Identifying and Estimating Perceived Returns to Binary Investments

Identifying and Estimating Perceived Returns to Binary Investments

Introduction

In this paper I describe a method for estimating distributions of perceived private returns to binary investments. These structural perceived returns estimates are of distributions of agents' compensating variation associated with a binary choice that condition on observables. This method complements program evaluation methods that estimate effects of specific policy shocks on binary choices by allowing for predictions of counterfactual policies that differ from past policies in magnitude or targeted population. For instance, h20 applies this method to estimate perceived returns to college, allowing for counterfactual predictions of targeted college attendance subsidies (and taxes) for diverse groups of individuals. Identification is achieved by assuming common agent knowledge of an identity that relates prices to returns, while also using instruments that are de facto known to agents, in the sense that they shift perceived prices the same amount that they shift actual prices, in addition to satisfying the traditional exclusion restriction.

This paper presents a special case of a general method for identifying the scale of binary choice models by assuming agent beliefs about a variable observed by the researcher and agent beliefs about the mapping between that variable and the perceived return latent variable. Existing work that makes such assumptions includes Chn05, who assume agent knowledge of their lifetime pecuniary return to college insofar as it is attributable to explanatory variables observed by the researcher, and dm18, who assume partial agent knowledge of trade revenues and agent knowledge of an estimated demand elasticity parameter. The present paper assumes partial agent knowledge of prices in the sense of dm18 while assuming agent knowledge that prices causally decrease returns dollar for dollar in accordance with an identity that relates profits, revenues, and costs. The use of this identity imposes a theoretical restriction on a structural parameter (the coefficient on price in the binary choice latent variable equation) without requiring its estimation by researchers or agents. Avoiding the assumption that agents obtain the same estimate of a parameter as researchers improves robustness to the concerns articulated by manski1993adolescent, manski2004measuring about the pitfalls of making incorrect assumptions on agents' knowledge of structural models.

The method in the present paper avoids assuming rational expectations on any model objects, instead assuming that the variation in prices associated with chosen instruments is known to agents regardless of whether agents are correct about prices on average. This makes it particularly attractive in applications where rational expectations assumptions in general are suspect, but the researcher can credibly argue that a particular price shock is nonetheless known to agents. Considering the example of college attendance, it is possible that exogeneous policy shocks may shift prices more than they shift perceived prices, as with Pell grants hansen1983impact, kane1995rising, they may shift perceived prices more than they shift prices, as with the Michigan HAIL policy dlmo18, or they may shift prices and perceived prices the same amount, as with the Social Security Student Benefit termination D03. Of these preceding sources of variation, only the last would be appropriate for estimating the model presented in this paper. In addition to college attendance, attractive targets for this method include healthcare, home purchases, R&D, and export decisions due to the substantial information frictions on prices in these settings.

In addition to considerations regarding the relative credibility of different assumptions on agent beliefs, applications also differ in data availability. The method described in this paper relies on cross-sectional data that contains binary choices on investments and prices associated with those investments. Methods that rely on rational expectations on ex post returns to investments require longitudinal data (without requiring data on prices), as in Chn05 and related research surveyed by Ch07. Meanwhile, inferring beliefs by eliciting them directly from agents requires surveys that contain this information, as in J10, Wz15, and bz18. The method described in this paper is thus useful in settings where there is no clear winner in terms of assumption validity, but when longitudinal data and data on agent perceptions in unavailable.

I describe how to estimate perceived returns when prices are known to agents and exogenous, and how to overcome violations of these conditions using instrumental variables. I compare performance of these methods with valid and invalid instruments across data generating processes that differ in the assumptions on agent knowledge of prices. In the most realistic settings, methods that make no use of instruments, or which use instruments that are correlated with agent misperceptions, perform poorly compared to those that use instruments that are de facto known to agents.

The plan of the rest of this paper is as follows. Section (ref) introduces the empirical model. Section (ref) describes the econometric strategy and the assumptions required for identification. Section (ref) evaluates the robustness of various methods and instruments to various empirical challenges in a series of simulated data exercises. Section (ref) concludes.

Model

I assume that agents choose whether to make an investment based on their beliefs about discounted net incomes and costs associated with choices, which I present as a two-sector generalized Roy r51 model. Agents choose to select the investment, $S_i=1$, or to not do so, $S_i=0$, which is observed by the researcher. I define $\widetilde{Y}_{1,i}$ as agent $i$'s perceived discounted present value of lifetime income associated with choosing the investment and $\widetilde{Y}_{0,i}$ as their perceived discounted present value of lifetime income associated with not doing so. I further define $\widetilde{C}_i$ as their perceived net present value cost of making the investment, which includes prices paid and nonpecuniary costs expressed in monetary values. Unlike common applications of the Roy model, none of $\widetilde{Y}_{1,i}$, $\widetilde{Y}_{0,i}$, and $\widetilde{C}_i$ are observed by the researcher for any individual because they represent agent perceptions.

I express the perceived potential incomes and costs for individual $i$ with the following linear-in-parameters production functions,

Here, $X_i$ are variables observed by the researcher that determine potential incomes and costs. The parameters $\{\beta\}$ capture the extent to which these variables drive beliefs about potential outcomes regardless of whether they are known to agents. $\widetilde{Price}_i$ is the agent's perceived price for the investment, which is known to agents but not to researchers. Importantly, it is assumed to only affect costs and has a coefficient that is normalized to unity. Finally, $\tilde{\epsilon}_{1,i}$, $\tilde{\epsilon}_{0,i}$, and $\tilde{\epsilon}_{Ci}$ represent idiosyncratic perceived returns to investment that are known to agents but not to the researcher.

I assume that agents maximize expected wealth independently of how they consume it, as in the case of perfect credit markets. It follows that the perceived net return/profit, $\widetilde{\pi}_i$, is sufficient to determine agents' decisions in accordance with the rule

I further assume that the definition of profit, $\pi_i \equiv Revenue_i-Cost_i$, is known to agents in the sense that it holds for their beliefs as well, such that

where $\widetilde{Revenue}_i$ denotes the agent's perceived income and $\widetilde{Cost}_i$ denotes the agent's perceived opportunity cost, which includes $\widetilde{Y}_{0,i}$. \footnote{I avoid denoting agents' beliefs with conditional expectations over realized values, as is common in the literature, to avoid the implication of rational expectations which follows from the law of iterated expectations.} It follows that the agent's decision rule can be expressed in terms of potential outcomes as

Defining the net marginal effects $\beta \equiv \beta_1-\beta_0-\beta_C$ and the net idiosyncratic component of perceived outcomes $\tilde{\epsilon}_i \equiv \tilde{\epsilon}_{1,i}-\tilde{\epsilon}_{0,i}-\tilde{\epsilon}_{Ci}$, we can combine ((ref)) with ((ref)) to write the perceived return latent variable as

Importantly, the assumptions given result in the latent variable being linear in perceived prices, with a marginal effect ($-1$) that is known to both agents and the researcher. \footnote{The researcher constraining the price coefficient to the value used by agents is key to identification, not the researcher or agents being correct about its value.} The expression of perceived returns as a latent variable in a binary choice problem with a single known marginal effect is the starting point of the estimation procedures described below.

Empirical Strategy

It follows from the model that latent perceived returns are identified by $\beta$, $\widetilde{Price}_i$, and $\tilde{\epsilon}_i$, given the observed $X_i$. The lack of observation of $\tilde{\epsilon}_i$ is a common problem that will be addressed with commonly used binary choice estimation techniques. In this section I will describe adjustments to these estimators that leverage the assumptions described above to permit identification of $\beta$ and the scale of the distribution of $\tilde{\epsilon}_i$ in the context of the researcher's failure to observe agents' perceived prices. To preface, these adjustments address challenges that arise due to perceived costs having a causal effect on perceived returns in the identity given in ((ref)).

The econometric methods described below establish conditions under which the assumed coefficient on perceived prices from ((ref)) exactly determines the marginal effect of realized prices on perceived returns in a binary choice model. Omitted variable bias and measurement error in prices as measures of perceived prices threaten the validity of this assumption. It follows that methods which address omitted variable bias and measurement error will validate the assumption on the marginal effect of realized prices on perceived returns. To clarify, consider the expression of agents' beliefs about prices used throughout this paper,

where the realized price, $Price_i$, is observed by the researcher, $\alpha$ gives the effect of explanatory variables on price misperceptions, and $\nu_i$ is the idiosyncratic component of agent $i$'s misperception of prices. Here, realized prices are assumed to increase agents' beliefs about prices at a known marginal rate of unity insofar as they are known to agents.

This expression allows us to present an empirically tractable version of perceived returns,

by substituting in prices observed by the researcher for agents' unobserved perceived prices and defining $\theta=\beta-\alpha{}$. \footnote{The distinction between the extent to which each control contributes to misperceptions in prices, $\alpha$, and to other components of perceived returns, $\beta$, is presented to emphasize that the methods in this paper are robust to systematic bias in perceptions associated with explanatory variables, even though they are not separately identified.} This representation presents the unexplained price misperception as an omitted variable, which will produce problems if $Price_i$ is correlated with $\nu_i$. Natural examples of problematic correlations between price misperceptions include agents systematically over-reacting or under-reacting to price predictors that are unobserved by the researcher. The extreme case of under-reaction is that in which an unobserved predictor of realized price variation is ignored by or unknown to agents altogether, which amounts to classical measurement error in realized prices as measures of perceived prices.

In what follows, I first consider a benchmark case in which unobserved components of price misperceptions are mean independent of realized prices and prices are uncorrelated with unobserved determinants of perceived returns. Though agents may be mistaken about prices, actual prices can stand in for perceived prices because any systematic price misperceptions are accounted for by observables. Second, I consider the case in which prices are correlated with unobserved price misperceptions and unobserved components of perceived returns. In this setting, instruments for observed prices that are uncorrelated with unobserved components of perceived returns will be needed to identify perceived returns. This case emphasizes the importance of choosing instruments that are de facto known to agents in addition to being exogenous for constructing credible counterfactuals relating to price changes.

Estimation with Known, Exogenous Prices

Here, I describe a benchmark procedure for estimating perceived returns with a simple adjustment to a common binary choice method. This procedure will provide consistent estimates of the perceived returns distribution under two assumptions that are likely to be violated in applications. First, this method assumes that prices and the unobserved component of perceived returns are uncorrelated. Second, it assumes that unobserved components of price misperceptions are mean independent of prices conditional on $X_i$, the simplest case of which is agents having perfect information on prices.

With the decision rule in ((ref)) and the expression of perceived returns in ((ref)), an assumption on the distribution of $-\nu_i{} +\tilde{\epsilon}_{i}$ is sufficient to consistently estimate perceived returns by maximum likelihood. I assume the composite unobserved component of perceived returns in ((ref)) is normally distributed as

The assumption of normality is chosen for convenience, and is not necessary for the estimation procedures in this paper. Defining $(\beta^*,\theta^*,{\gamma}^*)=(\frac{\beta}{\sigma},\frac{\theta}{\sigma}, \frac{1}{\sigma})$ for notational convenience, the probability of selection is given by

where $\Phi(\cdot)$ denotes the standard normal CDF.

The parameters $(\theta^*,{\gamma}^*)$ are the values that maximize the log-likelihood

The estimates of perceived returns are then given by

where imposing the constraint $\gamma^*=\frac{1}{\sigma}$ (rather than the standard constraint $\sigma=1$) is the only difference from a standard probit. Importantly, the assumption that $\gamma^*=\frac{1}{\sigma}$ is only valid under the assumptions described in Section (ref) when realized prices are uncorrelated with unobserved components of price misperceptions and perceived returns conditional on $X_i$. As this generally will not be the case, this assumption is not an innocuous normalization.

Estimation with Endogenous, Unknown Prices

Here, I describe a control function approach that addresses correlation between prices and unobserved components of perceived returns as well as arbitrary correlation between prices and misperceptions on prices. In Appendix (ref), I discuss a method developed by dm18 that performs well in this model when agents under-react to price variation, such as when they form rational expectations on prices based on a known price predictors and only a subset of price predictors are known to them. The method in this section uses an established estimator, but adds the assumption that instruments are uncorrelated with unobserved components of price misperceptions in addition to the more commonly invoked assumption that instruments are uncorrelated with other unobserved idiosyncratic components of perceived returns. This additional assumption contributes to credibility for predictions of responds to counterfactual price changes that are known to agents, without changing the asymptotic or finite sample properties of the estimator.

The control function approach uses the following system of equations, with reference to the expression of perceived returns in ((ref)),

where I have left unobserved price misperceptions and other unobserved components of perceived returns separate for clarity. Here, I introduce the instruments, $Z_i$, where $X_i \subset Z_i$, that are assumed to be conditionally uncorrelated with $-\nu_i+\tilde{\epsilon}_i$ and strongly correlated with observed prices. With some loss of generality, I will refer to instruments that satisfy this condition as “known and exogoneous” for brevity. \footnote{It is not necessary that agents know the instruments in $Z_i$, but only that they know the variation in prices that is attributable to $Z_i$. For example, agents need not know about a tax or subsidy shock to the price of investment, so long as they are aware of the change in price that arises from the policy shock. Furthermore, the language that instruments are known and exogenous suggests that $Cov(Z_i,\nu_i)=Cov(Z_i,\tilde{\epsilon}_i)=0$, while these are sufficient but not necessary for the less intuitive condition $Cov(Z_i,\tilde{\epsilon}_i-\nu_i)=0$, which accommodates the knife-edge case of the two sources of bias cancelling out.} With valid instruments, the price residual $u_i$ contains all components of prices that are correlated with idiosyncratic components of price misperceptions or other unobserved components of perceived returns.

Given the above, I estimate the following equation,

The first line follows directly from the representation of perceived returns in ((ref)). The second line substitutes in the linear projection of the composite error $-\nu_i{}+\tilde{\epsilon}_i$ on the first stage error $u_i$, wherein $\rho = \mathbb{E}[u_i(-\nu_i{}+\tilde{\epsilon}_i)]/\mathbb{E}[u_i^2]$ and $\xi_i$ is the residual when controlling for $u_i$. The third line substitutes the estimated residuals from the first stage regression of $Price_i$ on $Z_i$ in for their unobserved true values, generating a new error, $\zeta_i = \xi_i+(u_i-\hat u_i)\rho$. This new error will converge asymptotically to $\xi_i$, but will differ in small samples due to sampling error in the estimation of the residual from the first stage, $\hat u_i$.

To estimate perceived returns, I assume that the new error in the perceived returns control function expression is normally distributed,

noting that the variance of $\zeta_i$ will differ from that of $\tilde{\epsilon}_i$ if $\rho\neq0$. I estimate perceived returns using two-stage conditional maximum likelihood, following Rv88, while correcting for the inclusion of estimated regressors, following mt85, though other estimators will also provide consistent estimates. Defining $(\theta^*_\zeta,{\gamma}^*_\zeta,\rho^*_\zeta)=(\frac{\theta}{\sigma_\zeta},\frac{{1}}{\sigma_\zeta}, \frac{\rho}{\sigma_\zeta})$, the log-likelihood for the second stage of the control function approach is given by \footnote{As an closely-related alternative, we could perform a instrumental variables probit to obtain identical estimates of $\theta$. The control function method has the advantage of conditioning on the variation in prices that isn't used in identifying the effect on perceived returns, which permits more precise counterfactual predictions for policies that are targeted on observables. }

Estimates of perceived returns are obtained by plugging the estimated parameters and the assumed coefficient on perceived prices into the latent variable equation,

Simulations

In this section I apply the methods described above to simulated datasets to compare their performance. The important considerations involve agent beliefs about prices, price endogeneity, and instruments being known and/or exogenous to agents. Because the estimators used are standard, I stop short of performing full Monte Carlo simulations, instead comparing the performance instruments according to whether they are known or exogenous to agents within individual simulations. For additional simulations which compare the methods of this paper to the method of dm18, see Appendix (ref).

For the simulations, I use the following DGP,

where the nature of the covariance of $(Z_i,u_i,\nu_i,\tilde{\epsilon}_i)$ will determine the performance of various estimation approaches. Both the probit and the control function method will obtain estimates of $\beta$, while the probit will estimate

and the control function method will estimate

Each DGP is comprised of $N=10,000$ observations of agents whose decisions are governed by their perceived returns to investment.

Simulation with Known, Exogenous Prices

I begin with a well-behaved benchmark DGP that corresponds to the setting described in Section (ref). I generate data according to

I construct the instrument vector as $Z_i = [X_i \mbox{ } z_{1,i}]$ where $X_i$ includes only a constant, and $\alpha=0$ such that $\theta=\beta$. Finally, I set $\beta=1$ and $\delta=[0 \mbox{ } 1]'$. Although I set $Var(\nu_i) = 2$, I describe prices as known in this setting because the price misperception is uncorrelated with prices.\footnote{This setting is one in which agents are wrong about prices in ways that are unrelated to price determinants. This sort of price misperception is plausible in cases where prices change frequently according to a distribution that is de facto known to agents, such as frequently repeated investments.}

Table (ref) shows perceived returns estimates for one simulation of this DGP using the methods from Section (ref) and Section (ref). Figure (ref) shows the distributions implied by the estimates for each method. In this case, the lack of correlation between prices and unobserved components of perceived returns, including price misperceptions, means that both methods will provide consistent estimates of perceived returns.

Simulation with Unknown, Endogenous Prices

In this simulation, I consider a DGP that corresponds to the setting described in Section (ref) in which agents systematically misperceive prices in ways that not accounted for by observables, and prices are correlated with unobserved components of perceived returns. I also compare the performance of an instrument that is exogenous but unknown to one that is both known and exogenous. I generate data according to

I construct the instrument vector as $Z_i = [X_i \mbox{ } z_{1,i} \mbox{ } z_{2,i}]$ where $X_i$ includes only a constant, and $\alpha=0$ such that $\theta=\beta$. Finally, I set $\beta=1$ and $\delta=[0 \mbox{ } 1 \mbox{ } 1]'$.

In this case, there is positive correlation between $u_i$ and $\tilde{\epsilon}_i$ such that individuals who face idiosyncratically high prices also have high perceived returns, as may occur with price discrimination. Additionally, there is negative correlation between $u_i$ and $\nu_i$ such that individuals systematically underestimate the extent to which their price deviates from the average, as may occur if agents form rational expectations on prices conditional on an incomplete set of price determinants. Finally, this DGP includes two potential instruments; $z_{1,i}$, which is exogenous but not fully known to agents, as in the case of a poorly publicized policy shock, and $z_{2,i}$, which is both exogenous and known to agents.

Because $z_{1,i}$ is correlated with $\nu_i$, it is not a valid instrument for the purposes of this paper. For the control function estimates of $\rho$ and $\sigma_{\zeta}$, I use $u_{1,i}$ in place of $u_i$, where $u_{1,i} = z_{1,i}\delta_1+u_i$. In applications with many valid instruments, including different combinations of instruments will result in different estimates $\hat u_i$, $\hat \rho$ and $\hat \sigma_\zeta$, while nonetheless all returning consistent estimates of perceived returns. For comparisons between instruments, the complete distribution of perceived returns (succinctly described by the figures) and the estimated coefficients on $X_i$ will be correct for all valid instruments.

Table (ref) shows the estimates for one simulation of this DGP using both methods, and also using each instrumental variable individually. Figure (ref) shows the distributions implied by the estimates for each method. Because $z_{1,i}$ is correlated with misperceptions, it is not a valid instrument, and results in an estimated perceived returns distribution that is no better than that obtained when using no instruments.\footnote{For estimating instrument-specific intent to treat effects of prices on investment, which would be sufficient for determining the performance of a particular policy in the context of its actual implementation, instruments such as $z_{1,i}$ are valid. They nonetheless fail to provide credible insight into counterfactual policy changes that are well-publicized.}

Conclusions

In this paper I describe how to estimate perceived returns to investments by assuming agent knowledge of an intuitive identity and modestly altering common estimation techniques. The assumption on agent knowledge may be preferable to rational expectations or related assumptions in applications. I further describe the econometric challenges that arise from the assumption and how to overcome them with careful choice of instruments that are not only exogenous to agents, but are also de facto known to them.

This method is relevant in many empirical questions, especially those subject to substantial information frictions on prices such as such as college attendance, firm R&D, automobile purchases, home purchases, and healthcare. While the estimation techniques used in this paper are restricted to a probit and a control function probit, the general insights are relevant to more sophisticated models that involve responses to prices. Implementation of the identity relating perceived returns and prices used in this paper in the context of more sophisticated models, such as blp95 and its extensions, are left to future work.

In terms of policy implications, the methods described in this paper are relevant for constructing credible counterfactuals for well-publicized price changes, which are relevant for taxes and subsidies on investments including those associated with education and healthcare. The general insight is to avoid being too quick to assume that agents have rational expectations on model objects when alternative assumptions may be more defensible. Relatedly, the insights here also caution against extrapolating effects of counterfactual policies when the policy effects are estimated using a source of variation in prices that may not be known to agents. In practice, applied researchers should justify that sources of variation used for estimating treatment effects are known to agents just as they justify that they are exogenous to agents when making counterfactual predictions.

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