ECO 2901 EMPIRICAL INDUSTRIAL ORGANIZATION

Transcription

1 ECO 2901 EMPIRICAL INDUSTRIAL ORGANIZATION Lectures 3 & 4: Production Function Estimation Victor Aguirregabiria (University of Toronto) Toronto. Winter 2018 Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

2 Outline Outline 1. Introduction 2. Econometric issues 2.1. Measurement problems 2.2. Endogeneity: Simultaneity; Measurement error; Endogenous exit 3. Identification Assumptions and Estimation Methods 3.1. IV using input prices First order conditions for non-dynamic inputs Panel Data methods Olley and Pakes (1996) & Levinshon and Petrin (2003) 3.5. Ackerberg-Caves-Frazer (2015) critique 3.6. Gandhi Navarro-Rivers (2016) Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

3 Main References Outline Olley & Pakes (1996) was a break-through. Levinshon & Petrin (2003) also very influential Ackerberg, Caves & Frazer (2015) on identification issues and interpretation in OP & LP. Blundell & Bond (2000) dynamic panel data approach to PF Bond & Soderbom (2005) interesting insights on identification. Gandhi Navarro-Rivers (2013): First order conditions to deal with flexible inputs. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

4 Introduction Introduction Production functions (PF) are important elements in economics. Estimation of PFs plays a key role in empirical questions such as: - Productivity and its growth: measurement, heterogeneity (dispersion). - Missallocation of inputs. How allocation of capital and labor relates to TFP. - Estimation of Marginal Productivity of inputs; Estimation of Marginal Costs. - Technological change over time or across industries. Capital intensity. Skill labor intensity. - Evaluating the effects of adopting new technologies - Measuring learning-by-doing. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

5 Measurement and Econometric Issues Measurement and Econometric Issues (1) Measurement errors: M.e. in output (e.g., deflated revenue but not output); M.e. in capital; differences in quality of labor. (2) Specification: Functional form: Cobb-Douglas, CES, Translog, Leontief, NP? Constant vs Random coeffi cients? (3) Simultaneity: Observed inputs may be correlated with unobserved inputs or productivity shocks (e.g., TFP, managerial ability, capacity utilization, quality of land). This correlation introduces biases in some estimators. (4) Multicollinearity: Some inputs may be highly correlated if they are highly complementary. (5) Endogenous Exit/Selection: Firm exit from the sample is not exogenous. Endogenous exit can introduce selection bias in estimation of PF parameters. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

6 Measurement and Econometric Issues Measurement and Econometric Issues (2) First, we focus on Endogeneity problems and different solutions that have been proposed to deal with these problems. We discuss these issues in the context of a simple Cobb-Douglas PF. Some arguments and results can be extended to more general specifications of the PF. Second, we will study other issues: - Distinguishing Quantity-TFP and Revenue-TFP - Multiproduct firms - More flexible functional forms - Heterogeneous PF parameters Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

7 Model and Data Measurement and Econometric Issues Model and Data Panel data of N firms over T periods with information on output, labor, and capital (in logs): { y it, l it, k it : i = 1, 2,..., N ; t = 1, 2,...T } We are interested in the estimation of the Cobb-Douglas PF (in logs) [Simple extensions: other inputs; different technologies]: y it = β L l it + β K k it + ω it + e it ω it = unobserved inputs (for the econometrician) which are known to the firm when it decides K and L (e.g., managerial ability, quality of land, different technologies). e it = measurement error in output or shock affecting output that is unknown to the firm when it decides K and L. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

8 Measurement and Econometric Issues Simultaneity problem Simultaneity problem Marshack and Andrews (Ectca, 1944) presented one of the first descriptions of the simultaneity problem when estimating PF. y it = β L l it + β K k it + ω it + e it If ω it is known to the firm when he decides (k it, l it ), the observed inputs are correlated with the unobserved ω it and the OLS estimators of β L and β K are biased. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

9 Measurement and Econometric Issues Measurement error in inputs Measurement error in inputs Inputs, and especially capital, can be measured with error. Perpetual inventory method: depreciation and initial value of capital stocj are diffi cult to measure: k obs it = k it + e k it. Measurement error: attenuation bias in estimated coeffi cients. Absolute bias increases with the noise-to-signal ratio Var(ek it ) Var(kit obs ). Noise-to-signal ratio for capital can be substantially larger than in Var( eit k first differences than in levels: ) Var( kit obs ) >>> Var(ek it ) Var(kit obs. OLS in ) first-differences often generates very small (or even negative) capital coeffi cients (e.g., Griliches and Hausman, 1986, JoE). Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

10 Endogenous Exit Measurement and Econometric Issues Endogenous Exit Firms panel datasets are unbalanced, with significant amount of exits. We estimate the PF under selection: y it = β L l it + β K k it + ω it + e it if d it = 1 where d it is the indicator of the event "firm i is active in the market at period t". Surviving firms are not randomly chosen, e.g., they are more productivity and use more capital and labor than exiting firms. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

11 Measurement and Econometric Issues Endogenous Exit (2) Endogenous Exit The optimal exit/stay decision is: d it = 1 { V (k it, ω it ) 0 } V (k it, ω it ) is the value of the firm. Strictly increasing in k it and ω it. Since V (k it, ω it ) is strictly increasing in the two arguments, there is a cut-off value of productivity, ω (k it ), such that: d it = 1 { ω it ω (k it ) } and the threshold function ω is strictly decreasing in capital. Exit introduces correlation between error term of the PF and k it : E (ω it k it, d it = 1) = E (ω it k it, ω it ω (k it )) where λ (k it ) is decreasing in k it. = λ (k it ) Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

12 Measurement and Econometric Issues Endogenous Exit (3) Endogenous Exit λ (k it ) is the selection term. The PF can be written as: y it = β L l it + β K k it + λ (k it ) + ω it + e it where ω it {ω it d it = 1} λ (k it ) that, by construction, is mean-independent of k it. Ignoring the selection term λ (k it ) introduces bias in our estimates of the PF parameters. k it is negatively correlated with the selection term λ (k it ), and the selection problem tends to bias downward the estimate of the capital coeffi cient. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

13 Estimation Methods Input prices as IVs Estimation Methods: Input prices as IVs If input prices r it are observable, and under the assumption that cov (ω it, r it ) = 0, we can use them as instruments. This approach has several limitations/problems: (a) Input prices are not always observable. (b) If there is only one competitive input market in the population under study, there is not any cross-sectional variation in r. Time-series variation is not enough for identification. (c) When input prices have cross-sectional variation, it could be because inputs markets are not competitive and firms with higher productivity pay higher prices, i.e., cov (ω it, r it ) = 0, making input prices not a valid instrument. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

14 Estimation Methods First order conditions First order conditions for flexible inputs Suppose that labor is a perfectly flexible input and the firm is a price-taker in output and labor markets. Then, F.O.C. imply: P it Y it L it = W it For the Cobb-Douglas PF, this condition becomes: β L = W it L it P it Y it i.e., β L is identified by the wage bill-to-revenue ratio. In fact, this condition rejects this simple version of the model. Substantial sample variation in W it L it P it Y it. Either β L,it, or unobserved heterogeneity in cost of labor, or other assumptions do not hold. We will come back to this approach in Gandhi Navarro-Rivers (2013). Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

15 Estimation Methods Panel Data: Fixed Effects Panel Data: Fixed Effects [Mundlak, 1966] Suppose that we have firm level panel data with information on output, capital and labor: y it = β L l it + β K k it + ω it The Fixed Effects estimator (i.e., its consistency) is based on the following assumptions: (FE 1) ω it = η i + δ t + ωit (FE 2) var (l it l i ) and var (k it k i ) are > 0 (FE 3) ωit is not serially correlated (FE 4) ωit is NOT known to the firm when it chooses inputs at period t η i is interpreted as managerial ability, or a different technology that is constant over time. ωit represents weather or any other random, unpredictable shock. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

16 Estimation Methods Panel Data: Fixed Effects (2) Panel Data: Fixed Effects Under assumptions (PD-1) to (PD-4), the Fixed Effects estimator is consistent, i.e., OLS in the equation: (y it ȳ i ) = β L (l it l i ) + β K (k it k i ) + (ω it ω i ) Consistency of the FE (with fixed T ) requires the regressors x it = (l it, k it ) to be strictly exogenous. That is, for any (t, s): cov (x it, ω is ) = cov (x it, e is ) = 0 Otherwise, the FE-transformed regressors (l it l i ) and (k it k i ) would be correlated with the error (ω it ω i ). This is why Assumptions (FE-3) and (FE-4) are necessary for the consistency of the OLS estimator. In most applications, these are very restrictive conditions. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

17 Dynamic Panel Data Estimation Methods Dynamic Panel Data We can relax the assumption of strictly exogenous regressors, and estimate the PF using GMM in the equation in first differences (Arellano and Bond, 1993). y it = β L l it + β K k it + δ t + ω it We replace Assumption FE-4 with DPD-4 (DPD 4) Labor and capital are dynamic inputs: l it = f L (l i,t 1, k i,t 1, ω it ) and k it = f K (l i,t 1, k i,t 1, ω it ) Under these assumptions {l i,t j, k i,t j, y i,t j : j 2} are valid and useful instruments in the equation in first differences. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

18 Estimation Methods Dynamic Panel Data (2) Dynamic Panel Data Moment conditions for the GMM estimation of β L, β K, and δ t : E (l i,t j ωit ) = 0 for t = 3,..., T ; and j t 2 E (k i,t j ωit ) = 0 for t = 3,..., T ; and j t 2 E (y i,t j ωit ) = 0 for t = 3,..., T ; and j t 2 Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

19 Estimation Methods Dynamic Panel Data (3) Dynamic Panel Data Limitations of this approach: (a) It provides downward biased and imprecise estimates of β L and β K (see Blundell and Bond, 1999, 2001). Overidentifying restrictions are typically rejected. (b) The i.i.d. assumption on ωit is typically rejected: {x i,t 2, y i,t 2 } are not valid instruments. (c) Weak instruments problem: Arellano-Bond GMM estimator suffers of this problem in dynamic models where regressors in first differences are weakly autocorrelated. (d) First difference transformation also eliminates the cross-sectional variation and it is subject to the problem of measurement error in inputs. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

20 Estimation Methods Dynamic Panel Data Blundell and Bond (2001) BB propose two important extensions to the previous approach. Firsrt, ω it follows an AR(1) process: ω it = ρ ω it 1 + u it. The PF equation can be represented in semi-first differences as: (y it ρ y it 1 ) = β L (l it ρ l it 1 ) + β K (k it ρ k it 1 ) + η i + δ t + u it Second, the estimator is based not only on moment conditions in first differences (Arellano-Bond) but also on moment conditions in levels (Blundell-Bond). Under a stationarity condition, for j 1 : E ( l it j [η i + u it ]) = 0 E ( k it j [η i + u it ]) = 0 E ( y it j [η i + u it ]) = 0 Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

21 Estimation Methods Blundell and Bond (2001) Dynamic Panel Data In the equation in levels, (l it j, k it j, y it j ) for j 1 are valid instruments: (y it ρ y it 1 ) = β L (l it ρ l it 1 ) + β K (k it ρ k it 1 ) + η i + u it In the equation in first differences, (l it j, k it j, y it j ) for j 2 are valid instruments: ( y it ρ y it 1 ) = β L ( l it ρ l it 1 ) + β K ( k it ρ k it 1 ) + u it Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

22 Estimation Methods Blundell and Bond (2001): Results Dynamic Panel Data 509 manufacturing firms; Parameter OLS-Levels WG AB-GMM SYS-GMM β L (0.025) (0.030) (0.099) (0.098) β K (0.032) (0.033) (0.126) (0.074) ρ (0.006) (0.022) (0.073) (0.078) Sargan (p-value) m Constant RS (p-v) Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

23 Estimation Methods Dynamic Panel Data Bond and Soderbom (2005) Monte Carlo experiments Bond and Soderbom perform Monte Carlo experiments to study the actual identification power of ACs. They consider both deterministic and stochastic ACs. They simulate data from this model and estimate the PF using Blundell and Bond GMM method. (a) With deterministic ACs identification is weak when: (1) ACs are too low; or (2) ACs are too high; or (3) ACs of the two inputs are two similar (collinearity). With stochastic ACs identification results improve considerably. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

24 Estimation Methods Control Function Methods Control Function methods Olley & Pakes (1996; OP) and Levinsohn & Petrin (2003; LP) are control function methods. Instead of looking for instruments for K and L, we look for observable variables that can "control for" (or proxy) unobserved TFP. The control variables should come from a model of firm behavior. Note: Both OP and LP assume that labor is perfectly flexible input. This assumption is completely innocuous for their results. To emphasize this point, I present here versions of OP and LP that treat labor as a potentially dynamic input. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

25 Olley and Pakes (OP) Estimation Methods Olley and Pakes Consider the following model of simultaneous equations: (PF ) y it = β L l it + β K k it + ω it + e it (LD) l it = f L (l i,t 1, k it, ω it, r it ) (ID) i it = f K (l i,t 1, k it, ω it, r it ) (LD) & (ID): firms optimal labor and investment given state variables (l i,t 1, k it, ω it, r it ); r it = input prices. OP consider the following assumptions: (OP 1) f K (l i,t 1, k it, ω it, r it ) is invertible in ω it (OP 2) No cross-sectional variation in r it : r it = r t. (OP 3) ω it follows a first order Markov process. (OP 4) k it is decided at t 1: k it = (1 δ)k i,t 1 + i i,t 1 Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

26 Olley and Pakes (2) Estimation Methods Olley and Pakes OP method deals both with the simultaneity problem and with the selection problem due to endogenous exit. It doesn t deal with potential measurement error in inputs. OP method proceeds in two stages. First stage: estimates β L [Assumptions (OP-1) and (OP-2) are key]; and the second stage estimates β K [Assumptions (OP-3) and (OP-4) are key]. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

27 Olley and Pakes Estimation Methods First Stage Olley and Pakes Assumptions (OP-1) and (OP-2) imply that the investment equation is invertible in ω it : ω it = f 1 K (l i,t 1, k it, i it, r t ) Solving this equation in the PF we have: y it = β L l it + β K k it + fk 1(l i,t 1, k it, i it, r t ) + e it = β L l it + φ t (l i,t 1, k it, i it ) + e it This is a partially linear model. Parameter β L and functions φ 1 (.),..., φ T (.) can be estimated using semiparametric methods. A possible method is Robinson s method (1988). OP use an n th order polynomial to approximate the φ t functions. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

28 Olley and Pakes Estimation Methods First Stage Olley and Pakes This first stage is a "Control Function" method: instead of instrumenting the endogenous regressors, we include additional regressors that capture the endogenous part of the error term. We are controlling for endogeneity by including (l i,t 1, k it, i it ) as "proxies" of ω it. Key assumptions for the identification of β L : (a) Invertibility of f K (l i,t 1, k it, ω it, r t ) w.r.t ω it. (b) r it = r t, i.e., no cross-sectional variability in unobservables, other than ω it, affecting investment. (c) Given (l i,t 1, k it, i it, r t ), labor l it still has sample variability. Assumption (c) is key, and it is the base for Ackerberg-Caves-Frazer criticism/extension of Olley-Pakes approach. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

29 Olley and Pakes Estimation Methods First Stage Olley and Pakes Example (with lparametric linear investment func.): (PF ) y it = β L l it + β K k it + ω it + e it (Inverse ID) ω it = γ 1 i it + γ 2 l i,t 1 + γ 3 k it + γ 4 r it Then, y it = β L l it + (β K + γ 3 ) k it + γ 1 i it + γ 2 l i,t 1 + (γ 4 r it + e it ) Note that l it is correlated with r it. Therefore, we need r it = r t and include time dummies to control for r t in order to have consistency of the OLS estimator in this regression. Note also that to identify l it with enough precision we need not high collinearity between this variable and (k it, i it, l i,t 1 ). Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

30 Olley and Pakes Estimation Methods Olley and Pakes Second Stage Estimation of β K. It is based on the other two assumptions: (OP 3) (OP 4) ω it follows a first order Markov process. k it is decided at t 1: k it = (1 δ)k i,t 1 + i i,t 1 Since ω it is first order Markov, we can write: ω it = E [ω it ω i,t 1 ] + ξ it = h (ω i,t 1 ) + ξ it where ξ it is an innovation which is mean independent of any information at t 1 or before. And h(.) is some unknown function. φ it is identified from 1st step; and φ it = β K k it + ω it. Then, φ it = β K k it + h ( φ i,t 1 β K k i,t 1 ) + ξit Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

31 Olley and Pakes Estimation Methods Olley and Pakes Second Stage We estimate h(.) and β K by applying recursively the same type of semiparametric method as in the first stage of OP. φ it = β K k it + h ( φ i,t 1 β K k i,t 1 ) + ξit Suppose that we consider a quadratic function for h(.): i.e., h(ω) = π 1 ω + π 2 ω 2. Then: φ it = β K k it + π 1 ( φi,t 1 β K k i,t 1 ) + π2 ( φi,t 1 β K k i,t 1 ) 2 + ξ it It is clear that β K, π 1 and π 2 are identified in this equation. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

32 Olley and Pakes Estimation Methods Olley and Pakes Second Stage Time-to build is a key assumption for the consistency of this method. If investment at period t is productive, then the equation becomes: φ it = β K k i,t+1 + h ( φ i,t 1 β K k it ) + ξit k i,t+1 depends on investment at period t and therefore it is correlated with the innovation ξ it. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

33 Estimation Methods OP: Empirical Application Olley and Pakes US Telecom. equipment industry: Technological change and deregulation. - Elimination of barriers to entry; - Antitrust decisions against AT&T: The Consent Decree (implemented in 1984) > divestiture of AT&T. - Substantial entry/exit of plants. Data: US Census of manufacturers. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

34 Estimation Methods OP: Empirical Application Olley and Pakes Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

35 Estimation Methods OP: Empirical Application Olley and Pakes Going from OLS balanced panel to OLS full sample almost doubles β K and reduces β L by 20%. [Importance of endogenous exit]. Controlling for simultaneity further increases β K and reduces β L. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

36 Estimation Methods OP: Empirical Application Olley and Pakes Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

37 Estimation Methods Levinshon & Petrin (2003) Olley and Pakes The main difference with OP method is that LP use the demand function for intermediate inputs instead of the investment equation to invert out unobserved productivity. Two main motivations: - Investment can be responsive to more persistent shocks in TFP; materials is responsive to every shock in TFP. - In some datasets Zero Investment accounts for a large fraction of the data. At i it = 0 (corner solution / extensive margin) there is not invertibility between i it and ω it. Problems: loss of effi ciency; missing estimates of TFP for many observations. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

38 Estimation Methods Levinshon & Petrin (2003) Olley and Pakes They consider a Cobb-Douglas production function in terms of labor, capital, and intermediate inputs (materials): y it = β L l it + β K k it + β M m it + ω it + e it Investment equation is replaced with demand for materials: m it = f M (l i,t 1, k it, ω it, r it ) Assumption LP-1: f M (l i,t 1, k it, ω it, r it ) is invertible in ω it. They maintain OP-2 [No other unobservables; r it = r t ], OP-3 [Markov TFP], and OP-4 [Time-to-build]. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

39 Estimation Methods Levinshon & Petrin: First Step Olley and Pakes Least squares estimation of parameter β L and the nonparametric functions {φ t (.) : t = 1, 2,..., T } in regression equation: y it = β L l it + φ t (l i,t 1, k it, m it ) + e it φ t (l i,t 1, k it, m it ) = β K k it + β M m it + fm 1(l i,t 1, k it, m it, r t ) and fm 1 is the inverse function of f M with respect to ω it. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

40 Estimation Methods Levinshon & Petrin: Second Step Olley and Pakes The second step is also similar to OP s second step but in the model with the intermediate input. φ it is estimated in 1st step; and φ it = β K k it + β M m it + ω it. Then, φ it = β K k it + β M m it + h ( φ i,t 1 β K k i,t 1 β M m i,t 1 ) + ξit Important difference with OP: In this second step E (m it ξ it ) = 0, i.e., materials m it is endogenous. LP propose two approaches: - "unrestricted method": instrument m it with its lagged values [see GNR (2013) criticism]; - "restricted method": under statis input, price-taking: β M = Cost of materials/revenue. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

41 Estimation Methods LP: Empirical application Olley and Pakes Plant-level data from 8 different Chilean manufacturing industries: [Pinochet period]. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

42 Estimation Methods Olley and Pakes LP: Empirical Application. Var input shares Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

43 Estimation Methods LP: Empirical Application: Zeroes Olley and Pakes Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

44 Estimation Methods LP: Empirical Application: Zeroes Olley and Pakes Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

45 Estimation Methods Ackerberg-Caves-Frazer critique Ackerberg-Caves-Frazer critique Their criticism applies both to OP and LP. Here we present it in the context of OP. Main point: [1] If OP (or LP) assumptions are correct, then conditional on (k it, i it ) in OP [or (k it, m it ) in LP] there should not be any cross-sectional variation left in l it. Perfect collinearity between l it and φ t (k it, i it ). [2] In the data, we find that conditional on (k it, i it ) [or to (k it, m it ), or even to (l i,t 1, k it, m it )] there is still cross-sectional variation left in l it. This should be because the assumptions of the model do not hold. Identification may be spurious; estimation can be inconsistent... unless there are alternative assumptions that explain/allow for the not perfect collinearity between l it and φ t (k it, i it ) and imply consistency of these control function methods. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

46 Estimation Methods Ackerberg-Caves-Frazer critique Ackerberg-Caves-Frazer critique The state variables of the firm s problem are (k it, i it, r t ), then the firm s labor demand is: l it = f L (k it, ω it, r t ) And given that ω it = fk 1(k it, i it, r t ), we have that: ( l it = f L kit, fk 1(k ) it, i it, r t ), r t = G (k it, i it, r t ) Therefore, for (k it, i it, r t ) fixed, current labor l it should not have any sample variability. That is, if the assumptions in OP model hold, then there should be a deterministic relationship between l it and φ t (k it, i it ) and it should not be possible to identify β in the first step. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

47 Estimation Methods Ackerberg-Caves-Frazer critique Ackerberg-Caves-Frazer critique (2) In the data, we observe that there is not perfect collinearity between l it and φ t (k it, i it ). Two possible explanations: [1] Unobservable r it = r t that enter in the demand of both labor and investment OP and LP are inconsistent estimation methods. [2] Unobservable r it = r t that enter in the demand of labor but NOT in the investment decision OP and LP are consistent estimation methods. ACF consider arguments/assumptions that can generate scenario [2] and save OP / LP methods. The also propose an alternative method. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

48 ACF: Saving OP & LP Estimation Methods Ackerberg-Caves-Frazer critique Consider the model: (PF ) (LD) (ID) y it = β L l it + β K k it + ω it + e it ( l it = f L li,t 1, k it, ω it, rit L ) ( i it = f K li,t 1, k it, ω it, rit K ) where: (OP 1) (OP 2) (OP 3) (OP 4) (ACF 1) ( f K li,t 1, k it, ω it, rit K ) is invertible in ωit No cross-sectional variation in rit K : r it K = rt K. ω it follows a first order Markov process. k it is decided at t 1: k it = (1 δ)k i,t 1 + i i,t 1 var(r L it t, i it, l i,t 1, k it ) > 0 Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

49 ACF: Saving OP & LP Estimation Methods Ackerberg-Caves-Frazer critique (ACF 1) var(r L it t, i it, l i,t 1, k it ) > 0 Some economic assumptions that generate (OP-2) and (ACF-1): * Idiosyncratic shock in the price of labor that is i.i.d. over time occurs after the investment decision. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

50 Other identifying conditions: Quasi-fixed inputs ACF: A new method. Quasi-fixed inputs Consider a CD-PF with labor and capital as only inputs. Suppose that OP assumptions hold such that l it is perfectly collinear with φ t (l i,t 1, k it, i it ). If both capital and labor are quasi-fixed inputs, then it is possible to use a control function method in the spirit of OP or LP to identify/estimate β L and β K. Or in other words, this model has moment conditions that identify β L and β K (Wooldridge, EL 2009). Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

51 Other identifying conditions: Quasi-fixed inputs ACF: A new method. Quasi-fixed inputs In the first step we have: y it = β L l it + φ t (l i,t 1, k it, i it ) + e it = β L g t (l i,t 1, k it, i it ) + φ t (l i,t 1, k it, i it ) + e it = ψ t (l i,t 1, k it, i it ) + e it In this first step, we estimate ψ t (l i,t 1, k it, i it ) nonparametrically. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

52 Other identifying conditions: Quasi-fixed inputs ACF: A new method. Quasi-fixed inputs In the second step, given ψ it, and taking into account that ψ it = β L l it + β K k it + ω it, and ω it = h (ω i,t 1 ) + ξ it, we have that: ψ it = β L l it + β K k it + h (ψ it β L l it 1 + β K k it 1 ) + ξ it In this second step, l it is correlated with ξ it, but (k it, ψ it, l it 1, k it 1 ) are not, and (l it 2,k it 2 ) can be used to instrument l it. This approach is in the same spirit as the Dynamic Panel Data (DPD) methods of Arellano-Bond and Blundell-Bond. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

53 Other identifying conditions: Quasi-fixed inputs Other identifying conditions: Quasi-fixed inputs [4] This approach cannot be applied if some inputs (e.g., materials) are perfectly flexible. The PF coeffi cient parameter of the flexible inputs cannot be identified from the moment conditions in the second step. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

54 Other identifying conditions: F.O.C. for flexible inputs Other identifying conditions: F.O.C. for flexible inputs Klette & Grilliches (1996), Doraszelski & Jaumandreu (2013), and Gandhi, Navarro, & Rivers (2013) propose combining conditions from the PF with conditions from the demand of variable inputs. This approach requires the price of the variable input to be observable to the researcher, though this price may not have cross-sectional variation across firms. Note that in the LP method, the function that relates m it with the state variables is just the condition "VMP of materials equal to price of materials". The parameters in this condition are the same as in the PF. This approach takes these restrictions into account. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

55 Other identifying conditions: F.O.C. for flexible inputs Other identifying conditions: F.O.C. flexible inputs [2] For the CD-PF, with materials as flexible input, we have that: (PF ) y it = β L l it + β K k it + β M m it + ω it + e it (FOC ) p t p M t = ln(β M ) + β L l it + β K k it + (β M 1)m it + ω it The difference between these two equations is: ln(s M it ) = ln(β M ) + e it where sit M PM t M it P t Y it revenue. is the ratio between material expenditures and Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

56 Other identifying conditions: F.O.C. for flexible inputs Other identifying conditions: F.O.C. flexible inputs [3] The parameter(s) of the flexible inputs are identified from the expensiture-share equations. The parameter(s) of the quasi-fixed inputs are identified using the dynamic conditions described above. Gandhi, Navarro, & Rivers (2013) show that this approach can be extended in two important way: - To a nonparametric specification of the production function: y it = f (l it, k it, m it ) + ω it + e it. - To a model with monopolistic competition (instead of perfect competition) with and isolastic product demand. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

57 Other identifying conditions: F.O.C. for flexible inputs Other identifying conditions: F.O.C. flexible inputs [4] This approach relies on two strong assumptions. There is not any bias or missing parameter in the MC of the flexible input. Suppose that MC Mt = Pt M τ, then our estimate of β M will actually estimate β M τ. In its current version, this method cannot incorporate oligopolistic competition in the product market, only a limited form of monopolistic competition. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

58 Gandhi-Navarro-Rivers (2013) Gandhi-Navarro-Rivers (2013) Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

59 Endogenous exit Endogenous exit Endogenous exit implies that we have: φ it = β K k it + h (ω i,t 1 ) + ξ it if d it = 1 And we can write it as: φ it = β K k it + h(ω i,t 1 ) + λ(k it, l it 1, ω i,t 1 ) + ξ it where λ(ω i,t 1 ) is the selection term E (ξ it d it = 1, k it, l it 1, ω i,t 1 ). Under general conditions, this selection term is a function of the probability of staying in the market. Victor Aguirregabiria () Empirical IO Toronto. Winter / 60

60 Olley and Pakes Endogenous exit Endogenous exit The stay/exit decision is: d it = 1 { ω it > ω (k it, l it 1 ) } = 1 { ξ it > ω (k it, l it 1 ) h(ω i,t 1 ) = 1 { ξ it > ω (k it, l it 1 ) h ( ) } φ it 1 β K k it 1 = 1 { ξ it > g(k it, l i,t 1, φ i,t 1, k it 1 ) } The Prob of staying (Probit model): P it = E ( d it k it, l i,t 1, φ i,t 1, k it 1 ) And the equation controlling for selection is: φ it = β K k it + h ( φ i,t 1 β K k i,t 1 ) + λ (Pit ) + ξ it Victor Aguirregabiria () Empirical IO Toronto. Winter / 60