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Production function estimation using subjective expectations data
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The `production function' -- a representation of the process by which inputs are turned into outputs -- has long been an object of great economic interest (cobb_theory_1928,strom_production_1999). Production functions are critical to examining a wide range of topics including technological change, productivity dispersion, firm markups and the impact of policy. Research on these topics has gained greater salience in recent years, in part due to the productivity growth slowdown, particularly since the 2008-9 Global Financial Crisis. Before one can analyze such topics, however, it is necessary to consistently estimate a production function, which has proven no easy task.\footnote{Production functions have also found insightful applications beyond firm-level analysis to understand, for example, the impact of various types of inputs on child development (todd_specification_2003,cunha_formulating_2008). The measurement challenges and context there are somewhat distinct and the techniques used in estimation differ from those used in the analysis of firm production. In this context, attanasioetal2024 recently use subjective beliefs to examine parents' perception of the skills formation production function.}
Econometric research on production function estimation has had a renaissance in recent years.\footnote{For surveys see de_loecker_industrial_2021 or ackerberg_chapter_2007. Recent contributions include ackerberg_identification_2015, collard-wexler_productivity_2021, de_loecker_product_2011, de_loecker_prices_2016, doraszelski_rd_2013, doraszelski_measuring_2018, gandhi_identification_2020, orr_within-firm_2022, de_roux_estimating_2021 and valmari2023.} Estimation is complicated by a number of long-known issues, most notably the endogeneity of inputs: because a firm's productivity is unobservable and likely correlated with input choices, straightforward estimation methods such as OLS regression will be biased (marschak_random_1944,zellner_specification_1966).
Standard methods to deal with these problems included controlling for firm fixed effects mundlak_empirical_1961 by differencing and instrumenting with lagged input values anderson_estimation_1981, for example. Such approaches generally find implausibly low estimates for relevant parameters, especially on the output elasticity with respect to capital.\footnote{This has generally been thought to be because of the high persistence of the capital stock. Differencing removes all cross sectional information on capital, and much of the remaining time series variation may be measurement error. Moreover, lags will be poor predictors of the change in the capital stock, if the level of capital is close to a random walk.} blundell_gmm_2000 consider an alternative approach in this dynamic panel data literature by including lagged differences as instruments for the levels of factor inputs. This nonetheless requires conditioning on at least three consecutive time series observations on a firm, which in many empirical settings loses a considerable subset of data. Moreover, it relies on exact parametric specification of the productivity process and requires a strong stationarity assumption making the method potentially inappropriate for younger and fast-growing firms.
The drawbacks of these dynamic panel data methods have contributed to the popularity of an alternative suite of ‘proxy variable’ production function estimators that use a non-parametric function of various observables to control for unobserved productivity. The pioneers of this method were olley_dynamics_1996 (OP), who control for productivity using a flexible function of investment and capital that represents the inverse of firms’ optimal investment policy. The reasoning behind their approach is that if firms' investment policy function can be written as an invertible function of pre-determined capital and the persistent component of unobserved productivity, then the latter can be proxied with a flexible function of capital and investment. levinsohn_estimating_2003 (LP) instead propose using the inverse of firms’ material input demand as a proxy for productivity to address selection concerns implicit in the OP approach due to potentially high prevalence of zero investment among firms.
Noting that a production function’s labor input parameter is unidentified using the LP methodology under plausible assumptions, ackerberg_identification_2015 (ACF) outline a refinement on timing assumptions and proxy variable arguments to address this. Unlike LP, who rely on a material input demand function conditioned solely on capital, ACF propose controlling for unobserved productivity by inverting a material input demand function conditioned on labor as well as capital and then recovering both input elasticities in a second estimation stage. gandhi_identification_2020 (GNR) show that this suite of estimators may fail to identify the parameters of a ‘gross output’ production functions (i.e. one that includes materials as an input), and propose an alternative estimation strategy based on the implications of price-taking firms’ optimality conditions. bond_adjustment_2005 provide yet another alternative estimator, showing that in the presence of adjustment costs on all inputs the parameters of a Cobb-Douglas production function can be recovered by using lagged levels of inputs as instruments for current levels. This approach incorporates aspects of both the dynamic panel literature -- in using lags as instruments and specifying the productivity process -- and the proxy variable approach -- by relying on the implications of optimal firm input decisions to yield identification. However, simulation results show that their proposed method is sensitive to the form and magnitude of adjustment costs which, combined with the relatively numerous assumptions they require, may explain why it has not been widely deployed. More in-depth reviews of alternative production function estimation strategies are provided in ackerberg_chapter_2007 and de_loecker_industrial_2021.
Despite their differences, proxy methods such as OP, LP and ACF all rely on the existence of a strictly monotonic relationship between a firm’s (conditional) input demand and productivity -- an assumption justified with recourse to models of firms’ decisions that yield optimal policies satisfying monotonicity. The performance of these estimation methods is therefore threatened by any unobserved factor that violates the required relationship between productivity and the input used to generate its proxy, such as input adjustment costs or prices that vary across firms and optimization errors, and may suffer if relevant variables (e.g. factor prices) are omitted or unavailable.\footnote{gandhi_identification_2020 by contrast, explicitly require firms’ flexible input demands to be optimal and is therefore compromised by any factor causing deviations from optimality.}
This paper contributes to the literature on production function estimation by showing how data on firms’ perceptions of its future output and inputs can be used to recover consistent production function parameter estimators while relaxing assumptions on firms’ input demand policies. We leverage information in recent surveys that collect detailed information on firms' perceived probabilistic distribution of output (e.g. revenues) and inputs (e.g., employment, intermediates and capital expenditure) in the future. The intuition underpinning our approach is that firms’ expectations regarding future inputs and output contain information about their expected future productivity which, in turn, contains information about their current productivity. Unlike dynamic panel estimators, which require parametric specification of the productivity process, we require the relatively common assumption that persistent productivity follows a first-order Markov process -- an assumption imposed as well in OP, LP, ACF and GNR. Combined with assumptions that persistent productivity is unidimensional and that there is a monotonic relationship between current and expected future productivity, which are also imposed by standard proxy variable methods, firms’ expectations can be used to control for unobserved productivity and thereby recover consistent parameter estimates.\footnote{The requirement that persistent productivity be unidimensional confines us to a setting where productivity shocks are Hicks-neutral. While this is conventional in the literature, two notable exceptions that accommodate factor-augmenting technology shocks are doraszelski_measuring_2018 and demirer_production_2022. doraszelski_measuring_2018 relaxes the assumption of unidimensional productivity by trying to leverage data on firm-level input prices while demirer_production_2022 does so by imposing assumptions on firms’ input demands. While both these papers are valuable contributions and argue the importance of factor-augmenting productivity shocks, we believe our approach retains relevance given the dominance of the Hicks-neutral context among existing theoretical and empirical work.} Our proposed method is therefore similar to OP/LP/ACF as it requires a monotonic relationship between productivity and observables, but different as it leverages data on firms’ expectations rather than optimal input quantities. It is therefore robust to a range of factors -- such as unobserved firm-specific input prices and optimization error -- that would undermine alternative estimators by breaking the one-to-one link between input demands and productivity.
Monte Carlo simulations show that our proposed estimation method recovers precise estimates of production function parameters under a range of data generating processes. Notably, it retains consistency when firms' input decisions are subject to optimization error whereas other approaches generally do not. While our proposed estimator is undermined if firms' expectations exhibit certain (although not all) types of bias, we show our basic estimation algorithm can be extended to retain consistency in certain cases.
To test the empirical performance of our method, we leverage the UK's Management and Expectations Survey office_for_national_statistics_mesmanagement_2022. The MES records information on firms' inputs, output and their one-year-ahead expectations of these quantities between 2016 and 2020. We focus on three industries -- electronics manufacturing, wholesale and retail and restaurants -- and estimate industry-specific production functions using a range of methods. The estimates recovered using our proposed method are broadly similar for the electronics and retail production functions but differ in non-negligible yet plausible ways for the restaurant sector. To rationalize these results, we show material inputs are subject to particularly large within-year revisions among the restaurant sector, which suggests optimization is particularly hard for these firms and hence the LP and ACF monotonicity assumption less likely to hold. We use the alternative production function estimates to recover estimates of total factor productivity (TFP), and compare static and dynamic moments of its distribution across methods. We relate TFP estimates to firm performance and find estimates obtained using our proposed estimator are more positively associated with future employment growth than alternative estimators, particularly over a four-year horizon.
Combined with the Monte Carlo evidence, our empirical application demonstrates the utility of expectations data in the context of production function estimation and thereby contributes to a more general literature documenting the value of expectations data. Starting in the 1990s much of this literature's initial focus was on income dynamics with, for example, the seminal work by dominitz_using_1997, who demonstrate how surveys can be used to elicit subjective income expectations, and pistaferri_superior_2001, who shows the econometric benefits of such additional information as a means to separately identify permanent and transitory shocks to income. manski_measuring_2004 summarised these early advances and argued that data on expectations could be useful both as a means to relax and validate assumptions within various economics models. Of the subsequent work that has examined the value of subjective expectations data in a wide range of contexts, our work is related to gennaioli_expectations_2016, in that it demonstrates insights that can be gained from firms' expectations rather than those of individuals'. Our work is also related to recent and ongoing work by attanasio_modelling_2022, who return to the literature's early focus on subjective income expectations and show how such data can be used to estimate income processes in a flexible manner that relaxes commonly-imposed parametric assumptions. Similar to their work, we document that the additional information contained within data on subjective expectations allows one to relax particular assumptions that underpin conventional production function estimators and thus implicitly allows for more flexible models of firm behavior.
The remainder of the paper is as follows. In section (ref) we show how expectations data identify production function parameters, describe our proposed estimation methodology and compare it to other standard methods. Section (ref) outlines the Monte Carlo setup we use to compare alternative estimators and discusses the results across various data generating processes. Section (ref) describes the data we use in our empirical application, the results of which are described in section (ref). Section (ref) concludes.
Consider a general production function of the following form
where subscript $i$ denotes firm and subscript $t$ denotes time. Lower case letters denote logs, so $y$ is the log of output, $k$ is the log of capital, $l$ is the log of labor and $f(\cdot;\beta)$ is some general function of the two with parameters $\beta$, which captures the process by which they are combined during production.\footnote{For the general exposition of this subsection, output may either be value added (i.e. net output) or turnover (i.e. gross output). In the latter case, the omission of materials as an input can be justified under a Leontief model of production in which labor and capital are combined in a fixed proportion with materials ackerberg_identification_2015. In practice, the distinction will influence how data on firm expectations' are treated, which we return to in section (ref).} The variables $\omega$ and $\epsilon$ are unobserved by the econometrician. The variable $\omega$ represents idiosyncratic productivity that is known by the firm at the time period-$t$ input and investment decisions are made. In contrast, $\epsilon$ are unanticipated mean-zero disturbances representing productivity shocks, such as extreme weather events or machine failures, which only become observable to the firm after its period-$t$ decisions have been made. Alternatively, $\epsilon$ can represent mean-zero measurement error, which does not affect the firm but poses problems to the econometrician strom_production_1999.
Capital evolves according to
where $\delta$ is the depreciation rate and $I_{it-1}$ is investment.\footnote{Whereas we follow the literature in assuming “time-to-build”, since the MES also collects information on capital expenditures, it is conceivable that this assumption may also be relaxed along the lines of our derivations below.} Unobserved productivity $\omega$ follows a Markov process
where $\mathbb{E}[\xi_{it}|\Omega_{it-1}]=0$ and $\Omega_{it-1}$ represents the firm's information set at $t-1$. The information includes $k_{it-1},l_{it-1},\omega_{it-1},i_{it-1}$ (and thus $k_{it}$) but also additional variables such as input and output prices and demand factors.
Suppose firms form expectations about their period-$t+1$ production and inputs at the end of period $t$ conditional on their information set $\Omega_{it}\supset\{k_{it},l_{it},I_{it},\omega_{it},k_{it+1}\}$. It is reasonable to assume firms' expectations align with the actual production technology of equation ((ref)), which implies
where $F_{it}(l_{it+1})$ represents firm $i$'s subjective probability distribution over their next-period labor input given its information $\Omega_{it}$ and the second equality follows from the assumptions that $\mathbb{E}_{it}[\epsilon_{it}|\Omega_{it-1}]$ and $\mathbb{E}_{it}[\xi_{it}|\Omega_{it-1}]$ are equal to zero. We also append the subscripts $i$ and $t$ to highlight that the relevant variables are obtained with respect to the subjective probabilities reported by decision makers in firm $i$ at period $t$. Rearranging equation ((ref)) for $g(\omega_{it})$ obtains
Like other proxy variable approaches, we now require a monotonicity assumption. In our case this assumption is that the right hand side of equation ((ref)) is strictly increasing in $\omega_{it}$. Or, in words: given a firm's current (persistent) productivity there is a single level of productivity they expect next period and this single level can be uniquely inferred from their expectations about next-period output, labor and the deterministic level of next-period capital. This assumption is also required by ACF, who impose it in their assumption 2, and highlight it is also required by OP. Under the strict monotonicity assumption, $\omega_{it}$ can be recovered as
If output ($y_{it}$), inputs ($k_{it}$, $k_{it+1}$ and $l_{it}$), and beliefs ($\mathbb{E}_{it}[y_{it+1}|\Omega_{it}]$ and $F_{it}(\cdot)$) are observable, we can combine equations ((ref)) and ((ref)) to obtain a moment condition that can be used to recover the parameters of interest:
where $\Psi$ is some non-parametric function representing $g^{-1}(\cdot)$.
Consider for example the case of a Cobb-Douglas production function where $\beta = (\beta_0,\beta_k,\beta_l)$ and an AR(1) process for $\omega$ with auto-regressive parameter $\rho$ (i.e., $g(\omega)=\rho \omega$), then equation ((ref)) becomes
as long as $\rho \ne 0$. The model would then identify $\theta=(\beta,\rho)$ if, for example, $\mathbb{E}[x_{it}x_{it}^\top]$, where $x_{it}=(1,k_{it},l_{it},$ $\mathbb{E}_{it}[y_{it+1}|\Omega_{it}],k_{it+1},\mathbb{E}_{it}[l_{it+1}|\Omega_{it}])$, has full rank.\footnote{This is sufficient, but not necessary since there are four parameters in this specification.}
This example highlights that identification of $\theta$ by equation ((ref)) will depend on the specifications of the production function $f(\cdot; \theta)$, the Markov process encoded in $g(\cdot)$ and on the degree of variation observed in the data.
For more general specifications, one can establish that:
Proof. Since $y_{it} = f(x_{it}; \beta_0) + h_0(z_{it}) + \epsilon_{it}$, where $$h_0(z_{it}) = \Psi_0\left(\mathbb{E}_{it}[y_{it+1}|\Omega_{it}] - \int f(k_{it+1},l_{it+1}; \beta_0)dF_{it}(l_{it+1})\right).$$ Taking expectations conditional on $z_{it}$ on both sides and subtracting, one obtains that $$\underbrace{y_{it} - \mathbb{E}(y_{it}|z_{it}))}_{\equiv w_{it}} = \underbrace{f(x_{it}; \beta_0) - \mathbb{E}(f(x_{it}; \beta_0)|z_{it})}_{\equiv m(x_{it},z_{it};\beta_0)} + \epsilon_{it}.$$ Since $\mathbb{E}[\epsilon_{it}|\Omega_{it}]=0$ and $\{ x_{it},z_{it} \} \subset \Omega_{it}$, we have that $\mathbb{E}[\epsilon_{it}|x_{it},z_{it}]=0$ and $m(x_{it},z_{it};\beta_0)=\mathbb{E}(w_{it}|x_{it},z_{it})$. It thus uniquely solves $\min_{\tilde m(\cdot)} \mathbb{E}[(w_{it}-\tilde m(x_{it},z_{it}))^2]$ as long as condition ((ref)) is satisfied with positive probability, which implies that $\beta_0$ is identified. \newline \newline The function $\Psi_0(\cdot) \equiv g_0^{-1}(\cdot)$ is then identified since $$\underbrace{y_{it}-f(x_{it}; \beta_0)}_{\equiv \tilde y_{it}}=\Psi_0\left(\underbrace{\mathbb{E}_{it}[y_{it+1}|\Omega_{it}] - \int f(k_{it+1},l_{it+1}; \beta_0)dF_{it}(l_{it+1})}_{\equiv \tilde z_{it}}\right)+\epsilon_{it}.$$ Since $\tilde z_{it} \subset \Omega_{it}$, thus have that $\mathbb{E}[\epsilon_{it}|\tilde z_{it}]=0$ and $\Psi_0(\tilde z_{it})=\mathbb{E}(\tilde y_{it}|\tilde z_{it})$ and $g_0(\cdot)=\Psi_0^{-1}(\cdot)$. \qquad $\blacksquare$ \newline \newline The identification result generalizes ideas in robinson_root-n-consistent_1988, who deals with partially linear models where $f(\cdot; \beta)$ is linear. Condition ((ref)) is a conventional identification assumption used in the context of (nonlinear) least squares applied to the parametric function $m(\cdot;\beta)$, which can be obtained from $f(\cdot; \beta)$ and the observable distribution of $x_{it}$ given $z_{it}$. In fact, if $f(\cdot; \beta)$ is linear in parameters (e.g., Cobb-Douglas and translog), the result boils down to that in robinson_root-n-consistent_1988:
Proof. Since $f(\cdot; \beta)$ is linear in parameters we can represent it as $f(x_{it};\beta)=x_{it}^\top \beta$. The result obtains as Condition ((ref)) implies Condition ((ref)). Suppose that there exists $\beta \neq \beta_0$ such that
This then implies that $$\mathbb{E}\left\{ [ x_{it}-\mathbb{E}(x_{it}|z_{it})][ x_{it}-\mathbb{E}(x_{it}|z_{it})]^\top\right\}(\beta_0 - \beta)=0.$$ This means that $\beta_0 - \beta \neq 0$ is in the nullspace of $\mathbb{E}\left\{ [ x_{it}-\mathbb{E}(x_{it}|z_{it})][ x_{it}-\mathbb{E}(x_{it}|z_{it})]^\top \right\}$ thus implying that this matrix is singular. Hence, Condition ((ref)) implies Condition ((ref)) and the result follows from Theorem (ref). \qquad $\blacksquare$ \newline
This can also be obtained by directly applying the results in robinson_root-n-consistent_1988. As discussed there (see p.940), Condition ((ref)) prevents any element of $x_{it}$ from being almost surely perfectly predictable by $z_{it}$ in the least squares sense, although it does not preclude (nonlinear) functional relations among $x_{it}$ elements and identification is possible even if $x_{it}$ uniquely defines $z_{it}$, when the converse is not true.
Equation ((ref)) can be used to recover estimates of $(\theta,g)$ that are either fully or semi-parametric depending on whether one specifies the functional form of $g(\cdot)$. In the remainder of this paper, we follow a semi-parametric approach, which allows us to avoid imposing structure on $g(\cdot)$ and yields a novel estimation methodology. This section focuses on the case of Cobb-Douglas production technology to outline our proposed methodology, although it generalizes to other specifications (e.g. translog), which we examine in our empirical application.
Cobb-Douglas production implies:
Assuming $\Psi$ is a smooth function, equation ((ref)) is an example of a generalized additive model, early explorations of which were provided for instance by hastie_generalized_1986, and the partially linear model studied by robinson_root-n-consistent_1988 among others. Hastie and Tibshirani characterize the non-linear part of the model -- in our case, the $\Psi$ function -- as a weighted sum of unspecified smooth functions, the parameters and weighting of which can be recovered using (quasi-)maximum-likelihood-based estimation.\footnote{As in a linear regression, maximum likelihood based on normal errors amounts to least squares minimization here.}
There are two specific features of the model of equation ((ref)) that depart from the standard generalized additive model. First, we know that $\Psi$ is monotonic, which amounts to imposing constraints on derivatives of the smooth functions that comprise $\Psi$. The exact form of these constraints and the consequent optimization problem are derived by pya_shape_2015, who also present an algorithm to estimate such `shape constrained' general additive models that we deploy in our empirical application.
Second, the argument of the smooth $\Psi$ function is itself a function of the model's parameters. To address this issue, we take inspiration from friedman_projection_1981, who develop an iterative `backfitting' algorithm that recovers parameter estimates in additive models where the arguments of the smooth functions are linear functions of parameters (see also ichimura_chapter_2007). Adapting the algorithm to our setting yields the following iterative estimation procedure:
For more general production functions, one instead should use $Z_{ij} = \mathbb{E}_{it}[y_{it+1}|\Omega_{it}] - \int f(k_{it+1},l_{it+1}; \hat \beta)dF_{it}(l_{it+1}))$ in step 2 and $y_{it}=f(k_{it},l_{it}; \beta)+\Psi\left(Z_{ij}\right)+\epsilon_{it}$ in step 3. General treatments for the convergence of related procedures is examined, for example, in pastorelloetal2003 and dominitzsherman2005. We henceforth refer to this iterative algorithm as `NPR' and provide further implementation details in Appendix (ref).
Given the range of existing production function estimation methods, we emphasize three aspects that distinguish our NPR estimation algorithm.
First, unlike the widely-used proxy variable approaches of olley_dynamics_1996, levinsohn_estimating_2003 and ackerberg_identification_2015 (henceforth OP, LP and ACF respectively), it does not require that firm decisions be optimal.\footnote{In the standard models, assumptions over the information set and optimality of input choices generate the key econometric assumptions that the input demand equation is strictly monotonic in productivity and invertible (so there is only one scalar persistent unobservable). Whereas we do not impose those, it is nonetheless possible to include optimality conditions among the moments used in estimation if one so desires.} To see this note that the $\Psi$ function in equation ((ref)) plays a role analogous to the function $\Phi$ used by OP to represent firms' investment policy, or LP and ACF to represent firms' material input choice. In OP, LP and ACF, $\Phi$ is a function of current inputs used to control for current $\omega$. The success of this approach therefore hinges on the existence of a monotonic relationship between contemporaneous productivity and inputs, which is typically assumed with recourse to models of optimal firm behaviour that imply such relationships are monotonic and increasing. NPR, by contrast, requires the assumption that firms' expectations align with the true production technology but allows one to remain agnostic about how firms make their input decisions.\footnote{While the NPR algorithm requires firms' expectations align with the true production technology, the moment condition of equation ((ref)) may still yield a consistent estimator for $\beta$ in contexts where this does not hold owing to bias in firms' expectations. This is discussed in section (ref).} As confirmed by the Monte Carlo simulations discussed in section (ref), this distinction means NPR remains consistent when firms' decisions are subject to optimization error or when there are additional unobservable variables influencing input decision, whereas other proxy variable methods do not.\footnote{This feature also favours NPR over `index number' methods discussed by van_biesebroeck_robustness_2007, such as those proposed by solow_technical_1957 and hall_relation_1988, which derive equations expressing production function parameters as functions of observables under the assumption of optimal firm behaviour.}
A second point of distinction is that NPR can accommodate non-linear productivity dynamics, whereas the `dynamic panel' methods of blundell_initial_1998 and blundell_gmm_2000 typically require linearity. Such non-linearity is enabled by the flexible form of the $\Psi$ function at the core of the NPR method, although it is worth noting the monotonicity constraint required by NPR demands that $\omega$ follow a first order Markov process. In theory, a relative strength of dynamic panel methods is that they can be used in situations where $\omega$ follows a Markov process of higher order, but in practice this requires the researcher to correctly specify both the AR and MA components of the linear productivity process and requires a longer, and hence more selected, data panel. In practice, the use of longer lags as instruments typically generates estimation problems, especially on the capital coefficient as assets are highly persistent.
The final point is that NPR can identify the production function parameters $\beta$ from a single cross section of data. In principle, this also removes for example the requirement that the transition law ($g$) be homogeneous in time. While repeated observations of current-period and next-period inputs and outputs would accommodate more general models than that presented in section (ref), such as a production function with firm fixed effects,\footnote{See, for instance, attanasio_modelling_2022 for an elaboration on this point in the context of earnings dynamics where expectational data helps resolve important issues in dynamic panel data models, such as “Nickell bias”.} for this baseline -- which is standard in the literature -- a single observation per firm is adequate. Both the proxy variable and dynamic panel approaches, by contrast, require multiple observations per firm. Methods do exist for correcting for the selection that such sample restrictions introduce, but the absence of any such requirement for NPR is attractive.\footnote {blundell_initial_1998 discuss how selection may be controlled for by a firm fixed effects and olley_dynamics_1996 focus on a proxy variable approach. But the absence of an external instrument in the selection equation may pose identification issues for these approaches.}
A disadvantage of the NPR approach is that it requires data on firms' subjective expectations. Since the vast majority of production functions are estimated in logs, we require information of firms' subjective expectation distributions because a single value of firms' expected output, for example, would be insufficient to recover firms' expected log output. However, questions that provide such information are increasingly being included in firm surveys such as the Management and Organizational Practices Surveys (MOPS) buffington_management_2016, the Decision Maker Panel (DMP) by the Bank of England bloom_tracking_2017, the Survey of Business Uncertainty (SBU) by the Atlanta Federal Reserve Bank, the China Employer-Employee Survey (CEES) altigetal2022 and the UK Management and Expectations Survey (MES) office_for_national_statistics_mesmanagement_2022. As such data become increasingly available, we believe the three features of NPR discussed above bring notable advantages that warrant its addition to the established suite of production function estimators.\footnote{Several other surveys also collect firms' expectations about aggregate, macroeconomic variables (e.g. Survey on Inflation and Growth Expectations by the Bank of Italy or the Business Tendency Survey dovernetal2023). It is possible that those data offer additional, complementary information that can possibly be used as well for the estimation of production functions using moments that aggregate across firms. }
In our baseline case, when firms know the true production technology and their beliefs align with this, the law of motion for $\omega_{it}$ can be inverted to obtain equation ((ref)). This inversion is crucial to the NPR estimation algorithm and is analogous to the monotonicity condition that OP impose on the investment policy function and that LP and ACF impose on the material input policy function. Also known as the `scalar unobservable' assumption, it imposes a one-to-one mapping between firms' expectations and their current productivity.
Given the centrality of expectations to the NPR algorithm, it is important to consider whether and how expectational biases undermine the proposed approach. The first thing to note is that our suggested method can accommodate biased input expectations as long as such bias is also reflected in firms' expected output and vice-versa. If, for example, a firm is systematically optimistic in its sales forecasts we would require it to be similarly optimistic in its employment forecasts. The precise meaning of `similarity' in this context is governed by the production function. Specifically, in the case of a firm with over-optimistic output expectations, we require bias in the firm's employment expectations such that the integral of the production function with respect to expected labor equals the biased output expectation. For example, when the production function is Cobb-Douglas, equation ((ref)) gives: $$y_{it}=\beta_0-\beta_kk_{it}-\beta_ll_{it}+\Psi\left(\mathbb{E}_{it}[y_{it+1}|\Omega_{it}] - \beta_0-\beta_kk_{it+1}-\beta_l\mathbb{E}[l_{it+1}|\Omega_{it}]\right)+\epsilon_{it}.$$ When the bias in sales expectations, say $\texttt{bias}_{y,it}$, balances the bias in the employment expectations, say $\texttt{bias}_{l,it}$, so that $\texttt{bias}_{y,it}=\beta_l \texttt{bias}_{l,it}$, then one is still able to recover an unbiased expectation of firms' next-period productivity as the residual on the RHS of equation ((ref)). Furthermore, in this particular case, when production is Cobb-Douglas, it is possible to construct a `Wald'-type estimator of $\beta_l$ if one has the data to compare expectations of output and labor to their realised values. We do not pursue this in our main analysis because we present evidence that such bias is very minimal in our empirical context (see also bloom_well_2021). Nevertheless, we provide further exposition in Appendix (ref).
Since the vast majority of literature on managerial bias has focused on biases in output expectations, the ability of our method to accommodate this is encouraging. Biases in firms' productivity expectations, however, are more problematic. To see why, suppose firms' bias about their next-period productivity is captured by $\iota_{it}$ (positive values reflect optimism, negative values reflect pessimism), such that
Even if firms' expectations about output and inputs align with the true production technology, the presence of bias means
The bias term $\iota_{it}$ therefore violates the strict monotonicity assumption we require to recover $\omega$ since
Whether such bias is surmountable depends on its form. If bias is either time-invariant or a function of observables, we show in Appendix (ref) that one can embed the NPR estimation algorithm in an outer iterative estimation loop to recover estimates of firms' expected productivity bias. If the estimates of such bias are consistent, this in turn achieves consistency of the NPR production function parameters and we indeed show in the Monte Carlo simulations of Section (ref) that these extensions recover precise estimates. We are, however, unable to deploy the extended algorithm in our empirical setting since estimation of firms' bias requires a long enough panel of firms' forecast errors, which are unavailable in the data we use.\footnote{Although we have a reasonable panel for outputs and inputs, the MES subjective expectations data are two cross sections with limited longitudinal overlap}.
Another factor that undermines consistency of the NPR estimator is if the firm has imperfect knowledge of the production technology. In this case, deviations between the production function parameters perceived by firms and the true values create a `wedge' between expected output, the true production function evaluated at $\left(k_{it+1},\mathbb{E}_{it}[l_{it+1}]\right)$ and $g(\omega_{it})$, similar to the bias term $\iota_{it}$ of equation ((ref)). This is most clear in the context of Cobb-Douglas production. Suppose production is Cobb-Douglas but that firms' form expectations of next-period output based on incorrect knowledge of the production technology, such that
with
for $x \in \left(0,k,l\right)$. Here the $\upsilon$ terms capture firms' imperfect knowledge of the production technology. In this context, rearranging (ref) to isolate $\mathbb{E}_{it}[\omega_{it+1}|\Omega_{it}]$ gives
Similarly to when firms' biased expectations of their next-period productivity, imperfect knowledge of the production function creates a `wedge', $\Lambda\left(k_{it+1},\mathbb{E}_{it}[l_{it+1}|\Omega_{it}];\Upsilon_{i}\right)$, which is a function of next-period capital, expected next-period labor and a set of firm-specific parameters $\Upsilon_{i}=\left(\upsilon_{i0},\upsilon_{ik},\upsilon_{il}\right)$. This wedge breaks the one-to-one link between firms' expectations and their current productivity hence violates the scalar unobservable assumption required for consistency of the NPR estimator. In Appendix (ref) we demonstrate an extension to the baseline NPR estimator that recovers consistency in the presence of such imperfect knowledge. However, similarly to the bias-robust extensions this requires a panel of firms' expectations and therefore, given the data constraints of our empirical application, we do not pursue it further here.
Our baseline Monte Carlo setup follows that of ACF. The production function specification is Leontief in the material input:
where $\beta_0 = 1, \beta_K = 0.4, \beta_l = 0.6$ and $\beta_m = 1$. In our baseline analysis the productivity shock is assumed to follow an AR(1) process:
with $\rho = 0.7$. As pointed out by ACF, the LP estimator does not identify the production function parameters unless there is stochastic variation in firms' labor inputs, for example due to optimization error. We therefore focus on data generating processes featuring such variation, which we introduce in the same way as ACF by adding a mean-zero normally-distributed random variable to firms' optimal level of labor. In addition, we also consider the impact of optimization error in investment and materials (which ACF do not consider)\footnote{ACF's analysis considers the impact of measurement error in materials but this is distinct from optimization error as it does not affect output. Optimization error, by contrast, will affect output via the assumption of Leontief technology.}, which we incorporate in the same manner as labor.\footnote{Optimization errors in labor and investment are simulated from a mean-zero normal distribution with standard deviation 0.37, which matches the distributional assumption ACF make regarding the labor optimization error in their DGPs. By contrast, we simulate the optimization error in materials from a mean-zero normal distribution with a standard deviation of 0.185 (i.e. half of 0.37). We do this because the ACF estimator is vulnerable to this type of optimization error and failed to return any plausible estimates when simulations used the higher standard deviation.}
For each DGP, we use the closed-form solutions of the model to simulate data for 1,000 firms over 100 periods. Capital is initialised at zero and we only use data from the last 10 periods for estimation purposes, as by this time the capital stock appeared to have reached steady state. Further details of the environment and the data generating process (DGP) are given in Appendix (ref) (and the Appendix in ackerberg_identification_2015).
Table (ref) examines the performance of the estimators as various firm choices are subject to optimization error. We highlight three salient points. First, as expected, the OLS and OP parameter estimates are heavily biased across all DGPs.\footnote{Bias in the OP estimates is due to the presence of firm-specific capital adjustment costs, added by ACF to obtain across-firm variation in capital similar to that observed in their data.} Second, when optimization error affects labor only (first panel), LP and NPR all perform well, as does ACF when disregarding implausible estimates (i.e. when the values of the output elasticities are below zero or greater than one). As anticipated by ACF, LP (slightly) outperforms their proposed estimator in this environment in terms of precision. NPR improves on LP even more, achieving much greater precision on the capital coefficient.
Third, when optimization error affects material inputs, all estimators except NPR deteriorate. The LP estimator is vulnerable both to errors in labor and investment simultaneously (second panel), and labor and materials simultaneously (third panel). While ACF appears robust to errors in investment (if one ignores implausible estimates), it is compromised by errors in materials which violate the monotonicity condition ACF require between material input choices and productivity. This is compounded further when optimization errors to labor, investment and materials occur simultaneously, as shown in the final panel. In contrast to the other estimators, the NPR estimates remain both consistent and accurate throughout, although optimization error in materials reduces the precision of the NPR estimates, particularly for the capital coefficient.\footnote{ Despite not relying on material input data, NPR is affected by materials optimization error because of the assumption of Leontief production. When material optimization error is negative, the Leontief assumption means output will be determined by sub-optimally low materials and hence the specification of output as a function of labor and capital will be incorrect.} This is a clear demonstration of the observation made in subsection (ref) that NPR is robust to optimization error in inputs, whereas the other proxy variable estimators are not.
Table (ref) shows moments of parameter estimates obtained by applying the NPR estimator to data simulated under the `optimization error in $l$' scenario but with the addition of idiosyncratic shocks to firms' expectations. The parameters obtained by all estimators other than NPR are omitted from the table because they do not use the information contained in firms' expectations and are hence almost identical to the first panel of Table (ref). As explained in section (ref), NPR is robust to biased expectations over labor (the first row). This is because bias in expected inputs leads to bias in expected output according to the production technology, which means that a one-to-one mapping between expected outputs, inputs and productivity is preserved. The second and third rows, however, show that NPR loses consistency when there is bias in expected output or expected productivity. In these DGPs, the relationship between expected outputs and inputs is subject to two unobservables -- expected productivity and the bias shock -- and hence it is no longer possible to control for expected productivity using expectations data. The fourth row considers a similar DGP to that of the third row, although in this case the bias to firms' expected productivity is a function of a time-variant observable characteristic, $X_{it} \equiv \textrm{mgmt}_{it}$.\footnote{We use the label ${mgmt}_{it}$ because bloom_well_2021 find evidence that forecast biases are related to managerial quality.} Although this type of expectation bias again compromises the performance of the basic NPR estimator, application of a modified version of the NPR estimator described in detail in Appendix (ref) recovers consistency.\footnote{The bias-robust extension is discussed in `Case 3' of Appendix (ref). Management is simulated as a standard normal random variable drawn for each firm-period (i.e. ${mgmt}_{it}\sim N(0,1)$) and firms' expected productivity bias is simulated as -0.15 times management (i.e. $\iota_{it}=-0.15{mgmt}_{it}$).}
In summary, NPR is robust to bias in expected inputs and, while bias in expected output or productivity undermine performance of the basic estimator, an extension to the NPR estimation algorithm can accommodate these biases under certain assumptions. We do not consider the bias-robust versions of NPR further since, as detailed in Appendix (ref), they require adequate panel data on firms' expectations, which is not available in our empirical setting.
Our proposed methodology requires data on firms' log output, log inputs and one-period-ahead expectations of these quantities. We obtain this data from the Management and Expectations Survey (MES): a survey administered by the UK's statistical authority, and sent to a representative sample of non-financial private sector establishments.\footnote{Further details on the MES' sampling design are provided in bloom_well_2021.} The MES was designed to have broadly the same bank of questions as the Atlanta Fed SBU and US MOPS (see bloom_what_2019 for details) and was administered in 2017 and 2020, creating two `waves' of data. It can be linked to other business surveys and we exploit this property to match MES respondents to the Annual Business Survey (ABS, office_for_national_statistics_absannual_2023) enabling us to compare firms' subjective expectations to outturns.\footnote{Establishments with 250 or more employees are surveyed each year by the ABS. Businesses below this threshold are surveyed on a multi-year basis with surveyed businesses in any one year chosen as a stratified random sample from the ABS target population.} Both the 2017 and 2020 versions of the MES ask firms for their turnover, employment, capital expenditure and expenditure on intermediates (purchases of energy, materials and services) in the year of the survey and the previous year.\footnote{The survey specifies it should be completed by `the most senior person responsible for day-to-day operations', as a senior member of staff who is likely to have adequate knowledge of these quantities. This will correspond to the plant-manager or COO in most firms.} We use firms' reported investment in these years to build capital stocks via the perpetual inventory method, imputing base period capital from national accounts data on industry-specific capital stocks, apportioning industry totals among firms according to within-industry intermediate input shares.
Inclusion in the subsample of MES data we use in the majority of our analysis is conditioned on responding to all MES expectations questions and having adequate observations of turnover, capital, employment, intermediates and investment to implement the OP, LP and ACF estimators. To allow for parameter heterogeneity by sector, we confine attention to three industries defined using the UK's Standard Industrial Classification (SIC): `electronics' consists of firms that manufacture computer electronic and optical products or electrical equipment (SIC groups 26 and 27 respectively); `retail' consists of firms in the wholesale and retail trade except motor vehicles and motor cycles (SIC groups 46 and 47 respectively); and `restaurants' consists of firms who conduct food and beverage serving activities (SIC group 56).\footnote{These industries were selected as they are among the largest in the MES and provide examples of manufacturing as well as service activities. We have also estimated pooling across all industries.}
Table (ref) summarises the characteristics of the MES analysis subsample, across all industries (first panel), and the industries we focus on (subsequent panels). Relative to the combined sample, electronics firms are smaller in terms of inputs and output. Firms in the retail sector are relatively similar in size to the overall non-financial private sector economy, whereas restaurants are larger than average in terms of employment but smaller in terms of gross and net output indicating this sector is relatively labor intensive.
To elicit firms' expectations for the following year, the MES asks firms to report on five scenarios ranging from `lowest' to `highest'. Firms are asked for the value they expect each variable separately to take under each scenario in the next year and the likelihood of the scenario occurring.\footnote{In the 2017 MES, for example, the exact wording of the question regarding turnover expectations was: “Looking ahead to the 2018 calendar year, what is the approximate pound sterling value of turnover you would anticipate for this business in the following scenarios [Lowest, Low, Medium, High, Highest], and what likelihood do you assign to each scenario?”. This wording is very similar to that used in the US MOPS. An image showing the expectations question and its position in relation to questions on current and previous year turnover is given in Appendix (ref).} The 2017 MES used these questions to elicit firms' expectations of turnover (i.e. revenue), employment, capital expenditure and expenditure on intermediate inputs (energy, goods and services which we label “materials”), whereas the 2020 MES only asked firms about their expectations of turnover (i.e. revenue) and employment in order to limit survey length. In both years, expectations over monetary quantities were asked in nominal terms and we therefore conduct all analysis on a nominal basis and include a time dummy to allow for industry-year specific shocks (like output prices).
The 2017 MES was administered as a paper survey and, although firms were instructed to ensure the likelihoods assigned to the five scenarios summed to 100, some responses did not meet this criteria. In these cases the reported likelihoods were rescaled to sum to 100 if the total likelihood across the five scenarios was between 90 and 110. A small number of responses with a total reported likelihood outside of this window were discarded. This issue does not appear in the 2020 data as this wave of the MES was administered online and required respondents' reported likelihoods to sum to 100 before they could proceed to subsequent questions.\footnote{Both the 2017 and 2020 MES contain a limited number of what appear to be data entry errors in turnover and employment. We identify these by calculating ratios of turnover to employment and comparing these with equivalent ratios observed in the ABS. We identify spurious observations as those whose turnover-employment ratios differ across the MES and the ABS by a factor of two and with year-to-year growth in their MES turnover-employment ratio in the top 5% of the distribution. These observations are dropped from our analysis.}
The MES respondents report point-values for each of the five scenarios and their related probabilities. This is in contrast to the survey design implemented in, for example, dominitz_using_1997, which recovers households' subjective CDFs of one-year-ahead income by asking for the perceived likelihood that income next year will fall below a number of thresholds, where the thresholds are determined by first asking households for the minimum and maximum income they expect next year and splitting the interval into `bins'. Because of this discrepancy, it is not obvious how to use the MES' questions on subjective expectations to recover firms' subjective CDFs. One approach is to treat the scenario values as points on either a corresponding CDF or survival function on the stated support. In the CDF approach, for example, the cumulative likelihood for the `Medium' scenario would be taken as the sum of the likelihoods a firm reports against the `Lowest', `Low' and `Medium' scenarios. In the survival function approach, by contrast, the cumulative distribution function value for the `Medium' scenario would be taken as one minus the survival function, which is the sum of the likelihoods a firm reports against the `Highest', `High' and `Medium' scenarios. Experimentation with both approaches found the former created a lower mean expectation relative to a simple weighted sum across scenarios, while the latter created a higher mean expectation compared with a simple weighted sum across scenarios. We therefore estimate lognormal parameters for both approaches by choosing mean and variance parameters to minimize the sum of squared deviations between the fitted distribution and firms' reported scenario values and their corresponding CDF or survival function points.\footnote{Fitting a beta distribution or using absolute deviations as an objective yields similar results.} Firms' subjective CDFs are then characterised as $F_{it}(l_{it+1}) = \mathcal{N}(\bar{\mu_{i}},\overline{\sigma_i^{2}})$, where bars denote the averages across the CDF and survival function estimates.\footnote{In the Cobb-Douglas specification, one only requires expectations rather than the subjective CDF, which could alternatively be estimated as a weighted sum across scenario values using the reported likelihoods as weights. These weighted sums are very similar to those of the fitted lognormals and we proceed with the latter so that we can analyze non-linear production function specifications using the same subjective expectations as those used to examine the Cobb-Douglas case. An alternative would be to rely on “bounds” defined by the CDF and survival functions. We leave this for future research.}
Table (ref) summarises absolute deviations between firms' reported point values and those implied by the fitted lognormal CDFs. These differences are small across all variables for all industries implying the fitted lognormal distributions provide a good continuous approximation of the subjective distributions underlying firms' responses to the discrete MES expectations questions and compares favorably with the fit obtained in related works such as dominitz_using_1997 (see discussion in their section 3.5).\footnote{We confine analysis of firms' expectations here to turnover and employment as these are the key expectations variables required by the baseline NPR estimator. Similar results for firms' expectations of investment and intermediate inputs as reported in the 2017 MES are available from the authors on request.}
Our method of obtaining subjective distributions from the MES responses yields a distribution of lognormal parameters and implied moments across MES respondents. Table (ref) summarizes the medians of these parameters and a selection of moments across variables and samples.\footnote{Percentile statistics are prohibited from being exported from the secure server through which we access the MES and ABS data. Table (ref) therefore reports `fuzzy medians' of the firm-specific lognormal subjective distribution parameters, calculated as the mean value across the 50 observations closest to the median.} The average $\sigma$ parameter of the fitted subjective distributions indicates firms' uncertainty and shows that restaurants are slightly more uncertain on average about year-ahead turnover than firms in electronics or retail, and more substantially so regarding year-ahead employment. As well as greater within-firm uncertainty, the restaurants sector also exhibits greater across-firm variation in mean expectations. This can be seen in Figure (ref), which plots kernel densities of firms' expected mean growth rate for turnover and employment.
A subsample of firms in the MES were also surveyed in the ABS the year following their MES response, which allows us to compare their subjective expectations to actual outcomes. Table (ref) shows the absolute log difference between firms' expectations and outcomes separately by variable and year. On average, firms' expectations are generally good, with the forecast error being insignificantly different from zero. For employment the forecast error was zero for retail, 2% for electronics and -4% for restaurants. Electronics and retail industries also did reasonably well on turnover forecast (-1% and -5% respectively). Restaurants, however underestimated turnover by 18 log points (and 13 log points at the median).
In theory, as we elaborate in Appendix (ref), one can leverage expectation errors of the type summarized in Table (ref) as an additional source of identifying variation. This only holds, however, if firms' expectations about their future inputs are biased on average and if this bias is accompanied by bias in expected output of a magnitude that is consistent with the true production technology. We do not pursue this additional identification method in our empirical analysis as Table (ref) shows firms' mean expectation errors are insignificantly different from zero across all industries we consider.\footnote{When subjective beliefs and realisations are available, another possibility is to weight observations by the prediction quality (e.g. inversely proportional to forecast errors), leveraging observations with more accurate predictions. We are grateful to Moshe Buchinsky for this suggestion and leave it for future research.}
To evaluate the empirical performance of the NPR estimator, we implement a number of other popular production function estimators on the MES data separately for each of our focus industries. Tables (ref) show estimates for each industry.\footnote{All results discussed in this section are from a Cobb-Douglas gross output production function (i.e. one that takes turnover as the measure of output). Appendix (ref) contains an explanation of how one can construct expectations of (log) value added using expectations questions of the type contained in the MES for turnover and intermediates. We do not pursue estimation of a net production function however, as expected intermediates were only asked in MES 2017 leading to prohibitively small industry-specific samples. The number of firms in Table (ref) is comparable across all the methods but the number of observations is about half as small for the NPR estimator than for the other methods. This is because the proxy variable approaches require using lagged values of capital and investment, whereas NPR uses a single cross-section of data from each MES wave. The number of observations is less than twice the number of firms because some firms are surveyed in both the 2017 and 2020 MES.}
Across all industries, NPR returns estimates that are plausible whereas the first-differenced and fixed-effect OLS specifications (contained in columns `OLS FD' and `OLS FE' respectively), return coefficients that are considerably lower than the OLS levels coefficients with statistically insignificant capital coefficients. More specific comparisons between the various estimators differ by industry. Within the electronics industry, NPR returns a labor coefficient of 0.9 and a capital coefficient of 0.21, which are both insignificantly different from the results obtained via linear OLS and ACF. Taking this comparison at face value, it suggests that both labor and capital are subject to adjustment costs of such magnitude that labor cannot respond to within-period changes in persistent productivity (absolving the linear OLS estimates of bias), and that there is a one-to-one mapping between persistent productivity and intermediate inputs conditional on labor and capital (i.e. the ACF monotonicity condition holds). OP and LP meanwhile, return significantly lower capital coefficients, which may be due to the functional dependence issue highlighted by ACF.
Differences between NPR and ACF are larger for the retail industry estimates (0.80 vs. 0.66 for the output elasticity of labor), yet still insignificant. NPR and ACF return capital coefficient estimates of 0.16 and 0.17 respectively, which are significantly lower than the linear OLS estimate of 0.25. This suggests a different form of bias in the OLS estimate than the typical expectation that endogenous responses of labor to productivity cause upward bias the OLS labor coefficient and downward bias in the capital coefficient. OP and LP again return unlikely estimates with both the labor and capital coefficients appearing attenuated toward zero.
Estimates from the restaurant sector show NPR and linear OLS are statistically indistinguishable with NPR returning labor and capital coefficients of 0.82 and 0.26 respectively (compared to 0.82 and 0.21 for OLS levels). OP and LP again exhibit an attenuated labor coefficient while, unlike the results in the other industries, the ACF estimate appear implausible with a labor coefficient of 1.06 and capital coefficient insignificant at 0.05.
To what extent should we believe the NPR estimates instead of those obtained by the other methods? It is straightforward to discard the OLS FD and OLS FE results: the large attenuation observed across all three industries indicates across-firm variation is needed to yield credible estimates, which is a well-known observation. OP and LP also appear to perform poorly in all industries, possibly due to the functional dependence concern highlighted by ACF. In most cases the choice between NPR and OLS is moot, given the similarity of the estimates, although NPR has the advantage of being robust to simultaneity concerns. Note that there are more marked differences when a translog production function is estimated (see Appendix (ref) and below).
In the electronics and retail industries, NPR is also similar to ACF, which gives confidence the estimates are consistent. In the restaurant industry, however, its worth asking: why should one trust the NPR estimates over ACF? There are two points in support of the NPR results. First, the capital coefficient for ACF is very small (0.05) and insignificantly different from zero compared to a significant 0.26 for NPR. It is ex ante implausible to believe that capital does not matter at all in this sector.
The second point in support of NPR is that material input optimization appears less likely to hold in the restaurant industry than in either electronics or retail. Evidence to this point comes from a subsample of firms in the MES that were surveyed by the ABS. Unlike the ABS, which asks firms for annual values retrospectively after the year is complete, the MES was dispatched to firms in Fall-Winter of the survey year. This means the `current-year' values that firms were asked for (i.e. 2017 values for firms surveyed in MES 2017 and 2020 values for firms surveyed in 2020), were estimates made before the year was complete. Comparing levels of inputs and output that firms report in the MES survey year to the equivalent values observed in the ABS therefore gives indication of how unpredictable inputs are, and thus how hard they are to optimize. Table (ref) contains moments of the distribution of MES-ABS log differences separately by industry and variable. This shows turnover was relatively easy to forecast prior to year-end, with a median difference between MES reported values and the ABS equivalents of zero across all industries. For electronics and retail, employment was equally as predictable, it was only slightly less so in the restaurants sector with a median MES-ABS deviation of 3%. The predictability of intermediate inputs, however, varies more markedly across industries. Focusing on the median to avoid the influence of outliers, the average MES-ABS difference for intermediates is small for electronics and retail firms at 2% and 1% respectively. For restaurants, however, it is much larger at 30 log points. Under the assumption of Leontief technology, the fact that employment was relatively unchanged from the date at which restaurants were surveyed by the MES whereas intermediate inputs changed substantially implies intermediate inputs cannot be at their optimal level. As shown in Section (ref), this optimization error in intermediate inputs undermines ACF, owing to violation of the monotonicity condition they require, and we therefore view Table (ref) as further reason to doubt the ACF production function estimates for the restaurants industry in favour of the NPR estimates.\footnote{A third more technical point is to first note that the returns to scale implied by the ACF estimates are similar to those implied by the NPR estimates suggesting the ACF estimates may be affected by a global identification issue. ACF state:
}
While we have focused here on Cobb-Douglas production technology, the NPR estimator can also estimate parameters of a translog production function. Appendix (ref) describes the additional data preparation necessary for this alternate specification and presents estimates for our three industries along with alternative estimates obtained via various OLS estimators and the ACF method. Testing for Cobb-Douglas technology via a test that the additional translog parameters are jointly zero, the NPR estimates fail to reject Cobb-Douglas technology for the electronics and restaurant sectors. The average partial derivatives are not significantly different from the Cobb-Douglas equivalents and we therefore confine attention to the Cobb-Douglas results in subsequent empirical analysis for parsimony.
The results in the previous subsection show that the NPR estimator recovers production function parameter estimates that are robust to simultaneity concerns and are either more credible than or similar to alternative standard estimators. Equipped with such estimates we now demonstrate their utility and examine trends in total factor productivity (TFP). To examine trends in TFP, we take the parameter estimates obtained using the MES data and calculate TFP for the entire ABS sample as:
where in this and subsequent equations, $i$ denotes firm, $j$ denotes industry, and $t$ denotes year. $\hat{\beta}^{mj}$ represents production function coefficients obtained by implementing estimator $m$ on MES data for industry $j$. For parsimony and in light of the results of the previous subsection, we focus on comparing the NPR estimates to those obtained by OLS levels and ACF because the other estimators did not generally yield credible estimates.
We calculate TFP according to equation ((ref)) for all firms observed in our three industries in the ABS between 2010 and 2019 and de-mean TFP by year.\footnote{We focus on this period as the most recent decade around the MES survey years but the comparisons across estimators we highlight are qualitatively similar if one either uses a larger sample period of 2000-2019 or a narrower one of 2015-2019. We focus on the intermediate period to add credibility to the assumption of constant technology, which is imposed implicitly via our TFP calculations, while balancing statistical power due to sample size.} We relate firm performance to TFP using equations of the form:
where $y$ is one of several firm-level outcomes, $\tau$ is a year dummy, $\hat{a}^{m}$ the TFP estimate obtained using coefficients from estimator $m$ and $\epsilon$ is a mean-zero disturbance. The outcomes we consider are exit, defined as $t$ being the last year a firm is operational, and growth as measured by one- and five-year differences in the log of employment:
where $s\in(2,5)$.\footnote{We take $t+1$ as the base year of our growth measures to avoid endogeneity with the TFP estimates $\hat{a}^{m}_{it}$, since these are a linear function of $l_t$.}
Table (ref) contains the estimated $\hat{\pi}$ parameters from equation ((ref)) along with the mean values of the outcome variables and sample sizes. The first panel of the table presents results for the electronics industry. Higher TFP is associated with significantly lower firm exit and higher employment growth conditional on survival. The magnitude of association is similarly large across the NPR, OLS and ACF estimates of TFP, with a one standard-deviation increase in TFP reducing the probability of exit by around 1.6 percentage points, in comparison to an overall exit rate of 1%, and increasing one- and four-year employment growth by 0.01 and 0.04 log points in comparison to mean growth rates of 0.01 and 0.025 respectively. Similar associations are observed within the retail industry, although the positive association between TFP and five-year employment growth is stronger for the NPR and ACF TFP estimates than for OLS-estimated TFP.
The third panel of Table (ref) shows the restaurants industry exhibits greater variation across associations between firm performance and the various TFP estimates. Both NPR and ACF estimates of TFP are negatively associated with firm exit, which is somewhat surprising given the latter are somewhat implausible. This is in contrast to the OLS TFP estimate, which has no significant association with firm exit. All three TFP estimates are significantly positively associated with two-year employment growth but the strength of association varies from 0.012 with the OLS TFP estimate to 0.021 with the NPR TFP estimate. Despite being small in absolute terms, this difference is considerable and corresponds to 22% and 39% of the mean growth rate of 0.054. Point estimates of the association between TFP and employment growth over a four-year horizon indicate the association is again strongest for the NPR TFP estimate, although the smaller sample size means none of the point estimates or their differences are significant. Viewed together, these results suggest that failing to adequately account for input endogoeneity risks underestimating the degree of dynamic reallocation over a mid-range (e.g. four-year) horizon.
In this paper we have proposed a new production function estimation methodology that leverages data on firms' observable expectations, data which are becoming increasingly available across a range of countries. We show that such information enables one to relax the strong assumptions of firm input choice (strict monotonicity and invertibility with respect to a single scalar unobservable productivity term), that underpin currently-used proxy variable approaches in the olley_dynamics_1996 tradition. Moreover, our method can be implemented on a single cross section which has great attractions over these techniques (which require two or more periods), and dynamic panel methods such as those of blundell_gmm_2000, which assume linear productivity dynamics and require three or four consecutive time period observations per firm.
We present Monte Carlo simulations close to those in ackerberg_identification_2015 featuring forward-looking firms with heterogeneous quadratic adjustment costs in capital. We show that our proposed NPR estimator is robust to optimization error in firm inputs choices whereas other methods are biased when firms make optimization errors in their material input choices. We also demonstrate the NPR estimator can be extended to accommodate certain forms of bias in firms' expectations.
Implementing our proposed NPR estimator on UK data, we show it recovers parameter estimates that are either comparable to or more credible than those recovered by conventionally-used estimators. We use the various production function estimates to calculate TFP residuals and relate these to measures of firm performance. TFP is negatively associated with firm exit across all industries and production function estimators we consider while TFP estimates obtained using our proposed production function estimator are more positively associated with employment growth, especially over a mid-range horizon. This suggests that the extent of dynamic reallocation is particularly sensitive to adequately accounting for input endogeneity during production function estimation.
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