Equilibrium Counterfactuals

We incorporate structural modellers into the economy they model. Using the traditional moment-matching method, they ignore policy feedback and estimate parameters using a structural model that treats policy changes as zero probability (or exogenous) "counterfactuals." Estimation bias occurs since the economy's actual agents, in contrast to model agents, understand policy changes are positive probability endogenous events guided by the modellers. We characterize equilibrium bias. Depending on technologies, downward, upward, or sign bias occurs. Potential bias magnitudes are illustrated by calibrating the Leland (1994) model to the Tax Cuts and Jobs Act of 2017. Regarding parameter identification, we show the traditional structural identifying assumption, constant moment partial derivative sign, is incorrect for economies with endogenous policy optimization: The correct identifying assumption is constant moment total derivative sign accounting for estimation-policy feedback. Under this assumption, model agent expectations can be updated iteratively until the modellers' policy advice converges to agent expectations, with bias vanishing.


Introduction
The strength of structural methods relative to quasi-experimental methods is their (potential) ability to overcome the rational expectations critique of Lucas (1976) who described pitfalls in extrapolating econometric estimates across policy environments. For example, Blundell (2017) writes, "By specifying the parameters that describe the preferences and constraints of the decision-making process, structural models deliver counterfactual predictions. The ability to provide policy counterfactuals sets them apart from reduced-form models." The starting point for this paper is to note that, intent notwithstanding, traditional structural methods (e.g. moment calibration, simulated method of moments) violate rational expectations when the structural models serve their intended function of rigorously informing policy decisions.
To see this, consider that in specifying the decision problem of agents inside her model, the structural econometrician must specify government policy. Critically, it is customary to parameterize models according to status-quo policy (or to specify policy as an inviolable exogenous process). This standard practice violates rational expectations since the agents inside the model are treated as being ignorant of future endogenous policy changes despite the goal of the econometrician being to inform endogenous policy decisions.
What are the implications of such violations of rational expectations for structural econometrics, and what can be done about them operationally? To address these questions, we consider an economy in which "real-world" agents with rational expectations are placed alongside a structural econometrician who will give policy advice. That is, we develop a model that incorporates the structural modeler within it, consistent with the "communism of models" of Sargent (2005). The real-world agents are privately endowed with a policy-invariant parameter. 1 Knowledge of this parameter would be su¢ cient for the government to set policy at …rst-best. The econometrician will observe an empirical moment derived from agent actions which will serve as the basis for her parameter inference.
The baseline model ingredients are as follows. When the model opens, government policy is set at a pre-determined "status quo" value, denoted 0 . 2 Nature then draws the unknown parameter the real-world moment also varies indirectly due to the rational expectations of agents of feedback from the parameter inference (b u) to discretionary government policy (g(b u)). Thus, the real-world empirical moment can be expressed as m[u; g(b u(u))]. It is the indirect e¤ ect arising from joint estimation and policy control, m g g 0 b u 0 , that is generally omitted in structural estimation. Phrased di¤erently, the policy expectations of agents are not constants and not exogenous processes but are instead functions of the deep parameters being estimated.
There are three cases to consider. In the …rst case, the sign of the indirect e¤ect is opposite to that of the direct e¤ect. Here, the estimated parameter overshoots for u < u 0 and then undershoots for u > u 0 (and recall, u 0 justi…es the status quo policy 0 ). Intuitively, the modeler incorrectly treats small observed changes in the empirical moment to small changes in u because she here fails to account for the countervailing indirect e¤ect. In the second case, the indirect e¤ect is small in absolute value and has the same sign as the direct e¤ect. Here, the estimated parameter undershoots for u < u 0 and overshoots for u > u 0 . Intuitively, the modeler incorrectly treats large observed changes in the empirical moment to large changes in u because she here fails to account for the amplifying indirect e¤ect. In the third case, the indirect e¤ect is large in absolute value and has the same sign as the direct e¤ect. Here, the estimated parameter actually decreases with the true parameter, and it possible for an equilibrium to arise where the estimated parameter always has the wrong sign. The subtle intuition for this case is provided in the body of the paper.
We illustrate the potential quantitative signi…cance of these e¤ects by considering an econometrician whose objective is to infer bankruptcy costs using the canonical structural model of Leland (1994). In particular, we consider the recent cut in the corporate income tax rate implemented under the Tax Cuts and Jobs Act of 2017. Here the structural econometrician backs out implied bankruptcy costs from observed values of corporate interest coverage ratios. By assumption, the econometrician knows the underlying real technology but fails to impose the assumption of rational expectations on the part of …rms inside the model. In our calibrated example, this leads to an eight-fold overstatement of bankruptcy costs. Intuitively, …rms rationally anticipate a tax cut and thus choose low leverage in light of the low value of future debt tax shields. Neglecting this fact, the econometrician mistakenly infers that the low leverage stems from extremely high bankruptcy costs.
Importantly, we show that the nature of estimation must change radically when the government will (with positive probability) change policy based upon the structural econometrician's parameter estimates. That is, the econometric procedure must change as one moves from passive to active estimation. For example, with policy feedback, the standard moment monotonicity condition, which focuses on partial derivatives of moments, is neither necessary nor su¢ cient for correct structural parameter identi…cation. Rather, we show that total derivatives, cum policy feedback, are the correct moment selection criterion in the context of joint estimation and control exercises.
This implies that a moment that is informative (uninformative) under passive estimation may be uninformative (informative) under active estimation. That is, moment selection should vary according to whether estimation is active or passive.
Based on the preceding insights, we develop a simple algorithmic procedure for achieving unbiased parameter estimates and …rst-best government policy. Recall, the underlying source of bias was that the agents inside the structural model were treated as being ignorant of possible endogenous policy changes whereas real-world agents have rational expectations and understand that endogenous policy changes are positive probability events. Thus, there was a systematic gap between model agent beliefs and real-world agent beliefs, a gap left open using standard moment matching procedures. However, this gap can be closed by iterating on inference and policy advice. The econometrician starts iteration n with a provisional policy recommendation n . A corresponding parameter inference b u n is derived by matching the observed real-world moment with the n th iteration model-implied moment m(b u n ; n ). That is, in iteration n agents are treated as anticipating implementation of the provisional policy recommendation. Next, the implied optimal government policy g(b u n ) is computed and treated as the next provisional policy recommendation n+1 : Iteration proceeds until policy convergence/internal consistency. That is, a …xed-point is found where the policy advice supports the parameter inference, and vice-versa.
We turn now to other related literature. At core, our argument is related to the seminal paper by Hurwicz (1962) which o¤ers an early formal de…nition of structure in econometrics research.
According to Hurwicz, an equation can be said to be structural if it is invariant over the "domain of modi…cations anticipated." He writes: The concept of structure is relative to the domain of modi…cations anticipated. In particular, the structure is not necessarily de…ned for every domain W . Hence a certain equation of a system may be in structural form relative to some W 0 but not relative to W 00 . If two individuals di¤er with regard to modi…cations they are willing to consider, they will probably di¤er with regard to the relations accepted as structural.
The essence of our argument is that if real-world agents have rational expectations, and if the structural analysis is actually policy-relevant, the empirical moments targeted by modellers are not invariant over the domain of policy modi…cations considered by modellers.
In the spirit of our paper, Sargent (1987) sketched the existence of problems inherent in joint estimation and control under rational expectations: "There is a logical di¢ culty in using a rational expectations model to give advice, stemming from the self-referential aspect of the model that threatens to absorb the economic adviser into the model... That simultaneity is the source of the logical di¢ culties in using rational expectations models to give advice about government policy." These philosophical and logical di¢ culties apparently led Sargent to shy away from the use of macroeconometric models for the purpose of informing policy decisions. For example, Sargent (1998) states, "That's a hard problem. I don't make policy recommendations." 7 Building on the earlier work of Sims and Zha (2006), and Farmer, Waggoner and Zha (2009), Bianchi and Ilut (2017) show how such internal contradictions can be avoided provided one con…nes attention to a speci…c form of "counterfactual." In this approach, historical and prospective government policy is modeled as an exogenous Markov chain whose probability law the econometrician is not allowed to alter. Counterfactual analysis is then performed on particular historical periods by holding …xed non-policy shocks and then pretending as-if the realization of the Markov chain di¤ered from actual policy during the period of interest, e.g. a counterfactual shift from passive to active monetary policy during the '60s and '70s. Strictly speaking, such an approach precludes the econometrician giving advice that can actually alter policy, as well as analysis of novel policies.
After all, to escape the rational expectations trap here, the policy Markov chain must be treated as unalterable, otherwise the feedback bias we describe would emerge.
Also related to the present paper is work by Chemla and Hennessy (2019) showing that a bias arises when quasi-experimental evidence is used to inform endogenous policy decisions. Arguably, the present paper's critique is more problematic in that it is internal, taking models and agent ra- 7 Quoted in Sent (1998). tionality seriously, a goal shared by many structural econometricians. Another important di¤erence is that within the logic of a structural model, bias characterization is simpler. Finally, the present paper o¤ers a feasible algorithm for avoiding bias and achieving …rst-best policy, again within the logic of a structural model. Despite these di¤erences, the two papers share the message that the econometric tool-kit changes fundamentally as one moves from passive to active policy-relevant estimation.
Structural methods have been used across a wide variety of applied …elds. In their in ‡uential paper, Kydland and Prescott (1996)  The rest of the paper is as follows. Section 2 describes the economic setting. Section 3 characterizes the nature of bias if the econometrician fails to impose rational expectations. Section 4 shows how, under technical conditions, unbiased parameter inference and …rst-best government policy can be achieved through a fully-consistent application of rational expectations. In addition, Section 4 shows how traditional moment selection criteria are altered when one moves away from a pure estimation setting to a setting with joint estimation and control. Section 5 presents a quantitative example. Section 6 considers the possibility of sticky and extrapolative expectations for both the agents and the econometrician and provides a multivariate extension.

The Economic Setting
We consider a univariate parameter inference problem where the econometric model is exactly identi…ed. The …rst subsection describes timing and technology assumptions. The second subsection illustrates how the general framework maps to a speci…c applied econometric problem.
There is a real-world representative sample consisting of a continuum of atomistic agents ("…rms") privately endowed with a policy-invariant ("deep") structural parameter. Knowledge of this parameter is su¢ cient for the government to set policy optimally.
An econometrician will observe an empirical moment derived from the measured actions of the sample …rms. To …x ideas, one can think of the moment as being the sample mean of investment, new employees, R&D, or leverage. In practice, moments such as variance, skewness, or kurtosis may also be informative about deep …rm-level parameters. In the context of indirect inference, the moment can be the coe¢ cient obtained when …rm decision variables are regressed on observable covariates, such as the coe¢ cient on market-to-book (Q) in an investment regression. Examples are provided below.
The econometrician has developed a structural model and will match her model-implied moment with the observed empirical moment. Importantly, under conditions derived below, if the econometrician were to impose rational expectations in a fully internally consistent manner, this moment matching procedure would allow her to infer the true value of the deep parameter and the government would then be able to correctly determine the optimal policy.
The atomistic …rms are rational, forward-looking, and act non-cooperatively. Each atomistic …rm correctly understands it cannot change the moment observed by the econometrician by unilaterally changing its own action.
The deep parameter, denoted u, is common to all sample …rms. However, this assumption does not preclude …rm heterogeneity. For example, …rms may be identical ex ante but face idiosyncratic shocks ex post. Alternatively, …rms may face idiosyncratic shocks that alter their measured actions.
Finally, …rm-level parameters might be, say, multiples of a common aggregate parameter u, e.g.
i is a …rm-speci…c scalar known by …rm i: An alternative technological assumption, not adopted here, is that each …rm receives a noisy signal of the common parameter u. In such a setting, as in the present setting, parameter inference would need to account for feedback from inference to the policy variable.
The parameter u represents the realization of a random variable e u with cumulative distribution function with a strictly positive density on R with no atoms. The realized parameter u is privately observed by each of the sample …rms, but unobservable to the econometrician and the government. Below, b u(u) denotes an equilibrium parameter inference by the econometrician in the event that e u = u, with b u( ) denoting an equilibrium inference function.
Timing is as follows. When the model opens at time t = 0; the government policy variable is initially equal to the pre-determined status-quo 0 2 where the set of feasible government policies is ( ; ). Next, nature draws u according to the distribution function : Each sample …rm i then chooses an optimal pre-inference action i . This action can be multi-dimensional. The econometrician then observes the empirical moment m, which is derived from the pre-inference actions of the sample …rms. Next, the econometrician will attempt to match her model-implied moment with the empirical moment, resulting in parameter inference b u: The econometrician then reports b u to the government. All of these events take place at the initial time t = 0: Time is either discrete or continuous and the horizon can be …nite or in…nite. There is an independent stochastic process d such that for all t 0; d t 2 f0; 1g. Let The optimal pre-inference action of …rm i can be expressed as where the subscript i captures idiosyncratic shocks and the semi-colon separates variables from the constant 0 .
It is assumed that observation of a continuum of sample …rms is su¢ cient to ensure that any idiosyncratic shocks have no e¤ect on the observed moment, so that m can be expressed as m(u; ; 0 ): For brevity, the constant 0 will be suppressed and the empirical moment will be represented by the following mapping: The …rst argument in the moment function m is the unknown parameter u 2 R. The second argument in the moment function is anticipated discretionary government policy 2 .
The following assumption ensures the setting considered is seemingly-ideal. We next characterize how the moment varies with anticipated discretionary government policy. The function g : R ! represents optimal discretionary government policy. If the government had the ability to directly observe u, its optimal discretionary policy would be g(u). Of course, the sample …rms will have already chosen their pre-inference actions i . However, the government correctly understands that should it enjoy discretion, its policy choice , in addition to the parameter u, will determine the post-inference actions of the sample …rms and/or other agents in the economy, e.g. future generations of …rms. The function g represents the socially optimal u-contingent government policy in light of the relevant tradeo¤s. The following assumption is imposed.
Assumption 3. The optimal government policy g is a continuously di¤ erentiable strictly monotonic function mapping R onto .
The government is presumed to believe that the standard moment matching exercise will allow the econometrician to deliver a correct estimate of the unknown parameter. Critically, Assumption 1 would seem to imply that this con…dence is justi…ed. After all, the model moment function is equal to the empirical moment function, and the moment is monotone in the unknown parameter.
We have the following assumption.
Assumption 4. The government chooses discretionary policy optimally given its belief that for all From Assumption 4 it follows that for all u 2 R; the endogenous discretionary policy of the government is An alternative interpretation of condition (4) is that the function g represents equilibrium policy outcomes from an extensive form game in which the econometrician's parameter estimate is fed into the political process. This alternative interpretation would not alter the characterization of bias below, but would necessarily rule out characterization of the welfare consequences of biased parameter inference.
We posit that the real-world …rms form rational expectations. In particular, real-world …rms know that the government may enjoy policy discretion at some future date. They also know the government will place full faith in the econometrician's structural parameter estimate b u, and will then input this estimate into the policy function g: The following assumption formalizes this speci…cation of …rm beliefs.

Assumption 5 [Agent Rational Expectations].
For all u 2 R; real-world …rms correctly anticipate discretionary government policy, with The …rst equality in the preceding equation ensures that ( ) satis…es rational expectations. The second equality re ‡ects how discretionary government policy will actually be formed in equilibrium, with b u(u) being fed into g: E¤ectively, under rational expectations, the real-world …rms infer the econometrician's parameter estimate which allows them to correctly anticipate discretionary government policy.
From the preceding equation it follows that the empirical moment observed by the econometrician is: In reality, the post-inference government policy follows the stochastic process described in equation (1). The real-world sample …rms have rational expectations and understand this. However, we assume the econometrician departs from rational expectations by parameterizing her structural model according to the status-quo. We have the following assumption.

Assumption 6 [Status Quo Parameterization].
Firms inside the structural model anticipate that the status quo will be maintained even if the government enjoys policy discretion, with the belief Notice, by parameterizing her model according to the status quo, the econometrician implicitly treats the …rms as being unaware of her own activities and the policy function they are intended to serve, informing the government's discretionary decisions. Below we analyze the implications for parameter inference and government policy.
From the preceding discussion it follows that for all u 2 R; the structural econometrician's parameter estimate will be derived from the following inference equation The left side of the preceding equation is the real-world empirical moment. The empirical moment re ‡ects the fact that the sample …rms will choose their pre-inference actions optimally given the true parameter value u and their correct anticipation of discretionary government policy status quo parameterization (Assumption 6). The estimated parameter b u(u) is chosen so that the model implied moment is equal to the observed empirical moment.
Before proceeding, it is worthwhile to consider an alternative, more complex, motivation for the inference equation (8) since this equation serves as the foundation for all subsequent results regarding bias and bias correction. In particular, suppose instead the structural model does not treat government policy as …xed forever at 0 but instead treats government policy as an exogenous stochastic process as in, say, Keane and Wolpin (2002). To approximate such an inference approach within our framework, one can think of the structural model as treating government policy as an independent discrete-state Markov chain with one state, call it state 0, being the one real-world state in which the government will enjoy full policy discretion and follow the policy advice o¤ered by the econometrician. The structural model then incorrectly treats government policy in state 0 as being an exogenous parameter 0 while the real-world …rms understand that government policy in the discretionary state 0 will be endogenously set at g(b u(u)). The inference equation (8) still applies in such a setting, and consequently, so do all the results that follow below. Having said this, it is clear that the approach of Keane and Wolpin (2002), while violating rational expectations in exercises of joint estimation and control, still o¤ers an improvement over the common practice of treating policy as …xed forever at the status quo.

Example: Inferring Labor Adjustment Costs
At this stage it will be useful to …x ideas by considering a stripped-down example of the type of inference problem subsumed by our model. To this end, consider an econometrician who wants to The status quo features 0 units of regulation. The government will enjoy policy discretion with probability p > 0 and …rms anticipate units of discretionary regulation. The sample …rms make their hiring decisions before the policy uncertainty is resolved. Each "real-world" …rm solves the following linear-quadratic program: In the preceding equation, q > 0 represents the shadow value of an "installed" worker-the net present value of marginal product less wages. For simplicity, assume q is known to the econometrician. 8 The function N is, say, the normal cumulative distribution. This function is scaled by expected units of regulation. The term " i is mean-zero …rm-speci…c shock. In this way, the structural estimation allows for heterogeneity.
Imposing rational expectations, with = ; the econometrician observes the following empir- The econometrician chooses her parameter estimate so that the model-implied moment is just equal to the observed empirical moment. The inference equation (8) is: Rearranging terms in the preceding equation we …nd From the preceding equation it follows that That is, parameter inference is unbiased at point u if and only if the status quo is actually optimal at that point. The next subsection o¤ers a more general and precise characterization of bias.

Bias Characterization
This section characterizes the nature of parameter inference and associated policy outcomes if the structural model fails to impose the assumption that …rms have rational expectations.
Before proceeding, it will be convenient to express the di¤erential form of the inference equation.
In particular, under technical conditions derived below, there will exist a continuously di¤erentiable function b u( ) satisfying the inference equation (8). Assuming such a function exists, we have the following di¤erential form: The di¤erential form of the inference equation makes clear the potential for bias. The right side captures the econometrician's faulty inference procedure which is predicated upon the incorrect assumption that …rms expect the status quo to be maintained with probability 1. Thus, she incorrectly imputes any change in the observed moment to the direct e¤ect as captured by the partial derivative, m u : The left side of the preceding equation captures the true total di¤erential of the empirical moment with respect to u: If u is perturbed, there will be a direct e¤ect on the moment as captured by the …rst term, m u : In addition, the empirical moment will vary due to the rational anticipation of …rms that government policy will change based upon changes in the econometrician's parameter inference. This inference-policy feedback e¤ect is captured by the second term on the left side of the equation (m g 0 b u 0 ).
Let u 0 be the unique value of the parameter u at which a fully-informed government would …nd it optimal to implement the status quo policy 0 . That is Uniqueness of u 0 and invertibility follow from g being strictly monotone (Assumption 3).
The next proposition characterizes the realization(s) of the random variable e u at which parameter inference will be unbiased. Proof. Referring to the inference equation (7), it follows from the strict monotonicity of m in its Again referring to the inference equation (7), it follows from the strict monotonicity of m in its Finally if point u is a point such that parameter inference is unbiased and the status quo is optimal then it must be that From the strict monotonicity of g the unique point at which this occurs, u 0 : The intuition for the preceding result is as follows. At any realization of u other than u 0 , realworld …rms anticipate the government will implement a policy di¤erent from the status quo should it enjoy policy discretion. The real-world …rms then change their optimal behavior accordingly, leading to changes in the observed moment. However, under Assumption 6, the econometrician fails to take the inference-policy feedback e¤ect into account, leading to bias.
Having established parameter inference will only be unbiased at point u 0 , the next proposition provides insight into the nature of bias at all other u 2 R.
Proposition 2. Let the inference equation (7) be satis…ed at point u by b u(u). If m u m > 0, then Proof. There are four cases to consider. Suppose …rst m is increasing in both arguments. Then from the inference equation (7) it follows Suppose next m is decreasing in both arguments. Then Suppose next m is decreasing in its …rst argument and increasing in its second argument. Then Suppose …nally m is increasing in its …rst argument and decreasing in its second argument. Then The intuition behind the preceding result is as follows. Per Assumption 6, the econometrician's structural model incorrectly stipulates …rm beliefs at any u at which the discretionary government policy will di¤er from the status quo. This incorrect stipulation of beliefs leads to incorrect inference.
For example, taking the …rst part of the proposition, suppose the empirical moment function m is increasing (decreasing) in both arguments. Then if, say, (u) > 0 , the moment will be higher (lower) than would be inferred based upon the direct e¤ect m u ; causing b u to overshoot u. Taking the second part of the proposition, suppose m u > 0 and m < 0. Then if, say, (u) > 0 , the moment will be lower than would be inferred based upon the direct e¤ect m u ; causing b u to undershoot u.
The preceding proposition characterizes b u at a particular point u where the inference equation Notice, under the stated conditions, the term in square brackets in the preceding equation is strictly positive, implying the derivative of the function b u is positive. Finally, the last statement in the To illustrate the preceding lemma, and many that follow, it will be useful to de…ne a linear technology: where ; and are arbitrary nonzero constants. Under the linear technology, the inference equation (8) can be written as From equation (15) it follows that here 0 = u 0 . Using this fact, and rearranging terms in the preceding equation, the inference equation can be expressed as Under the conditions stated in Lemma 1, b u 0 is some constant in (0; 1): Lemma 1 leads directly to the following proposition. For all u < u 0 , b u(u) 2 (u; u 0 ) and for all u > u 0 , b u(u) 2 (u 0 ; u). If g is increasing, then u < u 0 implies (u) 2 (g(u); 0 ) and u > u 0 implies (u) 2 ( 0 ; g(u)). If g is decreasing, then u < u 0 implies (u) 2 ( 0 ; g(u)) and u > u 0 implies (u) 2 (g(u); 0 ).
Proof. The …rst statement in the proposition is from Lemma 1. Next note that b u 0 (u 0 ) 2 (0; 1): It follows that for u on the left neighborhood of u 0 ; b u(u) 2 (u; u 0 ) and for u on the right neighborhood Inspection of equation (14) reveals the intuition for the preceding proposition. Under the stated assumptions, the second term on the left side of the di¤erential form of the inference equation (14) dampens the sensitivity of the moment to changes in u-an e¤ect ignored by the econometrician.
She will then incorrectly impute the small changes in the moment to small changes in u: That is, b u will tend to have a slope less than unity, with b u overshooting for u < u 0 and undershooting for u > u 0 : These e¤ects are illustrated in Figures 1, 2 then there exists a continuously di¤ erentiable strictly monotone function b u( ) satisfying the inference equation (7) for all u 2 R.
Proof. De…ne the following candidate solution to the inference equation Since here b u(u 0 ) = u 0 ; the candidate solution satis…es the inference equation at u 0 (Proposition 1). Further, under the stated assumptions, the candidate solution has a well-de…ned derivative at all points, given in equation (17). Rearranging terms in equation (17), it follows that the candidate solution satis…es the di¤erential form of the inference equation (14) (22) is violated. Thus, b u must be strictly monotonic.
To take a speci…c example, if the conditions of Lemma 2 were to be satis…ed in the context of the linear technology (equation (19)), then it follows 6 = and the linear technology inference function (21) along with its derivative would be well-de…ned.
We have the following proposition. there exists a continuously di¤ erentiable strictly monotonic increasing function b u( ) satisfying the inference equation. For all u < u 0 , b u(u) < u and for all u > u 0 , b u(u) > u. If g is increasing then u < u 0 implies (u) < g(u) and u > u 0 implies (u) > g(u). If g is decreasing then u < u 0 implies (u) > g(u) and u > u 0 implies (u) < g(u).
then there exists a continuously di¤ erentiable strictly monotonic decreasing function b u( ) satisfying the inference equation. For all u < u 0 , b u(u) > u 0 > u and for all u > u 0 , b u(u) < u 0 < u: If g is increasing then u < u 0 implies (u) > 0 > g(u) and u > u 0 implies (u) < 0 < g(u): If g is decreasing then u < u 0 implies (u) < 0 < g(u) and u > u 0 implies (u) > 0 > g(u): Proof. From Lemma 2 there exists a continuously di¤erentiable strictly monotonic solution to the inference equation. From the …nal line in equation (18) it follows Considering this case, b u must be strictly monotone increasing. Moreover, on the left neighborhood of u 0 ; b u(u) < u and on the right neighborhood of u 0 ; b u(u) > 0. From the continuity of b u and Proposition 1 it follows that for all u < u 0 , b u(u) < u and for all u > u 0 , b u(u) > u: For the second part of the proposition, note that Considering this case, b u must be strictly monotone decreasing. It follows that for all u < u 0 , b u(u) > u 0 > u and for all u > u 0 , b u(u) < u 0 < u: The clauses pertaining to discretionary government policy follow from the fact that = g(b u): Inspection of equation (14) reveals the intuition for the …rst part of the preceding proposition.
Under the posited technologies, the policy feedback e¤ect causes the observed moment to be more sensitive to changes in u than is understood by the econometrician. She will then incorrectly impute large changes in the moment to large changes in u: That is, b u will tend to have a slope in excess of unity, so that b u undershoots for u < u 0 and overshoots for u > u 0 : In other words, the function b u(u) will cross the function u at the point u 0 from below.  Figure 4 shows how the econometrician will incorrectly impute large changes in the moment to large changes in u: Figure 5 shows the resulting single crossing of b u with u from below. Finally, since g has here been assumed to be increasing, Figure 6 shows the resulting policy undershooting for low values of u and overshooting for high values of u: The second part of the preceding proposition is illustrated most vividly by considering a particular example. To this end, consider the same linear moment m = u + but now assume g = 2b u; with u 0 = 0. That is, in the case being considered, discretionary government policy is more sensitive to the inferred value of the structural parameter. Equation (21)

Joint Estimation and Control under Rational Expectations
This section considers whether and how the econometrician can achieve unbiased parameter inference.

Avoiding Bias and Achieving Optimality
A natural to ask is whether it is possible to achieve unbiased parameter inference in the setting considered. Introspection suggests a ready solution. The underlying source of biased parameter inference in the preceding section was the failure of the econometrician to parameterize her model in a manner consistent with the rational expectations held by the …rms (Assumption 6). Therefore, achieving unbiased inference would seem to necessitate "parameterizing" expectations correctlywith the issue being that the policy expectation is correctly understood as a function, rather than a parameter. Indeed, we have the following lemma.
The second implication follows from the strict monotonicity of m in its second argument.
Of course, the government's ultimate objective is not to achieve unbiased parameter inference but rather to implement the optimal policy when it enjoys discretion. Therefore, the government would like to construct a rational expectations equilibrium predicated upon correct inference and …rms anticipating a speci…c endogenous outcome But a necessary condition for correct parameter inference to be feasible for all u is that the empirical moment be invertible. To this end, let We then have the following proposition.
For su¢ ciency, note It follows that in order for the econometrician to avoid bias and achieve …rst-best, she must replace the faulty inference equation (7) with the rational expectations inference equation Of course, the measured agents must understand the econometrician's procedure. Formally, in a rational expectations equilibrium there is no need for any agent to make a speech. Nevertheless, heuristically, in support of the postulated equilibrium, the econometrician could be understood as making the following speech to the …rms.
I the structural econometrician will correctly infer the true value of the parameter u from the observation of the moment m that your actions generate. Further, armed with my correct inference, the government will implement the optimal policy g(u) should it enjoy policy discretion. And now that I have made this speech to you, I know that you know I will do this, and so you should anticipate g(u) as the discretionary government policy and, thus, act accordingly.
To further aid intuition, it is useful to express the rational expectations inference equation (28) in di¤erential form: The left side of the preceding equation re ‡ects how the moment actually changes with u; and the right side re ‡ects how the structural model treats the moment as changing with u: The econometrician's structural model of …rm behavior now takes into account …rm expectations regarding policy recommendations, while the "counterfactuals" approach failed to do so.

Gallant and Tauchen Revisited
In the title to their important paper, Gallant and Tauchen (1996) pose a question often asked by structural modellers: "Which Moment to Match?"An overarching message of our paper is that the nature of econometric inference changes fundamentally if one is attempting joint estimation and control, rather than simply attempting estimation. This message carries over to moment selection.
To illustrate, consider an econometrician operating in a world with linear technologies, with two competing moments being considered candidates for matching. In particular, suppose the optimal government policy is u, where moments 1 and 2 have the following forms, respectively: According to the traditional moment selection criteria, moment 1 would be discarded since it violates the standard moment monotonicity condition (Assumption 1). In particular, according to the traditional moment selection criteria, moment 1 would be viewed as completely uninformative about the unknown parameter. In contrast, moment 2 would be viewed as informative about the unknown parameter.
But recall, the econometrician is engaged in an exercise of joint estimation and control, with the government attempting to achieve …rst-best. In this context, moment 1 is informative and moment 2 is uninformative. In particular, consider a conjectured rational expectations equilibrium with correct inference and …rst-best policy implementation. In such an equilibrium the two moments can be expressed as univariate functions of the unknown parameter. We have Notice, we have here a situation where without policy feedback, moment 2 is informative and moment 1 is uninformative. Conversely, with policy feedback, moment 2 is uninformative and moment 1 is informative. Strikingly, moment 2 can be highly informative about the true value of the unknown parameter solely due to its sensitivity to the governmental policy variable. Intuitively, as u changes, so too does governmental policy in equilibrium, and this causes …rm behavior to change in a manner informative about u: We thus have the following proposition.
Proposition 6. Monotonicity of the moment function m( ; ) is neither necessary nor su¢ cient for m to be informative about the unknown parameter with joint estimation of u and control of :

An Algorithmic Approach to Structural Inference
The objective of this section is to propose a practically feasible algorithm allowing the econometrician to iterate to (approximately) correct inference of u, leading to a rational expectations equilibrium in which policy approximates …rst-best, with (u) arbitrarily close to g(u): To this end, consider the following Algorithmic Inference Approach: Start iteration n 2 f1; 2; 3; :::g with a provisional policy recommendation n ; Recompute the provisional government policy as n+1 = g(b u n ); Iterate until (approximate) internal consistency, n+1 n < for arbitrarily small.
We then have the following proposition showing that if (equation (26)) is strictly monotonic, elimination of internal inconsistency is su¢ cient to ensure correct inference and optimal government policy.
Proposition 7. Let (equation (26)) be strictly monotonic. At the n-th iteration, let the structural model be parameterized assuming government will implement n should it enjoy policy discretion.
The resulting inference b u n will be equal to the true parameter u if and only if b u n rationalizes n so that policy convergence obtains with n = g(b u n ) n+1 : Proof. To establish su¢ ciency suppose n = g(b u n ). Under the stated conditions, the inference equation (7) can be rewritten as From monotonicity of , the unique value at which the observed moment matches the model-implied moment is the true u: To establish necessity, suppose n 6 = g(b u n ). It then follows from the moment matching equation and monotonicity of m in its second argument that Since m observed 6 = (b u n ) it follows b u n 6 = u: Of course, in practice, iteration will generally continue until approximate convergence. Therefore, it is interesting to evaluate the convergence properties of the preceding algorithm. Rather than do so numerically with arbitrary examples, we …rst consider below iterating on the preceding algorithm in the case of the linear technology. To begin, note that iterating on n values is equivalent to iterating on the u values that would justify them, e.g. u n+1 n+1 . Thus, from the statement of the algorithm: In the posited rational expectations equilibrium, with …rst-best policy conjectured by the …rms, the inference equation at iteration n + 1 is With the linear technology, the preceding equation can be expressed as follows Iterating on the preceding equation we have the following lemma which shows that the proposed algorithm will converge to the truth provided the policy feedback e¤ect is su¢ ciently weak relative to the direct e¤ect.
The algorithm converges to the true parameter u for all u 2 R for all starting points u 1 2 R if and only if < 1: In fact, Lemma 4 is a special case of a more general convergence condition which relies on bounding the policy feedback e¤ect, as we show next. From the mean value theorem, for each iteration n; there exists x n between b u n and u; and there exists g n between g (u) and n such that Applying the mean value theorem to the …nal term in the preceding equation, we know that for each iteration n there exists z n 2between u and b u n 1 such that Rearranging terms in the preceding equation, we …nd that at each iteration n Under the stated condition b u n converges to u:

Quantitative Example
This section considers an econometrician seeking to estimate unobserved costs of corporate bankruptcy based upon the …nancial policies adopted by corporations. Understanding the magni- Early models, such as that of Stiglitz (1973), failed to deliver interior optimal leverage ratios.
Lacking interior optimal leverage ratios, computational general equilibrium (CGE) models, e.g. Ballard, et. al (1985), posited exogenous …nancing rules. In the absence of closed models, public …nance economists such as Gordon and MacKie-Mason (1990) and Nadeau (1993) were forced into positing ad hoc costs of …nancial distress. In an important contribution, Leland (1994) showed how to develop a tractable logically closed model of capital structure for …rms facing taxation and costs of distress using contingent-claims pricing methods.
In this section, we use Leland's canonical framework to illustrate the magnitude of bias that can arise if the structural modeler fails to impose rational expectations. To this end, consider a government that is interested in setting the corporate income tax in a way that is optimal according to its objective function. The magnitude of …nancial distress costs is clearly relevant here since, as argued above, the magnitude of these costs determines e¢ ciency costs of corporate leverage, as well as having a bearing on the present value of corporate income tax collections.
With this economic setting in mind, consider a structural econometrician who will observe the …nancing policies adopted by a set of homogeneous …rms funding new investments during the preinference stage. 9 Speci…cally, the econometrician will measure the mean interest coverage ratio, as measured by the ratio of EBIT to interest expense. As shown below, this moment is directly informative about bankruptcy costs.
Consider …rst the decision problem of the …rms. Each …rm will choose a promised instantaneous coupon on a consol bond, denoted . The …rm will use the debt proceeds plus equity injections to fund a new investment, as is standard in project …nance settings. We assume parameters are such that the investment has positive net present value. Formally, the new investment has positive net present value if the value of the levered enterprise exceeds the cost of the investment.
Debt enjoys a tax advantage, with interest being a deductible expense on the corporate income tax return. Consequently, each instant it is alive, the project …rm will capture a gross tax shield equal to e ; with the variable e representing the corporate income tax rate that will be implemented just after the econometrician completes her parameter inference. The …rm must weigh this debt tax shield bene…t against costs of …nancial distress. In particular, in the event of EBIT being insu¢ cient to service the coupon, the …rm's debt will be cancelled and bondholders will recover the unlevered …rm value net of deadweight bankruptcy costs representing a fraction N (u) of unlevered …rm value. The function N here is the standard normal cumulative distribution function.
Suppose …rm EBIT follows a geometric Brownian motion with drift , volatility ; and initial value normalized at 1. The risk-free rate is denoted r: The objective is to maximize levered project value. Or equivalently, …rms maximize expected tax shield value minus expected default costs.
Letting represent the anticipated tax rate, …rms solve the following program where is the negative root of the following quadratic equation Note, the …rst term in the objective function captures tax shield value and the second term captures bankruptcy costs. E¤ectively, the tax shield represents an annuity that expires at the …rst passage of EBIT to the coupon from above. At this same point in time, bankruptcy costs incurred. This explains the presence of the term in the objective function, which measures the price at date zero of a so-called primitive claim paying 1 the …rst passage of EBIT to the coupon from above.
The …rst-order condition for the optimal coupon entails equating marginal tax bene…ts with marginal bankruptcy costs. In particular, the optimal coupon satis…es Rearranging terms in the preceding equation, it follows the optimal coupon is The moment observed by the econometrician, the mean interest coverage ratio, is 1= : Thus, in the present setting Notice, in this particular case, m u (u; ) > 0 and m (u; ) < 0: That is, the optimal interest coverage ratio is increasing in bankruptcy costs and decreasing in the tax rate.
Suppose now that the structural econometrician, who recommended the Trump tax cut, failed to impose the assumption that …rms have rational expectations ( = ). Speci…cally, suppose the econometrician treated the tax change as a counterfactual event and parameterized the model using the status quo tax rate. In the present context, the inference equation (7) (38) implies …rms observed during the inference stage will choose coupons equal to 13:63% times initial EBIT(=1). Future generations of …rms will adopt this same coupon rate. After all, under rational expectations, the inference stage …rms posit the same tax shield value as that which will actually be operative post-inference. In other words, no reaction will be apparent when one contrasts the behavior of the inference-stage …rms with the behavior of …rms post-inference.
The structural econometrician will here mistakenly predict that future generations of …rms will respond to the tax rate change by adopting a much lower coupon rate, failing to understand that the inference-stage …rms already responded rationally to the upcoming change. In particular, based upon an estimated bankruptcy cost equal to 43:4%(= 8:68 5%); equation (38) leads to a predicted coupon rate, call it b , equal to only 1:49% times initial EBIT. However, as shown above, the actual coupon rate after the tax rate change will be 13:63% times EBIT.
The faulty parameter inference leads to faulty predictions regarding …rm behavior after the policy change which in turn leads to a faulty assessment of policy tradeo¤s. To illustrate, note that the present value of tax collections per …rm in this economy is equal to the value of the perpetual stream of taxes on an unlevered entity minus the tax shield value. It follows that the actual and predicted present value of tax collections are, respectively That is, the actual present value of tax collections here will be 10.6% lower than predicted tax collections. Intuitively, the upward bias in estimated bankruptcy costs leads to a faulty prediction of low leverage leading to a faulty prediction of high corporate income tax collections.

Beyond rational expectations
The preceding sections followed the structural econometrics literature by assuming that agents had rational expectations while the econometrician believed that agents assumed the status quo policy would be maintained with probability 1. In this section, we relax these assumptions. Specifically, we allow agents to place a weight ! > 0 on (u) and a weight (1 !) on 0 . A weight ! < 1 can be viewed as allowing for sticky expectations while a weight ! > 1 may re ‡ect extrapolative expectations. 10 In addition to relaxing the assumption of real-world agent rational expectations (Assumption 5), we also relax our baseline model's Assumption 6 by allowing the structural econometrician to place a non-zero weight on optimal policy. To this end let w be de…ned as the weight that the econometrician places on agent anticipations of future optimal policy. Assuming linear technologies for ease of exposition, we write the following assumptions ticipate that (u) will be implemented with probability w > 0 while the status quo 0 will be maintained with probability (1 w ). Hence, the econometrician anticipates that agents expect The inference equation then becomes: where the left hand side of the preceding equation is the observed moment assuming …rms have expectations placing a weight ! on policy b u(u) and the right side is the model-implied moment re ‡ecting what the econometrician believes she observes. Of course, if w = !, there will be no bias.
We now consider w 6 = !. The above equation leads to In the interest of brevity, we con…ne attention to the cases where ! > w 0: Then, overshooting and undershooting will obtain in the same parameter regions as in our main model. 11 If = < 0 and 0 < ! w < 1; jb u(u) uj is lower than in our main model for all parameter values and will increase with (! w). In fact, the preceding analysis subsumes our main model as a special case in which ! = 1 and w = 0.
1 1 If ! < w, the econometrician will assume agents place higher expectations on future discretionary policy than they actually do. Overshooting (undershooting) will obtain in parameter regions where there is undershooting (overshooting) in our main model. While our moment matching condition is unchanged, we can write a modi…ed version of our algorithm whereby at each step n, The following modi…ed lemma obtains The algorithm converges to the true parameter u for all u 2 R for all starting points u 1 2 R if and only if (! w) < 1:

Multivariate Extension
The preceding sections considered an econometrician attempting to infer one unknown parameter, with the government controlling one policy variable. In this section, we consider a multivariate extension. For simplicity, linearity is assumed.
There are n u 1 unknown deep parameters, each with support on the real line. The realized vector is denoted u: The econometrician seeks to infer u based upon a vector m consisting of n u empirical moments. The government has n 1 policy tools, with the full-information optimal policy being g(u): The observed empirical moments are linear: m Au + B : In the preceding equation, A is an n u n u matrix of full rank with element ij denoting the moment i coe¢ cient on parameter u j . Matrix B is an n u n matrix with element ij denoting the moment i coe¢ cient on government policy variable j : The government policy vector is: = Kb u: In the preceding equation, K is an n n u matrix, with element ij denoting the policy i coe¢ cient on b u j .
Consider again the nature of bias that arises if the econometrician parameterizes government policy at the status quo 0 Ku 0 :

The inference equation is
Au + BKb u= Ab u + BKu 0 : The left side of the preceding equation is the observed moment assuming real-world …rms have rational expectations and the right side is the model-implied moment. Solving the preceding equation we obtain the multivariate analog of equation (21): From the preceding equation it follows u = u 0 ) g(u) = 0 ) b u = u: It follows from the preceding equation that in the multivariate setting u = u 0 is su¢ cient for absence of bias, but is not necessary. This is in contrast to the univariate case (Proposition 1) where u = u 0 was both necessary and su¢ cient for absence of bias.
Other implications of the linear multivariate bias equation (54)  With the preceding equation in mind, suppose 12 = 21 = 0: That is, the moment i coe¢ cient on parameter u j is 0 for i 6 = j: Here the traditional Jacobian formulation would suggest that the problem of inferring u 1 is separable from the problem of inferring u 2 . However, with policy feedback, it is apparent that the inference problems and biases are not separable, since b u 1 = u 1 + 1 [ 1 (u 1 u 10 ) + 2 (u 2 u 20 )] From the preceding equation it is apparent that even though b u 2 does not inform policy, b u 2 will nevertheless be biased so long as the government policy variable in ‡uences ( 2 6 = 0) the respective moment (here m 2 ) that is relied upon for inferring u 2 : It is also apparent that, in general, the existence of a moment that is independent of the government policy variable does not imply the absence of bias in any particular parameter estimate.
To see this, suppose all four elements of matrix A are positive. Suppose further that the government policy variable has no e¤ect on one of the moments, say m 2 , with 2 = 0: In this case, bias still emerges, with b u 1 = u 1 + 1 [ 1 (u 1 u 10 ) + 2 (u 2 u 20 )] Consider a norm kk on IR nu and consider // the subordinate norm on a n u n u matrix such that for any u 6 = 0; /M/ = sup u kMuk kuk : Then kMuk /M/ kuk and if /M/ < 1; c u n converges to u:

Conclusion
An asserted advantage of moment-based structural econometrics over reduced-form methods is that one can correctly identify policy-invariant parameters so that alternative policy options can be assessed. As we have shown, this approach, which generally treats policy changes as counterfactual zero probability exogenous events, violates rational expectations: agents inside the structural model should understand that policy changes are positive probability endogenous events which the econometric exercise in intended to inform. We examined the implications of this violation of rational expectations in moment-based econometric parameter inference which serves a policy function.
As shown, bias emerges unless the true value of the parameter justi…es the status quo. If instead a policy change is justi…ed, biased inference occurs. Finally, it was shown how rational expectations can be imposed in an internally consistent manner, yielding unbiased inference and optimal policy.
The more general point illustrated by our analysis is that econometric methods should vary according to whether the estimation is passive or active in the sense of in ‡uencing policy decisions.
Although the speci…cs of the transmission mechanism will di¤er, the essential problem highlighted by this paper is that with active estimation, future endogenous policy will be correlated with the causal parameters to be estimated. If agents have rational expectations, this channel will bias structural inference if the inference-policy feedback e¤ect is not taken into account. A potentially important direction for future research is to incorporate the policy control channel into the econometric tool-kit, especially as economists get closer to their goal of gaining the attention of policymakers.