifications: unobserved heterogeneity agent uncertainty optimization errors measurement error Different forms can have dramatically different implications for identification and estimation! 21

22 Framework for Structural Econometrics Models in IO Unobserved heterogeneity Situation where agents decisions depend on something the economist does not observe. Agent uncertainty Situation where agents decisions depend on something the agent does not (fully) observe. Note that in both cases the econometrician is ignorant. Still, they can have different implications. 22

23 Framework for Structural Econometrics Models in IO Example Cross-section data on firms consisting of output Q, total costs TC, and input prices p K and p L. Goal is to estimate α and β in Q i = A i L α i K β i. Suppose a regulator chooses a price pi r and that firms have different A i, the latter being observed by the firm and regulator but not by the econometrician. Assume inelastic demand. Firm chooses inputs to maximize π(k i, L i ) = p r i A i L α i K β i p Ki K i p Li L i. 23

24 Framework for Structural Econometrics Models in IO Firms produce in a cost minimizing way, so This means i K β i MP L = A iαl α 1 i MP K A i βl α i K β 1 = α β K i = p Li p Ki β α L i. K i L i = p Li p Ki. Substituting this into the production function gives [ ] pli β β [ ] Q i = A i p Ki α L i L α pli β β i = A i L α+β i, p Ki α and solving for L i gives L i = Q 1 α+β i A ( ) β 1 α+β pli i p Ki α+β ( β α ) β α+β 24

25 Framework for Structural Econometrics Models in IO The total labor cost p Li L i is then given by p Li L i = C L p γ Ki p1 γ Li Qi δ A δ i, where δ = 1/(α + β), γ = β/(α + β), and C L = (α/β) γ. Similarly, the total capital cost p Ki K i is given by where C K = (α/β) γ 1. p Ki K i = C K p γ Ki p1 γ Li Qi δ A δ i, The total cost function is therefore where C 0 = C L + C K. TC i = C 0 p γ Ki p1 γ Li Qi δ A δ i, 25

26 Framework for Structural Econometrics Models in IO Transforming this equation using natural logarithms gives ln TC i = ln C 0 + γ ln p Ki + (1 γ) ln p Li + δ ln Q i δ ln A i, which holds exactly. The efficiency differences are assumed to be i.i.d. positive random variables, so subtracting E[ln A i ] from the error term and adding it to the constant gives ln TC i = ln C 1 + γ ln p Ki + (1 γ) ln p Li + δ ln Q i δ ln u i, where ln C 1 = ln C 0 + E[ln A i ] and ln u i = ln A i E[ln A i ]. This equation can finally be taken to the data using OLS. 26

27 Framework for Structural Econometrics Models in IO Now suppose the firms (and the regulator) do not know the efficiency parameters A i either. Firms now choose inputs to maximize E[π(K i, L i )] = p r i E[A i L α i K β i ] p Ki K i p Li L i. First-order condition for expected profit maximization imply [ ] α p Ki L i = K i. β p Li Observed total costs are and do not depend on A i. TC i = α + β β p KiK i = α + β α p LiL i, 27

28 Framework for Structural Econometrics Models in IO This means L i = α β+α TC i/p Li and K i = β β+α TC i/p Ki, so Q a i = D 0 TC α+β i p β Ki p α Li A i. Final output produced Qi a logarithms gives does depend on A i.taking natural ln Q a i = ln D 0 + (α + β) ln TC i β ln p Ki α ln p Li + ln A i, which holds exactly. Researcher does not observe A i, so treat as random and move unconditional expectation again to the constant: ln Q a i = D 1 + (α + β) ln TC i β ln p Ki α ln p Li + η i, where η i = ln A i E[ln A i ] and D 1 = ln D 0 + E[ln A i ]. 28

29 Framework for Structural Econometrics Models in IO Optimization errors Failure of agents decisions to satisfy exactly first-order necessary conditions for optimal decisions. Measurement errors Occurs when the variable the research observes are different from those the agents observe. Straightforward way of converting a deterministic model into a statistical model. 29

30 Framework for Structural Econometrics Models in IO Steps left 1 selection of functional forms; 2 selection of distributional assumptions; 3 selection of an estimation technique; and 4 selection of specification test. 30