A Robust Approach to Estimating Production Functions: Replication of the ACF procedure

Transcription

1 A Robust Approach to Estimating Production Functions: Replication of the ACF procedure Kyoo il Kim Michigan State University Yao Luo University of Toronto Yingjun Su IESR, Jinan University August 2018 Abstract We study Ackerberg, Caves, and Frazer (2015) s production function estimation method using Monte Carlos simulations. First, we replicate their results by using the same procedure and confirm the existence, as noted by ACF, of a spurious minimum in the estimation. In the population, or when the sample size is large enough, this global identification problem may not be a concern because the spurious minimum holds only at the extreme values of the capital and the labor coefficients. However, in the finite samples, their estimator can produce estimates that may not be clearly regarded as the spurious ones. In our second experiment, we modify the ACF procedure and show that using additional lagged instruments or a sequential search help to obtain robust estimates. We also provide some arguments why such modifications help in the ACF setting. Keywords: Production Function, Control Function, Monte Carlo Simulation JEL Classifications: C14, C18, D24 Corresponding authors: yis17@pitt.edu Kyoo il Kim, kyookim@msu.edu; Yao Luo, yao.luo@utoronto.ca; Yingjun Su, 1

2 1 Introduction Estimating firm level productivity is a fundamental problem in economics. It suffers from many econometric issues such as measurement error, selection, and endogeneity of inputs. The fundamental challenge in estimating production function is that inputs and output are simultaneously chosen by the firms (Marschak and Andrews 1944). This simultaneity problem results in inputs being correlated with the unobserved productivity, leading to inconsistency of the production function estimates using least squares. In the modern literature Olley and Pakes (OP, 1996) and Levinsohn and Petrin (LP, 2003) propose a control function approach to address this simultaneity problem and their methods have been extensively used by empirical researchers. In particular, they utilize a proxy variable such as firm-level investment or intermediate input, which is strictly increasing in the unobserved productivity, being conditioned on the state variable such as capital input. Inverting this function gives a nonparametric function of capital and the proxy variable, which is used in the production function to control for the unobserved firm productivity. An important contribution by Ackerberg, Caves, and Frazer (ACF, 2015), however, shows a functional dependence problem in the first stage of LP procedure that regresses output on the labor input and the nonparametric function of other inputs. This functional dependence may result in a fundamental non-identification problem. 1 As an alternative, they propose inverting the input demand function that is conditional on state variables including the labor input as well and provide extensive simulation evidences to support their procedure. This paper replicates ACF s simulation results in both a narrow and a broader sense. First, we replicate their results by using the same setting with ACF. While the narrow replication produces identical results with ACF when we take the initial parameter values of estimation being the true values as in ACF, we also observe that their procedure yields a spurious minimum in the estimation, as discussed by ACF, when we add a grid search over different starting values deviated from true ones. This spurious minimum problem arises because the original ACF moment function has multiple solutions for their DGP in Monte Carlo simulations. As it is well noted by ACF (their footnote 16), their population moment condition holds both at the true parameter value and at a spurious parameter value such that the capital coefficient becomes zero and the labor coefficient becomes the sum of the true labor and the true capital coefficients. 2 One can interpret this 1 See also Gandhi, Navarro, and Rivers (2017) for related identification problems of the proxy variable approach in the production function estimation. 2 To provide some background to readers, here we cite ACF s footnote 16 in its entirety: As pointed out in a prior version of this paper, there is an identification caveat using our suggested moments in all three of these DGPs. More specifically, there is a global identification issue in that the moments have expectation zero not only at the true parameters, but also at one other point on the boundary of the parameter space where ˆβ k = 0, ˆβl = β l + β k, and the estimated AR(1) coefficient on ω equals the AR(1) coefficient on the wage 2

3 identification problem as a weak moment condition problem that yields multiple solutions. 3 Our Monte Carlos confirm that the simulated distribution of estimator has two modes - one at the true minimum and the other at the spurious minimum solution. In the population, or when the sample size is large enough, this global identification problem may not be a real concern because the spurious minimum holds only at the extreme values of the capital and the labor coefficients. In the finite samples, however, it becomes a relevant concern because the estimator, being driven by the spurious population minimum, can produce estimates that may not be clearly regarded as the spurious ones by empirical researchers. 4 In our second experiment, we modify the ACF procedure in various directions. We manage to find easy-to-implement modifications that yield estimates robust to the global identification problem in the finite samples. The first modification is using further lagged input variables as instruments in addition to the ACF instruments. 5 These instruments act as exclusion restrictions to strengthen the moment condition and help for identification. In the ACF setting, however, such further lagged inputs may not be relevant for the current input demand given the ACF instruments. Nevertheless, we argue that the lagged instruments such as the labor input lagged by two periods become relevant because they act like a proxy to missing factors such as input prices in the input demand function. The second modification is using a sequential search method. Although this search method can be computationally demanding, the method helps to break the nonlinearity of the ACF moment condition. We find, although the global identification problem still exists in the population moment condition, these modified ACF procedures are robust to the spurious minimum in the estimation, so they will be useful whether or not the spurious minimum problem is a serious concern to applied researchers. The rest of the paper is organized as follows. Section 2 outlines the production function estimation and the ACF procedure. Section 3 describes the Monte Carlo experiments and report the results. We conclude by discussing our findings in Section 4. The discussion includes possible explanations why such modifications would work in the ACF setting. process. One can easily calculate that at these alternative parameter values, the second stage moment equals the innovation in the wage process, which is orthogonal to k it and l it 1. This spurious minimum is a result of labor satisfying a static first order condition, and we suspect it would not occur were labor to have dynamic implications, nor when the alternative moments (29) are assumed. As such, we ignore this spurious minimum in our Monte Carlos. 3 We thank a referee for suggesting this interpretation of the problem. 4 This concern, whether it is large or small, will be more relevant if the standard error of the estimator is not small or if indeed the true capital coefficient is relatively small. 5 Wooldridge (2009) also used further lagged input variables as instruments in his moment functions of the LP setting. 3

4 2 Model and Estimation We briefly discuss Ackerberg, Caves, and Frazer (2015) s procedure to estimate production functions and we then develop our simulation study. ACF estimate the value-added production function y it = β 0 + β l l it + β k k it + ω it + η it where l it denotes the labor input for firm i at time t, k it denotes the capital input, ω it is unobserved state variable that impacts the firm s decisions on inputs and production level, and η it denotes a pure i.i.d. shock in the production. Because the labor and the capital input (or investment) decisions depend on ω it, the regressors l it and k it are potentially endogenous. To control for this endogeneity problem, OP and LP propose using a proxy variable to invert out the unobserved productivity ω it. For the proxy variable, OP use the investment and LP propose to use the intermediate input instead, because smaller firms often report zero investment. OP/LP make an implicit assumption on the timing of labor input in their proxy variable approach. ACF allow more flexible assumption on the timing of labor input. Following ACF we define the intermediate input s demand as m it = f t (l it, k it, ω it ). The approach requires f t (,, ) is strictly monotonic in ω it for all (l it, k it ) so that we have the inverse function ω it = h t (l it, k it, m it ). This setting allows l it to be determined before all or part of ω it is realized at time t. Substituting into the production function, we obtain y it = β 0 + β l l it + β k k it + h t (l it, k it, m it ) + η it = Φ t (l it, k it, m it ) + η it. In the first stage of ACF, we estimate the function Φ t (l it, k it, m it ) or its parameterization by minimizing the sample counterpart of the objective function E[(y it Φ t (l it, k it, m it )) 2 ]. When the control function is parameterized, the function Φ t (l it, k it, m it ) is estimated using a regression. In the second stage of ACF, we use the estimated control function Φ t and assume a first order Markov process of the productivity term as ω it = E[ω it ω i,t 1 ] + ξ it = g(ω i,t 1 ) + ξ it 4

5 to control for the endogeneity of inputs where ξ it denotes an innovation term. In particular, we use the conditional moments E[ξ it + η it I i,t 1 ] = E[y it β 0 β l l it β k k it ) g ( Φt 1 (l i,t 1, k i,t 1, m i,t 1 ) β 0 β l l i,t 1 β k k i,t 1 I i,t 1 ] = 0 where I i,t 1 denotes the firm s information at t 1. Based on timing assumptions of input decisions we can use instruments (e.g.) z it = (1, k it, l i,t 1, Φ t 1 (l i,t 1, k i,t 1, m i,t 1 )) if labor is to be chosen after time t 1 while we may use z it = (1, k it, l it, l i,t 1, Φ t 1 (l i,t 1, k i,t 1, m i,t 1 )) if labor is chosen at t 1. In practice, the conditional mean function in the Markov process is parameterized as g(ω i,t 1 ; β ω ). For example, researchers often use a simple autoregressive model as ω it = ρω i,t 1 + ξ it. Let w it = {(y it, l it ) z it } and write the moment condition: E[r(w it ; θ 0 )] = 0 where (1) r(w it ; θ) = z it (y it β 0 β l l it β k k it g( Φ t 1 (l i,t 1, k i,t 1, m i,t 1 ) β 0 β l l i,t 1 β k k i,t 1 ; β ω )) where θ = (β 0, β l, β k, β ω ) and θ 0 denotes the true parameter. Estimating the production parameters based on the moment condition (1) is a nonlinear optimization problem since the production parameters (β 0, β l, β k ) enter both in the production function and inside the function g( ). In our Monte Carlo study we focus on a simple setting for which ω it follows an autoregressive process as in ACF s Monte Carlos setting, so the moment condition for estimation is simplified to E[r(w it ; θ)] = E[z it (y it β 0 β l l it β k k it ρ ( Φ t 1 (l i,t 1, k i,t 1, m i,t 1 ) β 0 β l l i,t 1 β k k i,t 1 )] = 0. To reduce the dimension of the above non-linear search, ACF propose to concentrate out two parameters β 0 and ρ as follows (see Appendix A4 in ACF). For a trial value of the parameters β l and β k, ACF first construct an estimate for β 0 + ω it as β 0 + ω it (β l, β k ) = Φt (l it, k it, m it ) β l l it β k k it, where Φt (l it, k it, m it ) is obtained from the first stage. Then ACF estimates the AR(1) process by regressing β 0 + ω it (β l, β k ) on its lagged variable β 0 + ω i,t 1 (β l, β k ) and obtain the regression 5

6 residual as ˆξ it (β l, β k ). Finally, ACF estimate β l and β k using the concentrated moment condition E [ ˆξ it (β l, β k ) ( l i,t 1 k it )] = 0. (2) We run our Monte Carlo experiment based on this moment condition to estimate the production function parameters. 3 Monte Carlo Experiment We follow the Monte Carlo setup (DGP1) used by ACF to generate the data. Our design of the experiment is twofold. First, we estimate the production function by replicating ACF s original procedure but with varying initial values. Second, after observing the original ACF procedure is subject to a spurious minimum problem, we experiment with various modifications to their original procedure. 3.1 Replicating ACF We simulate 1000 datasets consisting of 1000 firms for 10 time periods following DGP1 in ACF. First, we estimate the production function by replicating the same ACF procedure. That is, we adopt the concentrated moment conditions proposed in the equation (41) of ACF (which is our equation (2)) with instruments (l t 1, k t ) to estimate β l and β k. In each replication, we plug in the true value (β l, β k ) = (0.6, 0.4) as the starting value for the search. For this experiment we are able to generate exactly the same results with ACF (See Table 1). As ACF note in their footnote 16 there is a global identification issue in their data generating process and they use the true value as the initial value to set aside this identification caveat in their Monte Carlos study. To measure how this spurious minimum problem influences the estimation in the finite samples, we experiment with using different sets of starting values, i.e. (0, 1), (0.1, 0.9), (0.2, 0.8),..., (0.9, 0.1) for the search of global minimum of the GMM objective function based on the moment condition (2). For each set of initial values, we obtain an estimate of (β l, β k ) that minimizes the objective function (i.e. local minimum). We then compare the values of these local minimums and obtain the estimate that yields the global minimum for each simulated dataset. Table 2-1 reports averages and standard deviations of the estimated coefficients that yield the global minimum from this procedure with 1000 repetitions. We also report the percentage of estimates that may be reasonably regarded as the spurious minimum out of the 1000 repetitions. Figure 1 clearly illustrates that the simulated distribution of estimator has two distinctive modes - one at the true minimum and the other at the spurious minimum. 6

7 In DGP1 firms are facing different wages that follow the AR(1) process ln(w it ) = 0.3 ln(w i,t 1 ) + ξ W it where the standard deviation of the log wage ln(w it ) is set to be constant over time and equal to 0.1 as in the original ACF s DGP. In this DGP, relative to a baseline in which all firms face the mean log wage in every period, the within firm, across-time, standard deviation of the log labor is increased by about 10% with this level of wage variation. In the next experiment we modify DGP1 by increasing the standard deviation of the log wage to 0.5 to see how the sensitivity to the spurious minimum changes for this DGP setting. For this DGP the within firm, across-time, standard deviation of the labor input is increased by about 50%. 6 The results for this alternative DGP1 are reported in Table 2-2. In particular the percentage of the spurious minimum indications (e.g.) for cases β l > 0.80 and β k < 0.20 increases from 8.5% to 39.1% with this alternative DGP1 for which we set the standard deviation of the log wage be higher. From these results we conclude that the spurious minimum problem is not negligible in the finite samples and the ACF procedure yields the estimates driven by the spurious minimum when we vary the initial values of estimation. Table 1: ACF Procedure with the true value as the initial value Coefficient # of repetition Mean Std. Dev. β l β k Note: True values of β l and β k are 0.6 and 0.4, respectively. Standard deviations reported are of parameter estimates across the 1000 replications. They are not standard errors of the estimator. Table 2-1: ACF Procedure with varying initial values (s.d. log wage=0.1) Coefficient # of repetition Mean Std. Dev. β l β k Percentage of spurious min β l > % β k < % β l > % β k < % β l > % β k < % 6 Although this wage variation in our modified DGP1 may be less realistic, our purpose of experimenting with this modified DGP is to observe the sensitivity of the ACF s original procedure yielding the spurious minimum solutions compared to the modified procedures below. 7

8 Table 2-2: ACF Procedure with varying initial values (s.d. log wage=0.5) Coefficient # of repetition Mean Std. Dev. β l β k Percentage of spurious min β l > % β k < % β l > % β k < % β l > % β k < % Figure 1: ACF Procedure with varying initial values 3.2 Experiment with Modified Procedures The goal of our Monte Carlo exercises in this section is to experiment with modified procedures based on ACF, which may potentially improve the original procedure. We explore two modifications. One is using additional instruments and the other is using a sequential search. 8

9 3.2.1 Using Additional Instruments In the following exercises, we add further lagged input variables such as l t 2 and k t 1 to the instrument set in the equation (41) of ACF (our equation (2)), i.e. we use the instruments (l t 1, l t 2, k t, k t 1 ). The further lagged input variables act as excluded instruments. We also add 1 to the instrument set. 7 This will ensure the regression residual ˆξ it (β l, β k ) to have the zero sample mean, which is the sample analogue of the innovation term ξ it having the zero mean. In all exercises here we use different sets of initial values, i.e. (0, 1), (0.1, 0.9), (0.2, 0.8),..., (0.9, 0.1), the same initial values used in the previous section. For our estimation with the over-identified moment conditions we use the continuously updated GMM estimator (CUE). 8 Results are reported in Table 3-1 and 3-2. Table 3-1 is from the DGP1 while Table 3-2 is from the alternative DGP1 for which the standard deviation of the log wage is increased to 0.5. Interestingly, the augmented procedure using additional instruments is robust to different wage processes and the standard deviation of the 1000 estimates with the alternative DGP1 is even slightly smaller than that of the estimates with DGP1, while the original ACF procedure was somewhat sensitive to the different DGPs. Adding the further lagged input variables in the instrument set obviously improves the estimates of the production function parameters. The means of the estimated coefficients are very close to the truth and the standard deviations of 1000 replicated estimates are also much smaller, indicating almost all estimates are driven by the true minimum of the population moment condition. Table 3-1: Modified ACF with augmented instruments (l t 1, l t 2, k t, k t 1 ) (s.d. log wage=0.1) Coefficient # of repetition Mean Std. Dev. β l β k Table 3-2: Modified ACF with augmented instruments (l t 1, l t 2, k t, k t 1 ) (s.d. log wage=0.5) Coefficient # of repetition Mean Std. Dev. β l β k In the appendix we discuss why this modification of the original ACF procedure helps for identification by further deterring the moment condition from producing the spurious minimum solution. 8 See e.g. Hansen, Heaton, and Yaron (1996) for the continuously updated estimator. In the earlier version of this paper we used an identity matrix as the weight for the GMM estimation. The CUE somewhat improves the performance of our modified procedures. We thank referees for this suggestion of using the CUE. 9

10 3.2.2 Using Sequential Search Algorithm Our point of departure for this modification is to observe that the minimizing criterion function based on the moment (2) is a highly nonlinear function of the parameters, due to the construction of the autoregressive regression residual ˆξ it (β l, β k ), and that it may have a nontrivial impact on the estimation. The reason is that both the regressand and the regressor in the autoregressive regression of the recovered productivity are functions of the parameters (β l, β k ). One possible solution is to break the simultaneity of both sides dependence on the parameters. To this end, we propose a sequential search algorithm as follows: First, fix the regressor β 0 + ω i,t 1 (β l, β k ) at different trial values (0, 1),..., (0.9, 0.1) for (β l, β k ) - the same initial values used in the previous sections- and then obtain the regression residual ˆξ it (β l, β k ) for which the unknown parameter (β l, β k ) now only enters the regressand β 0 + ω it (β l, β k ). Using this construction of ˆξ it (β l, β k ) we estimate the production function parameters from the moment condition (2) by taking the same sets of initial values that are used to construct the regressor of the autoregressive regression. This can easily extend to more granular choice of trial values for estimation but with more computational burden e.g. by increasing the trial values of the labor coefficient by 0.01 or even by starting from zero. Finally we select the estimated coefficients which deliver the global minimum among the set of estimates obtained with different trial values for each simulated dataset. In this experiment, we use the ACF instruments, i.e. (l t 1, k t ). Table 4-1 and Table 4-2 report the results. Similarly to the augmented procedure using additional instruments we find that the sequential procedure is robust to different wage processes in contrast to the original ACF procedure. For both DGPs we observe that the sequential search yields estimates that are very close to the truth and the resulting standard deviations are small, which indicates almost all estimates are driven by the true minimum of the population moment. Table 4-1: Modified ACF with sequential search (s.d. log wage=0.1) Coefficient # of repetition Mean Std. Dev. β l β k Table 4-2: Modified ACF with sequential search (s.d. log wage=0.5) Coefficient # of repetition Mean Std. Dev. β l β k

11 To summarize, from our Monte Carlo results, we find that the two modifications to the original ACF procedure, adding the further lagged inputs to the instrument set or using a sequential search method, help deterring the estimation procedure from yielding the spurious minimum solutions in the finite samples. Adding the further lagged inputs as instruments is an easy modification to the ACF procedure. We, however, note that the sequential search procedure running a search at fine grid points of production function parameters can become computationally demanding, especially when the number of inputs is not small in the production function. 4 Discussion Our Monte Carlo experiments confirm that the original ACF moments, for non-negligible counts of simulation runs, are subject to the spurious minimum in the estimation. ACF discuss this global identification problem in their footnote 16 (p. 2438). ACF note that their moments hold not only at the true parameter, but also at one other parameter value such that β k = 0, β l = β l0 + β k0, and the estimated AR(1) coefficient on the productivity ω it equals the AR(1) coefficient on the wage process of their DGP. Based on our Monte Carlo results, we find that this spurious minimum can occur in the finite samples unless the initial values are taken close to the true parameter values. Our Monte Carlo results suggest that we can mitigate this spurious miminum problem in the finite samples by adding the further lagged input variables to the instrument set, or using a sequential search. 4.1 Using further lagged labor input as an instrument While it is clear why the lagged capital input k i,t 1 is a relevant instrument due to the construction of capital being accumulated through investment, it is less obvious why the further lagged labor input is also a relevant instrument. The intuition behind using l i,t 2 as an additional instrument for the current labor input is as follows. In the ACF s DGP1, one of the labor input determinants is the wage such that l it = 1 (1 β l ) {β 0 + ln β l ln W it + β k k it + (ρ b ω i,t b σ2 )} (3) where W it denotes the wage and the productivity ω i,t 1 evolves to ω i,t b, at which point in time the firm chooses labor input, and σ 2 denotes the variance of the remaining innovation term (see their equation (37)). Because the log wage ln W it follows an AR(1) process, further lagged labor inputs become relevant for the current labor input. From our Monte Carlo experiments, we find l i,t 2 is a relevant instrument, which helps for estimation when it is augmented to the 11

12 original ACF moments. The usual first stage F-statistic of the lagged labor input l i,t 2 from the regression of the labor input l i,t including other observables as regressors is calculated as averaged over the 1000 repetitions, which clearly suggests l i,t 2 is a relevant instrument. A puzzle arises here because the ACF s labor input demand function implies the further lagged labor input l i,t 2 should not be relevant for the current labor input. To see this point note that the ACF s labor input is given by l it = q t (k it, ω it ), which can be written as 9 l it = q t (k it, f 1 t (l it, k it, m it )) = q t (k it, g(f 1 t 1(l i,t 1, k i,t 1, m i,t 1 )) + ξ it ) = q t (k it, g( Φ t 1 (l i,t 1, k i,t 1, m i,t 1 ) β 0 β l l i,t 1 β k k i,t 1 ) + ξ it ) where m it = f t (l it, k it, ω it ) denotes firm s intermediate input demand function and ω it = f 1 t (l it, k it, m it ) denotes its inverse. The second equality above uses the first order Markov process of the productivity ω it = g(ω i,t 1 ) + ξ it. Therefore, l i,t 2 should not be relevant for the current labor input given ( Φ t 1 (l i,t 1, k i,t 1, m i,t 1 ), k it, k i,t 1, l i,t 1 ) in the ACF setting. however, argue that l i,t 2 remains relevant for l it in the ACF s DGP because in the empirical inversion of the input demand (i.e. using Φt (l it, k it, m it )) the ACF s first stage regression does not include the wage as an regressor and because the log wage follows an autoregressive process as discussed above. We note that it is common practice, in the production function estimation using a proxy variable, to omit various other unobservable factors of the input demand, denoted by the t subscript which may indicate market/industry structure, input prices (Olley and Pakes 1996), or other aggregate shocks (Hahn, Kuersteiner, and Mazzocco 2017). In the ACF s DGP this missing factor t is the wage as in (3) and because of this omitted wage the lagged labor input l i,t 2 becomes a relevant instrument for l i,t when the individual firm s wage is not included in the input demand function. We, 4.2 Using a sequential search Our Monte Carlos also confirm that using a sequential search improves the ACF procedure. To gain an intuition behind this sequential search, note that we can rewrite the autoregressive regression to obtain the innovation term ˆξ it (β l, β k ) as y it β l l it β k k it = ρ { β 0 + ω i,t 1 (β l, β k )} + ξ it + η it (4) = ρ ( Φt 1 (l i,t 1, k i,t 1, m i,t 1 ) β l l i,t 1 β k k i,t 1 ) + ξ it + η it 9 Similar argument holds for the alternative timing assumption of the labor input, for which first, ω i,t 1 evolves to ω i,t b, at which point in time the firm chooses labor input as in ACF s DGP1. 12

13 because y it β l l it β k k it = β 0 +ω it +η it and E[β 0 +ω it +η it β 0 +ω i,t 1 ] = E[β 0 +ω it β 0 +ω i,t 1 ]. As it appears the GMM estimation using this equation and instruments like (2) is a non-linear regression problem since β l and β k enter as the coefficients of l it and k it as well as those of the lagged l i,t 1 and k i,t 1, respectively. Note, however, that given β l and β k being fixed for the regressors in the RHS of (4), identifying ρ and the remaining β l and β k coefficients in the regressand becomes a straightforward linear GMM regression problem, i.e. IV regression of y it on l it, k it, and Φt 1 (l i,t 1, k i,t 1, m i,t 1 ) β l l i,t 1 β k k i,t 1. Therefore, although it can be more computationally demanding due to the grid search, using a sequential search can help reduce the effects of nonlinearity in the GMM objective function by breaking the simultaneity of both sides dependence on the same parameters. 13

14 Appendix A Why including 1 in the instrument helps for identification Below we discuss why our modified moment condition that includes 1 in the instrument helps for identification in the ACF s Monte Carlos setup. Consider a Leontief production function as in ACF { } Y it = min e β 0 K β k it Lβ l it eω it, e βm M it e η it. In this Leontief production function setting we have β m + m it = β 0 + β k k it + β l l it + ω it (5) and hence substituting ω it - the inverse intermediate input demand function - into the valueadded production function we obtain y it = β 0 + β k k it + β l l it + ω it + η it = β 0 + β k k it + β l l it + [β m + m it β 0 β k k it β l l it ] + η it = β m + m it + η it. This implies the population equation Φ t (l it, k it, m it ) of the ACF s first stage becomes Φ t (l it, k it, m it ) = β m + m it. It follows that When we use Φ t (l it, k it, m it ) β k k it β l l it = β 0 + ω it. (6) β0 + ω it = Φt (l it, k it, m it ) β k k it β l l it for the autoregressive regression of the productivity in place of ω it as in the concentrated ACF procedure we should include the intercept as ( β0 + ω it ) = α 0 + ρ( β 0 + ω i,t 1 ) + ξ it (7) because the above regression becomes equivalent to ω it = ρω i,t 1 + ξ it only with α 0 = β 0 (1 ρ). This also makes the residual ξ it (β l, β k ) have mean zero by construction whether or not the intercept is actually equal to the true α 0 = β 0 (1 ρ). Below we argue that if we remove this constant term or equivalently if we run the AR(1) 14

15 regression with the productivity only, it helps for identification. To make this argument we first need to see how a spurious identification point arises in the original ACF moment condition. Let the spurious minimum be β k = 0 and β l = β l + β k = 1. Also write Φ it = Φ t (l it, k it, m it ). From (6) we then obtain Φ it β k k it β l l it = β 0 + β k k it (1 β l )l it + ω it. (8) Also note that from the optimal labor input in the ACF s DGP we have (1 β l )l it as (1 β l )l it = β 0 + ln β l ln W it + β k k it + (ρ b ω i,t b σ2 ) (9) where W it denotes the wage and the productivity ω i,t 1 evolves to ω i,t b, at which point in time the firm chooses labor input as in DGP1 (see their equation (37)). Plugging (9) into (8) we obtain Φ it β k k it β l l it = ln β l σ 2 /2 + ln W it ρ b ω i,t b + ω it = ( ln β l σ 2 /2) + ξit B + ln W it (10) because ω it = ρ b ω i,t b + ξit B. The equation (10) has two important implications. First the constant term ( ln β l σ 2 /2) is different from β 0 in (6). Second the regression residual from the AR(1) regression at the spurious minimum using Φ it β k k it β l l it is also different from ξ it in (7). Note that the wage follows an AR(1) process as ln W it = ρ W ln W i,t 1 +ξ W it and its innovation term ξit W is independent of (k it, l i,t 1 ). Therefore, the spurious parameter ( β k = 0, β l = 1) solves the ACF s moment condition as well as the true parameter, as discussed in the ACF footnote 16. Note that in the ACF s DGP we have β 0 = 0. Therefore, if we run the AR(1) regression without the intercept, it helps the moment condition to yield a true solution deterring the spurious minimum because the spurious minimum solution requires the AR(1) regression must have a non-zero intercept (or an intercept different from α 0 = β 0 (1 ρ)). Without the intercept the spurious innovation term ξ B it ρ W ξ B it + ξ W it in the spurious AR(1) regression using (10) would not have the mean zero, which becomes inconsistent with the true DGP. In our Monte Carlos experiment the AR(1) regression does not include the intercept, i.e. we estimate ω it = ρω i,t 1 + ξ it to obtain the residual. This regression without an intercept deters ξ it from having the zero mean at other parameter values than the true ones. Instead, we include 1 in the instrument so that we can ensure the innovation ξ it has mean zero at the true parameter values. This helps for identification away from the spurious minimum. If we included a constant in the regression, the residual would have zero mean for any parameter values that include the spurious solution. In general this constant β 0 in the production function is not known. Moreover, it is not 15

16 separately identified from the mean of the unobserved productivity. However, the sum of β 0 and the mean of the productivity can be estimated in the first stage of ACF procedure. From (6) note that this sum, whether they are zero or not, is obtained as the constant term of the function Φ t (l it, k it, m it ). Note that this sum is equal to β m in the Leontief production function (see the equation (5)), which can be easily estimated using OLS of y it on m it since y it = β m + m it + η it. After removing this constant from β 0 + ω it we then can use the regression ω it = ρω i,t 1 + ξ it to obtain the innovation term where ω it becomes a de-meaned productivity if the productivity does not have mean zero. 16

17 References [1] Ackerberg, D., K. Caves, and G. Frazer (2015), Identification Properties of Recent Production Function Estimators, Econometrica, 83, [2] Hahn, J., G. Kuersteiner, and M. Mazzocco (2017), Estimation with Aggregate Shocks, working paper. [3] Hansen, P., J. Heaton, and A. Yaron (1996), Finite-Sample Properties of Some Alternative GMM Estimators, Journal of Business and Economic Statistics, 14, [4] Gandhi, A., S. Navarro, and D. Rivers (2017), On the Identification of Gross Output Production Functions, working paper. [5] Levinsohn, J. and A. Petrin (2003), Estimating Production Functions Using Inputs to Control for Unobservables, Review of Economic Studies, 70, [6] Marschak, J. and W. Andrews (1944), Random Simultaneous Equations and the Theory of Production, Econometrica, 12, [7] Olley, S. and A. Pakes (1996), The Dynamics of Productivity in the Telecommunications Equipment Industry, Econometrica, 64, [8] Wooldridge, J. (2009), On Estimating Firm-Level Production Functions Using Proxy Variables to Control for Unobservables, Economics Letters, 104,