ECO 310: Empirical Industrial Organization Lecture 8 - Production Functions (II) Dimitri Dimitropoulos Fall 2014 UToronto 1 / 27 References I ABBP Section 2 I Olley and Pakes (1996). "The Dynamics of Productivity in the Telecommunications Equipment Industry." Econometrica, Vol. 64(6), pp. 1263-1297. 2 / 27 Olley and Pakes (1996, Ecta) 3 / 27 Olley and Pakes (1996) - Motivation I Olley and Pakes (OP) take a very di¤erent approach to solving the simultaneity problem inherent in production function estimation I OP use a control function method – instead of instrumenting the endogenous regressors, they include additional regressors that capture the endogenous part of the error term I Olley and Pakes begins with two premises: I Since we are using panel data, and observe the transition of each …rm from year to year, we should model the stochastic process for this. I Since the endogeniety problem stems from a …rm’s input demands, we should add structure to these input demands 4 / 27 Olley and Pakes (1996) - Motivation I Suppose we have a panel of J …rms observed over T years. I For each …rm j at time t, we observe (the logs of) output, labor and capital fqjt ; ljt ; kjt : j = 1; 2; :::J and t = 1; 2; :::T g I Consider the following model of simultaneous equations (PF) qjt = (LD) ljt = fL (wjt ; rt ; kjt ) (KD) ijt = fK (rt ; kjt ; !jt ) 0 + L ljt + K kjt + !jt + "jt I (LD) & (KD) rep. …rms’optimal decision rules for labor & cap. investment I We explicitly account for (LD) and (KD) as it is through these choices that the endogeneity problem makes itself present I However, we make no assumptions about the functional forms of fL or fK I We’ll rely on assumps. about the covariance structure of ljt ; kjt ; and !jt 5 / 27 Assumptions I Olley and Pakes assume: I Assumption 1. Common Capital Markets I Firms purchase capital in a common, competitive capital market I Since the capital market is competitive, no individual …rm has any in‡uence on the equilibrium price of capital. I And, since …rms opperate in a common capital market, the price of capital is the same for all …rms. I As such, there is no cross-section variation in capital prices rjt = rt for all j ,t 6 / 27 Assumptions I Olley and Pakes assume: I Assumption 2. Capital is Dynamic I There is a time-to-build assumption made about capital. I A …rm’s decision is about its level of investment (not its capital per se) I Speci…cally, capital is accumulated by through a dynamic investment process k jt = (1 )k jt 1 + ijt 1 where ijt is the rate of depreciation 1 is the …rm’s capital investment in period t 1 I However, while investment ijt is chosen in period t; it doesn’t become productive capital until period t + 1 I Thus, in period t; the quantiy of capital for the …rm is …xed. I Instead, taking as given the price of capital rt ; its current level of capital k jt ; and its productivity shock !jt ; the capital decison problem of the …rm in period t is to choose its investment ijt into capital for next period ijt = fk (rt ; k jt ; !jt ) where fk ( ) is the …rm’s optimal decision rule for capital investment 7 / 27 Assumptions I Olley and Pakes assume: I Assumption 2*. Labor is Non-Dynamic I Labor is assumed to be a completely variable input in production. I A …rm is free to choose whatever quantity of labor ljt it wishes in the current period I Taking as given the price of labor w jt ; its current level of capital k jt ; and its productivity shock !jt ; the labor decison problem of the …rm in period t is to choose its quantity of labor for current period production ljt = fL (w jt ; rt ; k jt ) where where fL ( ) is the …rm’s optimal decision rule for labor I Note: this assumption is not necessary for the Olley-Pakes estimator I In fact, Olley and Pakes do not actually use this assumption. I Rather, we include this assumption for completeness 8 / 27 Assumptions I Olley and Pakes assume: I Assumption 3. Investment is invertible in !jt I Firms with higher productivity are expected to invest more into capital accumulation. I Thus, the …rm’s optimal decision rule for capital investment fk (k jt ; rt ; !jt ) is strictly increasing in !jt I This means that the investment capital investment function fk (k jt ; rt ; !jt ) is invertible in its !jt argument ijt = fk (rt ; k jt ; !jt ) ! !jt = fk 1 (rt ; k jt ; ijt ) 9 / 27 Assumptions I Olley and Pakes assume: I Assumption 4. Productivity Shock follows a 1st Order Markov Process I The evolution of the productivity shock !jt productivity shock from year to year is determined by an exogneous stochastic process. I Speci…cally, !jt follows a 1st Order Markov Process I Thus, the current value of the productivity shock !jt is soley dependent on its one-period lagged value !jt = !jt 1 + jt where is a parameter with j j < 1 !jt jt 1 is the persistence in !jt between t is the innovation in the !jt between t 1 and t 1 and t 10 / 27 The Model I Suppose we have a panel of J …rms observed over T years. I For each …rm j at time t , we observe (the logs of) output, labor and capital fqjt ; ljt ; kjt : j = 1; 2; :::J and t = 1; 2; :::T g I Consider the following model of simultaneous equations (PF) qjt = (L) ljt = fL (wjt ; rt ; kjt ) (K) ijt = fK (rt ; kjt ; !jt ) 0 + L ljt + K kjt + !jt + "jt where (L) and (K) represent …rms’optimal decision rules for labor & capital I We explicitly account for (L) and (K) as it is through these choices that the endogeneity problem makes itself present I However, we make no assumption about the functional forms of fL or fK I Rather, we will rely on the covariance structure assumptions in (A1) - (A4) 11 / 27 The Model I In summary, our model of simultaneous equations (PF) qjt = (LD) ljt = fL (wjt ; rt ; kjt ) (KD) ijt = fK (rt ; kjt ; !jt ) 0 + L ljt + K kjt + !jt + "jt when (LD) & (KD) rep. …rms’optimal decision rules for labor & cap. investment I We make no assumptions about the functional forms of fL or fK I Instead, we assume I (A1) rt is common across all …rms I (A2) kjt is decided at t I (A3) ijt = fK (rt ; kjt ; !jt ) is invertible in !jt I (A4) !jt = !jt 1 + 1 with kjt = (1 )kj ;t 1 + ij ;t 1 jt 12 / 27 Estimation I The Olley Pakes estimator proceeds in two stages I Stage 1 I I I The …rst stage estimates L using a "control function" approach This stage relies on assumptions (A1) and (A3) Stage 2 I I The second stage estimates K ; taken as given the estimates from Stage 1. This stage relies on assumptions (A2) and (A4). 13 / 27 Estimation - Stage 1 I The …rst stage estimates I L using assumptions (A1) and (A3) Since, by (A3), ijt = fK (rt ; kjt ; !jt ) is invertible in !jt ; hence we can write !jt = fK 1 (rt ; kjt ; ijt ) I We can use this to substitute for !jt into (PF) qjt = 0 + L ljt + K kjt + fK 1 (rt ; kjt ; ijt ) + "jt or qjt = where (rt ; kjt ; ijt ) 0 + K L ljt + (rt ; kjt ; ijt ) + "jt kjt + fK 1 (rt ; kjt ; ijt ) I This last equation is a partially linear regression model I The parameter L and the function ( ) can be estimated using semiparametric methods 14 / 27 Estimation - Stage 1 I We face two di¤erent problem in estimating the function (rt ; kjt ; ijt ) 1. We do not observe the price of capital rt I 2. depends on rt ; which is not a variable observed in our data. I This is why we need assumption (A1) I Since rt is common across …rms, we can treat it as a year …xed e¤ect. I That is, we can control for the unobserved rt by a set of year-dummies I We write t (kjt ; ijt ) t (k jt ; ijt ) to make clear the dependence of on time has unknown functional form I Without a function form assumption about fK ; the function I Nevertheless, we can appeal to Taylor’s Theorem, which tells us that we can approximate any function using a high-order polynomial. I For example, a 2nd-Order of t = 0 + 1 k jt + is unknown t (k jt ; ijt ) i + 2 jt 3 k jt ijt + 4 k jt2 + i 2 + yeart 5 jt where yeart is a set of year dummies 15 / 27 Estimation - Stage 1 I Thus, the OP …rst stage regession qjt = L ljt + ( 0 + 1 kjt + i + 2 jt 3 kjt ijt + 4 kjt2 + i 2 + yeart ) + "jt 5 jt I This method is a control function method. I Instead of instrumenting the endogenous regressors, we include additional regressors that capture the endogenous part of the error term (i.e., proxy for the productivity shock). I By including a ‡exible function in (kjt ; ijt ), we are controlling for !jt I Thus standard estimation methods (e.g. OLS) yield us a consistent estimates of L 16 / 27 Estimation - Stage 1 I Thus, the OP …rst stage regession qjt = I L ljt + ( + 0 1 kjt + i + 2 jt 3 kjt ijt + 4 kjt2 + i 2 + yeart ) + "jt 5 jt The …rst stage provides us with: I an estimate of L I an estimate of t (kjt ; ijt ) I I as the coe¢ cient on ljt for each …rm j in each period t in particular, if we construct the …tted values b q jt then an estimate of t (kjt ; ijt ) bjt = b q jt b ljt L 17 / 27 Estimation - Stage 2 I The second stage estimates I K using assumptions (A2) and (A4) Recall, we de…ned t (kjt ; ijt ) I But fK 1 (rt ; kjt ; ijt ) = !jt I Thus 0 jt I = + 0 K kjt + fK 1 (rt ; kjt ; ijt ) + kjt + !jt Now, by (A4), !jt = !jt I K 1 + jt Using this to substitute in for !jt ; we have jt = 0 + K kjt + !jt 1 + jt 18 / 27 Estimation - Stage 2 I The second stage estimates I In principal, the period t equation jt I I I using assumptions (A2) and (A4) K = 0 + K kjt + !jt 1 + jt is a regression model through which we can estimate K However, in this equation !jt 1 is an unobservable variable However, a similar logic applies to jt in every period. In particular, in period t 1, we also have jt 1 = !j ;t 1 = 0 + K kj ;t 1 + !j ;t 1 kj ;t 1 or I j ;t 1 0 K Using this last expression to substitute in for !jt jt = 0 + K kjt + ( jt 1 0 1 into the period t equation K kjt 1) + jt 19 / 27 Estimation - Stage 2 I Thus, the OP second stage regession jt I = (1 ) 0 + K kjt + j ;t 1 K kj ;t 1 + jt Note: 1. We do not observe the jt ’s. But we do have consistent estimates of their values from Stage 1. So, our regression model b = (1 jt ) 0 + K kjt + bj ;t 1 K kj ;t 1 + jt 2. The parameters of this model enter non-linearly. So, we estimate this model by Non-Linear Least Squares 20 / 27 Aside - Nonlinear Least Squares I Consider the Nonlinear Regression Model yi = f (xi ; ) + "i where I I I xi is a vector of explanatory variables for the outcome yi is a vector of parameters f (xi ; ) is a function in which the parameters enter nonlinearly. I We have data fyi ; xi : i = 1; :::; ng; and wish to estimate the I Of course, we do not observe the error term "i I But, if we had an estimate of ; we could constuct an estimate of "i I The residuals e i = yi parameters f (xi ; b) 21 / 27 Aside - Nonlinear Least Squares I The Nonlinear Least Squares (NLLS) estimator of the parameters is vector tha minimizes the sum of of square residuals X 2 min [yi f (xi ; b)] b i I Generally, there is no anlytical solution for the NLLS estimator I Instead, we ask computer software to minimize the sum of squared residuals by searching numerically for the NLLS estimator. 22 / 27 Aside - Nonlinear Least Squares I Example 1. I Consider the Nonlinear Regression Model qi = 0 + 1 li + 2 ki + 1 2 l i k i + "i I We can estimate this model in STATA using Nonlinear Least Squares I The STATA syntax is nl ( q = fb0g + fb1g l + fb2g k + fb1g fb2g l I k ) Notes I The STATA keyword is nl I We type out our model explicitly, and enclose it in round brackets ( ) I Parameters are indicated using brace brakets { } I Variables are indicated using roman letters 23 / 27 Aside - Nonlinear Least Squares I Example 2. I To STATA syntax to estimate the OP second stage regression is nl ( t = (1 fr 0g)*fb0g + fbkg*k + fr 0g* where the one period lagged values of j ;t 1 by sort …rmid : generate t 1 + frog*fbkg*kt and kj ;t 1 were generated by t 1 = [_n-1] by sort …rmid : generate kt 1 = k[_n-1] 1 ) 24 / 27 Estimation - OP Algorithm I Is summary, the Olley and Pakes Alorithm 1. Stage 1. Estimate qjt = I I L ljt + ( L 0 using the model + 1 kjt + i + 2 jt 3 kjt ijt + 4 kjt2 + i 2 + yeart ) + "jt 5 jt Regress q jt on ljt ; k jt ; ijt ; k jt ijt ; k jt2 ; ijt2 ; and a set of year e¤ects. L is estimated as the coe¢ cient by ljt : 2. Construct estimates of the jt bjt = q bjt b ljt L bjt are the …tted values from Step 1. where the q 3. Stage 2. Estimate I I K b = (1 jt using the model ) 0 + K kjt + bj ;t Regress (nonlinearly) bjt on k jt ; bjt K 1 ; k jt 1 K kj ;t 1 + jt 1 is estimated as the coe¢ cient on k jt 25 / 27 Empirical Application I During the 1970s and early 1980s, technological change and deregulation caused a major restructuring of the telecommunication equipment industry. I Olley and Pakes apply their estimation algorithm to estimate the prameters of the production function for …rms in the telecommunications equipment industry. I They use their estimates to analyze changes that occurred in the distribution of plant-level performance between 1974 and 1987 26 / 27 Empirical Application - Results Estimates of Production Function Parameters - Table VI OLS Fixed E¤ects Olley Pakes (3) (4) (8) Labor 0.693 (0.019) 0.629 (0.026) 0.608 (0.027) Capital 0.304 (0.018) 0.150 (0.026) 0.342 (0.035) 27 / 27

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