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Estimating Individual Responses when Tomorrow Matters
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Economists often seek to assess how changes in the economic environment affect individual decisions. A leading example is the ex ante evaluation of policies that have not yet taken place. However, a key challenge is that, when the environment changes, individual decision rules are generally affected as well. In dynamic settings characterized by uncertainty, it is necessary to consider not only the immediate effect of the change but also its influence on expectations.
A common approach in applied work is to regress outcomes on covariates that one is interested in shifting in the counterfactual (e.g., under a new policy). Average partial effects based on regression estimates can be structurally interpreted as counterfactual policy effects under suitable conditions (stock1989nonparametric). However, underlying this interpretation is the assumption that the regression function remains invariant in the counterfactual. This invariance assumption can be restrictive in many settings where individuals' beliefs about the future matter.
Consider the introduction of a permanent income tax in a standard model of consumption and saving (see deaton1992understanding, for a textbook treatment). The effect of the tax can be estimated by regressing consumption on income (in logs), and by then computing an average partial effect associated with the tax change. However, such an effect is likely to be empirically misleading, since both current income and beliefs about future income will be affected by the tax. Not accounting for the change in beliefs will produce biased predictions of the effect of the tax, as emphasized by lucas1976econometric in his influential critique.
As a second example, consider the effect of a change in the weather process in a model of agricultural production. Suppose that farmers choose dynamic inputs (such as irrigation or a fertilizer) based on their forecasts of future weather. In addition to affecting contemporaneous weather conditions, a change in the weather process will affect farmers' beliefs about future weather, which may lead them to modify their input choices. Not accounting for farmers' adaptation will bias calculations of the impact of a change in the weather process (deschenes2007economic, burke2016adaptation).
In this paper, our aim is to study and estimate average partial effects in a dynamic framework that explicitly accounts for the role of individual expectations. In our intertemporal setup, individual beliefs are determinants of decisions, and they enter as additional state variables in the agent's decision problem. In this setting, we show how to assess the total effect of a counterfactual change by means of average partial effects calculations. In addition, we show how to decompose this total effect into a contemporaneous effect where beliefs are held fixed, and a purely dynamic effect that solely reflects the change in beliefs.
To implement this approach we rely on data on subjective expectations. Belief data are increasingly available in a variety of settings (manski2004measuring). Given estimates of subjective probabilities based on survey responses, we account for beliefs in the definition and estimation of average partial effects. There are many examples of the use of expectations data on the right-hand side of a regression. Our contribution is to show how to interpret the estimates of such regressions, and to provide conditions under which those can be used for counterfactual prediction.
To interpret regression-based average partial effects, we propose a structural dynamic framework where agents choose actions based on their beliefs about the future. Following a {semi-structural} approach, we use the framework to justify the use of average partial effects, yet we do not specify or estimate a structural model. As a result, the counterfactuals we focus on are restricted to changes in states of nature and beliefs about them, and our approach cannot answer other counterfactual questions related to changes in preferences or technology, for example.
In the structural framework that we outline, beliefs are time-varying state variables in the agent's decision problem. Variation in beliefs over time is crucial, since it allows us to control for preference heterogeneity, which we assume to be constant over time, by including individual fixed effects. {Variation in beliefs conditional on the other state variables is also key, in order to separately identify contemporaneous and dynamic effects.} We assume that current beliefs provide sufficient information to predict future beliefs, an assumption that we refer to as belief sufficiency. We show this assumption is compatible with various popular models of belief formation, with and without rational expectations, including various forms of learning.
The structural framework implies that the agent's decision rule is a function of exogenous state variables such as income or the weather, beliefs about them, and endogenous dynamic state variables {that depend on past actions}, such as assets or capital. We assess the effects of a counterfactual change by computing average partial effects which, unlike in the static case, account for changes in beliefs. Such effects correspond to well-defined structural counterfactuals under the assumption that the dynamic decision rule is invariant in the counterfactual. Hence, while we rely on a less restrictive invariance assumption than static average partial effects that do not allow for belief responses, a certain form of invariance is still needed to structurally interpret average partial effects in our setup.
To estimate average partial effects, we proceed in three steps that can be easily implemented given the availability of panel data on individual decisions and beliefs. In the first step, we estimate the subjective belief densities. This is straightforward in the case of beliefs about binary or discrete variables, in which case one can directly use the {empirical subjective} probabilities. For beliefs about continuous variables, to account for the fact that survey responses on subjective beliefs tend to be coarse, we assume that subjective densities depend on a low-dimensional parameter vector. In the second step, we estimate the regression function (i.e., the individual's decision rule). In the third step, we use these estimates to compute the impact on decisions of a counterfactual, given knowledge of how state variables and beliefs change under the counterfactual. Without additional assumptions, nonparametric identification is restricted to the empirical support of the conditioning variables. Moreover, the degree of individual heterogeneity that can be accounted for is limited by the length of the panel dimension.
{To use our approach for counterfactual analysis, the researcher needs to specify the values that current exogenous state variables and beliefs about them would take in the counterfactual. We focus on changes involving a transformation of exogenous variables, such as an income tax, and assume full pass-through to the exogenous variable and the associated belief. For example, under a permanent proportional tax of 10%, we assume that current income decreases by 10% and that the income belief density is shifted downwards by the same amount. We perform sensitivity analysis to assess the impact of violations of this assumption, and discuss how it could be relaxed with suitable data. }
As an empirical illustration, we study how consumption decisions depend on current income and beliefs about future income. We rely on Italian data from the Survey on Household Income and Wealth (SHIW), which contains panel data on respondents' probabilistic income expectations for two consecutive waves. We then use our approach to predict the impact of various counterfactual income taxes, involving transitory or permanent increases in marginal tax rates, and a change in the degree of progressivity of the tax. We assume that individuals fully incorporate the effects of the tax changes into their beliefs, and report sensitivity checks. We find that, conditional on current income, income beliefs shape consumption responses, and that they matter for predicting the effects of income taxes.
\paragraph{Related literature and outline.}
Subjective belief data are commonly included on the right-hand side of regressions. For example, guiso1999investment study how a firm's investment depends on its beliefs about future demand; hurd2004effects study the effects of subjective survival probabilities on decisions about retirement and social security claims; dominitz2007expected analyze how beliefs about equity returns affect portfolio choice; bover2015measuring studies how subjective expectations about future home prices affect car and secondary home purchases; and attanasio2019subjective study how parental investment in children is influenced by beliefs about the production function. We provide assumptions under which such regressions can be interpreted structurally and used for counterfactuals within a dynamic framework.
manski2004measuring (p. 1365) draws a distinction between expectations questions about unknown states of nature, which, combined with choice data, can be used to estimate econometric decision models, and questions about hypothetical choices under specified scenarios, which can be directly used to predict behavior. Our approach is designed for the first type of data (as in the examples mentioned in the previous paragraph), in the context of dynamic decision-making. This focus differs from a growing literature that relies on the second type of data, with the goal of providing methods for estimating heterogeneous treatment and policy effects using data on hypothetical choices (e.g., arcidiacono2020ex, giustinelli2019seate, briggs2024identification, meango2023identification, bernheim2022causal).
Our focus on the estimation of policy effects without a full structural model follows marschak1953economic, ichimura2000direct,ichimura2002semiparametric, and keane2002estimating,keane2002estimating2, among others; see also wolpin2013limits. In our approach, we rely on subjective belief data and do not assume rational expectations.
There is a growing literature on the combination of structural models and subjective belief data, see among others van2008social, delavande2008pill, van2012use, stinebrickner2014major, wiswall2015determinants, an2021dynamic, kocsar2022expectations, de2024evaluating, and keiller2024production; see also the recently released handbook on economic expectations (bachmann2022handbook). Our approach, which is tailored to specific counterfactuals, does not require to specify a full structural model.
Lastly, elicited beliefs about future income are increasingly available. Surveys with this information include the SHIW in Italy, the Survey of Economic Expectations and the Survey of Consumer Expectations in the US, the Survey of Household Finances in Spain, and the Copenhagen Life Panel in Denmark, among others. Previous contributions using income belief data include, among others, pistaferri2001superior, guiso2002empirical, and kaufmann2009disentangling, who use data on income expectations in the SHIW in combination with models of consumption and saving; stoltenberg2022consumption, who estimate a structural model with subjective income expectations using the same data; lee2023earnings, who use data on subjective expectations and earnings realizations in Denmark to estimate a model where agents have partial information about earnings shocks; attanasio2020euler, who combine data on subjective expectations with data on actual income and estimate an Euler equation for consumption; and arellano2024estimating, who model and estimate the dynamic process of subjective income expectations using data from India and Colombia.
The outline is as follows. In Section (ref) we introduce average partial effects for dynamic settings. In Section (ref) we describe a structural framework and discuss the interpretation of average partial effects in this context. We present two examples in Section (ref). We study identification and estimation in Section (ref), and we present our consumption application in Section (ref). Finally, in Section (ref) we describe some extensions of the approach. Replication files are available \href{https://drive.google.com/file/d/1RTX4eYAuRI1XpzOTFa15EhC7-GxrRC-Y/view?usp=sharing}{online}.
Suppose that a researcher has access to panel data on an individual outcome $y_{it}$ and some covariates $x_{it},z_{it}$, for {a large cross-section of individuals $i$ and some time periods} $t=1,...,T$. To fix ideas, we will refer to the case where $y_{it}$ denotes consumption, $x_{it}$ is income, and $z_{it}$ includes other determinants such as assets. In addition, we assume the researcher has data about individual beliefs. We denote $i$'s subjective density of $x_{i,t+1}$ at time $t$ as ${\Greekmath 0119}_{it}$, and in this section we suppose that the researcher observes ${\Greekmath 0119}_{it}$. In practice, we have in mind situations where data about respondents' probabilistic expectations are available. Eliciting such responses is becoming increasingly common, see manski2004measuring for a review. In Section (ref) we will describe how we use elicited belief data to construct an empirical counterpart of the subjective density ${\Greekmath 0119}_{it}$.
We postulate that, for some function ${\Greekmath 011E}_i$,
where ${\Greekmath 0122}_{it}$ has zero mean given $x_{it}$, ${\Greekmath 0119}_{it}$ and $z_{it}$. In the next section we will give conditions under which ((ref)) is obtained as the optimal decision rule for $y_{it}$ in a dynamic structural model. For example, in an intertemporal model of consumption and saving behavior, we will give conditions under which consumption $y_{it}$ depends, in addition to assets $z_{it}$ and current income $x_{it}$, on beliefs ${\Greekmath 0119}_{it}$ about next period's income $x_{i,t+1}$.
Suppose the researcher is interested in documenting the impact, {in period $t$}, of an exogenous change in $x_{it}$ to some other value $x_{it}^{({\Greekmath 010E})}$, which in turn is associated with a change in beliefs from ${\Greekmath 0119}_{it}$ to ${\Greekmath 0119}_{it}^{({\Greekmath 010E})}$. An example is a proportional tax, corresponding (in logs) to $x_{it}^{({\Greekmath 010E})}=x_{it}+{\Greekmath 010E}$. {More generally, one may consider a transformation $x_{it}^{({\Greekmath 010E})}={\Greekmath 010E}(x_{it})$, with ${\Greekmath 010E}$ some function, in which case the whole distribution of $x_{it}$ changes in the counterfactual.} Then, ${\Greekmath 0119}_{it}^{({\Greekmath 010E})}$ is the belief about future log income $x_{i,t+1}$ under the tax. {However, we assume that the other factors $z_{it}$, which are predetermined, are not affected at time $t$ under the counterfactual, although they may change in subsequent periods. Hence, the tax has two distinct effects on period-$t$ outcomes: a contemporaneous effect associated with the change in $x_{it}$, and a dynamic effect associated with the change in beliefs ${\Greekmath 0119}_{it}$.}
To account for both impacts of the policy, we define the {period-$t$} total average partial effect, or TAPE, as
We then further decompose this total effect as the sum of two terms: a contemporaneous APE (or CAPE), where beliefs are held constant, and a dynamic APE (or DAPE), which solely captures the change in beliefs. Formally, we decompose
{Note that these quantities measure the impacts of a policy introduced at time $t$ on outcomes at time $t$. In this paper we do not aim at recovering policy impacts on later outcomes, which would require additional assumptions.}
The structural framework in the next section will allow us to transparently discuss the assumptions needed to structurally interpret these average partial effects (TAPE, CAPE and DAPE). The framework has two main features. First, ${\Greekmath 0119}_{it}$ is sufficient to predict future beliefs ${\Greekmath 0119}_{i,t+1}$, as formally defined in Assumption (ref) in the next section. This implies that $x_{it}$, ${\Greekmath 0119}_{it}$, and $z_{it}$ are the state variables in the economic model (in addition to some shocks subsumed in ${\Greekmath 0122}_{it}$). This belief sufficiency assumption imposes restrictions on the belief formation process. However, we show it is satisfied in several popular models of beliefs.
Second, structurally interpreting the average partial effects requires ${\Greekmath 011E}_i$ to be invariant to the policy change. In the structural model, ${\Greekmath 011E}_i$ depends on preferences, discounting, the law of motion of $z_{it}$, and the law of motion of the beliefs ${\Greekmath 0119}_{it}$. Consequently, one will need to assume that none of these quantities varies under the policy change. Assuming that the law of motion of the beliefs, which we denote as ${\Greekmath 011A}_i$, is invariant requires that, while agents account for the impact of the change on their beliefs about $x_{i,t+1}$, the way they update their beliefs after period $t+1$ is unaffected. Under this assumption, ${\Greekmath 011A}_i$ is an individual “type” that is invariant to the change. We will see that this assumption is automatically satisfied in a popular version of the consumption example.\footnote{Relaxing this assumption is conceptually straightforward in our framework, by defining ${\Greekmath 0119}_{it}$ in ((ref)) as beliefs about a sequence of future $x$'s, $x_{i,t+1},x_{i,t+2},...,x_{i,t+S}$. However, doing so imposes stronger demands on the data. We will return to this point in Section (ref).}
It is informative to contrast our approach, which relies on the use of belief data and the dynamic decision rule ((ref)), to a static approach. Suppose instead that, for some function $g_i$,
where ${\Greekmath 0122}_{it}$ has zero mean given $x_{it}$ and $z_{it}$. A static average partial effect associated with the change in $x_{it}$ is
To interpret $\Delta_t^{\rm SAPE}$ as the average impact on outcomes when $x_{it}$ changes to $x_{it}^{({\Greekmath 010E})}$, one needs to assume that the function $g_i$ in ((ref)) remains constant (stock1989nonparametric). This invariance assumption is often implausible in applications where dynamics matter. Indeed, in many settings where the current value of $x_{it}$ changes, beliefs about future $x_{it}$'s, which are implicit in the function $g_i$, are likely to change as well. For example, under a permanent income tax, both current income and beliefs about future income change. In contrast, in our approach based on ((ref)), we require ${\Greekmath 011E}_i$ to be invariant in the counterfactual. Although this assumption is not without loss of generality (and we will discuss it further in the context of a structural framework in the next section), it is weaker than the assumption that $g_i$ in ((ref)) is invariant to the change. The key difference is that, unlike ((ref)), ((ref)) explicitly accounts for variation in beliefs.
Finally, note that, when beliefs matter in ((ref)), an approach based on ((ref)) is incorrect for two reasons. The first one is that beliefs ${\Greekmath 0119}_{it}$, which are generally correlated with $x_{it}$ (though not collinear with $x_{it}$), are omitted variables in ((ref)). Hence, not accounting for ${\Greekmath 0119}_{it}$ gives incorrect contemporaneous APE estimands in general. The second reason is that relying on ((ref)) makes it impossible to recover the total APE, and to decompose it into contemporaneous and dynamic APEs. Hence, when ((ref)) holds, $\Delta_t^{\rm SAPE}$ defined in equation ((ref)) is not economically interpretable in general.
In this section we describe a structural dynamic framework where individual decision rules take the form ((ref)), and we provide a structural interpretation for average partial effects.
Consider an individual $i$'s intertemporal decision making process in discrete time. In the presentation we first focus on a stationary infinite-horizon environment, and then show how to apply the framework to finite-horizon environments.
The timing is as follows. At the end of period $t-1$, the individual's information includes the history of exogenous state variables (i.e., states of nature) $x_{i,t-1},x_{i,t-2},...$, {which do not depend on past actions,} endogenous state variables $z_{i,t-1},z_{i,t-2},...$, {which depend on past actions,} actions $y_{i,t-1},y_{i,t-2},...$, and shocks (e.g., taste shocks) ${\Greekmath 0117}_{i,t-1},{\Greekmath 0117}_{i,t-2},...$. In addition, the individual may have observed other information, such as signals, that are relevant to her beliefs and future actions. Then, at the beginning of period $t$, $z_{it}$, $x_{it}$ and ${\Greekmath 0117}_{it}$ are realized and observed by the individual, and additional signals about future values $x_{i,t+1}$ may be observed as well. We denote the information set at that moment as $\Omega_{it}$. Given this information, the individual forms beliefs about $x_{i,t+1}$. Finally, she chooses the action $y_{it}$ based on the state variables in $\Omega_{it}$.
The individual's uncertainty about $x_{i,t+1}$ is represented by the subjective distribution of $$ \left(x_{i,t+1}\,|\, y_{it},\Omega_{it}\right),$$conditional on her information set $\Omega_{it}$, and possibly contingent on her potential action $y_{it}$.\footnote{Here $y_{it}$ denotes a potential action, contingent on which her beliefs are formed. In Assumption (ref) we will rule out that beliefs may be contingent on actions. We will study the case of contingent beliefs in Section (ref).} The belief distribution is subjective, and need not coincide with the realized distribution of $x_{i,t+1}$. In other words, we do not impose a rational expectations assumption. Our first assumption is that beliefs are not contingent on potential actions. Here and in the rest of this section, we use the shorthand $A\sim B$ to denote that $A$ and $B$ follow the same (subjective) distribution.\footnote{Throughout, densities are defined with respect to appropriate measures.}
We will refer to ${\Greekmath 0119}_{it}$, which is the individual subjective density of $x_{i,t+1}$, as the belief density, or simply as the “beliefs”. ${\Greekmath 0119}_{it}$ is an element of $\Omega_{it}$, and it is a random function. Assumption (ref) requires that beliefs about $x_{i,t+1}$, which are relevant to the choice of $y_{it}$, do not depend on $y_{it}$. In other words, beliefs are not contingent on potential actions. At the same time, Assumption (ref) allows past choices $y_{i,t-1},y_{i,t-2},...$ to influence current beliefs ${\Greekmath 0119}_{it}$. In Section (ref), we will outline a generalization of Assumption (ref) where agents have so-called “state-contingent” beliefs{; for example, beliefs about wages contingent on working in a particular sector}. The framework is unchanged in that case, except for the fact that ${\Greekmath 0119}_{it}$ then consists of a set of conditional densities ({of, e.g., wages}) indexed by potential action values $y$ {(e.g., sector participation)}.
We make the following assumption regarding belief updating.
{
We will refer to ${\Greekmath 011A}_i$ as the belief updating rule. Belief sufficiency, as stated by Assumption (ref), is a key condition in our framework. It requires that current beliefs ${\Greekmath 0119}_{it}$, along with $x_{it}$ and $x_{i,t+1}$, be sufficient statistics for $\Omega_{it}$ when predicting future beliefs. Moreover, Assumption (ref) requires that beliefs are not affected by past actions, which may be plausible in some settings. For example, in a consumption model there may be no feedback from past consumption choices to future income beliefs. However, in other settings, it may be important to allow future beliefs ${\Greekmath 0119}_{i,t+1}$ to depend on past actions $y_{it}$. This is allowed for by the following generalization of Assumption (ref).
\begingroup
\addtocounter{assumption}{-1} \endgroup
Assumptions (ref) and (ref) have similar implications in terms of policy rules, and both can be used to justify decision rules of the form ((ref)). {We will discuss belief sufficiency further below and show that it is consistent with a variety of belief formation processes.} }
Next, we make the following assumption regarding the endogenous state variables $z_{it}$.
{Assumption (ref) nests cases where $z_{i,t+1}={\Greekmath 010D}_i(z_{it},x_{it},y_{it})$ is non-stochastic, such as a standard budget constraint. Moreover, ${\Greekmath 010D}_i$ could additionally depend on ${\Greekmath 0119}_{it}$, $x_{i,t+1}$, or ${\Greekmath 0119}_{i,t+1}$, although we abstract from this dependence for conciseness.}
Lastly, we make the following assumption regarding the shocks ${\Greekmath 0117}_{it}$.
{The independence condition in Assumption (ref) is commonly made in structural models where alternative-specific taste shocks are serially uncorrelated. The presence of serially correlated time-varying unobservables would invalidate this assumption.}
{In this environment, we will focus on counterfactuals involving changes in exogenous variables $x_{it}$ and beliefs ${\Greekmath 0119}_{it}$, associated with counterfactual values $x_{it}^{({\Greekmath 010E})}$ and ${\Greekmath 0119}_{it}^{({\Greekmath 010E})}$, respectively. We assume that $x_{it}^{({\Greekmath 010E})}={\Greekmath 010E}(x_{it})$ is a deterministic transformation of $x_{it}$. For example, ${\Greekmath 010E}(\cdot)$ is a tax schedule (e.g., proportional or progressive), or a transformation of temperature (e.g., a mean shift).\footnote{Note that the assumption that $x_{it}$ responds fully to ${\Greekmath 010E}(\cdot)$ is consistent with our framework where $x_{it}$ is an exogenous state variable.}
Further, we assume that counterfactual beliefs are equal to the beliefs under the transformation ${\Greekmath 010E}$; that is, that ${\Greekmath 0119}_{it}^{({\Greekmath 010E})}$ is the subjective density of $$ \left({\Greekmath 010E}(x_{i,t+1})\,|\,\Omega_{it}\right),$$ which is simply the density of the transformed random variable ${\Greekmath 010E}(x_{i,t+1})$ for $x_{i,t+1}\sim {\Greekmath 0119}_{it}$. Consider as an example a permanent 10% proportional tax change, where ${\Greekmath 010E}(x)=x-0.10$ (in logs). We assume that beliefs are equal to ${\Greekmath 0119}_{it}^{({\Greekmath 010E})}(x)={\Greekmath 0119}_{it}\left(x+0.10\right)\equiv {\Greekmath 0119}_{it}^{({\Greekmath 010E},full)}(x)$. This amounts to assuming full pass-through of the tax onto the beliefs, which holds if individuals think of the change as being permanent. In Subsection (ref) we will return to this point, and introduce a sensitivity analysis approach where we vary individuals' expectations about the counterfactual remaining in place in the future. In that case, ${\Greekmath 0119}_{it}^{({\Greekmath 010E})}$ will be a mixture between ${\Greekmath 0119}_{it}^{({\Greekmath 010E},full)}$ and the baseline ${\Greekmath 0119}_{it}$. Also, note that while here the tax only affects mean beliefs, other ${\Greekmath 010E}$ transformations may affect the entire belief density.
}
We now illustrate that our belief sufficiency conditions, Assumptions (ref) and (ref), are consistent with several models of belief formation in economics, see pesaran2006survey for references.
\paragraph{Latent components.} As a first example, suppose that agents have rational expectations, and that $x_{it}={\Greekmath 0111}_{it}+{\Greekmath 0122}_{it}$ where ${\Greekmath 0111}_{it}$ follows a homogeneous first-order Markov process, and ${\Greekmath 0122}_{it}$ is independent of ${\Greekmath 0111}_{it}$ with a stationary distribution. Suppose that agent $i$'s information set at time $t$ is $$\Omega_{it}=\{x_{it},x_{i,t-1},...,{\Greekmath 0111}_{it},{\Greekmath 0111}_{i,t-1},...\}.$$ An example is a permanent-transitory specification of the income process, as in our consumption example in Section (ref). Note that ${\Greekmath 0119}_{it}$, which is the conditional density of $x_{i,t+1}$ given $\Omega_{it}$, coincides with the conditional density of $x_{i,t+1}$ given ${\Greekmath 0111}_{it}$. Given that ${\Greekmath 0111}_{it}$ follows an exogenous and homogeneous first-order Markov process, this implies that Assumption (ref) is satisfied. However, note that Assumption (ref) generally fails in this model if ${\Greekmath 0111}_{it}$ is not first-order Markov.
\paragraph{Learning (exogenous beliefs).} As a second example, suppose that $x_{it}={\Greekmath 010B}_i+{\Greekmath 0122}_{it}$. Suppose that agents do not know ${\Greekmath 010B}_i$, and that they try to learn it given the observations $x_{it}$. Suppose in addition that ${\Greekmath 0122}_{it}$ is i.i.d. Gaussian, and that agents are Bayesian decision-makers with Gaussian priors about ${\Greekmath 010B}_i$ and rational expectations. We show in Appendix (ref) that belief sufficiency, as stated by Assumption (ref), holds. This follows from the form of the updating equations for the posterior mean and variance of ${\Greekmath 010B}_i$, see ((ref))-((ref)) in Appendix (ref). Note that this example does not allow for learning from past choices, since beliefs are exogenous.
\paragraph{Learning (endogenous beliefs).} As a third example, consider a case where there are two possible choices $y_{it}=1$ and $y_{it}=0$. Suppose that the agent observes $x_{it}={\Greekmath 010B}_i+{\Greekmath 0122}_{it}$ no matter what action she chooses, and that she observes an additional signal $s_{it}={\Greekmath 010B}_i+v_{it}$ only when choosing $y_{i,t-1}=1$. Suppose in addition that $({\Greekmath 0122}_{it},v_{it})$ is Gaussian {and i.i.d., that ${\Greekmath 0122}_{it}$ and $v_{it}$ are independent}, and that agents have rational expectations and have a Gaussian prior about ${\Greekmath 010B}_i$. We show in Appendix (ref) that {Assumption (ref) is satisfied.} This again follows from the form of the updating equations for the posterior mean and variance of ${\Greekmath 010B}_i$, which here are conditional on the past action $y_{i,t-1}$; see ((ref))-((ref)) in Appendix (ref) for the case $y_{i,t-1}=1$. {Moreover,} in this example, beliefs are endogenous in the sense that they are affected by past choices.\footnote{{Note that} beliefs are not state-contingent in this example, and Assumption (ref) holds. We will show in Section (ref) that our framework can be extended to allow for state-contingent beliefs, and we will provide a learning model as an illustration.} {Hence, while Assumption (ref) holds, Assumption (ref) is not satisfied in this example.}
{To see a case where Assumption (ref) fails, consider the same setup but now with $v_{it}$ an AR(1) process, so signals $s_{it}$ are serially correlated. In this case, we show in Appendix (ref) that ${\Greekmath 0119}_{i,t+1}$ is not independent of $s_{it}$ conditional on current beliefs ${\Greekmath 0119}_{it}$ and other state variables. In this example, beliefs ${\Greekmath 0119}_{it}$ are not sufficient for future beliefs ${\Greekmath 0119}_{i,t+1}$, since signals have predictive power for future beliefs conditional on current beliefs and other state variables. Hence, Assumption (ref) does not hold.}
\paragraph{Adaptive expectations.} Our setup is also compatible with some models of non-rational expectations. As an example, consider a simple model of adaptive expectations, where mean beliefs evolve as
armona2019home refer to individuals with ${\Greekmath 0115}_i>0$ as “extrapolators”, to those with ${\Greekmath 0115}_i=0$ as “non-updators”, and to those with ${\Greekmath 0115}_i<0$ as “mean reverters”. Assumption (ref) is satisfied if ((ref)) holds, and, say, beliefs are normally distributed with constant variance ${\Greekmath 011B}_i^2$. More generally, Assumption (ref) is consistent with models of adaptive expectations where the entire belief density ${\Greekmath 0119}_{it}$ depends on ${\Greekmath 0119}_{i,t-1}$ and $x_{it}$.
This discussion provides several examples of belief formation models where belief sufficiency, as stated by Assumption (ref) {or Assumption (ref)}, holds. Under either assumption, along with Assumptions (ref), (ref) and (ref), the vector $(x_{it},{\Greekmath 0119}_{it},z_{it},{\Greekmath 0117}_{it})$ contains all the relevant state variables when making the decision. An advantage of our approach is that, since beliefs ${\Greekmath 0119}_{it}$ are state variables, we can study counterfactuals that account for changes in beliefs without the need for a full-fledged structural model.
Let $u_i(y_{it},x_{it},z_{it},{\Greekmath 0117}_{it})$ denote period $t$'s contemporaneous payoffs.\footnote{Here ${\Greekmath 0119}_{it}$ are not payoff-relevant. However, the nonparametric decision rule in ((ref)) will remain the same if payoffs $u_i(y_{it},x_{it},{\Greekmath 0119}_{it},z_{it},{\Greekmath 0117}_{it})$ depend on ${\Greekmath 0119}_{it}$.} Here the action may be continuous or discrete, so our framework covers structural dynamic discrete choice models as well as models with continuous choices. It also covers settings with vector-valued actions, including mixed discrete-continuous choices (e.g., bruneel2022discrete). {We consider a standard setup where individuals maximize the expected discounted sum of utilities, with a constant discount factor ${\Greekmath 010C}_i$.} The individual solves the infinite horizon program $$(y_{i,1},y_{i,2},...)=\limfunc{argmax}_{(y_1,y_2,...)}\, \mathbb{E}\left[\sum_{t=1}^{\infty}{\Greekmath 010C}_i^{t-1}u_i\left(y_{t},x_{it},z_{it},{\Greekmath 0117}_{it}\right)\right],$$ where the expectation is taken with respect to the process of $x_{it},{\Greekmath 0119}_{it},z_{it},{\Greekmath 0117}_{it}$ for given values $(y_1,y_2,...)$, as prescribed by Assumptions (ref), (ref), (ref), and (ref).
Let $V_i(x,{\Greekmath 0119},z,{\Greekmath 0117})$ denote the value function associated with any given state $(x,{\Greekmath 0119},z,{\Greekmath 0117})$. Bellman's principle then implies\footnote{Here the integral in $(x_{t+1},{\Greekmath 0119}_{t+1},z_{t+1},{\Greekmath 0117}_{t+1})$ is taken relative to an appropriate measure.}
{We assume that the policy rule for actions is a measurable function of state variables, that is,}\footnote{ {{See Chapter 9 in stokey1989recursive for a formal analysis.}}}
for some function ${\Greekmath 011E}$. Then, let $${\Greekmath 011E}_i\left(x_{it},{\Greekmath 0119}_{it},z_{it}\right)=\int {\Greekmath 011E}\left(x_{it},{\Greekmath 0119}_{it},z_{it},{\Greekmath 0117}_{it},{\Greekmath 011A}_i,u_i,{\Greekmath 010C}_i,{\Greekmath 010D}_i,{\Greekmath 011C}_i\right){\Greekmath 011C}_i({\Greekmath 0117}_{it})d{\Greekmath 0117}_{it}$$ denote the average decision rule with respect to the shocks ${\Greekmath 0117}_{it}$. It follows from Assumption (ref) that\footnote{We treat ${\Greekmath 011A}_i,u_i,{\Greekmath 010C}_i,{\Greekmath 010D}_i,{\Greekmath 011C}_i$ as non-random quantities. {That is, in our setup agents are assumed to know their preferences and discount factor, the law of motion of $z_{it}$, the belief updating rule, and the distribution of shocks.}} $${\Greekmath 011E}_i\left(x_{it},{\Greekmath 0119}_{it},z_{it}\right)=\mathbb{E}\left[{\Greekmath 011E}\left(x_{it},{\Greekmath 0119}_{it},z_{it},{\Greekmath 0117}_{it},{\Greekmath 011A}_i,u_i,{\Greekmath 010C}_i,{\Greekmath 010D}_i,{\Greekmath 011C}_i\right)\,|\, x_{it},{\Greekmath 0119}_{it},z_{it}\right].$$ Hence, ((ref)) holds for ${\Greekmath 0122}_{it}=y_{it}-{\Greekmath 011E}_i\left(x_{it},{\Greekmath 0119}_{it},z_{it}\right)$, which has zero mean given $x_{it},{\Greekmath 0119}_{it},z_{it}$. In this framework, ${\Greekmath 011E}_i$ in ((ref)) can thus be interpreted as the individual's decision rule averaged over the shocks ${\Greekmath 0117}_{it}$.\footnote{It is straightforward to include additional state variables in ((ref)), under the assumption that beliefs about them are constant and invariant in the counterfactual. Accounting for additional state variables can be empirically relevant, and we will include a number of such variables as controls in our application.} {Note that we have derived ((ref)) under Assumption (ref), but the same expression obtains under Assumption (ref).}
Lastly, the setup is readily adapted to a finite horizon environment. In this case, $t\in\{1,...,T_i\}$, and the Bellman equation ((ref)) becomes, for $t< T_i$ and some terminal value $V_{i,T_i}$, {
where the transitions ${\Greekmath 011A}_{it}$ between ${\Greekmath 0119}_{it}$ and ${\Greekmath 0119}_{i,t+1}$ are time-specific, and ${\Greekmath 010D}_{it}$ is the density of $z_{i,t+1}$ conditional on $z_{it},x_{it},y_{it}$.} Actions then take the form
where the dependence of ${\Greekmath 011E}$ on $i$ and $t$ stems from the presence of $u_i$, ${\Greekmath 010C}_i$, ${\Greekmath 011C}_i$, the terminal value $V_{i,T_i}$, and the ${\Greekmath 011A}_{is}$ and ${\Greekmath 010D}_{is}$ in all periods $s\geq t$. Hence, by including $t$ (i.e., age) in $z_{it}$, ((ref)) takes the same form as ((ref)).
Structurally interpreting an average partial effect as the effect of a counterfactual change requires ${\Greekmath 011E}_i$ to remain invariant in the counterfactual. We now discuss this invariance condition.
Keeping $u_i$ and ${\Greekmath 010C}_i$ constant requires assuming that $u_i$ (such as preferences) and ${\Greekmath 010C}_i$ (discounting) are invariant to changes in the environment. This is a common assumption in dynamic structural models. Invariance of the density of taste shocks ${\Greekmath 011C}_i$ is also commonly assumed. In turn, keeping ${\Greekmath 010D}_i$ constant requires assuming that the process through which past actions and states feed back onto future $z_{it}$ values is invariant in the counterfactual. When $z_{it}$ is a stock that depreciates over time or an asset with some return, for example, this requires assuming away the presence of general equilibrium effects through which the return or the depreciation rate might change in the counterfactual.
In addition, as our framework makes clear, structurally interpreting average partial effects generally requires assuming that the belief updating rule ${\Greekmath 011A}_i$ remains constant in the counterfactual. A change in ${\Greekmath 011A}_i$ corresponds to a steady-state or “long-run” counterfactual where the entire process of $x_{it}$, as perceived by the agent, changes. In our setup, we allow for policies or other counterfactuals to affect beliefs ${\Greekmath 0119}_{it}$, yet we assume that the belief updating rule ${\Greekmath 011A}_i$ is an individual characteristic that remains unaffected. In Section (ref) we will describe how to extend the approach to account for beliefs over multiple horizons, hence making the invariance assumption about ${\Greekmath 011A}_i$ less restrictive. Our focus on counterfactuals involving changes in $x_{it}$ and ${\Greekmath 0119}_{it}$, while ${\Greekmath 011A}_i$ is kept constant, can be viewed as an intermediate case between a static counterfactual where only $x_{it}$ varies, and a long-run, steady-state counterfactual where the entire long-run belief process, including the belief updating rule ${\Greekmath 011A}_i$, is allowed to vary.\footnote{To identify such long-run counterfactuals {in a regression-based approach, without taking a stand on all aspects of the structural model,} one would need to recover the effect of the belief updating rule ${\Greekmath 011A}_i$ on decisions. This would require the availability of empirical counterparts for ${\Greekmath 011A}_i$, as well as suitable cross-sectional exogeneity assumptions (or a valid instrument for ${\Greekmath 011A}_i$). Both conditions would impose strong demands on the data. In particular, ${\Greekmath 011A}_i$ is a subjective process perceived by the agent, which is not directly informed by responses to subjective expectations questions (since ${\Greekmath 011A}_i$ need not coincide with the process of realized beliefs ${\Greekmath 0119}_{it}$).}
{Lastly, in addition to ${\Greekmath 011A}_i$ being invariant, a separate requirement of our approach to compute average partial effects is knowledge of the values $\left(x_{it}^{({\Greekmath 010E})},{\Greekmath 0119}_{it}^{({\Greekmath 010E})}\right)$ in the counterfactual. Our baseline implementation is based on a full pass-through assumption.}
In this section, we describe two examples of our framework. In the first one, we consider a model of consumption, savings, and income, with the aim to assess the effects on consumption of a change in the income process. In the second example, we outline a model of agricultural production that allows farmers to adapt to the weather, with the goal to document the effects of current and expected weather. Both examples fall into the class of structural models that we introduced in the previous section. However, the validity of {our approach does not depend on the details of these specific examples.}
In the first example, we consider a standard incomplete markets model of consumption and saving behavior. For simplicity, we focus on infinite-horizon environment, as in chamberlain2000optimal, although the analysis can easily be adapted to a life-cycle environment.
In the model, $y_{it}$ is household $i$'s log consumption in period $t$, and household utility over consumption is $u_i(y_{it},{\Greekmath 0117}_{it})$, where $u_i$ is an increasing utility function and ${\Greekmath 0117}_{it}$ are i.i.d. taste shocks with density ${\Greekmath 011C}_i$. Household $i$'s discount factor is ${\Greekmath 010C}_i$. Log income $x_{it}$ and beliefs ${\Greekmath 0119}_{it}$ about $x_{i,t+1}$ are exogenous, and Assumptions (ref) and (ref) hold. Households can self-insure using a risk-free bond with constant interest rate $r_i$, and assets $z_{it}$ follow
where $w_{it}=\exp(x_{it})$ and $c_{it}=\exp(y_{it})$ denote income and consumption, respectively. As in ((ref)), the (log) consumption rule takes the form\footnote{In a finite-horizon environment, ${\Greekmath 011E}$ contains time $t$ (i.e., age) as an additional argument, as in ((ref)). }
As a specific example for the income process perceived by the agent, consider a permanent-transitory model (e.g., hall1982sensitivity):
where $u_{it}\sim{\cal{N}}(0,{\Greekmath 011B}_{iu}^2)$ and $v_{it}\sim{\cal{N}}(0,{\Greekmath 011B}_{iv}^2)$ are independent over time and independent of each other at all leads and lags. At time $t$, the agent observes $x_{it}$ and ${\Greekmath 0111}_{it}$, but neither $x_{i,t+1}$ nor ${\Greekmath 0111}_{i,t+1}$. In this case, we have
where ${\Greekmath 0127}$ is the standard Gaussian density, and Assumption (ref) holds. In this specific example, only the mean of ${\Greekmath 0119}_{it}$ varies over time and its variance is constant.
Suppose we wish to assess the impact on consumption at time $t$ of a proportional income tax $T(w)=(1-\exp({\Greekmath 010E}))w$ introduced at time $t$, where recall that $w=\exp(x)$ denotes household income. Under the tax, log income is thus $x^{({\Greekmath 010E})}=x+ {\Greekmath 010E}$. Suppose households believe the tax will remain in place in the future, and they fully adjust their beliefs to the tax, {as described in Subsection (ref)}. When ${\Greekmath 0119}_{it}$ is given by ((ref)) in the absence of the tax, implementing the tax will lead to the new beliefs $${\Greekmath 0119}_{it}^{({\Greekmath 010E})}(\widetilde{x})=\frac{1}{\sqrt{{\Greekmath 011B}_{iu}^2+{\Greekmath 011B}_{iv}^2}}{\Greekmath 0127}\left(\frac{\widetilde{x}-{\Greekmath 0111}_{it}- {\Greekmath 010E}}{\sqrt{{\Greekmath 011B}_{iu}^2+{\Greekmath 011B}_{iv}^2}}\right).$$ Hence, the tax affects both current log income and the perceived conditional mean of future log income.
In this model, a proportional tax does not affect the belief updating rule ${\Greekmath 011A}_i$.\footnote{Indeed, the introduction of the tax is isomorphic to a change in the permanent component, from ${\Greekmath 0111}_{it}$ to ${\Greekmath 0111}_{it}^{({\Greekmath 010E})}={\Greekmath 0111}_{it}+ {\Greekmath 010E}$. Moreover, the distribution of $(x_{i,t+1},{\Greekmath 0111}_{i,t+1})$ given $(x_{it},{\Greekmath 0111}_{it})$ does not change under the tax.} Hence, the total APE fully captures the effect of the tax on consumption. In this case, the contemporaneous APE corresponds to the effect of a purely transitory tax at $t$ that will disappear at $t+1$; equivalently, it is the effect of a ${\Greekmath 010E}$-shift in the transitory income shock $u_{it}$. In turn, the dynamic APE can be interpreted as the effect, {on period-$t$ outcomes,} of a tax that is announced at $t$ and will be implemented at $t+1$.\footnote{The DAPE in ((ref)) is evaluated at income $x^{({\Greekmath 010E})}$ after the tax, so that the CAPE and the DAPE add up to the TAPE. It is also possible to compute an alternative DAPE evaluated under income $x$ before the tax, ${\widetilde{\Delta}_t^{\rm DAPE}({\Greekmath 010E})=\mathbb{E}\left[{\Greekmath 011E}_i\left(x_{it},{\Greekmath 0119}_{it}^{({\Greekmath 010E})},z_{it}\right)-{\Greekmath 011E}_i(x_{it},{\Greekmath 0119}_{it},z_{it})\right]}$.} Lastly, the total APE, which is the sum of the contemporaneous and dynamic APEs, corresponds to the effect of a ${\Greekmath 010E}$-shift in the permanent income shock $v_{it}$.
The model in this subsection relies on specific assumptions about the income process, information, and beliefs. However, those assumptions could be incorrect; for example, agents might have different beliefs about future income. It is important to note that, in our approach, and in our empirical application in Section (ref), we do not assume that the consumption model with permanent-transitory income beliefs describes the data. {Irrespective of the details of the structural model, average partial effects can be interpreted as the structural effects of a counterfactual tax under the conditions we provide, including invariance of the belief updating rule ${\Greekmath 011A}_i$.}
In the second example, we consider a model of agricultural production with costly investment. Output $q_{i,t+1}=g_i(x_{i,t+1},k_{i,t+1})$ depends on the weather $x_{i,t+1}$ and on a dynamic input $k_{i,t+1}$ (such as capital). The weather $x_{it}$, and farmer $i$'s beliefs ${\Greekmath 0119}_{it}$ about $x_{i,t+1}$, satisfy Assumptions (ref) and (ref). The farmer can invest $y_{it}$ in the dynamic input $k_{it}$ at a cost $c_i(y_{it},{\Greekmath 0117}_{it})$, for some i.i.d. cost shifters ${\Greekmath 0117}_{it}$ with density ${\Greekmath 011C}_i$. The dynamic input follows the law of motion $k_{i,t+1}=(1-{\Greekmath 010E}_i)k_{it}+y_{it}$. The farmer decides on $y_{it}$ after observing today's weather $x_{it}$ and her beliefs ${\Greekmath 0119}_{it}$ about tomorrow's weather, but before observing $x_{i,t+1}$. Lastly, the instantaneous profit in period $t$ is $q_{it}-c_i(y_{it},{\Greekmath 0117}_{it})$, and the farmer's discount factor is ${\Greekmath 010C}_i$.
The state variables of the decision problem are $x_{it}$, ${\Greekmath 0119}_{it}$, $k_{it}$, and ${\Greekmath 0117}_{it}$, and, under suitable regularity conditions, the optimal investment rule takes the form
for some function ${\Greekmath 011E}$. Substituting ((ref)) into the output equation, output in period $t+1$ can thus be written as
for some function $\widetilde{{\Greekmath 011E}}$. The presence of ${\Greekmath 0119}_{it}$ in ((ref)) and ((ref)) reflects that the farmer may {adapt} to the prospect of harmful weather in the future by investing today.\footnote{Farmers' adaptation has been studied in the literature using various approaches. burke2016adaptation rely on a long-difference approach to account for farmers' responses to a changing climate. shrader2020improving proposes a framework to account for adaptation in a model where, in contrast with our dynamic framework, the firm's current choice does not affect outcomes (i.e., profit) in later periods. See also dell2014we and keane2020climate. Other approaches in the literature rely on specific aspects of the production model, such as envelope condition arguments (hsiang2016climate, lemoine2018estimating, gammans2020reckoning).}
{In this application, one may be interested in studying investment, through the policy rule ((ref)), or in studying an outcome that depends on investment, such as output in ((ref)).} {In particular, Equation} ((ref)) motivates regressing output on current and past weather and on {past} weather beliefs. Exploiting changes over time in $x_{it}$ and ${\Greekmath 0119}_{it}$, within farmer, is robust to the presence of individual heterogeneity. As an application, one can estimate our belief-augmented average partial effects to assess the impact of a change in the weather process that affects both weather realizations and weather beliefs. In this case as well, structurally interpreting the total APE as reflecting the total effect of such a change relies on the assumption the belief updating process ${\Greekmath 011A}_i$ is invariant. While this assumption may be tenable in the short or medium run, the total APE will not capture the full impact of long-run changes in the climate under which ${\Greekmath 011A}_i$ could be affected.
In this section we study identification and estimation of average partial effects based on ((ref)).
Our approach to the measurement of beliefs ${\Greekmath 0119}_{it}$ relies on data about respondents' expectations. It is increasingly common to elicit responses in a probabilistic manner, by asking respondents to report their subjective probabilities about future events (see manski2004measuring). Responses to questions about subjective probabilistic expectations provide information about some features of ${\Greekmath 0119}_{it}$. Typically, the responses can be interpreted as some functionals $m_{it}=m({\Greekmath 0119}_{it})$, such as the mean, variance, or some other moments of ${\Greekmath 0119}_{it}$. {We assume that such data are available for a sample of individuals $i=1,...,n$ and time periods $t=1,...,T$.} In this section, we abstract from measurement error in responses. However, we will account for measurement error in our empirical application.
When beliefs concern a binary variable $x_{i,t+1}\in\{0,1\}$ (e.g., job loss), the subjective probability ${\Greekmath 0119}_{it}(1)=\Pr(x_{i,t+1}=1\,|\, \Omega_{it})$ provides all the required information in the sense that, under Assumption (ref) or (ref), it is a sufficient statistic for decisions. {One can thus directly use the elicited subjective probability in our approach.} However, when beliefs are about a continuous variable, such as income in our application, the subjective density ${\Greekmath 0119}_{it}$ is a function. At the same time, expectations data are often coarse. A common strategy in such a case is to assume that ${\Greekmath 0119}_{it}$ belongs to a parametric family. For example, in the 1995 and 1998 waves of the SHIW in Italy, respondents are asked about the minimum and maximum earnings that they expect to receive if employed in the following year, together with the probability that their earnings will be below the mid-point between those two values. kaufmann2009disentangling assume that income beliefs follow a triangular distribution conditional on employment.
We will assume that ${\Greekmath 0119}_{it}$ is parametrically specified; that is, that there exists a finite-dimensional vector ${\Greekmath 0112}_{it}$ such that
where ${\Greekmath 0119}(\cdot; {\Greekmath 0112})$ is known given ${\Greekmath 0112}$. When $x_{it}$ is binary or discrete, this assumption is without loss of generality.\footnote{Note that a special case of our parametric assumption is ${\Greekmath 0112}_{it}=m_{it}$. In this case, the key assumption is that the mapping ${\Greekmath 0119}\mapsto m({\Greekmath 0119})$ is injective, so that $m_{it}$ uniquely determines ${\Greekmath 0119}_{it}$.} However, when $x_{it}$ is continuous the assumed parametric family may be misspecified. {In Appendix (ref), we discuss how one could relax the parametric specification on ${\Greekmath 0119}_{it}$ with rich enough data on beliefs.}
We impose the following mean independence condition,
Note that ((ref)) is satisfied in the structural framework of Section (ref). To enhance the plausibility of this condition in applications, one can control for additional time-varying regressors (which can be interpreted as additional state variables), as well as for time-invariant fixed-effects. We will account for both factors in our empirical application.\footnote{In certain applications, ((ref)) may not be plausible, but one may have access to instruments $w_{it}$ (e.g., instruments that exploit some policy variation in sample) such that $\mathbb{E}[{\Greekmath 0122}_{it}\,|\, w_{it}]=0$. Identification of ${\Greekmath 011E}_i$ then requires suitable relevance conditions (see newey2003instrumental).}
Given ((ref)), ((ref)). and ((ref)), we have
It follows that, in an environment with a growing number of time periods (i.e., $T$ tends to infinity), the individual-specific decision rule ${\Greekmath 011E}_i(x,{\Greekmath 0119},z)$ is identified for all $x$, ${\Greekmath 0119}$, $z$ in the empirical support of $x_{it}$, ${\Greekmath 0119}_{it}={\Greekmath 0119}(\cdot;{\Greekmath 0112}_{it})$, and $z_{it}$. {It is worth emphasizing that, in order to separately identify the contemporaneous effect of $x_{it}$ and the dynamic effect of ${\Greekmath 0119}_{it}$, it is crucial that beliefs ${\Greekmath 0119}_{it}$ vary over time conditional on $x_{it}$ and $z_{it}$. Such empirical variation reflects changes in the agent's information set $\Omega_{it}$ over and beyond the changes in the covariates $(x_{it},z_{it})$ that the econometrician observes.}
In many empirical settings, however, belief data are only available on a short panel. In that case, the individual-specific function ${\Greekmath 011E}_i$ is no longer identified. We follow the literature on nonlinear panel data models and impose structure on heterogeneity via a latent variable, or “type”, ${\Greekmath 010B}_i$. Specifically, we assume that, for a function ${\Greekmath 011E}$ and a latent variable ${\Greekmath 010B}_i$, we have
In the structural model of Section (ref), the type ${\Greekmath 010B}_i$ could index primitive parameters such as preferences, for example.
{ A simple specification of ((ref)) is based on the additive model
where ${{\Greekmath 011E}}$ is common across individuals, and ${\Greekmath 010B}_i$ is an additive individual fixed effect. Specification ((ref)) imposes that, while the partial effects associated with changes in income or income beliefs may vary with $x_{it}$, ${\Greekmath 0119}_{it}$, and $z_{it}$, they are common across individuals within a cell $(x_{it},{\Greekmath 0119}_{it},z_{it})$. At the same time, ((ref)) allows consumption levels to differ among individuals. Under suitable exogeneity assumptions,\footnote{For example, if $(x_{it},{\Greekmath 0119}_{it})$ are strictly exogenous and $z_{it}$ are predetermined, one can replace ((ref)) by
} identification of ${{\Greekmath 011E}}$ can then be based on moment restrictions (e.g., arellano1991some). Note that, in short panels, ${\Greekmath 010B}_i$ is not identified, however its value is not needed to recover average partial effects. We will report estimates based on ((ref)) in our application to the SHIW, where the panel dimension is limited to two consecutive periods. However, a drawback of an approach based on ((ref)) is that it seems difficult to justify additivity based on structural assumptions, in the spirit of Section (ref).}
{ Identifying and estimating a non-separable model of the form ((ref)) raises three challenges: the presence of the latent variable ${\Greekmath 010B}_i$, the nonlinearity of the function ${\Greekmath 011E}$, and the fact that the $z_{it}$'s depend on past actions, hence are not strictly exogenous in a panel data sense. The literature has only recently begun to analyze these three issues simultaneously (see bonhomme2025moment). One avenue to tackle these challenges is to suppose, in addition to beliefs ${\Greekmath 0119}_{it}$ being strictly exogenous, that the law of motion of $z_{it}$, as represented by ${\Greekmath 010D}_i$, is the same for all individuals. This assumption of a homogeneous feedback process simplifies the model structure, as shown by kasahara2009nonparametric and bonhomme2023identification, and as we illustrate in Subsection (ref). In effect, the researcher can proceed as if $z_{it}$ were strictly exogenous, despite their dependence on past actions. Common techniques for identification and estimation of mixture models with strictly exogenous covariates can then be used, for example based on a finite-type assumption that is popular in structural models. Moreover, while the plausibility of the homogeneity assumption is context-specific, it appears natural in our consumption application provided agents face a common budget constraint, e.g., a common interest rate.}
{Consider a counterfactual change ${\Greekmath 010E}$, leading to $\left(x_{it}^{({\Greekmath 010E})},{\Greekmath 0119}_{it}^{({\Greekmath 010E})}\right)$. Average partial effects require knowledge of those counterfactual values. In the absence of data on those, a possibility is to assume that individuals fully incorporate the effect of the change in $x_{it}$ and ${\Greekmath 0119}_{it}$, as we outlined in Subsection (ref).}
{To implement this assumption in practice, we suppose that beliefs remain in the same parametric family in the counterfactual.} Hence, for some parameter ${\Greekmath 0112}_{it}^{({\Greekmath 010E})}$, $${\Greekmath 0119}_{it}^{({\Greekmath 010E})}={\Greekmath 0119}\left(\cdot;{\Greekmath 0112}_{it}^{({\Greekmath 010E})}\right).$$ Then, we propose to set
where the expectation is with respect to the baseline belief density, $x_{i,t+1}\sim {\Greekmath 0119}(\cdot;{\Greekmath 0112}_{it})$.\footnote{That is, $ {\Greekmath 0112}_{it}^{({\Greekmath 010E})}={\limfunc{argmax}}_{{{\Greekmath 0112}}}\,\,\int \log\left({\Greekmath 0119}\left({\Greekmath 010E}(x);{{\Greekmath 0112}}\right)\right){\Greekmath 0119}(x;{\Greekmath 0112}_{it})dx$.}
{As an example, consider the introduction of a permanent proportional income tax. Let $x_{it}$ denote log income without the counterfactual tax, and let $x_{it}^{({\Greekmath 010E})}=x_{it}+{\Greekmath 010E}$ denote log income net of the tax. Suppose ${\Greekmath 0119}_{it}$ is normal with mean ${\Greekmath 0116}_{it}$ and variance ${\Greekmath 011B}_{it}^2$, so ${\Greekmath 0112}_{it}=({\Greekmath 0116}_{it},{\Greekmath 011B}_{it}^2)$. Under ((ref)), ${\Greekmath 0119}_{it}^{({\Greekmath 010E})}$ remains normal under the tax, with mean and variance ${\Greekmath 0112}_{it}^{({\Greekmath 010E})}=({\Greekmath 0116}_{it}+{\Greekmath 010E},{\Greekmath 011B}_{it}^2)$.}
Lastly, given actual and counterfactual values of $x_{it}$ and ${\Greekmath 0119}_{it}$, when ${\Greekmath 011E}_i$ is identified on the empirical support, average partial effects (TAPE, CAPE and DAPE) are all identified, provided the support of covariates in the counterfactual lies within their empirical support. In short panels, ${\Greekmath 011E}_i$ may not be identified. However, under the additive specification ((ref)), the average partial effects are similarly identified provided the common function ${\Greekmath 011E}$ is identified (since ${\Greekmath 010B}_i$ cancels out in the definitions of the TAPE, CAPE and DAPE).
For estimation we proceed in three steps.
\paragraph{First step.} First, we estimate the parameters ${\Greekmath 0112}_{it}$ that govern the belief density. Assuming that subjective expectations responses $m_{it}=m({\Greekmath 0119}_{it})$ are available, a minimum-distance estimator solves $$\widehat{{\Greekmath 0112}}_{it}=\underset{{\Greekmath 0112}}{\limfunc{argmin}} \,\,\, d\left(m_{it},m({\Greekmath 0119}(\cdot;{\Greekmath 0112}))\right),$$ where $d$ is some distance function (e.g., Euclidean). Under the assumption that beliefs are elicited without error, i.e., $m_{it}=m({\Greekmath 0119}_{it})$, this step involves no sampling uncertainty.
\paragraph{Second step.}
In the second step, we estimate ${\Greekmath 011E}_i$ as the conditional expectation function in ((ref)). Various approaches are available. For example, stock1989nonparametric proposes a partially linear semiparametric approach. {There are also various ways of incorporating unobserved heterogeneity. In the application, which is based on a two-period panel, we will report results based on two approaches.}
{In the first approach, we assume that ${\Greekmath 011E}_i$ is additive in latent heterogeneity ${\Greekmath 010B}_i$ as in ((ref)), and we rely on an linear specification in a basis of functions:
where $P_r$ is a family of functions, such as polynomials, and $R$ is the number of terms.} {Given observations $y_{it},x_{it},z_{it}$ and estimates $\widehat{{\Greekmath 0112}}_{it}$, for $i=1,...,n$ and $t=1,...,T$, we estimate the ${\Greekmath 010B}$ coefficients using penalized least squares regression:
We will contrast two choices for the penalty term: no penalty (i.e., $\limfunc{Pen}({\Greekmath 010B})=0$) so the estimator is simply OLS, and an $\ell^1$ penalty (i.e., $\limfunc{Pen}({\Greekmath 010B})={\Greekmath 0115}\sum_{r=2}^R|{\Greekmath 010B}_{r}|$) corresponding to the Lasso estimator.}
{In the second approach, we rely on the non-separable model ((ref)), and we assume that ${\Greekmath 010B}_i\in\{1,...,K\}$ takes a finite number of values. We postulate a parametric model for $y_{it}$ given $x_{it},{\Greekmath 0112}_{it},z_{it}$, indexed by a parameter ${\Greekmath 010C}$, as well as a parametric specification for ${\Greekmath 010B}_i$ given $x_i=(x_{i1},...,x_{iT})$, ${\Greekmath 0112}_i=({\Greekmath 0112}_{i1},...,{\Greekmath 0112}_{iT})$, and $z_{i1}$, indexed by ${\Greekmath 0111}$. As discussed in the previous subsection, under the assumption that households face a common budget constraint, estimation can be based on the quasi log-likelihood function
Notice that the law of motion of $z_{it}$, which does not depend on ${\Greekmath 010B}_i$ under homogeneous feedback, does not appear in this expression.\footnote{Denoting the (homogeneous) feedback process as $f(z_{it}\,|\, z_{i,t-1},x_{i,t-1},y_{i,t-1})$, the log-likelihood function is
where the second term on the right-hand side does not depend on $({\Greekmath 010C},{\Greekmath 0111})$. }}
\paragraph{Third step.}
Lastly, in the third step we estimate counterfactuals. Under Assumption ((ref)) we plug in the estimates $\widehat{{\Greekmath 0112}}_{it}$ and $\widehat{{\Greekmath 010B}}_{ir}$, and the counterfactual values $x^{({\Greekmath 010E})}_{it}$ and $\widehat{{\Greekmath 0112}}^{({\Greekmath 010E})}_{it}$, in the APE formulas. For example, again focusing on specification ((ref)), we estimate the total APE, {averaged across periods $t=1,...,T$}, as
with analogous expressions for the contemporaneous and dynamic APEs. Notice that the fixed-effects $\widehat{{\Greekmath 010B}}_{i1}$ cancel out in ((ref)). $\widehat{\Delta}^{\rm TAPE}({\Greekmath 010E})$ is a standard multi-step estimator, for which inference methods are available (e.g., newey1994large). When including a large number $R$ of terms in the expansion and relying on a penalty for regularization, plug-in estimators such as ((ref)) may be biased. To address this issue, in our application we implement the double Lasso method of belloni2014inference for estimation and inference (see Appendix (ref) for details).
{ In the second approach where we relax ((ref)) and assume that types are discrete, we use the estimated type probabilities to construct empirical counterparts to the TAPE, CAPE, and DAPE, averaged across periods. For example, in the case of the total APE we compute
where $(\widehat{{\Greekmath 010C}},\widehat{{\Greekmath 0111}})$ maximize $L({\Greekmath 010C},{\Greekmath 0111})$ in ((ref)). }
In this section we apply our approach to empirically study how consumption depends on current and expected income, and to conduct various tax counterfactuals.
The Italian Survey on Household Income and Wealth (SHIW) is a cross-sectional survey that collects information on annual consumption, disposable income, and wealth of Italian families. Since 1989, it includes a panel component. We use the 1989--1991 waves and the 1995--1998 waves, which include questions about income expectations asked to a subsample of households.
The expectations questions differ in both sets of waves. However, as we show in Appendix (ref), the results are qualitatively similar when analyzing the waves separately, so we pool them together to increase power. In 1989 and 1991, individuals are asked about the probability their income growth will fall within a set of predetermined intervals. In 1995 and 1998, individuals are asked the maximum and minimum amounts they expect to earn if employed, and the probability of earning less than the mid-point between the maximum and minimum. We assume beliefs about log income in the following year follow a normal distribution. In Appendix (ref) we describe our approach to estimate the mean ${\Greekmath 0116}_{it}$ and standard deviation ${\Greekmath 011B}_{it}$ of the beliefs for each individual and time period, which follows arellano2021income. We will also comment on robustness checks obtained under different assumptions and estimation strategies.
We focus on employed household heads, while excluding the self-employed. Our cross-sectional sample with information on beliefs has 7,796 household-year observations, and our panel sample with data on beliefs in two consecutive waves for the same head has 1,646 observations. In Appendix Tables (ref) and (ref) we report descriptive statistics about income expectations questions. In Appendix Table (ref) we provide descriptive statistics about income, consumption, assets, and the estimated means and variances of log income beliefs. Belief questions are about individual income, while consumption, assets, and current income are reported at the household level. We will account for this discrepancy in our construction of average partial effects, and we will also report estimates that control for spousal beliefs when available. Another issue with the belief data in the SHIW is that expectations questions about income in the next 12 months are asked a few months after the end of the calendar year. We will return to this issue in the next subsection. As a preliminary validation check for the expectations questions, in Appendix Table (ref) we document that beliefs have explanatory power for future log income, even conditional on current log income and other controls, in line with what kaufmann2009disentangling found for the 1995-1998 waves.
Based on our first approach, we estimate several versions of the following regression of log consumption:
where $y_{it}$ is log consumption, $x_{it}$ is log income, ${\Greekmath 0112}_{it}$ contains the mean and variance of income beliefs, and $z_{it}$ include log assets as well as a variety of controls (including age, household composition, and a wave indicator).\footnote{Using log assets discards 3.5% of our panel data sample (see Appendix Table (ref)). We have conducted robustness checks without that restriction and obtained similar results.} We will later also present results based on our second approach under a non-separable model with finite types.
We show our main estimates in Table (ref), where we estimate equation ((ref)) by OLS in first differences in both sets of waves. In the table we show standard errors clustered at the household level.\footnote{Standard errors in Table (ref) do not account for the estimation of the means and variances of beliefs, in line with our baseline assumption that beliefs are elicited without error. We will study the impact of measurement error in beliefs on our estimates at the end of this subsection.} The results in columns (2) and (3) show that the mean of log income beliefs influences consumption decisions significantly over and beyond current income, while the variance of the beliefs has an insignificant effect.
It is also interesting to compare the estimates in column (2) with those in column (1) that do not account for beliefs. When including beliefs, the coefficient of family income decreases from $0.58$ to $0.44$. This finding is consistent with the presence of an upward omitted variable bias in column (1).
In column (4) of Table (ref), we interact the mean income beliefs with current income. While the estimates suggest the effect of the mean belief tends to be larger for higher-income households, the interaction effect is only marginally significant. Lastly, in column (5) we add the variance of beliefs and its interaction with income. We find small differences compared to column (4), with insignificant coefficients associated with the variance of beliefs.
In addition to these specifications we also estimate {two other models: a flexible model with additive heterogeneity using the Lasso, and a non-separable model with finite types. We use those models to estimate average partial effects (see Subsection (ref)).}
In Appendix (ref) we report a series of robustness checks. Our main estimates are obtained using a particular approach to construct the mean and variance of log income beliefs. We first probe the robustness of our estimates to different assumptions about the distribution of beliefs, and to different construction methods for the mean and variance of beliefs. The results reported in Appendix Table (ref) show only minor differences compared to our baseline estimates.
While consumption and income correspond to households, the income beliefs questions correspond to individual income. In the baseline results we only use the beliefs of household heads (and adjust our counterfactual calculations). In a robustness check we control for spouses' beliefs about their own income in the consumption regression. The results, also reported in Appendix Table (ref), are again very similar to our main estimates.\footnote{In unreported results, we also controlled for individual income interacted with beliefs, finding similar results.}
The estimates in Table (ref) are obtained by pooling two sets of waves, 1989--1991 and 1995--1998. Economic conditions, as well as the belief elicitation strategies, differ between these two periods. As a robustness check, we report estimates for the two sets of waves separately. The results, reported in Appendix Table (ref), show general qualitative agreement and some quantitative differences between the two periods, with a stronger effect of beliefs in 1995--1998.\footnote{Appendix Table (ref) also reports results controlling for the respondent's subjective probability of being employed in the following year, when available.}
Lastly, although assets are important determinants of consumption, their measurement in the SHIW is imperfect. Indeed, respondents are asked about end-of-year assets, while the state variable in the consumption function is beginning-of-period assets. We assess the robustness of our results in this dimension in two ways. First, following stoltenberg2022consumption we construct an alternative measure of assets by subtracting yearly savings from end-of-year assets. A concern with this specification in our context is that savings in the SHIW are constructed by netting out consumption expenditures from total income, so measurement error in consumption might bias our regression coefficients. Given this, we also report the results of a second specification where we do not include any control for assets. In addition to these checks, we also report results based on an IV strategy that relies on first-period assets and income as instruments for current assets. All the results for current income and income beliefs that we report in Appendix Table (ref) are overall quite similar to our main estimates.
A possible concern with the estimates in Table (ref) is measurement error in belief data. To explore this issue, we focus on the 1989--1991 waves. In those two waves, individuals are asked to distribute 100 balls into 12 bins, corresponding to different intervals of beliefs about log income growth. Assuming log income growth beliefs to be normally distributed, a simple model of the responses is that individuals draw 100 i.i.d. values from their normal belief distributions, and put those in the bins.
However, this simple model does not provide a good approximation to individuals' responses in the SHIW. Indeed, by simulating income beliefs responses from the model, we document that, if they were indeed drawing 100 values, respondents would be reporting a larger number of bins than they do in the data (specifically, $3.61$ bins on average according to the model compared to $1.75$ in the data). The results of this comparison are presented in Appendix Table (ref).
As an alternative model, we postulate that individuals only draw $M<100$ values. We interpret these values as $M$ income growth “scenarios” that the respondent contemplates before giving her answer. The simulations reported in Appendix Table (ref) show that, when $M$ is of the order of 5 or 10 draws, instead of 100, the predicted number of bins reported by the individuals is much closer to the data.
Given this model of measurement error, for any given $M$ we implement a “small-${\Greekmath 011B}$” approximation (e.g., evdokimov2022simple), and use it to bias-correct our regression estimates. While different $M$ values can imply very different belief responses, we find that the resulting coefficient estimates vary little across values of $M$. We provide details about this approach in Appendix (ref) and report the main results in Appendix Figure (ref). At the same time, we acknowledge that, while this sensitivity analysis exercise is reassuring, it relies on a specific model of measurement error, and our ability to entertain other models is limited by the short panel dimension available in the SHIW.\footnote{{{One reason for measurement error}} could be experimenter demand effects, as studied by mummolo2019demand and de2025subjective.}
Lastly, a possible source of measurement error specific to the SHIW, and not captured by the model we have just outlined, relates to the timing of the expectations questions. As pointed out by pistaferri2001superior, since income and consumption refer to the previous calendar year, yet expectations are asked a few months after the end of the year, one needs to assume that individuals do not update their information sets during these few months.\footnote{Alternatively, one could instead follow a structural approach and specify a complete structural model of consumption choices and belief formation. stoltenberg2022consumption propose such an approach and find that income beliefs, corrected for the timing discrepancy within the structure of their model (which assumes rational expectations), have larger effects on consumption than the original beliefs.}
We now use our framework, and our estimates of the consumption function, to assess the effects of a counterfactual income tax on consumption. We assume that the tax schedule takes the parametric form $T(w_g)=w_g-{\Greekmath 0115} w_g^{1-{\Greekmath 011C}}$, where $w_g$ denotes gross income (e.g., benabou2002tax). To define a baseline level of the tax, we rely on the estimates obtained by holter2019tax for Italy, averaged over family characteristics in our sample.
We consider three counterfactuals, corresponding to changes in the ${\Greekmath 0115}$ and ${\Greekmath 011C}$ parameters that index the tax schedule. In the transitory tax and permanent tax counterfactuals, we increase the average tax by 10 percentage points by decreasing ${\Greekmath 0115}$, only for one period in the former case and in all subsequent periods in the latter. In the regressivity counterfactual, we set the parameter ${\Greekmath 011C}$ to its value in the French tax system (which is somewhat less progressive than the Italian one) while at the same time decreasing ${\Greekmath 0115}$ such that the tax change is neutral in terms of total tax revenue.
To estimate the effects of the counterfactuals we compute average partial effects. We report estimates of TAPE, CAPE, and DAPE obtained using linear regression (see Table (ref)), as well as estimates obtained using the Lasso. For the latter, we rely on the double/debiased Lasso method introduced by belloni2014inference, based on interactions and powers of the covariates up to the third order. In the calculations for the permanent tax and regressivity counterfactuals, we assume that individuals fully adjust their beliefs to the new tax; i.e., we implement the formula in ((ref)). We report point estimates and standard errors based on the bootstrap in Appendix Table (ref).
The top panel in Figure (ref) shows average partial effects based on the estimates from column (5) in Table (ref), while the bottom panel corresponds to estimates based on the Lasso. On the left graphs we show the effects on log consumption of a 10% transitory tax. The overall effect based on OLS is $-0.049$, and it is very similar according to the Lasso. In addition, in both specifications there is only moderate variation along income quantiles (indicated on the x axis).
On the middle graphs we show the effect of a 10% permanent tax. Note that the contemporaneous average partial effect (CAPE) coincides with the effect of a transitory tax (compare with the left graphs). Beyond this contemporaneous effect, we find sizable dynamic effects. The dynamic APE (DAPE), which reflects the impact of a changes in beliefs, contributes an additional -0.024 according to OLS, and $-0.028$ according to the Lasso. The total change in consumption, which is approximately $-0.073$ in both specifications, is less than the 10% decrease in income, as is expected if households are only partially insured against income changes (blundell2008consumption). Moreover, the estimates from both specifications indicate that dynamic effects are larger for higher-income households.
Lastly, on the right graphs we show the effect of a revenue-neutral decrease in the progressivity of the tax. While the total effects averaged over all households are relatively small (around $-0.011$), they show substantial heterogeneity along the income distribution: reducing progressivity tends to favor the rich, and it hurts the log consumption of the poor proportionally more. The estimates of OLS and the Lasso are very similar. However, in this case estimates are less precise, see Appendix Table (ref). As in the other two counterfactuals, we observe that the contemporaneous and dynamic effects of the tax have the same sign.
{We next report on estimates based on a parametric non-separable model with finite-type heterogeneity, as in ((ref)). The covariates specification is as in Table (ref). In addition, we let type probabilities depend on an intercept, average income, and average mean beliefs across the two periods. The belief coefficient, the coefficient of log family income, and the intercept, are all allowed to vary with the latent type in the main equation. We model error terms in the consumption equation to be i.i.d. normal (with a variance that does not depend on the type), and the type probabilities as following a multinomial logit specification. We report results for $K=2$ (which is the optimal number of types according to the Bayesian Information Criterion) and $K=3$ types. We also estimated specifications with $K=4$ but found estimates to be more unstable. We describe how we deal with multiple local optima of the likelihood function in Appendix (ref).}
{In Figure (ref) we report the average effects associated with the three tax counterfactuals in a model with types, aggregated across types. In Appendix Tables (ref) and (ref) we report parameter and APE estimates and standard errors. The patterns in Figure (ref) are qualitatively similar to the ones based on a model with additive heterogeneity, see Figure (ref). Quantitatively, the dynamic APE tends to be smaller in the non-separable models, especially in the two-types specification. Moreover, the contemporaneous APE are somewhat larger than in the additive specification.}
{However, the aggregate numbers shown in Figure (ref) mask important heterogeneity. To see this, we show average effects by latent types in Figure (ref) (and report the corresponding point estimates and standard errors in Appendix Tables (ref) and (ref)). Under the permanent tax, for the two-types model, one of the types has a larger contemporaneous effect ($-0.09$ versus $-0.06$), yet virtually no dynamic effect. This type accounts for slightly more than a third of households. The three-types model shows even more heterogeneity: one of the types still has a large contemporaneous effect and non dynamic effect, but the other two show different patterns. In particular, type 1, which accounts for 19% of households, has a low contemporaneous effect ($-0.01$) and a large dynamic effect ($-0.06$). These differences could reflect behavioral heterogeneity, e.g., between “hand-to-mouth” consumers and “permanent income” consumers, although sharpening this interpretation would require a structural model.}
{It is interesting to compare these estimates to average partial effects calculations that do not account for the role of beliefs, that is, which rely on model ((ref)) under additive heterogeneity, yet exclude the belief-related covariates.} In that case, the average consumption effect over all households of a 10% permanent income tax is $-0.065$. This is larger than the contemporaneous effect ($-0.049$) in Figure (ref), consistently with beliefs being an omitted yet relevant regressor in the specification without beliefs. However, this is lower than the total effect in the same figure that accounts for both contemporaneous and dynamic margins ($-0.073$). These differences underscore the need to account for beliefs when computing average partial effects. In addition, note that an estimation method that does not include beliefs cannot account for the difference in impact between a permanent tax and a transitory one.
Lastly, it is worth emphasizing that two conditions are needed in order to interpret the average partial effects in Figures (ref), (ref) or (ref) as structural tax counterfactuals. The first one is that individual beliefs respond one-to-one to the tax. {By varying the parameter ${\Greekmath 0118}$ in ((ref)), we can predict tax effects under different assumptions about belief responses, in the spirit of sensitivity analysis. Our baseline scenario corresponds to the model with additive heterogeneity under full pass-through, that is, assuming that individuals assign probability one to the counterfactual remaining in place in the next period. In Appendix Figure (ref) we show sensitivity results to different values of this probability, $(1+{\Greekmath 0118})^{-1}$. Note that, when ${\Greekmath 0118}$ changes, the CAPE remains unchanged, as it captures changes in consumption exclusively due to changes in current income. On the other hand, changes in ${\Greekmath 0118}$ do affect the DAPE. Interestingly, the dynamic APE remains substantial when individuals assign a $50\%$ probability to the tax not remaining in place next period.}
The second condition is that the belief updating rule ${\Greekmath 011A}_i$ is invariant under the tax. When tax changes have a long-lasting effect, changes in ${\Greekmath 011A}_i$ may occur and induce a third margin of response, beyond contemporaneous and dynamic effects (i.e., beyond CAPE and DAPE). While this third margin may be small or zero in certain cases (as in the permanent-transitory model with a proportional tax, see Subsection (ref)), accounting for it may be important in other cases. The extension to beliefs over longer horizons that we outline in Section (ref) provides a possible way forward.
{In this last part of the section, we illustrate how the structural approach and our semi-structural approach relate to each other in the context of a consumption model.} For this purpose, we simulate a large sample from a life-cycle model of consumption and savings based on Kaplan_Violante_2010, where identical, risk-averse households save to smooth consumption while facing borrowing constraints. {We entertain two different processes of belief formation. We use this exercise to compare and contrast the structural and semi-structural approaches to counterfactual prediction.}
{Relative to the model we presented in Subsection (ref), we make several changes.} First, we impose no borrowing. Second, we specify two different processes for households' expectations. In the first case, we assume that expectations are rational, and coincide with ((ref)). In the second case, we still assume that ((ref)) describes the realized income process, but we specify households' expectations as adaptive, similarly to ((ref)). In both cases, income beliefs, which are key state variables in the model, can be summarized by their time-varying means, which follow a first-order Markov process jointly with log income. Except for having different expectations processes, the two models have exactly the same structure and primitive parameters. See Appendix (ref) for details. The structural model has no time-invariant household heterogeneity.
Under both versions of the model, we compute the true effect of a 10% permanent proportional income tax, and we decompose it under the model into a contemporaneous effect due to current income and a dynamic effect due to beliefs. Then, we compare these counterfactual predictions with our average partial effects (TAPE, CAPE, and DAPE), which we obtain by estimating consumption regressions in the simulated sample. Since the model has a finite horizon, the consumption function ${\Greekmath 011E}$ is age-dependent, and we proxy for this dependence by controlling for age and its square in the regressions. Note that, as we discussed in Subsection (ref), the belief updating rule ${\Greekmath 011A}_i$ is invariant under the counterfactual in the rational expectations version of the model. In the adaptive expectations version we assume that invariance is satisfied as well. We provide details about the model, parameter values, and calculation of counterfactuals in Appendix (ref).
We report the counterfactual calculations in Table (ref). We use a large number of simulated draws, so that variability due to the simulation is negligible. Focusing first on the version with rational expectations (in the left panel), the model predicts a decrease in log consumption of $-0.097$, which is almost one-for-one with the tax increase, as is expected in this model, and a large part can be attributed to a change in beliefs. The semi-structural predictions, which do not rely on the knowledge of the structure and the parameter values of the structural model but are computed using regressions, come close to these numbers. We report the results of three specifications, where we control for linear, quadratic, or spline functions of log assets, and all of them give comparable results in this case.
Turning next to the version with adaptive expectations (in the right panel), the model predicts a smaller effect of the tax ($-0.062$), given the expectations process that we assume. {As a result, a researcher incorrectly assuming rational expectations in this setting, even if she had recovered the other primitive parameters of the model, would overestimate the effect of the tax. This illustrates that, when using a structural approach to predict counterfactuals, correctly specifying belief formation is key. In contrast, our semi-structural approach does not require knowledge of the belief formation process (e.g., rational or adaptive expectations). Indeed, the right panel in Table (ref) shows that the semi-structural predictions, which do not rely on correct specification of the model (including the belief formation part of the model), again come close to the tax effects, albeit in this case only when the regression specification is flexible enough (i.e., quadratic or spline).}\footnote{This reflects the fact that the linear approximation to the consumption policy rule is less accurate in the structural model with adaptive expectations than in the model with rational expectations. In Appendix Figure (ref) we report the policy rules at several ages, for both rational and adaptive expectations. In Appendix Table (ref) we present the tax counterfactual results for different ages.}
We discuss possible extensions of our framework, and conclude with a discussion of implications for belief data collection.
A key assumption in our framework is that, while beliefs about next period's state variables change in the data and counterfactual, the belief updating rule ${\Greekmath 011A}_i$ is time-invariant in sample and invariant to the counterfactual change. This assumption can be relaxed by introducing beliefs over multiple horizons.
If one had access to data on the sequence of beliefs about $x_{i,t+1},x_{i,t+2},...$ into the far future, accounting for those as determinants of the decision, and shifting them in the counterfactual, would provide valid predictions without the need for an invariance assumption about some ${\Greekmath 011A}_i$ process. To go one step in this direction, one can elicit beliefs over multiple horizons $x_{i,t+1},x_{i,t+2},...,x_{i,t+S}$ (as in kocsar2025workers), and account for variation in those beliefs in estimation and counterfactuals.
To describe such an approach, let us replace Assumption (ref) by the following, for some $S\geq 1$:
and denote the corresponding conditional density as ${\Greekmath 0119}_{it}(x_{i,t+S},...,x_{i,t+1})$. In this case, ((ref)) becomes {
}where ${\Greekmath 0119}_{t}^{(1)}$ denotes the marginal of ${\Greekmath 0119}_{t}$ corresponding to period-$t+1$ outcomes. This implies that equation ((ref)) is satisfied for the ${\Greekmath 0119}_{it}$ corresponding to ((ref)). Hence our approach is unchanged, except for the use of a multivariate subjective belief density.
It is interesting to allow for “state-contingent” beliefs, where beliefs are contingent on {potential choices} $y_{it}$, and Assumption (ref) does not hold. For example, in a model of occupational choice, individual income beliefs contingent on occupational choice may be available (e.g., patnaik2020role, arcidiacono2020ex). In that case, our framework is unchanged except for the fact that the state-contingent beliefs enter as arguments in the decision rule.
To see this, suppose for simplicity that actions $y_{it}$ belong to a finite set ${\cal{Y}}$ with $n$ elements. {In this case, one can define ${\Greekmath 0119}_{it}=\{{\Greekmath 0119}_{it}(\cdot;y)\, :\, y\in {\cal{Y}}\}$ to be a set of $n$ densities where, for all $y\in{\cal{Y}}$, ${\Greekmath 0119}_{it}(\cdot;y)$ is the subjective density of $\left(x_{i,t+1}\,|\, y_{it}=y,\Omega_{it}\right)$.} With this new definition of ${\Greekmath 0119}_{it}$, and the associated change in the definition of ${\Greekmath 011A}_i$ in {Assumption (ref)}, the framework is unchanged relative to Section (ref). In particular, the decision rule is still given by ((ref)), so actions depend on the $n$ belief densities ${\Greekmath 0119}_{it}(\cdot;y)$.
As an example of a model with state-contingent beliefs, suppose $x_{it}={\Greekmath 010B}_i+{\Greekmath 0122}_{it}(k)$ when $y_{i,t-1}=k$, for $k\in\{0,1\}$.\footnote{This is equivalent to assuming the individual only observes $x_{it}(k)={\Greekmath 010B}_i+{\Greekmath 0122}_{it}(k)$ when $y_{i,t-1}=k$. As an extension, ${\Greekmath 010B}_i$ may also depend on $k$ (for example, ${\Greekmath 010B}_i$ may represent a vector of occupation-specific abilities), and $x_{it}(k)={\Greekmath 010B}_i(k)+{\Greekmath 0122}_{it}(k)$. In that case, the updating formulas ((ref))-((ref)) need to be adjusted to vector-valued ${\Greekmath 0116}_{it}$ and matrix-valued ${\Greekmath 011B}^2_{it}$. See arcidiacono2025college for an example.} Suppose in addition that ${\Greekmath 0122}_{it}(k)\sim {\cal{N}}(0,{\Greekmath 011B}_{{\Greekmath 0122}_i(k)}^2)$, independent across $i$ and $t$, and that agents are Bayesian with a normal prior on ${\Greekmath 010B}_i$. {At the beginning of period $t$, the posterior distribution of ${\Greekmath 010B}_{i}$ when $y_{i,t-1}=k$ is then} ${\cal{N}}({\Greekmath 0116}_{it},{\Greekmath 011B}_{it}^2)$, where ${\Greekmath 0116}_{it}$ and ${\Greekmath 011B}^2_{it}$ are functions of $k$ satisfying
{When deciding to choose $y_{it} \in \{0,1\}$, $i$ also considers the belief distribution about the yet unobserved $x_{i,t+1}$, which is the sum of two independent normal variables. The first, ${\Greekmath 010B}_i$, has an expected distribution with mean and variance given by ((ref)) and ((ref)), respectively. The second variable, ${\Greekmath 0122}_{i,t+1}(j)$, has zero mean and a variance that depends on her current (not yet taken) choice, $y_{it}=j$.} We then define beliefs as ${\Greekmath 0119}_{it}=({\Greekmath 0119}_{it}(0),{\Greekmath 0119}_{it}(1))$, where ${\Greekmath 0119}_{it}(j)$ is the normal density with mean ${\Greekmath 0116}_{it}$ and variance ${\Greekmath 011B}_{it}^2+{\Greekmath 011B}_{{\Greekmath 0122}_i(j)}^2$ for $j\in\{0,1\}$. It follows from ((ref))-((ref)) that {Assumption (ref)}, for these beliefs ${\Greekmath 0119}_{it}$, is satisfied.
In this paper we provide a method to account for the role of individual expectations in assessing the impact of policies and other counterfactuals. {Our approach is justified} under dynamic structural assumptions, yet implementing the method does not require full specification and estimation of a structural model. A key input to our approach is the use of data on subjective beliefs. Belief elicitation is an active research area. Our approach motivates more work on this front, in several directions.
First, in this section we have shown the usefulness of eliciting belief responses over multiple horizons, and how to incorporate such beliefs to our approach. Research along this line (see, e.g., kocsar2025workers) should be particularly useful to understand dynamic responses under less restrictive invariance conditions.
Second, we have discussed the usefulness of collecting data on state-contingent beliefs (e.g., patnaik2020role, arcidiacono2020ex), and shown that such data can easily be incorporated into our approach. We have also discussed the usefulness of eliciting beliefs under counterfactual policy scenarios (e.g., roth2023effects), to directly measure how beliefs may or may not change in a counterfactual situation.
Lastly, we have highlighted the usefulness of having longitudinal information on individual beliefs. While many data sets with elicited beliefs such as the SHIW have a panel component, the panel dimension often tends to be short, which {puts constraints on} the researcher's ability to allow for individual heterogeneity. Collecting longer longitudinal information {exhibiting more variation in beliefs over time} is important for harnessing the power of belief data.