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Under-Identification of Structural Models Based on Timing and Information Set Assumptions
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A number of recent papers address endogeneity in structural models using a panel data identification strategy based on timing and information set assumptions. \ A preeminent example is estimation of production functions with endogenous input choice - in this context, timing and information set assumptions are used for identification in two distinct lines of literature: estimation methods using a proxy variable approach (e.g. Olley and Pakes (1996), Levinsohn and Petrin (2003), and Ackerberg, Caves, and Frazer (ACF, 2015)), and estimation based on what are often called dynamic panel methods (e.g. Blundell and Bond (2000)). \ The methods proposed in these literatures have been used in thousands of papers, both in the context of production functions and in many other structural models. In this paper, we study an interesting identification problem that can arise in these methods. \ This under-identification issue has been pointed out more narrowly in prior papers (see footnote 16 of ACF and Kim, Luo, and Su (KLS, 2019)) which concern the proxy variable approach. \ In this paper, we illustrate how this is a more widespread issue in estimation methods that rely on timing and information set assumptions. \ In particular, we show that the under-identification problem can also arise in the context of dynamic panel methods. \ Because dynamic panel methods have a relatively simple representation, we then use this dynamic panel context to further study the under-identification problem and suggest possible solutions.
The identification problem we study is unusual. \ While typical identification problems involve regions of non-identification, this identification problem does not - in our simple models, the moment conditions are zero at the true parameters and non-zero locally around the true parameters, but also at a second distinct point in the parameter space. \ So, the model might be described as being locally identified (around the true parameters) but not globally identified. \ There is a quadratic aspect to the model and moment conditions that leads to the two possible solutions. \ This form of the non-identification also impacts some of the solutions we propose for the issue - in particular it means that we can leverage sign restrictions to resolve it.
Before proceeding, we discuss a couple of important issues. \ First, we illustrate the under-identification problem with perhaps the simplest possible dynamic panel model. \ In particular, our basic model does not include fixed effects, which are relatively common in the empirical literature using these techniques. \ This is mainly to aid in our initial illustration of the problem and solutions. \ Later in the paper, we show that the under-identification problem also exists in dynamic panel models with fixed effects.
Second, much of our discussion is couched in the production function context. This is because production functions are perhaps the most frequent empirical context for the application of these identification strategies based on timing and information set assumptions. \ However, it is important to note that there are many empirical applications of dynamic panel methods (and timing/information set assumptions more generally) in other scenarios. One interesting example is a set of recent work on demand estimation with potential endogenous product characteristics, e.g. Sweeting (2009), Grennan (2013), Lee (2013), and Sullivan (2017). Another example is some recent work on hedonic price regression models with a time-varying unobserved attribute that is correlated with observed attributes and follows a persistent time-series process, e.g. Bajari, Fruehwirth, Kim, and Timmins (2012) and Bishop and Murphy (2019). This under-identification issue and potential solutions we propose are potentially relevant for all this work as well.
Our findings are also related to Sentana (2015), who also studies \textquotedblleft finite\textquotedblright\ under-identification problems. While he considers general methods to efficiently estimate all observationally equivalent parameters that are consistent with an underlying true model, we are instead interested in singling out the true parameter by imposing an additional assumption in our specific model. \ Our point is that enforcing this additional assumption selects the true parameter that is consistent with the DGP, while the moment conditions typically used in practice yield multiple solutions. Sentana (2015)'s examples of finite under-identification are also distinct from ours, and an important contribution of our paper is to illustrate the under-identification in this particular model - a type of model that has been very widely used in the two large literatures cited above.\footnote{ One of the examples in Sentana (2015) is derived from the dynamic panel literature (e.g. Arellano and Bond (1991)), but it concerns the model
and an under-identification issue involved in identifying $\alpha _{1}$ and $ \alpha _{2}$, which arises when $\alpha _{1}+\alpha _{2}=1$. This is distinct from the model we study, which is used in the empirical literature to assess the impact of an (partially) endogenous $X_{it}$ on $Y_{it}.$ And as noted in the text, our goal is to study how one can resolve the identification problem (both in theory and in practice) by imposing additional economic restrictions, unlike Sentana (2015) who studies the full properties of the finite identified set.}
The paper proceeds by first illustrating the under-identification problem in what is perhaps the simplest possible dynamic panel model. We then broaden our scope to consider the problem more generally, showing how it can manifest itself in more general models (e.g. dynamic panel models with fixed effects, different data generating processes for the right-hand-side endogenous variables, and models with multiple right-hand-side endogenous variables). We then turn to potential solutions to the problem, in part based on specifying a model for the endogenous right-hand-side variables (e.g. inputs in the production function context) and enforcing an assumed sign restriction. We end by using lessons from our specific solutions for this simple example to propose more general advice for researchers using these methods (both proxy variable and dynamic panel approaches) in practice - in an effort to hopefully minimize the problems stemming from this under-identification result.
Consider a panel regression model
where $x_{it}$ is correlated with error term $\omega _{it}$ but not with $ \eta _{it}$. Our leading example is production function estimation, in which $y_{it}$ is the output, $x_{it}$ is an input, $\omega _{it}$ is a serially correlated unobserved productivity shock, and $\eta _{it}$ is, e.g. measurement error in output that is often assumed to be uncorrelated across time. \ For now we assume $\omega _{it}$ follows a first-order autoregressive process
where $\xi _{it}$ is the innovation components of this process. A simple identification restriction that has been used in the production function literature involves assuming the following timing and information set conditions.
This is a simple form of a \textquotedblleft timing and information set\textquotedblright\ assumption that has been used to address the endogeneity of $x_{it}$ in ((ref)). Loosely speaking, the assumption can be interpreted as following from an economic model where (i) firms do not observe $\omega _{it}$ until $t$, and (ii) firms endogenously choose $ x_{it}$ at $t$ - thereby implying (with some additional, more technical assumptions) that future innovations in $\omega _{it}$, i.e. $\xi _{i,t+1}$, are uncorrelated with $x_{it}$.\footnote{ Also note why this is not only a timing assumption, but also an assumption about firms' information sets at various points in time. \ For example, consider an alternative information set assumption where firms observe the shocks $\omega _{it}$ one period ahead of time, i.e. $\omega _{i,t+1}$ is observed at $t$. \ Since, in general, firm choice of inputs $ x_{it}$ may have dynamic implications (e.g. capital, labor with any sort of adjustment costs), those choices could optimally depend on $\omega_{i,t+1}$, implying that $\xi_{i,t+1}$ would no longer be uncorrelated with $x_{it}$. On the other hand, suppose $\omega_{i,t+1}$ continues to be observed at $t$, except with an alternative timing assumption where firms must commit to their choice of $x_{it}$ at $t-1$. This case brings us back to $\xi_{i,t+1}$ being uncorrelated with $x_{it}$ once again.}
For estimation, one also needs to make an assumption on the measurement error in output, typically $E[\eta _{it}|I_{it}]=0$. \ Given this, one can consider estimating the model parameters $(\alpha ,\beta ,\rho _{\omega })$ by $\rho _{\omega }$-differencing the production function
Since, by assumption, $\xi _{it}$, $\eta _{it}$, and $\eta _{i,t-1}$ are all conditional mean independent of $I_{i,t-1}$, it follows that
This is a very simple example of a type of “dynamic panel” moment condition that has been used to estimate many production functions (see, e.g., Blundell and Bond (2000) and Ackerberg, Caves, and Frazer (2015) for a discussion of how the dynamic panel approach is related to proxy variable approaches). Note that this model is simpler than those used in many actual applications of dynamic panel methods. In particular, dynamic panel approaches often include a fixed effect $\alpha _{i}$ in the production function. This typically requires a second differencing to form usable moment conditions.\footnote{ Moreover, some dynamic panel models include lagged $y$ (i.e. models with state dependence) instead of the serially correlated $\omega$ in our model. The nature of these alternative models is somewhat different. Including lagged $y$ directly in the structural model means the persistence in $y$ is causal (lagged $y$ causes current $y$), while in the models we study, the persistence originates from the persistent unobserved state $\omega$. The models we study (without lagged $y$) are used more often in the production function literature\ (as well as the demand and hedonic literature cited in the introduction), as there is typically more concern about serially correlated unobserved productivity shocks than true state dependent phenomena (although there are exceptions, e.g. a production function with learning-by-doing has causal state dependence).} Our simplified model ((ref)) excludes fixed effects for expositional purposes. In a moment we show that our under-identification result also applies to a model with a fixed effect.
We know that the moment conditions ((ref)) equal zero at the true parameter values. \ But we now show that these moment conditions may also equal zero at another set of parameter values. \ To study this, we need to consider a complete model of the data generating process (DGP) for the endogenously chosen $x_{it}$. \ Note that the typical paper in this literature does not specify a full DGP for $x_{it}$ - this is because an explicit DGP for $x_{it}$ is not needed to form the moments conditions ((ref)) and attempt to estimate $(\alpha ,\beta ,\rho _{\omega })$ . Specifying a form of the DGP for $x_{it}$ is important for understanding the identification issue we study. As we will discuss in the next section, this issue is ubiquitous under various forms of the DGP.
We start with what might be the simplest possible model of an endogenous $ x_{it}$ that is correlated with productivity shock $\omega_{it}$:
This allows a firm's choice of $x_{it}$ to depend on productivity shock $ \omega _{it}$ as well as another persistent unobserved market factor $\kappa _{it}.$ Analogous to $\omega _{it}$, we allow $\kappa _{it}$ to follow a first-order autoregressive process
where $u_{it}$ is the innovation component of the process. For example, if $x_{it}$ measures labor input, then $\kappa _{it}$ can be interpreted as a wage factor that is persistent over time.\footnote{Serial correlation in productivity and/or input price shocks is widely allowed in the literature. Empirical studies that find strong persistence in firm input prices include Alonso-Borrego and Arellano (1999) and Grieco, Li, and Zhang (2016). But it is worth noting that while we allow $\omega _{it}$ and $\kappa _{it}$ to be serially correlated, we do not require it.} Analogous to how the innovation term in the $\omega _{it}$ process satisfies $E[\xi _{i,t+1}|I_{it}]=0$, we assume that the innovation term in the $ \kappa _{it}$ process satisfies $E[u_{i,t+1}|I_{it}]=0$. \ This is consistent with a model in which firms observe $\omega _{it}$ and $\kappa _{it}$ prior to choosing $x_{it}$, but who do not observe $\xi _{i,t+1}$ and $u_{i,t+1}$ (and thus $\omega _{i,t+1}$ and $\kappa _{i,t+1}$) until the next period.
We now show that with this DGP for $x_{it}$, the moment conditions ((ref)) generally have an additional zero - one where $\rho_{\omega}=\rho _{x}^{0}$, i.e. where the productivity persistence parameter is not equal to its true value, but instead equal to the persistence parameter of the input process. This identification issue has been noted in the context of the proxy variable literature on estimating production functions (see footnote 16 of ACF and discussions in Kim, Luo, and Su (2019)). KLS (2019) studies this pseudo-solution using Monte Carlo simulations in the context of the proxy variable literature. \ In this paper, we show that this under-identification is a more general issue that also applies to the dynamic panel literature. \ Studying the issue in the dynamic panel approach is particularly useful because the framework is simpler in some respects, and this allows us to better understand the nature of the under-identification problem and to propose more general approaches that we think might help mitigate the problem.
Our derivation in the Appendix shows that, at this alternative parameter vector $(\alpha,\beta,\rho )=(\alpha ^{0}-\frac{\pi^{0}}{\theta^{0}} ,\beta^{0}+\frac{1}{\theta^{0}},\rho_{x}^{0})$, our simple dynamic panel moment condition essentially recovers the innovation in the input demand shock $\kappa _{it}$ (i.e. $u_{it}$) rather than the innovation in the productivity shock $\omega_{it}$ (i.e. $\xi_{it}$). So given that we assume innovation in $\kappa_{it}$ has the same properties as innovation in $\omega_{it}$, our moment condition also equals zero at this pseudo-solution. \ This is perhaps the simplest illustration of the under-identification issue we study, though we develop more intuition for it momentarily.\footnote{ Note that this pseudo-solution still exists even if we include the unobservables $\omega_{it}$ and $\kappa _{it}$ in the information set $ I_{it}$. In general, the identified set may depend on the information available in data and thus the number of periods observed. In practice, it may also depend on the chosen instruments in the estimation.}
Because the two zeros of the moment condition arise at different values of $ \rho$, one may think that the identification issue does not arise when $ \rho_{\omega }^{0}=\rho _{x}^{0}$. While this is not the focus of the paper, we show that in this case the identification problem is even worse - in particular, in this case there is an entire region of non-identification. To see this argument from the input demand equation ((ref)), note that if $\rho _{\omega }^{0}=\rho _{x}^{0}=\rho ^{0}$, then
In other words, when $\rho _{\omega }^{0}=\rho _{x}^{0}$, the differencing $ x_{it}-\rho x_{i,t-1}$ dispenses with all remaining information in $x_{it}$ other than the innovation terms, which are orthogonal to $I_{i,t-1}$. It follows that, for an arbitrary $(\alpha ,\beta )$, moment equation ((ref)) at $(\alpha ,\beta ,\rho _{\omega })=(\alpha ,\beta ,\rho ^{0})$ satisfies
where the second and third equalities come from substituting in the true DGP from ((ref)) and ((ref)), respectively, for which we impose $\rho _{\omega }^{0}=\rho _{x}^{0}=\rho ^{0}$. The last result holds since, given the timing and information set assumptions, the unobservables $ \xi _{it}$, $u_{it}$, $\eta _{it}$, and $\eta _{i,t-1}$ are orthogonal to the information available at time $t-1$. In other words, the moment equation has arbitrarily many solutions of $(\alpha ,\beta )$ that satisfy
Therefore, moment condition ((ref)) does not contain any information on $\beta $ when $\rho _{\omega }^{0}=\rho _{x}^{0}$. Intuitively, this is similar to the non-identification result for time-invariant variables in a fixed effects panel model. In any case, our focus is on the existence of the pseudo-solution that arises from moment condition ((ref)) in a general setting. Therefore, for the remainder of the paper, we avoid this more severe problem by assuming:
which, again, is analogous to ruling out a time-invariant $x_{it}$ in a fixed effects model. In fact, this assumption has testable implications.\footnote{ One can show that $x_t$ has an AR(2) reduced form under Assumption (ref) but an AR(1) reduced form otherwise. See the previous version of this paper for more details.}
Returning to our main point, when $\rho_{\omega }^{0}\neq \rho _{x}^{0}$ we have verified that the basic moment condition of the dynamic panel approach may fail identification by yielding two solutions: one at the true AR(1) parameter of the productivity shock and the other at the AR(1) parameter of the input demand shock. It is important to note that this identification problem is an issue with moment condition ((ref)). It is not necessarily a problem of the full model, i.e. there is not necessarily an identification problem if one were to impose all the restrictions of the model, e.g. those relating to higher order moments. In fact, Hu and Shum (2012) use deconvolution techniques to show non-parametric identification of a first-order Markov model that is similar to what we study. However, their results rely on higher order moments. For example, to identify a first-order Markov model, Hu and Shum (2012) require more than 2 time periods (in some cases up to 5 time periods) of data, i.e. they utilize the strong restrictions the first-order Markov assumption places on data in non-consecutive time periods. In our view, even though the full model might be identified, it is important to study identification problems based on ((ref)) because the vast majority of the empirical literature using timing and information set assumptions (e.g. that following Olley and Pakes (1996) and Blundell and Bond (2000)) utilizes conditional moments similar to ((ref)).
Momentarily we consider a possible solution to this problem that utilizes additional moments based on the DGP of $x_{it}$, benefiting from the fact that, in the production function context (and likely in many other contexts), one may be willing to make assumptions about the sign of $\theta $ , i.e. the direction of endogeneity. But first we examine the extent to which this pseudo-solution problem manifests itself in more general models.
The above analysis identified a pseudo-solution in the context of a simplified dynamic panel model, combined with an assumed DGP for $x_{it}$ that was also very simple. A natural question to ask is to what extent this pseudo-solution exists in more general dynamic panel models. Is this pseudo-solution a “knife-edged” phenomenon that only occurs with a very specific production function or productivity process, or that only occurs in models where the productivity and the factor input shocks have the same structure (e.g. both AR(1))? Or is it a more general phenomenon?
To address these questions, we now consider different or more general models - either in terms of the structural function of interest, the unobserved productivity or input shock processes, or the timing assumptions of the model. While we are only able to derive theoretical results for a subset of these alternative models, in other models we use simulations to study their objective functions (with very large datasets to approximate their asymptotic values).
In some of our examples we can show that the exact same pseudo-solution identified above persists in the more general model we study. In others, there is still a pseudo-solution but at a different (though “nearby”) point in parameter space. In others, there are multiple pseudo-solutions that we can locate. In still others, the pseudo-solution that sets the moments exactly equal to zero disappears, but there is a nearby parameter that sets the moments “close-to-zero” and generates a local minimum of the GMM objective function. While in this last case the model is formally identified, we still think it important to note because the existence of local minima can generate challenges in numeric optimization (and the guidance we provide later in the paper could also help overcome these numeric challenges). Notably, in one of our extensions - one where we strengthen the timing assumptions of the model - the pseudo-solution appears to completely disappear.
Traditional dynamic panel models also include fixed effects in the main equation, e.g.
Here, we show that this model also exhibits the pseudo-solution identified above. \
In model ((ref)), forming usable moments requires an additional difference to remove the fixed effect $\alpha _{i}$. \ More specifically, the $\rho _{\omega }^{0}$-difference
contains $\alpha _{i}$, so an additional first difference is typically done to obtain
Since this eliminates $\alpha _{i}$, one can proceed to estimate $\beta ^{0}$ with a moment condition based on this, e.g.
To show that this moment condition may also have the under-identification problem, again consider a DGP for input $x_{it}$. \ Since the DGP for $ y_{it} $ includes a fixed effect, it is natural to include one here as well, e.g.
Following the previous argument, substitute the inverted production function into
and $\rho _{x}^{0}$-difference to obtain
First differencing this equation results in
and given that the unobservable terms in this equation are assumed to be mean independent of $I_{i,t-2}$, this implies
Comparing ((ref)) to ((ref)), it is clear that there will be a pseudo-solution to moment condition ((ref)) in the same location as the model without a fixed effect, i.e. where $\rho _{\omega }=\rho _{x}^{0}$ and $\beta =\beta ^{0}+\frac{1}{\theta ^{0}}$. Note that this argument does not depend on the existence of a fixed effect in the input demand equation, i.e. there is a pseudo-solution even if $\pi _{i}=\pi$. \
We next examine the under-identification problem where we have multiple variable inputs in the production function context or, more generally, multiple endogenous regressors. Consider a panel regression model with two endogenous regressors $x$ and $z$,
where
Here, $\wp _{it}$ and $\kappa _{it}$ denote two persistent market factors -- such as input prices -- and they follow autoregressive processes $\wp _{it}=\rho _{z}^{0}\wp _{i,t-1}+v_{it}$ and $\kappa _{it}=\rho _{x}^{0}\kappa _{i,t-1}+u_{it}$, respectively. \ Note that both inputs depend on both factors -- this is typically the case in practice as the optimal choice of one variable input typically depends on the prices of all variable inputs.
On the one hand, exploiting the AR(1) process of productivity $\omega _{it}=\rho_{\omega}^0\omega _{i,t-1}+\xi_{it}$, and $\rho_{\omega}^0$ -differencing leads to
which can be used for constructing the following moment condition, as a generalization of moment condition (ref),
On the other hand, consider a linear combination of the two endogenous inputs
where the expression on the right-hand side follows from the two input functions. This shows that we can write a linear combination of two inputs as a linear function of two sources of persistence (one for productivity and one for market factor) -- similar in structure to the one input in the benchmark model. Solving for $\omega _{it}$ and plugging it into the output equation (ref) leads to a similar structure of the output function, except that $\omega _{it}$ is replaced by $\wp _{it}$ and the coefficients on inputs are different from the true production function parameters. Notice that this is similar in structure to the alternative expression of moment condition ((ref)) we derived in the Appendix for the benchmark model.
Therefore, following similar arguments as before, we can derive a spurious solution of moment function ((ref)) by taking $\rho_z^0$ -differencing due to the autoregressive process of $\wp_{it}$, instead of the productivity. As the result, this spurious solution is at the dynamic parameter $\rho_\omega=\rho_z^0$ of the persistence market factor $\wp_{it}$ , not the productivity. Similarly, a different linear combination $ \delta_\wp^0 x_{it} - \theta_\wp^0 z_{it}$ leads to another spurious solution of moment ((ref)). This solution is at the dynamic parameter $\rho_\omega=\rho_x^0$ of the persistence market factor $ \kappa_{it}$.
In summary, we conclude that when the production function has multiple inputs, the dynamic panel moments may yield the same number of spurious solutions as the number of flexible inputs that have persistent shocks.
The under-identification problem also still exists when we have both flexible and dynamic inputs. Consider a panel regression model with a flexible input $x$ and a dynamic input $z$ like capital as follows
where the flexible input function is given by
and persistent market factor $\kappa _{it}$ follows the AR(1) process $ \kappa _{it}=\rho _{x}^{0}\kappa _{i,t-1}+u_{it}$ as before.
Solving for $\omega _{it}$ in the input function and plugging it into (ref) leads to a similar structure of the alternative expression of the output function, except that $\omega _{it}$ is replaced by $\kappa _{it}$ and the coefficients on inputs are different from the true production function. Following similar arguments as before, we can derive a spurious solution of the dynamic panel moment due to this similar structure where the spurious solution is at the dynamic parameter $\rho _{\omega }=\rho _{x}^{0}$ of the persistent input factor $\kappa _{it}$. Note that we do not expect a spurious identification problem with respect to the dynamic input $z_{it}$. In other words, if $\beta^0 = 0$ in the above model so there is only a dynamic input $z_{it}$, we do not expect a spurious solution. This is because the input demand function for a dynamic input like $z_{it}$ will depend on factors other than the shocks, e.g. $z_{it-1}$ and/or further past $z_{it}$'s.
In the above three extensions to our basic model, we were able to analytically show that the pseudo-solution (or pseudo-solutions) continues to be a potential problem. In this section, we consider some additional extensions. While we are unable to analytically show the existence of pseudo-solutions in these extensions, we are able to use simulations to show that they often continue to exist, at least in neighborhoods of our benchmark model. In particular, we consider relaxing the assumed linearity of the input demand function and the AR(1) process for unobserved shock $ \kappa _{it}$ in the input demand function.
It is important to note that we continue to assume that $\omega _{it}$ follows an AR(1) process. This means that our basic dynamic panel moment condition ((ref)) continues to be zero at the true parameter value. \ What we are focused on is not the zero at the true parameter value, but whether there continues to be a pseudo-solution (or pseudo-solutions) at other parameters, and the location of such pseudo-solutions. \ There is no need to do multiple Monte Carlo replications to study this -- it is easier to simply boost the number of observations (to 200,000) to simulate the “asymptotic” objective function and examine the location of zeros. In the benchmark case we know there are “asymptotic” zeros at the true parameter, set to $\beta=\beta^0 =0.6$, and at the pseudo-parameter $\beta =\beta^0+1/\theta^0=0.6+1/1=1.6$ (since in our benchmark model, we set $ \theta^0=1$).\footnote{Our benchmark case for these simulations uses the same parameter values as used in the Monte-Carlo experiments, see Section 5 for more details.} In the alternative models, we know there is an “asymptotic” zero at the true parameter $\beta =0.6$. The question is whether there are pseudo-solutions - either at $\beta =1.6$ or elsewhere.
First, we consider an alternative model where the input function is a non-linear function of the productivity process, i.e.:
where we set $\theta _{1}^{0}$ = $1$, as in the benchmark model, and vary $ \theta _{2}^{0}$. Figure (ref) shows the basic dynamic panel objective function (based on the moment condition ((ref))), for both the benchmark model (without nonlinearity, i.e. $\theta_{2}^{0}=0$) and this alternative model where we set $\theta _{2}^{0}=0.5$ and $1$.\footnote{ For ease of comparison, we rescale objective functions such that they are equal at $\beta =0$ in Figures (ref)-(ref).} For easy visualization, we display the objective function as a function of just $ \beta$ by concentrating out the constant term and AR(1) parameter.\footnote{ To concentrate out $\rho_{\omega}$ and the constant, for a candidate $\beta =\widetilde{\beta}$, we estimate $\rho_{\omega}$ and the constant with the linear equation
using instrumental variables with $(y_{i,t-2}-\widetilde{\beta} x_{i,t-2})$ as an instrument for $(y_{i,t-1}-\widetilde{\beta} x_{i,t-1}).$ So that this objective function illustrates exact zeros (rather than “close to zeros”), we use only $x_{i,t-1}$ as the instrument for the concentrated moment condition for $\beta$.} The objective functions for the alternative model also have pseudo-solutions, at $\beta$ quite close to the pseudo-solution in the benchmark model. Interestingly, to the right of the pseudo-solution, the objective function becomes flatter than the benchmark case, especially when $\theta_{2}^{0}=1$. However, the main finding is that the pseudo-solution does not appear to disappear when the $ x_{it}$ process is not linear in $\omega _{it}$.
Second, we consider a model where there is a nonlinearity in the first-order Markov process for $\kappa _{it}$. Specifically, we consider:
When we set $\theta _{2}^{0}=0$, $\frac{e^{\theta _{2}^{0}\left\vert \kappa _{i,t-1}\right\vert }}{1+e^{\theta _{2}^{0}\left\vert \kappa _{i,t-1}\right\vert }}=0.5$ $\forall \kappa _{i,t-1}$, so this model is equivalent to the benchmark case (where we set the AR(1)\ parameter = $0.5$ ). When $\theta_{2}^{0}$ $>0$, the Markov process is non-linear. One interpretation of this is that the extent to which $\kappa _{i,t-1}$ depreciates depends on the level of $\left\vert \kappa _{i,t-1}\right\vert $ . Note that since $\left\vert \kappa _{i,t-1}\right\vert $ is positive, the heterogeneous depreciation rate in this model is between 0.5 and 1.\ Figure (ref) shows the (concentrated) objective function for this model when $\theta _{2}^{0}=0.5$ and $0.75$. Again, the pseudo-solution persists, but moves down relative to the benchmark model. In these examples, the pseudo-solutions are at approximately 1.46 and 1.51 for $\theta _{2}^{0}=0.5$ and $0.75$, respectively. Interestingly, the effect of $\theta _{2}^{0}$ on the location of the pseudo-parameter is non-monotonic. This appears to be because changing $\theta _{2}^{0}$ affects multiple things in the model - e.g., it simultaneously changes both the average level of depreciation and its heterogeneity.
Third, we change the $\kappa _{it}$ process in a different way - keeping linearity but allowing it to follow an AR(2)\ process instead of an AR(1)\ process, i.e.
Starting with{\ the benchmark case ($\rho _{1}^{x}=0.5$ and $\rho _{2}^{x}=0$ ) we consider alternative models where $\rho _{2}^{x}$ is set to various values between 0 and 1. Figure (ref) shows the (concentrated) objective functions and indicates that the existence of the pseudo-solution may depend on the value of $\rho _{2}^{x}$ in this alternative model. When $ \rho _{2}^{x}$ is close to 0, i.e. the $\kappa _{it}$ process is close to an AR(1) model, the pseudo-solution persists, as shown in the left panel of Figure (ref) for $\rho _{2}^{x}=0.02$. When we gradually increase $ \rho _{2}^{x}$, the pseudo-solution disappears from the region. However, the right panel suggests distinct patterns when we increase $\rho _{2}^{x}$ further. For $\rho _{2}^{x}=0.2$ and $0.3$, we spot a wide flat region where the truth and the potential pseudo-solutions seem to be indistinguishable. When $\rho _{2}^{x}$ becomes large enough, i.e. $\rho _{2}^{x}\geq 0.4$, a pseudo-solution reappears but at a lower $\beta $, at approximately 1.17. And there is even an additional pseudo-solution when $\rho _{2}^{x}=0.4$ to the left of the truth, at approximately 0.16. Therefore, the AR(2) process may complicate the identification of the model but for a wide range of $\rho _{2}^{x}$, the pseudo-solution is still a potential concern.}
Lastly, we consider adding an additional i.i.d shock to the input process (in addition to the AR(1) $\kappa _{it}$). In other words, we assume the following input process
where $\epsilon _{it}$ is i.i.d over time (and distributed normally in our simulations). Essentially this is a specification where the input process (conditional on $\omega _{it}$) follows a particular type of ARMA process. \ With this process it is straightforward to show theoretically that at the pseudo-parameters $(\alpha^{0}-\frac{\pi ^{0}}{\theta ^{0}},\beta^{0}+ \frac{1}{\theta^{0}},\rho_{x}^{0})$, the dynamic panel moment condition is non-zero - one can simply substitute the inverted input process into the production function to get
and $\rho _{x}^{0}$ difference, resulting in
where $\tau _{it}=-\frac{1}{\theta ^{0}}u_{it}+(\eta _{it}-\rho _{x}^{0}\eta _{i,t-1}) -\frac{1}{\theta ^{0}}(\epsilon _{it}-\rho_{x}^0\epsilon _{i,t-1})$. Because of the additional $ \epsilon _{it}$ terms (and since $E[\epsilon _{i,t-1}|x_{i,t-1}]\neq 0$), $ E[\tau _{it}|I_{i,t-1}]\neq 0$ (unless $\rho _{x}^{0}\neq 0$). \ Thus, the moment condition is non-zero at the pseudo-parameters identified above ($ \beta =1.6$ with the parameters used in our simulations).
Of course, this is just one point in parameter space. One might be concerned that, as in the prior examples, there might still be a pseudo-solution, just at a different parameter value. \ Simulations show that this does not always appear to be the case. Figure (ref) shows the (concentrated) objective function for the benchmark case ($\epsilon $ = 0) and in models that add $\epsilon _{it}$ to the input process with different standard deviations, i.e. $\sigma _{\epsilon }$ is set to between 0.1 and 1 with a stepsize equal to 0.1. Interestingly, the left panel shows that when the variance of the shock is small, a pseudo-solution appears to persist - increasing somewhat from the $\beta =1.6$ location in the benchmark case. On the other hand, when we further increase $\sigma _{\epsilon }$, the pseudo-solution appears to disappear from the region, as suggested by the right panel of Figure (ref). Nonetheless, around the location of the baseline pseudo-solution there is still a local minimum of the objective functions in these models with larger $\sigma _{\epsilon }$. In practice, these local minima like this can also be problematic for numeric optimization, so it does not completely eliminate the potential issue.
In summary, these models suggest that the existence of pseudo-solutions is not a knife-edge result that relies on the very specific input demand equation in our benchmark model. While the exact locations of these pseudo-solutions (or local minima in some cases) differ from the benchmark model, the general patterns are similar - in particular, the pseudo-solutions are typically at a larger $\beta $ than the true parameter $ \beta ^{0}=0.6$. Obviously, this is only a small set of alternative models and there could very well be other alternative models without pseudo-solutions, especially as one moves parameters further away from the benchmark model. In other experiments, we have found that many things can happen - for example, in the following model (where the curvature of the non-linearity in $\kappa _{it}$ is reversed
we see multiple pseudo-solutions, one slightly above 1.6, and one at approximately $\beta =1.95$ (Figure (ref)). \ But overall, the results hint that this is a general issue - the $\rho $-differencing moment (ref) of the dynamic panel approach (and likely of related quasi-differencing approaches, e.g. the proxy variable approaches of OP/LP/ACF) creates a quadratic form indeterminacy that seems to exist relatively generally.
Interestingly, we have found one class of modeling assumptions that does appear to remove the pseudo-solution very generally. \ Recall Assumption 1, the timing and information set assumption that the moment conditions of our benchmark model are predicated on. \ Again, this can be interpreted as following from an economic model where 1) firms do not observe $\omega _{it}$ and $\kappa _{it}$ until $t$, and 2) firms choose $x_{it}$ at $t$ and hence implying (with some additional assumptions) that future innovations in $ \omega _{it}$ and $\kappa _{it}$, i.e. $\xi _{i\tau }$ and $u_{i\tau }$ for $ \tau >t$, are uncorrelated with $x_{it}$. \
Some work in this literature has been willing to make such timing (and information set) assumptions stronger. \ For example, in OP/LP/ACF procedures, it is often assumed that firms need to commit to their capital choice at $t-1$. \ In demand models considering endogenous product characteristics, it is sometimes assumed that those product design decisions are made one or more periods before the product is sold. \ If one assumes that $x_{it}$ is chosen at $t-1$ (and one continues to make the assumption that firms do not observe $\omega_{it}$ and $\kappa_{it}$ until $t$) then information set $I_{it-1}$ in Assumption 1 additionally includes $x_{it}$. \ This means that ((ref)) implies an additional orthogonality condition that can be used for estimation (since $\xi_{it}$ is now mean independent of $x_{it}$). Intuitively, since $x_{it}$ is chosen at $t-1$, innovations that are not observed until after that choice, e.g. $\xi _{it}$ are orthogonal to it.
In the dynamic panel literature, this assumption is often referred to as $ x_{it}$ being “predetermined”. \ Ackerberg (2020) details how these timing and information set assumptions can be strengthened or relaxed -- an important point is that it is not only the timing of when $x_{it}$ is chosen that matters but also what the firm knows at that time. \ For example, if $ x_{it}$ is chosen at $t-1$ but one assumes that firms observe $\omega _{it}$ and $\kappa _{it}$ also in $t-1$ (one period earlier), then one does not obtain additional moment restrictions.
So, consider an alternative model where firms make input decisions at time $ t-1$ as a function of their information set $I_{i,t-1}$, e.g.
Note that $\rho _{\omega }^{0}\omega _{i,t-1}$ is a conditional expectation of $\omega _{it}$ at time $t-1$. \ In a production function context, it makes sense that a firm would base their $x_{it}$ decision on their expectation of $\omega _{it}$, but this assumption is not necessary as our derivation is for arbitrary $\theta ^{0}$. Since $x_{it}$ is chosen at $t-1,$ it is now a function of $\kappa _{i,t-1}$ instead of $\kappa _{it}$ (because, like $\omega _{it}$, $\kappa _{it}$ has not been observed yet). \footnote{ Working with $\kappa _{it}$ instead of $\kappa _{i,t-1}$ does not change our result (i.e. firms knowing $\kappa _{it}$ at time $t-1$). What matters for our result is firm $i$ knowing only $\omega _{i,t-1}$ at time $t-1$, not $ \omega _{it}$. It does not matter for $\kappa _{it}$. In some sense it is just a relabeling of $\kappa _{it}$. More generally, what is important for a stronger timing/information assumption to rule out the pseudo-solution is the unobserved factor that enters both the productivity function and the input function, not for the unobserved factor(s) that only enters the input function.}
Following the arguments of the benchmark case, we can consider the basic dynamic panel moment evaluated at the pseudo-parameters $(\beta,\rho _{\omega },\alpha )=(\beta ^{0}+\frac{1}{\theta ^{0}},\rho _{x}^{0},\alpha ^{0}-\frac{\pi ^{0}}{\theta ^{0}})$. \ The first part of the derivation from Section 2 stays the same:
However, $(x_{it}-\rho _{x}^{0}x_{i,t-1})$ is now different. \ Using ((ref)), we have
and note also that we can write
Substituting these into ((ref)), we get
Importantly, because $x_{it}$ is chosen at $t-1$, it is in $I_{i,t-1}$. \ And because $x_{it}$ is chosen as a function of $\omega _{i,t-1}$ and $ \kappa _{i,t-1}$, it is clearly correlated with $u_{i,t-1}$ and $\xi _{it-1}$ . \ Hence, as long as $x_{it}$ is utilized in the formation of moment conditions, the pseudo-solution can be ruled out.\footnote{ This does not necessarily preclude possible pseudo-solutions at other points in the parameter space, but we have not found any in our simulations (nor have we found local minimums).} Of course, at the true parameters, the dynamic panel moment condition is
and since $\xi _{it}$ is not in $I_{i,t-1}$ and hence not correlated with $ x_{it}$, this continues to hold. \
In conclusion, if one is willing to make stronger timing assumptions, it appears that the pseudo-solution problem we have identified disappears. \ Presumably this is also the case if one is willing to make even stronger timing assumptions, e.g. where $x_{it}$ is chosen at some $t-\Delta $ where $ \Delta $ $>1$. \ On the other hand, and as evidenced by the empirical literature, these are often assumptions that researchers are unwilling to make, such as in the case of variable inputs in the production function context.
We now turn to strategies to resolve this under-identification problem. First, we focus on the benchmark model from Section 2 (i.e. linear $x_{it}$ process with AR(1) shock) and derive a system of reduced form equations that provides an alternative representation of the under-identification problem. Second, we show how, under a sign restriction on the structural parameter of the input process, we can use this reduced form representation to form an identification strategy. Estimation follows straightforwardly. Third, we use the intuition from this simple model to propose several practical strategies in more general models.
We now re-visit the baseline full model in Section 2 (equations ((ref)),((ref)),((ref)) and ((ref))). These equations imply the following system of “reduced-form” equations for $y_{it}$ and $x_{it}$, where the explanatory variables are dated at period t-1 or before\footnote{We thank Victor Aguirregabiria for suggesting this alternative representation.}
and where $e_{it}^{y}=\beta (\theta \xi _{it}+u_{it})+\xi _{it}+\eta _{it}-\rho _{\omega }\eta _{i,t-1}-\beta \theta (\rho _{\omega }-\rho _{x})\eta _{i,t-1}$ and $e_{it}^{x}=\theta \xi _{it}+u_{it}-\theta (\rho _{\omega }-\rho _{x})\eta _{i,t-1}$. The six reduced-form parameters in ((ref))-((ref)) relate to the six structural parameters $(\beta,\theta,\rho_{\omega},\rho _{x},\alpha,\pi )$ according to the equation system (ref) in the Appendix.
It is straightforward to consistently estimate the reduced-form parameters in equations ((ref)) and ((ref)). According to our assumptions, the two error terms are mean independent of $x_{i,t-1}$, but not of $y_{i,t-1}$ (since $y_{i,t-1}$ is a function of $\eta _{i,t-1}$). However, under our assumptions, $y_{i,t-2}$ is a valid instrument for $ y_{i,t-1}$ and can be used for IV estimation. Given IV estimates of the reduced form parameters $\pi $, we can recover the structural parameters $ (\beta ,\theta ,\rho _{\omega },\rho _{x},\alpha ,\pi )$ by inverting the system of parameter equations ((ref)). However, analogous to our derivations in Section 2, there are two solutions of the mapping from the reduced form parameters to the structural parameters. Notably, the values of $\theta $ at the two solutions are:
and the Appendix shows, for each of these possible $\theta$'s, there is a solution of the mapping from the reduced form parameters into the remaining structural parameters. The fact that ((ref)) imply that the $\theta$'s at the two possible solutions are guaranteed to have opposite sign will be useful to us momentarily.\footnote{This reduced form approach generalizes to the case of multiple endogenous variables. In this case, the mapping from the reduced form to the structural parameters is again from $R^{d}$ to $R^{d}$ (where d is the number of structural parameters) but now contains multiple quadratic equations, and, as illustrated in Section 3.2, there is typically more than one spurious solution.}
This reduced form representation is an alternative way of expressing the under-identification issue studied in this paper - one can easily verify that for the parameters $(\beta,\rho_{\omega},\alpha )$, the relationship between the two solutions\ are exactly the same as that described by Theorem 1. \ However, we think that this reduced form setup is theoretically interesting in itself because it shows that the under-identification issue can be alternatively represented in terms of a mapping from reduced form to structural parameters (rather than multiple solutions to a set of sample moments). \ Notably, this result means that for essentially any dataset, if one estimates this reduced form model (derived from the structural model ((ref)),((ref)),((ref)) and ((ref))), there will necessarily be two solutions. \ In other words, there will be two solutions regardless of the true DGP underlying the data. \ This seems counter to some of the results in Section 3, where we showed that for some DGPs (e.g. with stronger timing assumptions), the standard dynamic panel moments appear to not have multiple solutions. \ This is because there is a difference between the exact moments used in Sections 2 and 3 versus those used in the reduced form approach - those in the reduced form approach necessarily impose the input choice equation for all the moment conditions. \ This result also points to the pervasiveness of the potential under-identification problem in these types of models - one can get under-identification either due to the specific moments used (as in this section), or due to the DGP (as illustrated in Sections 2 and 3 using more traditional moments).\ While the latter might be most important in practice, the former also seems important to be aware of since researchers often experiment with different sets of moments in these types of problems.
This reduced form representation is also useful because it provides estimates of the parameters of the process determining $x_{it}$, i.e. $ (\theta,\rho_{x},\pi)$ and we next show how this, plus a potential sign restrictions on one of these parameters, provides one potential solution to the under-identification problem.
The above derivation indicates that the two solutions for $\theta $ produced by the reduced form approach necessarily have the opposite sign. \ Recall that $\theta ^{0}$ measures how the endogenous $x_{it}$\ depends on the unobserved term $\omega _{it}$, i.e. $x_{it}=\pi ^{0}+\theta ^{0}\omega _{it}+\kappa _{it}$. Thus, its sign represents the direction of the endogeneity problem. In many cases one may be willing to a priori sign $\theta ^{0}$ (e.g., in the production function context, firm profit maximization would likely imply $\theta ^{0}>0$; in other econometric contexts, one might be willing to sign the endogeneity problem (e.g., Manski and Pepper (2000)). In these cases, the reduced form representation provides a simple two-step method to consistently estimate the structural parameters, i.e. estimate ((ref)) and ((ref)) using IV estimation, use the mapping between the reduced form and the structural parameters to calculate the two sets of structural parameters, and choose the appropriate one given the sign restriction on $\theta ^{0}$.
In fact, a sign restriction can help resolve the under-identification problem even when we only consider moment condition ((ref)), as more commonly done in the traditional dynamic panel approach. In Section 2, we showed that both $A=(\alpha ^{0},\beta ^{0},\rho _{\omega }^{0})$ and $ B=(\alpha ^{0}-\frac{\pi ^{0}}{\theta ^{0}},\beta ^{0}+\frac{1}{\theta ^{0}} ,\rho _{x}^{0})$ solve the conditional moment condition (ref) given the information set $I_{it}$. Since element-wise differencing $A-B$ gives $(-\frac{\pi ^{0}}{\theta^{0}},-\frac{ 1}{\theta ^{0}},\rho _{\omega }^{0}-\rho _{x}^{0})$, the difference between the two solutions for $\beta $ depends only on $\theta^{0}$ - the structural parameter in the input process. Therefore, if we find both solutions to the moment condition, knowledge of the sign of $\theta^{0}$ tells us which of the two solutions is the correct one. In other words, if we assume $\theta^{0}>0$, then the coefficient vector with the lower estimate of $\beta ^{0}$ is the true solution (and vice-versa).\footnote{ In some cases, it may be more appealing to make sign restrictions on other parameters, e.g. on $\beta $. Unlike $\theta $, there is no guarantee that the true and pseudo-solutions for $\beta $ have opposite sign. However, they can have opposite signs (if $\beta ^{0}$ and $\theta ^{0}$ have opposite signs and $\left\vert \frac{1}{\theta ^{0}}\right\vert >\left\vert \beta^{0}\right\vert)$, and this could provide the basis for identifying the pseudo-solution. In any case, we should be aware that the key to utilizing sign restrictions is finding both minima first.}
Sections (ref) and (ref) suggest a reduced form approach together with a sign restriction to solve the under-identification problem in our simple model. However, this relies on the specific DGP for $x_{it}$ given by ((ref)) and ((ref)). Given that the previous section suggests that these dynamic panel and proxy variable approaches have similar identification issues with more general DGPs for $x_{it}$, a natural question is what to do in those cases. In particular, suppose a researcher is not willing to impose ((ref)) and ((ref)) and instead wants to allow for (or model) a more general $x_{it}$ process. \ Obviously, it is difficult for us to give exact advice given the many possible models that researchers may construct. However, we feel our solution in the simple case provides some constructive advice on how researchers might reduce the likelihood of ending up at a pseudo-solution in more complex models.
One possibility would be to specify the DGP for $x_{it}$ up to parameters and follow the logic of Sections (ref) and (ref) to derive an analogous system of estimating equations. Given the intuition above, imposing assumptions on the direction of the relationship between $x_{it}$ and $\omega _{it}$, analogous to the sign restriction on $\theta $ used above, could resolve the under-identification problem.
However, suppose one is unwilling to commit to assuming a specific model for $x_{it}$. This is perhaps more natural given the history of this literature - one nice aspect of both the dynamic panel and proxy variable methods is that they do not rely on fully specifying a parametric DGP for $x_{it}$ (though often they do make some assumptions on these DGPs, e.g. the scalar unobservable assumption used in the proxy variable literature). \ We also believe our findings lead to constructive practical advice in this situation. In particular, in our simple model (and in “nearby” more complex models), the pseudo-solution for $\beta$ is greater than the true $\beta^{0}$ (this is based on $\theta ^{0}>0$, the reverse is true when $\theta^{0}<0$). This means that we know the direction in which we might err. In particular, if one has found a candidate estimate $\widehat{\beta}$ (and believes $\theta^{0}>0$), one might want to be extra diligent in continuing to search for minima at parameters where $\beta <\widehat{\beta}$. This is because if one has erroneously found the spurious zero, the non-pseudo-solution for $\beta$ should be below $\widehat{\beta}$. One should of course always be diligent in numeric search to avoid local minima, but the results here suggest one should be extra concerned about local minima in one specific direction relative to a candidate estimate $\widehat{\beta}$.
Another possibility is to estimate the model based on only the traditional dynamic panel moment condition ((ref)), i.e. without specifying a DGP for $x_{it}$, but to then do some ex-post testing of that solution. In particular, given a candidate estimate $(\widehat{\alpha}, \widehat{\beta},\widehat{\rho}_{\omega})$ based on moment condition ((ref)), one can construct the residuals
Again, assuming one is willing to sign $\theta$, one could examine the relationship between $x_{it}$ and this residual, e.g. through a regression or correlation analysis. Although this regression cannot be interpreted causally because of the existence of $\eta_{it}$ in the constructed residual, the measurement error tends to attenuate the coefficient.\footnote{ Note that this would be easier in the proxy variable methods, as a by-product of the assumptions guaranteeing invertibility in those models allows one to separately identify $\widehat{\omega _{it}}$ and $\widehat{ \eta_{it}}$.} \ So, at least in models “close” to our simple model, one would expect that the sign of this relationship at the pseudo-solution would be the reverse of what one would expect. \ In other words, if one believes that $\theta>0$, an indication of a possible pseudo-solution would be finding a negative correlation between $\widehat{\omega _{it}+\eta _{it}}$ and $x_{it}$.\footnote{ Note that this is related to the practice of comparing these coefficient estimates to those of standard OLS, i.e. assessing signing the bias of OLS. This is because the sign of the OLS bias is related to $\theta $. So the possibility of pseudo-solutions provides another rationale for doing this, i.e. if one is at the pseudo-solution, the sign of the OLS bias may be the opposite of what one expects it to be.} Alternatively, one can impose the sign restriction as a moment inequality: $E[x_{it}(y_{it}-\alpha ^{0}-\beta ^{0}x_{it})]\geq 0$.
Lastly, one might consider starting by estimating a model that is more robust to the pseudo-solution problem but is not necessarily an appropriate model. \ Initial estimates from this misspecified model can be used as starting values for estimating the correct model to reduce the likelihood of ending up at a pseudo-solution in the correct model. \ Our above analyses suggest at least two possibilities for this. \ First, even if one does not believe the stronger timing assumption of Section (ref), one might estimate that model anyway. Then one could use the estimates as starting values for a model with the (correct) weaker timing assumption. \ The hope is that the stronger timing assumption does not overly bias the parameters, and thus keeps the numeric search procedure from straying to a pseudo-solution. \ \ Second, one could assume input choice ((ref) ) and use our reduced-form procedure and the sign restriction to recover structural parameters, even if one does not believe that the input choice equation is exactly specified. \ One could again use the resulting estimates as starting parameters for an estimation procedure that does not impose a specific input choice equation. \ Again, the hope is that while those starting parameters may be inconsistent, they may be close enough to the true minima to be helpful at getting the numeric search procedure to get the true solutions as long as the input process is not too far away from the linear input demand model.
Of course, these informal suggestions are not foolproof in avoiding pseudo-solutions, and in some datasets with even more complex $x_{it}$ processes, there may not even be pseudo-solutions. \ However, we feel that being aware of the possibility of such pseudo-solutions and this guidance for avoiding them can be useful for applied researchers.
We have studied a potential identification problem in methods that use timing and information set assumptions to resolve endogeneity problems in structural models. These methods have been applied widely, both in the production function context and elsewhere. Prior work calls attention to this identification problem in a fairly narrow context. We show that the identification problem appears to be endemic across a broad set of models using these types of assumptions, and demonstrate how to resolve it using a sign restriction and additional modelling. We then suggest ways of extracting intuition from that solution to more complex empirical settings.
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