Structural Econometric Modeling in Industrial Organization Handout 1

Structural Econometric Modeling in Industrial Organization Handout 1 Professor Matthijs Wildenbeest

16 May 2011

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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, 2007.

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Background on Empirical IO

• Structural versus nonstructural econometrics • Constructing structural models • Framework for structural econometrics models in IO

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Structural versus Nonstructural Econometrics Example: auctions Suppose we observe winning bids, y = {y1 , . . . , yT }, in a large number of T similar auctions, as well as the number of bidders in each market, x = {x1 , . . . , xT }. 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

Structural versus Nonstructural Econometrics

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Structural versus Nonstructural Econometrics

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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) =

θ2 x y θ2 x+1



θ1 θ2 (x − 1) θ2 (x − 1) − 1

θ2 x ,

so that the expected value of the winning bid given the number of bidder is   θ2 x θ1 θ2 (x − 1) . E (y |x, θ) = θ2 (x − 1) − 1 θ2 x − 1 7

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.

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Compare the predictive performance of two competing theories. 8

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.”

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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.

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Constructing Structural Models Example Cross-section data on output, Qi , labor inputs, Li , and capital inputs Ki . Estimate the regression ln Qi = θ0 + θ1 ln Li + θ2 ln Ki + i , by ordinary least squares (OLS). Error term i necessary because right hand side variables do not perfectly explain log output. Interpretation? • Best Linear Predictor (BLP) of ln Qi given a constant, ln Li

and ln Ki : 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

Constructing Structural Models Only structure from economics not enough to estimate (logarithmic transformation) of Cobb-Douglas production function Qi = ALαi Kiβ : we have to add an error term as well: Qi = ALαi Kiβ exp i . Where does the error term come from? If 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

Constructing Structural Models

Linear regression model y = α + xβ + . 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.

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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.

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Constructing Structural Models Nonexperimental data raises significant modeling issues. Estimating the demand curve qtd = γ0 + γ1 pt + γ2 x1t + 1t by OLS only gives consistent estimates of the demand curve parameters if price pt and a demand shifter like income x1t are uncorrelated with the error 1t . If we perform experiments where we randomly select prices and observe the quantity demanded this will work. Same for the supply curve qts = β0 + β1 pt + β2 x2t + 2t , where x2t is now a supply shifter like input prices. 15

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 qts = qtd , we could apply instrumental variable techniques to get consistent estimates of the simultaneous equation model.

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Constructing Structural Models Simultaneous equations models

When dealing with endogeneity it is important to think about a “complete” simultaneous equations model.

Example Researcher estimates: pi = POPi θ1 + COMPi θ2 + i , where pi is the price in market i, POPi is population size, and COMPi is a dummy for whether the firm faces competition. Has this equation a structural meaning? Could be: θ2 measures effect of competition on prices.

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Constructing Structural Models Simultaneous equations models

Problem: COMPi is likely to depend on pi : COMPi = POPi γ1 + pi γ2 + ηi . Therefore COMPi will be correlated with i , so OLS will give inconsistent estimates of θ2 . Possible solution: use average income Yi as instrument for COMPi , since one can argue Yi is correlated with COMPi but not with i . Statistical rationale.

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Constructing Structural Models Simultaneous equations models

To be completely convincing two things need to be done: 1

explain why Yi is not part of pi .

2

make the case that Yi is part of COMPi .

Therefore, specify the complete system: pi COMPi

= POPi θ1 + COMPi θ2 + i ; = POPi γ1 + pi γ2 + Yi γ3 + ηi .

This requires the researcher to think carefully about the economic model underlying the simultaneous system of equations.

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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)

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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!

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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.

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Framework for Structural Econometrics Models in IO Example Cross-section data on firms consisting of output Q, total costs TC , and input prices pK and pL . Goal is to estimate α and β in Qi = Ai Lαi Kiβ . Suppose a regulator chooses a price pir and that firms have different Ai , the latter being observed by the firm and regulator but not by the econometrician. Assume inelastic demand. Firm chooses inputs to maximize π(Ki , Li ) = pir Ai Lαi Kiβ − pKi Ki − pLi Li .

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Framework for Structural Econometrics Models in IO Firms produce in a cost minimizing way, so Ai αLα−1 Kiβ α Ki pLi MPL i = = = . β−1 α MPK β Li pKi Ai βLi Ki This means Ki =

pLi β Li . pKi α

Substituting this into the production function gives  Qi = Ai

pLi β Li pKi α

β

Lαi

 = Ai

pLi β pKi α

β

Lα+β , i

and solving for Li gives 1 α+β

Li = Qi

−1 α+β

Ai



pLi pKi

 −β   −β α+β β α+β α 24

Framework for Structural Econometrics Models in IO The total labor cost pLi Li is then given by γ 1−γ δ −δ pLi Li = CL pKi pLi Qi Ai ,

where δ = 1/(α + β), γ = β/(α + β), and CL = (α/β)γ . Similarly, the total capital cost pKi Ki is given by γ 1−γ δ −δ pKi Ki = CK pKi pLi Qi Ai ,

where CK = (α/β)γ−1 . The total cost function is therefore γ 1−γ δ −δ TCi = C0 pKi pLi Qi Ai ,

where C0 = CL + CK . 25

Framework for Structural Econometrics Models in IO Transforming this equation using natural logarithms gives ln TCi = ln C0 + γ ln pKi + (1 − γ) ln pLi + δ ln Qi − δ ln Ai , which holds exactly. The efficiency differences are assumed to be i.i.d. positive random variables, so subtracting E [ln Ai ] from the error term and adding it to the constant gives ln TCi = ln C1 + γ ln pKi + (1 − γ) ln pLi + δ ln Qi − δ ln ui , where ln C1 = ln C0 + E [ln Ai ] and ln ui = ln Ai − E [ln Ai ]. This equation can finally be taken to the data using OLS.

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Framework for Structural Econometrics Models in IO Now suppose the firms (and the regulator) do not know the efficiency parameters Ai either. Firms now choose inputs to maximize E [π(Ki , Li )] = pir E [Ai Lαi Kiβ ] − pKi Ki − pLi Li . First-order condition for expected profit maximization imply   α pKi Ki . Li = β pLi Observed total costs are TCi =

α+β α+β pKi Ki = pLi Li , β α

and do not depend on Ai . 27

Framework for Structural Econometrics Models in IO This means Li =

α β+α TCi /pLi

and Ki =

β β+α TCi /pKi ,

so

−β −α Qia = D0 TCiα+β pKi pLi Ai .

Final output produced Qia does depend on Ai .Taking natural logarithms gives ln Qia = ln D0 + (α + β) ln TCi − β ln pKi − α ln pLi + ln Ai , which holds exactly. Researcher does not observe Ai , so treat as random and move unconditional expectation again to the constant: ln Qia = D1 + (α + β) ln TCi − β ln pKi − α ln pLi + ηi , where ηi = ln Ai − E [ln Ai ] and D1 = ln D0 + E [ln Ai ].

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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.

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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.

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