Estimating the Production Function when Firms Are Constrained

Transcription

1 Estimating the Production Function when Firms Are Constrained Ajay Shenoy February 16, 2018 First Version: 30 July 2016 Abstract Behind many production function estimators lies a crucial assumption that the firm s choice of intermediate inputs depends only on observed choices of other inputs and on unobserved productivity. I show that this assumption fails when firms are constrained or face other market distortions. I derive a test for the assumption, which is rejected in several industries. I show that when firms are constrained a simple autoregressive estimator becomes viable and is complementary to choice-based methods, in that it requires choices to be distorted. I propose criteria for choosing between estimators, which in simulations yields lower error than either estimator alone. (JEL Codes: C52, D24) Preliminary: Click here for the latest version University of California, Santa Cruz; at azshenoy@ucsc.edu. Phone: (831) Postal Address: Rm. E2455, University of California, M/S Economics Department, 1156 High Street, Santa Cruz CA, I am grateful to Salvador Navarro for providing code, data, and encouragement. I also thank Dan Ackerberg for helpful comments and suggestions. I thank Liam Rose for excellent research assistance. This paper benefited greatly from the suggestions of Natalia Lazzati, Alan Spearot, and seminar participants at U.C. Santa Cruz, Stanford, SIDE, Michigan State, NYU, Yale, Brown, BU, Columbia, NEUDC (Tufts), and Dartmouth.

2 2 AJAY SHENOY 1 Introduction Central though it is to any model of the economy, the production function is one of the hardest primatives to estimate. A more productive firm one that gets aboveaverage output from each worker it hires and each machine it installs is expected to use more of both inputs. Productivity also, by definition, raises output even holding inputs fixed. But the researcher does not observe and cannot control for productivity, making any attempt to estimate the causal effect of using more labor or capital vulnerable to omitted variable bias. Starting with the work of Olley and Pakes (1996), a host of new methods have addressed this problem by exploiting the information contained in the firm s choice of inputs. Proxy methods like that of Ackerberg et al. (2015) use the choices of the firm to infer its productivity. They assume that, conditional on its labor and capital (which may or may not be chosen optimally), a more productive firm always uses more intermediate inputs. These inputs are a proxy for productivity, which is no longer an omitted variable. Meanwhile first-order methods such as Gandhi et al. (2013) assume the firm chooses its level of intermediates optimally. By combining the production function with the first-order condition for intermediates, Gandhi et al. (2013) derive an estimating equation that does not contain productivity. Though powerful, these choice-based methods rely on equally powerful assumptions. I show that these assumptions need not hold when a firm s choice of intermediates is distorted by market imperfections for example, a credit constraint. Proxy methods require a one-for-one relation between productivity and the choice of intermediates. If some firms lack credit or cannot find suppliers, two otherwise identical firms may use different levels of intermediates. First-order methods require the choice of intermediates to be optimal, meaning the marginal product equals the price. That need not hold if a firm is constrained. Unfortunately, constraints and other market distortions are widespread in industries across the world. According to the 2006 World Bank Enterprise Survey, 32 percent of Central American firms and 37 percent of Chilean firms find getting electricity to be a serious or very serious obstacle. Nearly 60 percent of Zimbabwean firms have idle production capacity because inputs are unavailable. Half suffer power failures and 80 percent lack financing. According to the Economics Research Forum s survey of firms, 25 percent of Egyptian firms and 26 percent of Tunisian firms say that getting raw materials is a severe constraint. Roughly half in both countries say

3 ESTIMATING THE PRODUCTION FUNCTION WHEN FIRMS ARE CONSTRAINED 3 the cost of raw materials is a constraint. Given that most have little access to financial services, it is hard to imagine they are able to choose the optimal level of inputs. Even in the U.S. many firms are constrained. The 2007 Survey of Business Owners reports that among firms that shut down, 10 percent did so because they lack access to credit. Taken together these surveys suggest market distortions are too widespread to ignore. This paper s first contribution is to show why constraints and other distortions may create identification problems for choice-based methods. To be clear, the issue is not whether the choices of capital or labor are distorted no method discussed in this paper requires either to be chosen optimally. It is whether, conditional on capital and labor, the choice of intermediates is distorted. Any distortion undermines what Ackerberg et al. (2015) call the Scalar Unobservable assumption, which states that the choice of intermediates is a function of only capital, labor, and productivity. A distorted choice will depend on not only these but on the cause of the distortion. As some version of the Scalar Unobservable assumption is crucial to all choice-based methods, its failure may render them inconsistent. This insight motivates the paper s second contribution, a test for this crucial assumption. The Scalar Unobservable assumption implies the cost of intermediates as a share of the firm s output is a function of only the choices of capital, labor, and intermediates used in production. After controlling for a nonparametric function of these inputs no other variable known to the firm in year t 1 or earlier should be informative. But if the firm is constrained or its choice is otherwise distorted, lags of these inputs may be informative about the constraint, which in turn is informative about the cost share of intermediates. Testing for whether these lags are informative is in effect a test of the Scalar Unobservable assumption. I apply the test to the sample of manufacturing firms in Chile and Colombia studied by Gandhi et al. (2013). The test rejects the Scalar Unobservable assumption in many industries, suggesting it cannot be taken for granted. The paper s third contribution is to show that a simplified dynamic panel estimator may be used when firms are constrained. This estimator assumes productivity is autoregressive (rather than an arbitrary Markov process), but does not assume a Scalar Unobservable. To be clear, the estimator itself is not new (see, for example, Ackerberg et al., 2015). Rather I show that this estimator sidesteps the need for any choice assumptions that may be undermined by market distortions. I also show that this estimator, though estimated by nonlinear GMM, has a close connection to two-

4 4 AJAY SHENOY stage least squares. As I describe below, this connection is useful because it is possible to test whether the estimator is well-identified using standard weak instruments statistics. Such statistics are valuable because when firms are relatively unconstrained, the autoregressive estimator at least when used to estimate a gross rather than a valueadded production function is weakly identified. In the absence of constraints or other distortions, the firm s choice of intermediates has no exogenous variation independent of its choices of the other inputs. But market distortions induce variation in its choice that can then be used for identification. The same distortions that undermine choice-based methods are crucial for the consistency of the autoregressive estimator, implying the methods are complementary. Showing how constraints and distortions induce this complementarity is the fourth and most important contribution of this paper. I show that the bias of Gandhi et al. (2013) is strictly increasing in the severity of the constraints. Meanwhile, a key weak identification statistic for the autoregressive method is bounded above by an expression that captures how informative the instruments are about the distortion. Finally, I show that the problem of weak identification (and thus complementarity) arises only when estimating a gross production function. As long as productivity is autoregressive and the lags of capital and labor are informative about their current levels, the autoregressive method will be well-identified when used to estimate a valueadded production function. Monte Carlo simulations confirm there is complementarity when estimating a gross production function. As the severity of constraints increases, the performance of the autoregressive method improves even as that of the Gandhi-Navarro-Rivers deteriorates. Measures of strong identification improve with the severity of constraints, confirming that weak identification undermines the autoregressive method unless firms are constrained. The principle of complementarity suggests that if it is possible to choose between Gandhi-Navarro-Rivers and the autoregressive estimator, it may be possible to construct estimates that are valid regardless of whether firms are constrained. The paper s final contribution is to propose criteria by which to make that choice. The criteria are 1) the test for constraints, 2) a common-sense restriction that the chosen estimator should yield positive elasticities for all inputs, and 3) a weak identification statistic for two-stage least squares. Though such a statistic depends on the unknown autocorrelation coefficient in the stochastic process for productivity, I show

5 ESTIMATING THE PRODUCTION FUNCTION WHEN FIRMS ARE CONSTRAINED 5 that the autoregressive estimate of this coefficient is consistent even when the production function is weakly identified. This estimate can be used, for example, to construct the rk statistic proposed by Kleibergen and Paap (2006). The autoregressive method should only be used when this rk statistic is large. Simulations suggest that using these three criteria to choose between Gandhi-Navarro-Rivers and the autoregressive estimator achieves the best of both worlds. The error of the selection method is lower than that of either method alone. 2 The Problem of Identification under Constraints 2.1 Review of Choice-Based Methods Consider a firm that produces output at time t by combining capital K t, labor L t, and real expenditure on intermediate inputs M t using a gross production function F. (I consider a value-added production function below.) I assume output depends on real expenditures of intermediates denoted in units of the output good. 1 Firms differ in their productivity, which most choice-based methods assume is Hicks neutral. 2 Assumption 1 (Hicks Neutrality) The productivity of the firm is Hicks neutral and has two parts: ω t, which is persistent, and ε t, which is unknown when inputs are chosen and is independent across time and firms. Gross output is Y t = e ωt+εt F (K t, L t, M t ) (1) Since all the methods discussed in this paper are nonparametric it is necessary to assume the production function is smooth. 3 I also assume it is increasing and concave, which ensures the optimal choice of M is well-defined and increasing in 1 Like capital, the term intermediate inputs is a catch-all for many different inputs (e.g. fuel, electricity, and raw materials), meaning there is no unambiguous definition for the level of intermediate inputs. If the researcher is willing to somehow define a price of intermediates, it would imply a level of intermediates. The researcher might further assume that output depends on this implied level of intermediates rather than expenditures. I show in Appendix C.2 that in this case the main results of the paper require little or no modification. 2 For example, Olley and Pakes (1996); Levinsohn and Petrin (2003); Wooldridge (2009); Gandhi et al. (2013); Ackerberg et al. (2015). 3 It is common to estimate the production function using a log-sieve approximation that is, to approximate the log of the function with a polynomial in the logs of its arguments. That approach assumes the production function has a power series representation of some order p, which is equivalent to assuming smoothness of order p.

6 6 AJAY SHENOY K, L and ω. Though only smoothness is necessary for my main results, as I explain below the other conditions on F are useful in tying together the assumptions needed for first-order methods like Gandhi et al. (2013) and proxy methods like Ackerberg et al. (2015). Assumption 2 (Production) The production function F is smooth. It is also concave and increasing in each of its arguments. It is also standard to make some assumption about timing. 4 To be precise, assume there are several quasi-fixed inputs that are chosen before intermediates. Throughout the paper I assume these inputs are capital and labor, though it is straightforward to allow for other inputs. It is not necessary to make any assumptions about whether capital and labor are chosen optimally as long as the choice is made using only information known to the firm. But it is necessary to assume neither choice is perfectly flexible: Assumption 3 (Dynamic Capital and Labor) K t and L t are at least partly determined at t 1 or earlier. In their empirical application, Gandhi et al. (2013) assume that both capital and labor are completely determined at t 1. This stronger assumption is useful, though in fact unnecessary for their method (or any other method discussed in this paper). As long as capital and labor are at least somewhat dynamic if there is an adjustment cost or time to build their lags are informative about their current levels, making the Gandhi et al. (2013) approach viable. But for consistency I make the same assumption as Gandhi et al. (2013) in my simulations; the assumption actually favors their method and is thus conservative. 5 Having chosen its capital K t and labor L t, the firm now chooses its expenditure on intermediate inputs M t. Though not crucial for the autoregressive method derived in Section 4, this timing is crucial for the choice-based methods: 4 As Ackerberg et al. (2015) explain, the production function is identified only if labor and capital are chosen before intermediate inputs, if there are i.i.d. shocks to the price of labor or output (but not productivity) after inputs are chosen but before labor is chosen, or if there is i.i.d. optimization error in the choice of labor. I assume the first of these as it is easiest to model and seems plausible. 5 Under this timing assumption the choice-based methods can use current capital and labor as instruments, but the autoregressive method of Section 4 cannot. Under the weaker assumption that capital and labor are chosen with some information about ω t the choice-based methods would have to use the same set of instruments as the autoregressive method.

7 ESTIMATING THE PRODUCTION FUNCTION WHEN FIRMS ARE CONSTRAINED 7 Assumption 4 (Timing) M t is chosen after the other inputs at a point when ω t is known but ε t is still unknown. The method of Gandhi et al. (2013) assumes the firm s choice of intermediates is optimal, meaning the firm solves max M t E εt [e ωt+εt F (K t, L t, M t ) M t ] (2) by setting the expected marginal product of intermediate expenditure equal to the price. Here I have assumed for simplicity that M t is defined as expenditures on intermediates, meaning the price is normalized to 1 (I show how to relax this assumption in Online Appendix C.2). Then the method assumes Assumption 5 (Optimal Choices) Firms choose M t to satisfy 1 = E[e εt ]e ωt F M (K t, L t, M t ) (3) To be clear, the method makes no assumptions about whether capital and labor or any other inputs are chosen optimally. But conditional on however much capital and labor the firm has, its choice of intermediates must be optimal. Gandhi et al. (2013) then exploit the assumption of Hicks Neutrality, which implies that productivity enters both the production function and the righthand side of the first-order condition multiplicatively. Dividing (3) by realized output (1) and multiplying by M t gives M t Y t = F M(K t, L t, M t )M t E[e εt ]e εt (4) F (K t, L t, M t ) The lefthand side is simply the cost share of intermediates, which is observable. 6 Gandhi et al. nonparametrically estimate this share regression in logs to recover the right-hand side of (4), which is the elasticity of output with respect to intermediate inputs M and a nuisance term. Since ω t has been purged from (4) the estimate is consistent. Then the residual ˆε t is a consistent estimate of the shock ε t. Divide the predicted value from the share regression by M and the sample average of eˆεt to 6 Gandhi et al. (2013) define the share as P M M t Y t, as they assume it is possible to calculate a separate real price and real level of intermediates, and that output depends on the real level rather than the real expenditure. By contrast, Ackerberg et al. (2015) in their exposition take the approach used here. I show in Appendix C.2 that applying the definitions of Gandhi et al. (2013) would leave the main results unchanged.

8 8 AJAY SHENOY isolate F M (K t,l t,m t) F (K t,l t,m t). Integrate this ratio with respect to M to recover (the log of) the production function F up to a constant of integration C (K t, L t ). Though the integral is now known, the production function cannot be extracted from it unless C (K t, L t ) is known. Let I t denote the integral and define Y t = y t I t ε t, where y t = log Y t. Since y t log F (K t, L t, M t ) = ω t + ε t (5) the known part of productivity can be written as ω t = Y t + C (K t, L t ) (6) Now Gandhi et al. (2013) make another assumption common in this literature: Assumption 6 (Markov Productivity) The known shock follows a first-order Markov process. For some function Ψ, productivity at t can be written as ω t = Ψ(ω t 1 ) + η t, where η t is unknown before time t. Then Y t + C (K t, L t ) = Ψ(Y t 1 + C (K t 1, L t 1 )) + η t (7) Since this is a Markov process the innovation in productivity η t does not depend on capital K t 1 or labor L t 1. As Gandhi et al. (2013) assume capital and labor were chosen at t 1 before η t is known, the converse is also true. Then capital and labor are uncorrelated with η t, making them valid instruments that can be used to estimate C nonparametrically. Combined with the estimate of I t this estimate of C gives the production function. An alternative to the first-order approach is the proxy variable approach of Ackerberg et al. (2015), who build on the work of Olley and Pakes (1996), Levinsohn and Petrin (2003), and Wooldridge (2009). Ackerberg et al. estimate a value-added production function F (K t, L t ) rather than a gross production function. Unlike Gandhi et al. they need not assume the choice of intermediates is optimal, only that it is strictly increasing in productivity and depends only on productivity, labor, capital, and other observables. They drop Assumption 5 and instead assume: Assumption 7 (Monotonicity) All else equal, the choice of intermediates M t is strictly increasing in ω t.

9 ESTIMATING THE PRODUCTION FUNCTION WHEN FIRMS ARE CONSTRAINED 9 Assumption 8 (Scalar Unobservable) The choice of intermediate inputs is M t = M(K t, L t, ω t ) for some smooth function M( ). Given our assumption that F is concave and increasing in its arguments, Optimal Choices implies these two assumptions, but the converse need not hold. If either of these two assumptions fails, Optimal Choices must also fail. It is the second of these assumptions the Scalar Unobservable assumption that is the focus of this paper. Under these assumptions productivity can be written as ω t = M 1 (K t, L t, M t ) for some function M 1. After controlling for capital and labor, intermediates are a proxy for productivity. Define f(k t, l t ) = log F (e kt, e lt ) and m 1 (k t, l t, m t ) = M 1 (e kt, e lt, e mt ) where k t, l t, and m t are the logs of K t, L t, and M t. The log of value-added output y t can be written as y t = f(k t, l t ) + m 1 (k t, l t, m t ) + ε t = Φ(k t, l t, m t ) + ε t (8) The term Φ(k t, l t, m t ) can be estimated nonparametrically, giving a consistent estimate of log f(k t, l t ) + ω t. By the Markov Productivity assumption, y t f(k t, l t ) Ψ (ˆΦ(kt 1, l t 1, m t 1 ) f(k t 1, l t 1 ) ) = η t + ε t (9) which is uncorrelated with (k t, l t, m t 1, k t 1, l t 1,...). These variables are instruments that can be used to estimate the value-added production function by the generalized method of moments. 2.2 The Scalar Unobservable Assumption Fails When Firms Are Constrained But suppose firms cannot choose their intermediate inputs freely. To be precise, suppose that each firm has a constraint Z t on its choice of intermediates. 7 The constraint may be a function of capital (which may be offered as collateral) and one or more other terms S 1 t, S 2 t,..., S I t. These terms, some or all of which may be unobserved, might comprise retained earnings, the wealth of the entrepreneur, or her 7 The firm s choices of capital and labor may or may not be constrained. Neither the choice-based methods nor any of the other methods discussed in this paper require assumptions about how other inputs are chosen.

10 10 AJAY SHENOY political connections to state-run banks. 8 Append to the production function (1) and the firm s optimization (2) the following conditions : M t Z t = Z(K t, S 1 t,..., S I t ) (10) S i t = Γ i (S i t 1, S i t 2,...) + v i t for all i = 1,..., I (11) where vt i is a serially independent shock. Equation 11 states that the other components of the constraint follow stochastic processes that need not have the Markov property. Let λ be the Lagrange multiplier on (10). The new first-order condition is 1 + λ(k t, St 1,..., St I,...) = E[e εt ]e ωt F M (K t, L t, M t ). (12) It is immediately clear that Assumption 5 of Optimal Choices fails whenever λ > 0 that is, whenever any firms are constrained. Rearrange this expression and invert F M for M t to show that the level of intermediate inputs is now M t = Ṁ(K t, L t, ω t, λ t ) = M(K t, L t, ω t, St 1,..., St I ) (13) which depends on more than one unobservable: productivity ω t and one or more terms {S j t }. Assumption 8, the Scalar Unobservable assumption, also fails. One might be tempted to treat the constraint as a second unobservable and seek an additional flexible input, as proposed in Ackerberg et al. (2007). This approach uses a bivariate proxy function to recover both productivity and the constraint. But even assuming a second flexible input exists, the bivariate equivalent of the Monotonicity Assumption would fail. A firm that is constrained could not choose strictly higher inputs for a higher level of productivity, and a firm that is unconstrained would not choose higher inputs for higher levels of the constraint. The firm s choices cannot be used to recover productivity. 8 The constraint could also be a function of other observed terms like labor or last year s output.

11 ESTIMATING THE PRODUCTION FUNCTION WHEN FIRMS ARE CONSTRAINED Generalization: Imperfections Other than Constraints Throughout the paper I treat the suboptimal choice as having been caused by a credit constraint. The focus on constraints is largely for the sake of exposition. All of the important results hold for any unobserved feature of the firm or the economy that gives some firms easier access to intermediates. If some firms get an unobserved discount on their inputs because they are regular customers; or if some firms suffer periodic power cuts because their town elected a mayor of the opposition party; or if some firms are new to the industry and have not found enough suppliers to meet their needs; then the Scalar Unobservable assumption fails. The results that follow are no less valid. To make this point more formally, let τ t denote a market distortion that may vary across firms. The distortion drives a wedge between the marginal products of firms that appear similar. Then (12) is modified to 1 + τ t = E[e εt ]e ωt F M (K t, L t, M t ). (14) As long as τ is a function of unobserved terms that cause the actual choice of m t to deviate from the undistorted optimum, all of the important results that follow will hold. 9 3 A Specification Test: Does the Scalar Unobservable Assumption Hold? 3.1 Approach Using arguments similar to those used to derive Equation 8, it is easy to see that Assumptions 1, 4, 7, and 8 imply that there exists a function ξ(k t, l t, m t ) such that y t = ξ(k t, l t, m t ) + ε t (15) where y t may be either gross output or value-added output depending on whether the researcher is estimating a gross production function F or a value-added produc- 9 The sole exception is the inequality in Equation 27 of Proposition 2. Though the expression for the bias of the estimate is valid regardless of the source of the market imperfection, the sign of the bias may or may not be negative.

12 12 AJAY SHENOY tion function F. Let s M t = log(m t /Y t ) denote the log of the share of intermediates in output. Multiply both sides of (15) by 1 and add m t to both sides: s M t = ξ(k t, l t, m t ) ε t (16) where ξ(k t, l t, m t ) = m t ξ(k t, l t, m t ). Alternatively, if we assume Optimal Choices we could derive this expression directly by taking logs of Equation 4. Equation 16 implies the systematic variation in the share of intermediates is a function of only k t, l t, and m t. To be precise, after controlling for these inputs the residual variation in the share is simply ε t, which is uncorrelated with any variable known before time t. But if markets are imperfect say, if firms are constrained and Equation 12 holds then the cost share is also a function of the Lagrange multiplier. Take logs of both sides of Equation 12, and define Λ t = log(1 + λ t ). Then instead of Equation 16, the share would be s M t = ξ(k t, l t, m t ) Λ t ε t (17) Let r t be a vector of instruments orthogonal to η t, meaning anything known to the firm before period t. The elements of r t need not all be from the same period as long as they are all from t 1 or earlier. Then conditional on Assumptions 4 and 7, one simple test of the Scalar Unobservable assumption is to estimate the semiparametric regression s M t = ξ(k t, l t, m t ) + r t ϱ ε t (18) and test the hypothesis ϱ = 0. (In practice it may be useful to control for variables beyond just the nonparametric term ξ( ) such as year dummies. ) If the Scalar Unobservable assumption holds, then Equation 16 holds, implying the true value ϱ is zero. But if firms are constrained or face other market imperfections, then Equation 17 implies ϱ captures the partial correlation between r t and Λ t after controlling for k t, l t, m t. Rejecting that ϱ = 0 suggests the Scalar Unobservable assumption fails. The consequences of its failure are especially clear for the method of Gandhi et al. (2013). Equation 18 is similar to the log share regression used to estimate the elasticity of intermediates, and as I show in Section 5.2 the bias of this estimate can be written directly as a function of moments and conditional moments of Λ t. But a rejection also has consequences for the method of Ackerberg et al. (2015) and other proxy

13 ESTIMATING THE PRODUCTION FUNCTION WHEN FIRMS ARE CONSTRAINED 13 methods. Equation 18 was derived from Equation 8, which is the first stage of a proxy estimator. As I show in Appendix C.1 the constraint can be thought of as inducing non-classical measurement error in the choice of intermediates, in that the actual m t deviates from the optimal m t. One potential concern with this test is that measurement error in k t, l t, m t will bias estimates of ξ. Then ξ would inadequately control for the productive capacity of the firm, potentially causing r t to be informative about s M t because it is informative about the true values of k t, l t, m t. But measurement error in the inputs would immediately imply that the Scalar Unobservable assumption fails. Recall that (18) was derived from what is essentially the first stage of Ackerberg-Caves-Frazer, which is based on the idea that observed m t is invertible in k t, l t, ω t. Invertibility fails if firms that make different choices of intermediates appear to make the same choice because of measurement error. In other words, if the test rejects it is still the case that the assumptions of the choice-based methods fails even though market imperfections are not what cause the rejection. Nevertheless, I show in Online Appendix C.3 how the test can be adapted not to reject when there is white noise measurement error, and how to adapt the autoregressive estimator of Section 4 and the selection procedure of Section 7 in this case. Finally, as implied in Section 2.3 everything derived here still holds if the suboptimal choice arises through an imperfection other than a constraint. Define Λ t = log(1 + τ t ) and (17) follows from Equation Evidence of Imperfections I run the test on samples constructed from the Chilean and Colombian census of manufacturers: Chilean Manufacturing: I use exactly the same dataset as Gandhi et al. (2013), which is the Chilean manufacturing census used in Ackerberg et al. (2006) and expanded by Greenstreet (2007). Colombian Manufacturing: I use exactly the same dataset as Gandhi et al. (2013), which is the Colombian manufacturing census. 10 Figure 1 shows the p-value from the test when run on each industry studied in Gandhi et al. (2013). I approximate ξ(k t, l t, m t ) with a log-sieve polynomial of degree 10 I am grateful to Salvador Navarro for giving me the files and code needed to reproduce the data for both Chile and Colombia.

14 14 AJAY SHENOY 2,3,...,8. 11 I include k t 1, l t 1, m t 1, k t 2, m t 2, k t 2 m t 2 in r t. (This is the same set of variables used in the simulations of Section 6.2). All of the tests reject at the 5 percent level in all five of Chile s industries. The p-values suggest there is little difference between a degree-2 and a degree-8 polynomial (or any degree in between). The results are similar for all but one of Colombia s industries. But as described in Section 7, it is better to hold the test to the higher 1 percent standard. By this standard the test still rejects in all of Chile s industries, but not in Colombia s apparel or food products industries. The test still rejects for most of the log-sieve polynomials (especially the higher degree polynomials) in textiles and fabricated metals. As the test rejects at even the 1 percent level in most of these industries, the Scalar Unobservable assumption should not be taken for granted. But in several industries the test does not reject at the 1 percent level, suggesting it is equally unwise to assume the assumption always fails. The test can be used to distinguish between cases where the assumption is reasonable or unreasonable. 4 An Autoregressive Estimator that Does Not Require Any Choice Assumptions Ackerberg et al. (2015) first noted that if productivity follows a simple autoregressive process (as made formal below) there is a link between their method and the linear dynamic panel estimator (for example, Arellano and Bond, 1991; Blundell and Bond, 1998). My argument is that a version of the dynamic panel estimator what I call the autoregressive estimator conforms to all the assumptions of the choicebased estimators but does not require any choice assumption. That makes it a natural alternative to choice-based methods when firms are constrained. I then show that although this estimator is nonlinear, it closely resembles a twostage least squares estimator. This resemblance is useful in showing the complementarity between the autoregressive estimator and that of Gandhi et al. (2013). 11 As Gandhi et al. (2013) measure physical output and rescale the level of intermediates by a price index, I define s M = P M t M t /(P Y t Y t ) as they do. In addition to the terms in Equation 18, I also control for year dummies. This is a simple way to deal with potential measurement error in the price of intermediates, which would otherwise be negatively correlated with the choice of intermediates. Running the test without year dummies makes little difference.

15 ESTIMATING THE PRODUCTION FUNCTION WHEN FIRMS ARE CONSTRAINED 15 Figure 1 Firms are Constrained in Many Chilean and Colombian Industries Chile Colombia Food Products Food Products Textiles Textiles Apparel Apparel Wood Products Wood Products Fabricated Metals Fabricated Metals p-value p-value Note: I run the test on each industry studied in Gandhi et al. (2013), approximating ξ(kt, lt, mt) with a log-sieve polynomial of degree 2,3,...,8. This figure reports the p-values for each test. The top row of the dot plot is the degree 2 test, and each subsequent row raises the degree by 1. All tests are based on standard errors clustered by plant, and additionally control for year dummies.

16 16 AJAY SHENOY 4.1 The Autoregressive Assumption The choice assumption cannot be dropped without imposing a new assumption in its place. This new assumption imposes linearity on the Markov process that governs productivity: Assumption 9 (Autoregressive Productivity) The known shock follows a first-order autoregressive process. For some parameters ( ω, ρ), productivity at t + 1 can be written as ω t = ω + ρω t 1 + η t, where η t is unknown before time t. Though this assumption is stronger than Markov Productivity (Assumption 6), there is no bulletproof means of judging how much stronger. A flawed but nevertheless informative approach is to plot the estimates of Ψ(ω t 1 ) from Gandhi et al. (2013). Since the estimate need not be valid if the estimator is biased, these plots must be taken with caution. But they give some sense of how the Gandhi-Navarro- Rivers estimator itself assesses the nonlinearity of the Markov process. Figure 2 makes such plots for the Chilean industries after re-centering ˆΨ(ω t 1 ) vertically and horizontally by the mean of ˆω t. The domain is restricted to slightly more than two standard deviations of the estimated distribution of productivity for the most volatile of the five industries. Each plot graphs the estimated function (solid line) against the best-fitting linear approximation (dashed line). In all cases the estimate is close to linear. The most nonlinear of the five industries is 311, Food Products. I thus take the Markov process for Industry 311 as a test case for how the selection procedure of Section 7 handles a violation of Assumption Procedure Let f(k t, l t, m t ) = log F (e kt, e lt, e mt ) denote the log of the gross production function. Under Autoregressive Productivity, gross output may be rewritten as y t = f(k t, l t, m t ) + ω t + ε t = f(k t, l t, m t ) + ω + ρω t 1 + η t + ε t = f(k t, l t, m t ) + ω + ρ(y t 1 f(k t 1, l t 1, m t 1 )) + η t ρε t 1 + ε t }{{} ν t (19)

17 ESTIMATING THE PRODUCTION FUNCTION WHEN FIRMS ARE CONSTRAINED 17 Figure 2 Estimates of the Markov Process for Productivity for Chilean Industries from Gandhi et al. (2013) Industry 311 Industry 321 Industry 322 Productivity Lagged Productivity Productivity Lagged Productivity Productivity Lagged Productivity Industry 331 Industry 381 Productivity Lagged Productivity Productivity Lagged Productivity Note: The estimates are re-centered vertically and horizontally by the mean of ˆω t. Gandhi et al. (2013) approximate the Markov process as a third-order polynomial in ω t 1. Equation (19) can be estimated using generalized method of moments. If ε t is only measurement error, under the timing assumptions of Section 2.1 any function of (k t, l t, k t 1, l t 1, m t 1,...) is uncorrelated with the combined error term ν t. If ε t is not measurement error but a true shock to revenue it might affect investment and hiring. Then ρε t 1 may be correlated with k t and l t, ruling these out as instruments. I make this more conservative assumption in the simulations that follow. Since m t may be correlated with η t, and y t 1 is correlated with ρε t 1, neither is an instrument. 12 In the simulations below I follow Gandhi et al. (2013) in approximat- 12 If labor and capital are chosen with some information about ω t it is still valid to use the instruments chosen when ε t is not measurement error (all lags of capital, labor, and intermediates). However, Gandhi-Navarro-Rivers would have to drop k t and l t from its list of instruments, leaving the two methods with the same set of instruments. This is why Assumption 4, the Timing Assumption, favors

18 18 AJAY SHENOY ing f(k t, l t, m t ) with a second-order translog polynomial. I instrument with the (demeaned) lags of each term of the polynomial, and the second lags of m, k, and their interaction. 13 As noted in Online Appendix C.3, if there is reason to expect substantial (white noise) measurement error in the observed inputs then the list of instruments should include only lags of degree 2 or higher. As noted above, this estimator is simply a linear dynamic panel estimator that assumes there are no firm-level fixed effects. I exclude firm-level fixed effects because most out-of-the-box choice-based methods do not allow for them. 14 However, it is easy to allow for them by directly applying a dynamic panel estimator. Finally, it is straightforward to instead estimate the value-added production function with the estimating equation y t = f(k t, l t ) + ω + ρ(y t 1 f(k t 1, l t 1 )) + ν t (20) where y t is now value-added output. Now only lags of k t and l t are needed as instruments (though lags of m t may also be used to increase precision). In the what follows I focus mainly on estimating a gross production function, as that is the more challenging case. Many of the limitations of the autoregressive estimator are not present when estimating a value-added production function (see Section 5.3). 4.3 Link to Two-Stage Least Squares Though the autoregressive estimator is technically estimated using nonlinear GMM, it behaves very much like a version of two-stage least squares. To see why, consider the simple case where the production function is Cobb-Douglas: F (K t, L t, M t ) = K π k t L π l t Mt πm. Suppose for a moment that ρ is known, and define ρ (x) = x t ρx t 1. Then (19) can be rewritten as y t = π k k t + π l l t + π m m t + ω + ρ(y t 1 π k k t 1 π l l t 1 π m m t 1 ) + ν t y t ρy t 1 = ω + π k (k t ρk t 1 ) + π l (l t ρl t 1 ) + π m (m t ρm t 1 ) + ν t Gandhi-Navarro-Rivers over the autoregressive method. 13 That is, let the vector of instruments be r t = {k t 1, l t 1, m t 1, k 2 t 1, k t 1 l t 1, k t 1 m t 1, l 2 t 1, l t 1 m t 1, m 2 t 1, k t 2, m t 2, k t 2 m t 2 } 14 Gandhi et al. (2013) do show how to extend their method to include an additive firm-level fixedeffect.

19 ESTIMATING THE PRODUCTION FUNCTION WHEN FIRMS ARE CONSTRAINED 19 ρ (y) = ω + π k ρ (k) + π l ρ (l) + π m ρ (m) + ν t (21) These ρ ( ) terms are observed and thus can be treated as outcomes or regressors. One could estimate π k, π l, π m by running two-stage least squares using r t to instrument for ρ (k), ρ (l), ρ (m). The GMM estimator does something similar to this. Suppose f is estimated using a log sieve polynomial. Define X t as a matrix where each column is one term of this polynomial (k t, l t, m t, kt 2, k t l t, ) and each row is an observation. Let π be the coefficients of the sieve polynomial. Let r nt be the vector of instruments for observation n, and let R t be the matrix where row n equals rnt. T Let y t be a vector of outcomes. The GMM estimator solves [ min y t X t π ρ(y t 1 X t 1 π) π,ρ ] T Rt W R T t [ ] y t X t π ρ(y t 1 X t 1 π) (22) Solving the first-order condition for π gives ˆπ = [ ˆρ (X) T R t W R T t ˆρ (X) ] 1 ˆρ (X) T R t W R T t ˆρ (y) (23) while ˆρ is estimated by solving the nonlinear equation 0 = (y t 1 X t 1 ˆπ) T R t W R T t [ ] y t X t ˆπ ρ(y t 1 X t 1 ˆπ) (24) after substituting (23) for ˆπ. If ρ were known and W = (Rt T R t ) 1 then the expression for ˆπ would simply be the two-stage least squares estimator. What the GMM estimator effectively does is solve the nonlinear equation (24) for ρ and plug this estimate into (23). If the GMM estimate of ρ is consistent then (23) would converge to the two-stage least squares estimate. Stock and Wright (2000) lay out the conditions under which certain parameters of a GMM estimator are consistent even under weak identification asymptotics (to be precise, when they follow a drifting process that assumes identification is local to zero). The following proposition, which is proven in Appendix A.1, shows that these conditions hold:

20 20 AJAY SHENOY Proposition 1 Let N denote the sample size. Define the matrix Ξ(z) = ( N ) n=1 N r ntzn T E[r nt zn T ] N and let Ξ N = [ ] Ξ(y t ) Ξ(x t ) Ξ(y t 1 ) Ξ(x t 1 ) be a block matrix. Assume 1. Ξ N d Ξ N (0, Σ Ξ ) 2. E[r nt ω n,t 1 ] = Υ 0 Then under the weak identification asymptotics of Stock and Wright (2000), the autoregressive estimate for ρ is consistent (regardless of whether the other parameters are weakly identified). The first assumption holds as long as the appropriate central limit theorem holds for output and for the inputs of production. The second assumption holds as long as the choices of inputs depends in some way on current or past productivity (as ω t 1 depends on ω t 2, and so on). 15 The consistency of ρ implies there is a close connection between the autoregressive and two-stage least squares estimates of π. This connection is useful because the effect of weak instruments on two-stage least squares has been longer studied and is better understood than the effect of weak identification on nonlinear GMM. For example, Stock and Yogo (2005) show that under some assumptions, weak instruments bias two-stage least squares towards ordinary least squares. There are also diagnostic statistics (e.g. Stock and Yogo, 2005; Kleibergen and Paap, 2006) common in the applied literature that can be used, as I show below, to assess the severity of weak instruments. 15 The assumption behind weak identification asymptotics that the moment conditions follow a drifting process that makes them local to zero can never be proven but is standard in the literature. Much as any proof of consistency uses a thought experiment in which N goes to infinity, the drifting assumption is a useful approximation.

21 ESTIMATING THE PRODUCTION FUNCTION WHEN FIRMS ARE CONSTRAINED 21 5 Complementarity in Estimating the Gross Production Though it may seem clear that the success of a choice-based estimator hinges on whether firms are constrained, this section argues that at least when estimating a gross production function the same is true of the autoregressive estimator. While choice-based methods grow more biased as constraints tighten, the autoregressive estimate of the gross production function becomes less biased. This complementarity arises because although constraints undermine the identification assumptions of choice-based methods, they strengthen the instruments used for identification in the autoregressive estimator. 5.1 Intuition: Cobb-Douglas Case Though the exclusion restriction of the autoregressive estimator holds regardless of whether the Scalar Unobservable assumption fails, the rank condition essentially requires it to fail. This ironic feature is the key to its complementarity with Gandhi- Navarro-Rivers. It is easiest to see this rank failure when production is Cobb-Douglas and choices are optimal. Let F (K t, L t, M t ) = K π k t L π l optimal choice of intermediates is m t = t Mt πm. After de-meaning, the 1 1 π m [ω t + π k k t + π l l t ] (25) Let r t = [r 1 t,..., r J t ] T be a vector of valid instruments. The parameter π M is identified only if a change in the value of π m induces changes in the moment conditions E[ν t r 1 t ] = = E[ν t r J t ] = 0 that are linearly independent of those induced by changes in the other parameters π k, π l. (In an ordinary least squares regression, this assumption is equivalent to saying the regressors are not perfectly collinear.) A small change in π m changes the j-th moment condition by E[ν t r j t ] = E[r j t (ρm t 1 m t )] π m [ = E r j 1 { } ] t (ρω t 1 ω t ) + π k (ρk t 1 k t ) + π l (ρl t 1 l t ) 1 π m

22 22 AJAY SHENOY [ = E r j 1 { } ] t η t + π k (ρk t 1 k t ) + π l (ρl t 1 l t ) 1 π m = 1 1 π m E[r j t η t ] + π k 1 π m E[ν t r j t ] π k + π l 1 π m E[ν t r j t ] π l where, as before, all variables are de-meaned. By the exclusion restriction r j t is uncorrelated with η t, implying the first term is zero. But then any change in the moment condition induced by π m is perfectly collinear with those induced by π k and π l, implying π m is not identified. As shown in Section 5.2, this result generalizes to any nonparametrically estimated production function. The intuition is similar to the functional dependence critique raised in Ackerberg et al. (2015) and Gandhi et al. (2013). The optimal choice of intermediates is perfectly determined by the choices of other inputs and productivity, which depends on its own lag and an unpredictable innovation. After controlling for this lag and the other inputs, the only remaining variation is the innovation. Since this variation cannot be used for identification without violating the exclusion restriction, there is no way to identify the effect of intermediates on output. But if firms are constrained the unobserved elements that determine the constraint S 1 t, S 2 t,..., S I t will induce additional systematic variation in the choice of intermediates. As I show below, this fact creates an indirect complementarity between Gandhi-Navarro-Rivers and the autoregressive estimator. 5.2 General Result When firms are potentially constrained the Lagrange multiplier on the constraint its mean, variance, and correlation with the instruments governs the relative bias of Gandhi-Navarro-Rivers and the autoregressive estimator. The complementarity between them rests on the fact, embodied in two theorems below, that the Lagrange multiplier has opposite effects on their consistency. First, formalize the notion of constraints by modifying Assumption 5 as follows: Assumption 10 (Constrained Optimal Choices) Define Λ t = log(1 + λ t ), where λ t is a Lagrange multiplier that gives the shadow cost to the firm of being unable to choose M t optimally. Then the choice of the firm satisfies Λ t = ω t + log E[e εt ] + log F M (K t, L t, M t ) (26)

23 ESTIMATING THE PRODUCTION FUNCTION WHEN FIRMS ARE CONSTRAINED 23 As noted earlier, the market imperfection need not be a constraint. It could be some other distortion τ t that drives a wedge between the marginal products of firms. Redefine Λ t = log(1 + τ t ) and all of the following results (except the sign of the bias in Proposition 2) still hold The Bias of Gandhi-Navarro-Rivers Increases with Constraints The following proposition, which is proven in Appendix A, shows that Gandhi-Navarro- Rivers grows more biased as constraints tighten: Proposition 2 Let Λ t = Λ t E[Λ t k t, l t, m t ]. Then the average bias of the Gandhi- Navarro-Rivers estimate of the elasticity of intermediates is (E[Λ t ] + log E[e Λ t ]) (27) In the special case where Λ t N (0, σλ 2 ) the bias is simply (E[Λ t] + σλ 2 /2). Finally, if the market imperfection is caused by constraints like that shown in the constrained optimization (12), then the bias is less than or equal to zero (with equality holding if and only if firms are unconstrained). The proposition implies that an increase in either the mean of Λ t or the variability of Λ t, the residual from a nonparametric regression of Λ t on (k t, l t, m t ), increases the absolute value of the bias. If the distortion is a hard constraint (as opposed to a more general market imperfection), the bias is always downward. Constraints make output appear less responsive to intermediates than in truth. Since the second stage of Gandhi-Navarro-Rivers relies on a consistent estimate of the elasticity of intermediates, the other parameters in particular, the elasticities of capital and labor will also be biased The Bias of the Autoregressive Estimator Decreases with Constraints Like any GMM estimator, the autoregressive estimator is under-identified if the Jacobian of the moment conditions is singular at the true value of the parameters. Section 5.1 shows that the Jacobian is singular in the special case where firms are unconstrained and the production function is known to be Cobb-Douglas. In practice the production function is unknown and must be estimated nonparametrically, and the fraction of firms that are constrained will be neither 0 or 1, but lie somewhere

24 24 AJAY SHENOY in between. In that case the Jacobian of the moment conditions is not singular, but it may be near-singular. Though in principle the estimator is identified, in practice such weak identification will bias nonlinear GMM much as weak instruments bias two-stage least squares (Stock and Wright, 2000). This problem arises for the autoregressive estimator when constraints are relatively slack. It can be shown that one measure of near-singularity, the smallest singular value of the Jacobian of the moment conditions, is bounded above by a statistic that is increasing in the severity of constraints and equals zero when firms are unconstrained. 16 But the statement of this proposition is not intuitive. 17 It is more intuitive to exploit the link between the autoregressive estimator and two-stage least squares (see Section 4.3). The severity of constraints, as I show below, places an upper bound on the size of a common measure of weak instruments. Assume that log F (K t, L t, M t ) and log F M (K t, L t, M t ) each has a polynomial sieve approximation in logs: log F (K t, L t, M t ) = log F M (K t, L t, M t ) = a 0 1,a0 2,a0 3 a 1 1,a1 2,a1 3 A 0 a 0 1,a0 2 3k a0 1,a0 t l a0 2 t m a0 3 t A 1 a 1 1,a1 2,a1 3k a1 1 t l a1 2 t m a1 3 t (28) Let x t be a vector of the terms of the sieve approximation, and let ẋ t be a vector of the terms in x t excluding m t. As in Section 4.3, assume for now that ρ is known and denote ρ (x) = x t ρx t 1. Consider the two-stage least squares regression that instruments ρ (x) with r t. One measure of the joint strength of the instruments is the Cragg-Donald statistic, which is analogous to the first-stage F-statistic but may be used when there are multiple endogenous regressors. The following proposition, proven in Appendix A.3, links the Cragg-Donald statistic to the log-lagrange multiplier Λ t : Proposition 3 Let V be the matrix of residuals from a regression of ρ (x) on the matrix of instruments, and let Σ V be the variance matrix. Let ρ ( ẋ) and ρ ( Λ) be the predicted values from a regression of ρ (ẋ) and ρ (Λ) on the instruments (as would 16 This approach follows in the spirit of Wright (2003), who derives a test for under-identification based on the distance between the Jacobian and the closest matrix that is not of full column rank; and of Kleibergen and Paap (2006), who propose a widely used test for weak instruments based on the singular value decomposition. 17 I am nevertheless happy to provide both proposition and proof upon request.