metadata: authors: 0: author: Sampi Bravo,James Robert Ezequiel
1: author: Jooste,Charl
2: author: Vostroknutova,Ekaterina
count: World
docna: 0: docna: Identification Properties for Estimating the
            Impact of Regulation on Markups and Productivity
docty: Policy Research Working Paper
subtopic: International Trade and Trade Rules,Competition Policy,Competitiveness and Competition Policy,De Facto Governments,Democratic Government,State Owned Enterprise Reform,Public Sector Administrative and Civil
            Service Reform,Economics and Finance of Public
            Institution Development,Public Sector Administrative & Civil
            Service Reform,Macroeconomic Management,Governance Diagnostic Capacity Building,Economic Forecasting,Trade Policy
teratopic: Macroeconomics and Economic Growth,Governance,Private Sector Development,Public Sector Development,International Economics and Trade
abstracts: cdata!: This paper addresses several
            shortcomings in the productivity and markup estimation
            literature. Using Monte-Carlo simulations, the analysis
            shows that the methods in Ackerberg, Caves and Frazer (2015)
            and De Loecker and Warzynski (2012) produce biased estimates
            of the impact of policy variables on markups and
            productivity. This bias stems from endogeneity due to the
            following: (1) the functional form of the production
            function; (2) the omission of demand shifters; (3) the
            absence of price information; (4) the violation of the
            Markov process for productivity; and (5) misspecification
            when marginal costs are excluded in the estimation. The
            paper addresses these concerns using a quasi-maximum
            likelihood approach and a generalized estimator for the
            production function. It produces unbiased estimates of the
            impact of regulation on markups and productivity. The paper
            therefore proposes a work-around solution for the
            identification problem identified in Bond, Hashemi, Kaplan
            and Zoch (2020), and an unbiased measure of productivity, by
            directly accounting for the joint impact of regulation on
            markups and productivity.
repnme: repnme: Identification Properties for Estimating
            the Impact of Regulation on Markups and Productivity
display_title: Identification Properties for
            Estimating the Impact of Regulation on Markups and Productivity
listing_relative_url: /research/2021/01/32767934/identification-properties-estimating-impact-regulation-markups-productivity
topicv3: Impact of Trade m1326020 1192,State Owned Enterprises m1326323 721,Trade m1326924 698,Competition Policy m1326982 406,Market Failures m1327149 1068
publishtoextweb_dt: 2021-01-21T00:00:00Z
historic_topic: Macroeconomics and Economic Growth,Governance,Private Sector Development,Public Sector Development,International Economics and Trade
origu: EFI-LCR-MTI-MacroFiscal (ELCMU)
no_of_pages: 41
author_keys: 0: author_key: 1496952
1: author_key: 1368476
2: author_key: 687122
model_extractions: Policy Research Working Paper 9523 Identification Properties for Estimating the Impact of Regulation on Markups and Productivity _James Sampi Charl Jooste Ekaterina Vostroknutova_ Macroeconomics, Trade and Investment Global Practice January 2021
Policy Research Working Paper 9523

# **Abstract**

This paper addresses several shortcomings in the productivity and markup estimation literature. Using Monte-Carlo simulations, the analysis shows that the methods in Ackerberg, Caves and Frazer (2015) and De Loecker and Warzynski (2012) produce biased estimates of the impact of policy variables on markups and productivity. This bias stems from endogeneity due to the following: (1) the functional form of the production function; (2) the omission of demand shifters; (3) the absence of price information; (4) the violation of the Markov process for productivity; and (5) misspecification when marginal costs are excluded in the estimation. The paper addresses these concerns using a quasi-maximum likelihood approach and a generalized estimator for the production function. It produces unbiased estimates of the impact of regulation on markups and productivity. The paper therefore proposes a work-around solution for the identification problem identified in Bond, Hashemi, Kaplan and Zoch (2020), and an unbiased measure of productivity, by directly accounting for the joint impact of regulation on markups and productivity.

This paper is a product of the Macroeconomics, Trade and Investment Global Practice. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at jsampibravo@worldbank.org, cjooste@worldbank.org, evostroknutova@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# Identification Properties for Estimating the Impact of Regulation

# on Markups and Productivity<sup>_∗_</sup>

James Sampi<sup>1</sup> , Charl Jooste<sup>2</sup> , and Ekaterina Vostroknutova<sup>3</sup> 1 World Bank, Vrije Universiteit Amsterdam January 25, 2021

**JEL code** : C150, C310, L510, 0470

**Keywords** : Productivity, Markups, Regulation, Quasi-Maximum Likelihood

> _∗_ We are grateful to Chad Syverson, Philippe Aghion, Siem Jan Koopman, and Andrew Harvey for their helpful and constructive comments. We are also grateful to Martin Rama, Jorge Araujo and to the participants of Authors’ Workshop for ”Competition, Productivity, and Growth in Latin America” report at the World Bank, for a helpful discussion. Corresponding authors: jsampibravo@worldbank.org, cjooste@worldbank.org, evostroknutova@worldbank.org.
# **1 Introduction**

The objective of this paper is to establish a causal relationship between regulation, market power, and productivity. We contribute to the growing literature on how distortions affect productivity (see Restuccia and Rogerson (2013), RR) by emphasizing one specific channel relevant for both developing and developed countries. In particular, we develop a new econometric approach to measuring how regulation can create distortions that transmit mainly through idiosyncratic prices and costs, thus generating a misallocation of inputs between firms. As a result, such regulations affect market structure (through markups) and create output distortions (e.g. productivity).

Market structure and regulations are central to the efficiency with which resources are allocated across firms, industries, and sectors. The seminal work of Hsieh and Klenow (2009) shows that market rigidities (distortions) can prevent reallocation of inputs from low-marginal product firms to high-marginal product firms. These distortions can affect productivity through three channels, which are conceptually different but not fully independent: technology (within-firm), selection (entry/exit), and allocation (between-firm), as in Olley, Steven and Pakes (1996), and Melitz and Polanec (2015). Such distortions can include regulation, discretionary provisions such as preferential treatment of firms (including state-owned enterprises), and market imperfections such as monopoly power, market frictions, and enforcement of property rights (RR 2017). In particular, regulation can include tariffs and taxes that are applied differentially, product market regulations that restrict size or limit market access, factor market regulations that have uneven implications for different firms, etc.

This paper is not the first attempt to analyze the regulatory impacts on productivity and markups. A series of papers argue that greater competition makes firms more efficient (Nickel (1996); Olley, Steven and Pakes (1996); Sivadasan (2009); and Melitz (2003)). In terms of markups and regulation, Fabrizio, Rose and Wolfram (2007) show that there 2
are efficiency gains from replacing regulated monopolies with a market based structure. An interesting result from Genakos, Koutroumpis and Pagliero (2018) shows that certain markup regulations (such as markup ceilings) tend to create collusion for wholesalers, and hence an abolishment in regulation will lead to a decrease in retail and wholesale prices - once again the emphasis is on better competition regulation. Backus (2019) is a particularly interesting paper as it considers an example of an industry where measured productivity responds directly to competition, possibly through specialization or managerial input effects, as opposed to selection or reallocation channels.

For policy analysis it is thus critical to back out an unbiased estimate of the effects of regulation on both markups and productivity. This requires either an unbiased measure of productivity and an unbiased measure of markups (both are unobservable to the econometrician), or an estimator that can measure the policy response using the variation in productivity and markups, as opposed to the level.

Obtaining an unbiased estimate for markups would help to distinguish firms with market power (De Loecker, Eeckhout and Unger (2020)), explain price rigidities in the business cycle (Blanchard and Gali (2016)); discern channels of productivity growth (Aghion, Bloom, Blundell, Griffith and Howitt (2005); Aghion, Bergeaud, Boppart, Klenow and Li (2019); Hall (2018)); assess market distortions (Chari, Kehoe and Mcgrattan (2007)); study cyclical variations in the unemployment rate (Gali, Smets and Wouters (2011)) and labor shares in GDP (Autor, Dorn, Katz, Patterson and Van Reenen (2017); De Loecker et al. (2020)); and identify sources of business cycle variability (Chari et al. (2007); Bils, Klenow and Malin (2018), Hong (2018); among others).

The literature relies mainly on two methods to estimate markups: the degree of excess prices relative to marginal cost using both the primal and dual approach (Hall (1988)), and more recently, the method by De Loecker and Warzynski (2012) (DLW, 2012 henceforth) using the ratio of the output elasticity to the factor share in production.<sup>1</sup> A standard ap1Another approach is Diewert and Fox (2008) using revenues divided by total costs.

3
proach for estimating output elasticities and backing out productivity is Ackerberg, Caves and Frazer (2015) (ACF, 2015 henceforth). While these methodologies have led to significant improvements in understanding productivity and markups, they are not without criticism (see discussions in Flynn, Ghandi and Traina, 2019; Doraszelski and Jaumandreu (2019); Bond, Hashemi, Kaplan and Zoch (2020)). An application by Kealy, Pujolas and Sosa-Padilla (2019) shows that the impact of trade-liberalization on productivity is very different when different methodologies (OLS, ACF, Gandhi, Navarro and Rivers (2020), Levinsohn and Petrin (2003a)) are compared using the same data set.

In this paper we discuss various criticisms and suggest an estimator to deal with them. The primary concern is endogeneity when attempting to estimate the impact of regulation on productivity or markups. Five important aspects explain this endogeneity: **1) An incorrectly assumed production function** . This error emerges when the econometrician estimates the production function under an erroneous assumed functional form. The discrepancy between the estimated output elasticity and its true value is correlated with productivity, and therefore with regulation; **2) Regulation acting as a demand shifter** , which induces the error in the estimated markups to be correlated with regulation, which is due to the omission of demand shifters in the first stage of ACF when approximating the production function; **3) Absence of price information** , causing the estimated output elasticity to converge towards one as described by Bond et al. (2020), thus generating similar issues as in 1); **4) The violation of the Markov process for productivity** , which emerges if regulation affects average firm productivity, thus conflicting with the Markov process defined in ACF; and finally, **5) The omission of marginal costs for backing out productivity** , where the residual of the estimated productivity in ACF is correlated with markups, and therefore, regulation. These constraints make any policy inference on the impact of competition regulation questionable.

We establish a methodology that allows for a consistent estimate of the impact of change in regulation on markups, without having to measure the level of the markup. This involves 4
another important aspect for obtaining unbiased estimates of productivity. Noting that there exists a relationship between productivity and markups (Baqaee and Fahri (2020)), the estimator will need to jointly account for the impact of regulation on both unobservables. The proposed approach is similar to Doraszelski and Jaumandreu (2019), but ours is suited for a wider set of concave production functions, which is ex-ante unknown to the econometrician. The structural system of equations is estimated via a Quasi-Maximum Likelihood (QMLE) method. Monte-Carlo simulations show that DLW methodology yields biased results when estimating the impact of regulation on markups and when the production function differs from the Cobb-Douglas case. The ACF method produces a downward a biased coefficients for all production function specifications. In terms of our proposed QLME method, both the markup and productivity effects are unbiased, although the efficiency of the estimator declines in short panels.

The contributions to the literature are three-fold: (1) A work-around solution for the identification problem when using the revenue elasticity to recover markups, (2) an improvement in estimating an unbiased measure of productivity by (3) directly accounting for the joint impact of regulation on markups and productivity.

The remainder of the paper is organized as follows. Section 2 introduces the theoretical framework for estimating productivity and markups, and discusses the sources of potential biases. We introduce our methodological approach in Section 3, while Section 4 describes the implementation of the econometric procedure. Section 5 presents the Monte-Carlo simulation results. Finally, Section 6 concludes.

# **2 Framework**

The literature (see for example De Loecker and Warzynski (2012)) has focused on estimating the impact of regulation change on productivity/markups by imposing the following 5
regressions: where _ωjt_ stands for productivity in logs (assuming the production function is in logs), and _μjt_ establishment markups. The time-invariant firm-specific dummies are captured by _φj_ and _τj_ , while time-variation dummies are denoted by _φt_ and _τt_ for productivity and markups, respectively. Finally, _zjt_ stands for the change of regulation, although it can be generalized for any demand shifter. The exclusion of an AR(1) component does not bear on the analysis in the following sections.

# **2.1 Cost minimization problem**

Firms compete in the market, in a way that is unknown to the econometrician. Firm _j_ produces output _Qjt_ in period _t_ with a given amount of capital _Kjt_ , while selecting the amount of labor _Ljt_ and material inputs _Mjt_ . The production function is represented as follows: where _ωjt_ is the Hicks-neutral productivity measure, and _εjt_ accounts for shocks unanticipated by firms at the time of selecting inputs. The productivity estimate _ωjt_ is observed by the firm but not the econometrician. As described in ACF, the firm decides on its input allocations for _Ljt_ and _Mjt_ conditional on the expected realization of _ωjt_ . Inputs and productivity are contemporaneously correlated. Furthermore, the function _F_ ( _., ._ ) is continuously differentiable of class _C_<sup>2</sup> on all its inputs. Finally, the disturbance _εjt_ is uncorrelated over time and uncorrelated with the inputs.

Firms are assumed to minimize the cost of production for a given output level by choos6
ing variable inputs, which can be represented by The associated first-order conditions for the cost minimization problem are: Note that the price is a markup over marginal cost ( _Pjt_ = _μjtMCjt_ ), where _Pjt_ is the relevant output price, and _μjt_ the firm markup. Following Hall (1988, 1990), we use the cost minimization conditions in Equation 5 to infer the markup level.

where _SXjt_ is the input share to revenue of the amount spent on _Xjt_ , where _Xjt_ = _{Ljt, Mjt}_ . The output elasticity with respect to input _Xjt_ is _ρXjt_ =<sup>_X_</sup> _Qjt_<sup>_<u>jt</u>_</sup> _∂X∂Qjjtt_<sup>.Importantly, a consistent</sup> estimation of Equation 6 requires a consistent estimate for the output elasticity. In the special case of a Cobb-Douglas production function, the output elasticity collapses to a static coefficient _βX_ with respect to the associated parameter of the _Xjt_ input in the production function. We can re-write the output elasticity in a more generic form: where _gX_ ( _X jt_ ) is a continuously differentiable function of class _C_<sup>2</sup> , with _X jt_ representing the inclusion of all inputs _{Xjt}_ . The function _gX_ ( _X jt_ ) maps the curvature of the production function _F_ ( _., ., ._ ) and is hence not restrictive to a Cobb-Douglas functional form. In the 7
applied literature, the estimation of _βX_ relies on the procedure introduced by ACF (mainly Cobb-Douglas), while DLW (2012) for markups is generalized to a translog production function.

# **2.1.1 Estimating productivity and markups: ACF and DLW**

The ACF (2015) procedure relies on a two-step estimation approach using generalized method of moments (GMM). In the first step, ACF assume that the material input demand be expressed in terms of other production inputs, including productivity, through the function _m_ ( _._ ). Productivity is obtained by inverting this function. Specifically, The latter is a direct implication of imposing a monotonic increasing relationship between productivity and material inputs (Ass. 4c in ACF, ”Scalar unobservable”). The first step minimizes the distance between output and an unknown function Φ( _._ ), which is obtained by replacing _ωjt_ into the production function.

Assuming _E_ ( _εjt|It_ ) = 0 from Equation 3, we obtain

where Φ is approximated by polynomials.

The second step relies on the following Markov process assumption for productivity: 8
where _h_ ( _._ ) is a continuous function of _ωjt_ . The second step is therefore summarized as where Φ<sup>ˆ</sup> _j,t−_ 1 is estimated in the first step. ACF estimates the production function parameters from the moment condition in equation 11. The productivity estimate can thus be formulated as follows To estimate markups, DLW (2012) apply ACF methodology with a slightly different Equation 6: ˆ where the error term is the residual from the first stage in ACF, _εjt_ = log _Qjt −_ Φ<sup>ˆ</sup> _jt_ , and _S_ =<sup>_WjtLjt_thewagebillshareinrevenues.InthecaseofaCobb-Douglasproduction</sup> _Xjt PjtQjt_<sup>is</sup> function, the output elasticity _ρXjt_ is _βX_ . Under a translog value added production function, the output elasticity for labor, for example, is given by Notice that even when assuming a translog production function, it still requires a consistent estimate of _βX_ from the second step in the ACF procedure.

# **2.2 Endogeneity issues**

The estimation of productivity and markups following Equations 12 and 13 are the central expressions for assessing the effects of regulation on those variables. We discuss three potential biases in estimating the effects of regulation on markups, and two biases for productivity. Doraszelski and Jaumandreu (2019) also discuss several issues when estimating 9
productivity and markups.

The estimated productivity and markup from Equations 12 and 13 can be re-written to illustrate the sources of biases: Here we notice that an omitted variable, such as regulations that are not considered in the first stage, will result in a non-zero outcome in the brackets. The regressions of 1 and 2 will be biased.

**1) Erroneously imposing the production function on data** . To see this problem, we can express the difference between the estimated output elasticity and its true value from Equation 7 as follows:

if _gX_ ( _X jt_ ) does not pertain to the family of CES production functions, of which CobbDouglas is one, then expression 17 differs from zero. As an example, if the econometrician incorrectly assumes a Cobb-Douglas functional form, expression 17 converges to _−_ log _gX_ ( _X jt_ ). Because Equation 1 expresses productivity as a function of regulation, where productivity is correlated with production inputs, the expression _−_ log _gX_ ( _X jt_ ) will also be correlated with regulation.

**2) Regulation acting as a demand shifter** . This error occurs when the regulation variable is omitted when approximating the production function in the first stage of the ACF procedure. As an example, consider that regulation affects productivity via changes in innovation or an improvement in managerial skills (both of which are independent of the input allocation decisions), then the estimated error _ε_ ˆ _jt_ includes regulation. It implies that

10
_E_ [ _zjt_ ( _εjt − ε_ ˆ _jt_ ) _|It_ ] = 0. This problem is less of a concern if _zjt_ is included in the first stage of ACF.

**3) In the absence of price data (a problem in many firm-level surveys)** , the output elasticity is inconsistent and tends to become the revenue elasticity. This implies that the estimated markup of Equation 13 converges towards one (see Bond et al. (2020) for an in-depth discussion). Considering the revenue elasticity instead of output, Equation 6 becomes

where _ρ_<sup>_R_</sup> _Xjt_<sup>standsforrevenueelasticity.Thisestimatoriswidelyappliedintheliterature</sup> (see for example Diez, Leigh and Tambunlertchai (2018)). Notice, however, that markups are either increasing when the revenue elasticity is constant, but labor shares are falling, or when labor shares are held constant, but the revenue elasticity increases.<sup>2</sup> The revenue elasticity can be expressed as follows where _ρ_<sup>_P_</sup> _Xjt_<sup>=</sup> _∂Q_<sup>_∂Pj_</sup> _jt_<sup>_t_</sup> _<u>QPjtjt</u>_<sup>represents the price elasticity of demand, and</sup><sup>_ρXjt_is output elasticity</sup> as described in Equation 7. To illustrate the problem raised in Bond et al. (2020), consider the following firm profit maximization problem: where _C_ ( _._ ) is the cost function and the marginal cost is expressed as _MCjt_ =<sup>_∂C_</sup> _∂Q_<sup><u>(</u></sup><sup>_<u>Q</u>_</sup> _jt_<sup>_<u>jt</u>_</sup><sup><u>)</u>.</sup>

> 2Other explanations for higher markups not accounted for directly here include fixed and sunk costs (and thus lower marginal costs), network effects, monopsony power, and mergers and acquisitions (Berry, Gaynor and Morton (2019)).

11
The first order condition for this problem is given by By combining the first order conditions of the minimization and maximization problems and replacing the output elasticity with the revenue elasticity in the equation for calculating markups, we obtain This result shows that the estimated markup, using the revenue elasticity, should always converge to one. Therefore, the true value of the markup cannot be recovered when using revenue data instead of data on physical quantities. In addition, Bond et al. (2020) show that markup estimates cannot be obtained when: (1) revenue/sales data are deflated by a sectoral deflator (data for physical quantities are necessary to obtain this estimate); (2) there exists residual demand or when demand is heterogenous; (3) using a large cross-section but short-time series assuming firms are heterogeneous.

**4) The Markov process for productivity is violated** , when regulation affects average firm productivity as formulated in ACF.

From Equation 1, we can express the conditional mean of productivity as follows By replacing Equation 23 with the expected value of Equation 15, we obtain _E_ (Φ _jt −_ ˆΦ _jt|It_ ) = _xjt_ , which is correlated with regulation _zjt_ . This problem can be handled by reformulating the second step in ACF such that

12
Notice that this formulation implies a joint estimation of production function parameters and the regulation effects on productivity.

**5) The omission of marginal costs for backing out productivity** , implies that the estimated residual of productivity in ACF is correlated with markups, and therefore, regulation.

Using Equation 5, we can re-write the optimal labor input demand as: where _MCjt_ = _Pjt/μjt_ .

If the econometrician observes the firm’s marginal cost, then, the polynomial approximation in the first step of ACF should include it, where the polynomial function is Φ<sup>_μ_</sup> _jt_<sup>.</sup> Productivity in ACF can be formulated such that where Φ<sup>_μ_</sup> _jt_<sup>is the polynomial approximation that includes marginal cost, and Φ</sup><sup>_jt_the standard</sup> approximation in ACF as per Equation 9. Because Φ _jt −_ Φ<sup>_μ_</sup> _jt_<sup>isafunctionof</sup><sup>_μjt_,itimplies</sup> that _E_ [ _zjt_ (Φ _jt −_ Φ<sup>_μ_</sup> _jt_<sup>)] = 0.Thus,Equation 2 will generate biased results.This issue can be</sup> handled by estimating Equations 2 and 9 jointly.

Overall, the aforementioned issues require a joint-estimation framework to assess the impact of regulation on both productivity and markups. It should be noted that while it is possible to estimate the impact of regulation on markups, it is not possible to uncover the level of markups when price data are not available.

13
# **3 Alternative approach**

In this section we develop a consistent framework for estimating the effects of regulation on productivity and markups, without imposing an explicit functional form for the production function, or knowing the functional form ex ante.

In order to capture the effects of regulation on productivity, we consider the following extension of the ACF Markov process assumption.

**Assumption 1.** _(First Order Markov) Productivity shocks evolve according to the following distribution_

_this distribution is known to firms and stochastically increasing in ωjt and zjt. In addition, the information set It includes current and past productivity shocks but it does not include future shocks._ Note that Assumption 1 states that the impact of regulation transmits through prices affecting productivity/efficiency directly, but does not impact firms’ allocation decisions immediately. This class of distortions is commonly assumed in the literature, whereby distortion impacts productivity via between-firm misallocation, rather than within-firm decision (see RR 2008), by creating heterogeneous changes in prices faced by individual<sup>3</sup>

> 3Note that this does not impose specific constraints on the kind of regulations that can be studied, as long as they produce idiosyncratic distortions to prices. For example, hiring/firing cost, size dependent policies or otherwise regulations with differential impact on classes of firms (e.g. taxes and tariffs/quotas); differential enforcement of informality; factor markets regulations that affect allocation of labor between firms or credit constraints, would be acceptable examples. Following Restuccia and Rogerson (2008), the channels through which misallocation can be created (within, between, and selection) are not fully independent from each other, e.g. a regulation that affects market structure and therefore mainly acts through the between channel can also affect within-firm productivity and entry. Furthermore, regulation is assumed to be an exogenous process.

14
In what follows, productivity evolves according to By replacing Equation 28 into Equation 3, and combining Equations 2, 6 and 7, we obtain the following system of equations<sup>4</sup> where _Fjt_ stands for _F_ ( _Kjt, Ljt, Mjt_ ).

The joint estimation of the system in Equation 29 addresses the endogeneity issues raised in the previous section. In addition, this framework does not require the monotonic increasing relationship between productivity and material inputs as in ACF (Ass. 4c, _”Scalar unobservable”_ ). Although this approach still does not recover the level of markups, it provides consistent estimates of the impacts of regulation on markups. Recently, Doraszelski and Jaumandreu (2019) proposed to estimate a similar but more restrictive system of equations by imposing a specific functional form for production and the output elasticity. Specifically, the output elasticity is assumed to be constant over time. Thus, the simplification in Doraszelski and Jaumandreu does not address the error from **imposing the production function** . In this regard, the system of equations in expression 29 is a more general approach that is suited for a wide number of production functions. Notice that Doraszelski and Jaumandreu focus on recovering unbiased values for the production function parameters, and hence the **level** of the markup, while our primary objective is to recover the **impact of regulation policies on productivity and markups** .

The system of equations in 29 still requires an approximation for the functions _Fjt_ and _gX_ ( _X jt_ ).

> 4Notice that the omission of the AR(1) component on the markups equation does not affect the results.

15
**Proposition 1.** _(Production function approximation) If the function Fjt is continuously differentiable of class C_<sup>2</sup> _in all its inputs, then, the production function_ log _Fjt is approximated by Taylor expansions, where the coefficient of the linear component associated to the input_ log _Xjt is its output elasticity ρXjt (see Equation 7) evaluated at the point_ (log _Xj_ )<sup>_∗_</sup>

_where Rjt is the residual term, while θX and δXY are parameters associated with the Hessian of the function_ log _Fjt evaluated at the point_ (log _Fj_ )<sup>_∗_</sup> _. See Appendix A for the proof._ Notice that the identification of _gX_ ( _X jt_ ) is a sufficient condition for recovering the value of _βX_ , thus estimating the parameters pertaining to the production function consistently. As an example, in the case of a Cobb-Douglas production function, the estimated value of _gX_ ( _X j|_ (log _X_ ) _∗_ ) should be equal to one. Similarly, Kmenta (1967) points out that the function _gX_ ( _X j|_ (log _X_ ) _∗_ ) should equal one in the case of a CES production function when the elasticity of substitution is close to zero. If (log _Fj_ )<sup>_∗_</sup> becomes the time average of the function log _Fjt_ , then, the approximation is a demeaning sequence.

**Proposition 2.** _(Approximation for gX_ ( _._ ) _) If function gX_ ( _._ ) _mimics the curvature of Fjt for any input Xjt such that_<sup>_∂gX_</sup> _∂X_<sup><u>(</u></sup> _Xjtjt_ <u>)</u> _>_ 0 _and_<sup>_∂_2</sup><sup>_<u>g</u>_</sup> _∂X_<sup>_X_</sup><sup><u>(</u></sup> _jtX_<sup>2</sup> _<u>jt</u>_ <u>)</u> _<_ 0 _, then, there is a function ψX_ ( _._ ) _such that gX_ ( _._ ) _and F_ ( _._ ) _are topologically equivalent. By imposing a continuous linear mapping of ψX on gX_ () _, such that ψX_ ( _._ ) _is represented by a function equivalent to the inverse of F_ ( _._ ) _, we obtain_

16
_where γ is a parameter that controls for the curvature of the function gX_ () _, while γo accounts for the level of the output elasticity. As an example, in the case of a Cobb-Douglas production function, the value of γ should be estimated to be close to zero. See Appendix B for the proof._ Finally, the demeaned (controlling for fixed effects) system of equations is summarized as: ̄ ̄ where _xt_ stands for _xt −_<sup>�</sup> _t_<sup>_xt_.Thevariable</sup> log _F jt_ is approximated using Equation 30. Demeaning the system prevents us from recovering the value of _γo_ , thus making it impossible to back out the value of the output elasticity (therefore markups) when the DGP process is a CES production function with elasticity of substitution significantly different from zero, or any other type of unknown production function. However, the approximation in Equation 31 will provide enough information on the curvature of the output elasticity regardless of its value in levels, thus the system in Equation 32 should produce consistent estimates for _τ_<sup>_μ_</sup> and _φ_<sup>_ω_</sup> , independently of the functional form of the production function.

# **4 Econometric estimation**

Our joint estimation method relies on the Quasi-Maximum Likelihood (QMLE) approach, which guarantees the consistency of the estimator under special circumstances (for multiple endogenous explanatory variables see Imbens and Wooldridge (2009)), in addition of providing more accurate statistical significance for parameter estimates than any other moment condition method such as the Generalized Method of Moments (GMM), see Fuhrer, Moore and Schuh (1995). A similar approach is developed by Wooldridge (2009) for estimating 17
production function parameters, but unlike Wooldridge, ours relies on a broader systems identification. Another advantage of QMLE is that we do not need to specify the full jointlikelihood density function, which allows us to back out a consistent estimator even if our data generating distributional assumptions may be mis-specified, or in this case relaxing some of the assumptions regarding the variance covariance matrix.<sup>5</sup> Relaxing the covariance matrix allows us to introduce explicitly the correlation coefficient between error terms. Specifically, the system of equations in 32 can be estimated even when considering _E_ ( _ε_<sup>_μ_</sup> _jt_<sup>(</sup><sup>_ε_</sup> _jt_<sup>_ω_+</sup><sup>_εjt_)</sup><sup>_′_)=</sup><sup>_σ_=0.Theintroductionof</sup><sup>_σ_issuitedwhenother</sup> unobservable variables such as labor-augmenting productivity or labor adjustment costs are not accounted for. This may induce measurement error in labor, and wages, thus affecting the residual term in both equations of system 32. Labor-augmenting productivity is widely considered as a factor that increases marginal productivity and wages. The correlation _σ_ also accounts for factors such as consumer preferences and product quality that increase output and the markup without affecting regulation, productivity or production inputs. In addition, _σ_ = 0 accounts for the fact of using deflated revenues instead of output quantities. Specifically, the process of deflating nominal variables introduces an extra term that depends on price, thus markups. Therefore, the econometrician should expect the error term in both equations of system 32 to be correlated.

We define the vector _et_ , such that where _et_ = _{ε_<sup>_μ_</sup> _t_<sup>_, εω_</sup> _t_<sup>+</sup><sup>_εt}′_, while Σ =</sup><sup>_E_(</sup><sup>_ete′_</sup> _t_<sup>) and Θ is a vector of parameters that summarizes</sup> all the coefficients in system 32. The objective therefore is to estimate Θ via numerical 5For a comparison to GMM, dynamic panels in terms of consistency and efficiency see (Phillips (2018).

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optimization of the log-likelihood function given by where _lt_ = log _pe_ (0 _,_ Σ; Θ). In the special case of a multivariate normal distribution, the log-likelihood function is given by Notice that researchers may want to consider a Student’s t-distribution for short samples. Indeed, Harvey and Luati (2014) argue that the Student’s t-distribution can render the dynamics more robust to incidental influential observations and outliers. Nevertheless, it is well known that in the presence of outliers the QMLE may lead to biased outcomes, see Lange, Little and Taylor (1989) for a detailed discussion.

In addition, researchers in the productivity literature have mostly focused on estimating the production function parameters using GMM estimations.<sup>6</sup> Nevertheless, Back and Brown (1992) prove that an optimal ”well-behaved” GMM estimator is equivalent to the maximum likelihood estimator (MLE).<sup>7</sup> The following theorem summarizes this statement.

**Theorem 1.** _(from Back and Brown (1992)) Assume Gθ is first-moment continuous at_ Θ0 _. Then the maximum likelihood estimator (MLE),_ Θ _M , for the parametric problem in Equation 34 has the same asymptotic distribution as the optimal GMM estimator for the following problem_

_where P is the class of probability distributions, {x} is a sequence of random vectors dis-_

> 6A non-exhaustive list includes Ackerberg, Chen, Hahn and Liao (2014), Ackerberg et al. (2015), Biesebroeck (2007), Blundell and Bond (2000), Brynjolfsson and Hitt (2003), Doraszelski and Jaumandreu (2019), Grieco, Li and Zhang (2016), Klette (1999), Levinsohn and Petrin (2003b), among others.

> 7We define ”well-behaved” as ’consistent and uniformly asymptotically normal’ in a defined sample.

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_tributed according to P_ 0 _, with P_ 0 _a particular element of P ._ Therefore, for a given GMM estimator that satisfies Equation 36, there is an equivalent optimal ’well-behaved’ MLE estimator associated. Building on Back and Brown (1992), the equivalent MLE must satisfy the following conditions

- (i) The probability limit of Θ _M_ say _L_ ( _P_ ) exists and satisfies

where _x_ is the vector of exogenous variables in the system of equations 32, and the function _e_ () is the linear representation of _et_ as a function of the dependent and exogenous variables.

- (ii) There exits a constant _κ_ ( _P_ ) such that

where _E_ the expectation term with respect to _P_ , and _{_ Θ<sup> ̃</sup> _M }_ a sequence of MLE estimators.

- (iii) Defining the estimated covariance Σ _M_ , the limit Σ = lim _M −→∞_ Σ _M_ exists. Moreover,

Notice that for short samples (e.g. very small _N_ and _T_ ) the three conditions aforementioned are hardly simultaneously satisfied. In this case, neither the QMLE nor GMM estimators lead to consistent outcomes.

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# **5 Monte-Carlo simulation**

We construct a hypothetical data set with different number of firms, _N_ = _{_ 100 _,_ 500 _}_ . We also control for different sample sizes, _T_ = _{_ 5 _,_ 10 _}_ . In addition, we consider the cases of correlated ( _c_ = 0 _._ 64) and uncorrelated ( _n_ ) error terms. We repeat the exercise for a Cobb-Douglas case and two CES cases.

The data set assumes a labor coefficient ( _βL_ ) equal to 0.7. The first CES case assumes an elasticity of substitution equal to 0.2, and the final CES case assumes an elasticity of substitution equal to 0.5. The main parameters of interest are the estimates for the impact of regulation on productivity, _φ_<sup>_ω_</sup> , and the estimates for the impact of regulation on markups, _τ_<sup>_μ_</sup> . These are calibrated 0.1 and - 0.1, respectively.

We perform 500 repetitions and compare the proposed QMLE methodology with the standard ACF and DLW methods for estimating productivity and markups, respectively. Note that while QMLE allows us to estimate the regulatory effect on productivity and markups directly in the system, we have to run an independent regression of regulation on the backed-out productivity and markups.

It is also important to note that for a better comparison, we assume that ACF and DLW know the production function ex ante (which is impossible in reality), while our proposed QLME method does not need this information, but directly estimates the curvature of inputs.

In the CES case, the estimate of the labor coefficient in output using the QLME method becomes irrelevant. Notice that it will be equal to _βXgX_ ( _X jt_ ) where the function _gX_ ( _._ ) should be estimated to be close to one in the Cobb-Douglas case, but remains unknown otherwise.

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# **5.1 Simulation design**

Our Monte-Carlo setup is similar to that in Ackerberg et al. (2015), based on Syverson (2001) and Van Biesebroeck (2007). The labor equation is determined by the first condition of Equation 5. The model parameters, in most cases, are borrowed from Ackerberg et al. (2015). The production function is assumed to follow either a Cobb-Douglas or a CES data generating processes (DGPs). For the Cobb-Douglas case, we simulate while, for the CES case with _βL_ = 0 _._ 7 and _βK_ = 0 _._ 3. Productivity, _ωjt_ , follows an AR(1) process inclusive of regulation.

with _φ_<sup>_ω_</sup> = 0 _._ 1, and _ε_<sup>_ω_</sup> _jt_<sup>are normally distributed with mean zero and a 0.1 standard deviation.</sup> Meanwhile, _zjt_ is a dummy variable, equal to 1 or 0, and is randomly assigned to firms. The stock of capital evolves according to _Kjt_ +1 = (1 _− δ_ ) _Kjt_ + _Ijt_ with _δ_ = 0 _._ 05. Optimal firm investment is based on maximizing the expected (discounted) value of future profits, with convex capital adjustments costs. Based on Angelopoulou and Kalyvitis (2012), we simplify the investment equation for the Cobb-Douglas case as follows 22
while, for the CES case it is where _ψI_ , the adjustment cost parameter, is a vector of random numbers independently normally distributed with 0.1 mean and 0.005 standard deviation. The wage equation follows an AR(1) process described by where _ρ_<sup>_W_</sup> = 0 _._ 3, and _gjt_<sup>_W_isanindependentlynormallydistributedvariablewithrandom</sup> mean between 0 and 0.1, and 0.05 standard deviation.

The markup equation is expressed as with _τ_<sup>_o_</sup> = 0 _._ 9, and _τ_<sup>_μ_</sup> = _−_ 0 _._ 1. While _ε_<sup>_μ_</sup> _jt_<sup>isanormallydistributedvariablewithzeromean</sup> and 0.1 standard deviation. Importantly, it is also assumed to be correlated with _εjt_ .

Additionally, we consider the following condition for the material input demand, which is a key variable for the ACF method Finally, we do not want our results to depend on arbitrary initial conditions. Therefore, we start with zero initial value and simulate data 100 periods ahead, then we consider the latest _T_ observations as our final sample.

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# **5.2 Results**

The Monte-Carlo results for the Cobb-Douglas case are summarized in Table 1, while Tables 2 and 3 present results for the CES case.

For the Cobb-Douglas production function case, the QMLE method estimates _γ_ to be close to zero, capturing the curvature of the production function with precision, regardless of the sample size and the number of yearly observations. Both DLW and QLME approaches pick up the impact of regulation on markups, _τ_<sup>_μ_</sup> , with all parameters close to the true value of -0.1. However, the ACF method produces a downward bias for the estimated impact of regulation on productivity, _φ_<sup>_ω_</sup> , while the proposed QLME method gets very close to the true value of 0.1. In addition, both QLME and ACF methods yield an upward biased estimate for the labor coefficient _βL_ .

The results are robust to sample size, except in the case where we only observe a few years (the case where _T_ = 5). While the QLME picks up the mean estimate, it is unable to do so significantly. These results highlight the estimation error using ACF and DLW in the simplest case, when the econometrician knows the production function. While the error in terms of estimating the impact on markups is negligible, the error when estimating the regulatory impacts on productivity is a concern.

The Table 2 summarizes the results when we deviate from a Cobb-Douglas case. The <u>1</u> elasticity of substitution is _σ_<sup>_CES_</sup> = 1 _−ρ_<sup>= 1</sup><sup>_._25.TheACFandDLWestimationsarebased</sup> on a translog production function. Here the estimates in DLW and ACF improve over the Cobb-Douglas case in terms of measuring the regulation impact on productivity, but worsen in terms of markup estimates. This trade-off can be partly explained by the implicit relationship between the elasticity of substitution and markups.<sup>8</sup> The DLW method biases upwards the effect of regulation on markups, while ACF underestimate the regulatory impact on productivity. Likewise, in most cases, the presence of correlated errors worsens

> 8Note that the elasticity of substitution is a function depending on production inputs, thus, whenever the markups affect input allocations, both variables will be correlated.

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the estimates. In terms of the proposed QLME method, both the markup and productivity effects are estimated unbiasedly. Note that the estimator is asymptotically consistent in the numerical estimation. For small _T_ and small _N_ the variance in the estimator increases. In addition, the estimator is able to pick up the production function specification - _γ_ is much larger than zero.

<u>1</u> Table 3 summarizes the results when the elasticity of substitution is _σ_<sup>_CES_</sup> = 1 _−ρ_<sup>=2.</sup> In this case, the ACF and DLW methods are unable to estimate the true coefficient of regulation on markups and productivity, respectively. In addition, the presence of correlation between error terms worsens the DLW significantly for markups. In contrast, the QLME method is able to pick up the true coefficient in the majority of cases. The small sample problem in _T_ makes inference hard in all cases. _T_ = 5 is deliberately chosen to be small to test the strength and weaknesses of the results. The small _T_ makes inference hard and thus researchers should be cautious when interpreting results on small firm-level panels.

In addition to estimating the impact of regulation on productivity and markups, researchers are typically interested to compare market power between industries, either by using the markup in levels or its dispersion within firms in a given sector. Because of the absence of price information and other problems related to endogeneity as discussed in section 2.2, we expect that any estimate on the markup level would be biased (see Bond et al. (2020)). Figure 1 presents the simulated markup distribution derived using a Cobb-Douglas production function. We compare it with the distribution of the estimated markups using the DLW and the QMLE methods. The results suggest that none of the methods can recover the precise level of markups. The DLW method is biased upwards relative to the QMLE method. Research that attempts to explain how the markup level changes over time using sales revenue data will be inaccurate. However, the distribution is similar when centered. With enough variation in the data, one would be able to estimate the impact of regulation on the change in markups, implying that policy inference is still possible and hence represents a work-around to the problem raised in Bond et al. (2020).

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We also compare the distribution of the demeaned markups for the CES production function. Notice that our proposed QMLE method is unable to recover the labor coefficient parameter _βL_ when _ρ_ differs significantly from zero. Thus, we can only compare the demeaned sequence of markups instead of the level. Figure 2 shows the results when considering _ρ_ = 0 _._ 5. Results suggest that the centered distribution changes markedly for the DLW case. The variance is much larger and the mass of the distribution is not equal to zero. The estimated impact of regulation on markups will thus be biased. In this case, one cannot uncover either the level or the impact of other variables on the changes in markups. However, the QLME distributional result is similar to the simulated data - hence policy inference is possible using QLME, but not DLW.

# **6 Conclusions**

Uncovering the effects of regulation policies on productivity and markups using the methods of Ackerberg et al. (2015) and De Loecker and Warzynski (2012) leads to several endogeneity issues and produces biased estimates which are sensitive to: (1) the functional form of the production function; (2) the omission of demand shifters; (3) the absence of price information; (4) the violation of the Markov process for productivity; and (5) misspecification when marginal costs are excluded in the estimation process. Mindful of these constraints, we use regulation as a pure exogenous process to identify variation in markups. Given the hard constraint on data availability (i.e. unobservable prices and unknowable production functions), we establish a methodology that allows for a consistent estimate of the impact of change in regulation on change in markup.

Building on the work of Doraszelski and Jaumandreu (2019) and Ackerberg et al. (2015), we identify unobservable productivity and markups in a joint framework by exploring the connection to regulations. Quasi-Maximum Likelihood (QMLE) is used for backing out the structural parameters of the system. A Monte Carlo exercise shows improvements in 26
obtaining the estimates of key policy variables on productivity and markups. The QLME framework can approximate several functional forms for the production function, and is thus more general than what is used in the literature. The results show that the estimate of the impact of policy parameters on productivity is biased when using Ackerberg et al. (2015), but consistent when using the QLME approach. When data exhibit a production function other than Cobb-Douglas, the estimates of De Loecker and Warzynski (2012) are biased, while the QLME is still able to back out the regulatory impact on markups. It is important to note that none of the methods is able to obtain accurate estimates of the level of markups when the data do not contain prices. However, inference is still possible on the policy impact of regulation using the QLME approach.

Our main contribution is therefore a methodology that offers a work-around on the identification problems raised in Bond et al. (2020) and an improved estimator of productivity. Another important contribution is the direct estimation of the impact of regulations on markups and productivity. This method is therefore more general, and has potentially a wider set of applications, compared to the existing methodologies.

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