Rationalizing rational expectations: Characterizations and tests

D'Haultfœuille, Xavier; Gaillac, Christophe; Maurel, Arnaud Article Rationalizing rational expectations: Characterizations and tests Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: D'Haultfœuille, Xavier; Gaillac, Christophe; Maurel, Arnaud (2021) : Rationalizing rational expectations: Characterizations and tests, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 12, Iss. 3, pp. 817-842, https://doi.org/10.3982/QE1724 This Version is available at: https://hdl.handle.net/10419/253611 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Quantitative Economics 12 (2021), 817–842 1759-7331/20210817 Rationalizing rational expectations: Characterizations and tests Xav i e r D’Haultfoeuille CREST-ENSAE Christophe Gaillac CREST-ENSAE and TSE Arnaud Maurel Department of Economics, Duke University, NBER, and IZA In this paper, we build a new test of rational expectations based on the marginal distributions of realizations and subjective beliefs. This test is widely applicable, including in the common situation where realizations and beliefs are observed in two different data sets that cannot be matched. We show that whether one can rationalize rational expectations is equivalent to the distribution of realizations being a mean-preserving spread of the distribution of beliefs. The null hypothesis can then be rewritten as a system of many moment inequality and equality constraints, for which tests have been recently developed in the literature. The test is robust to measurement errors under some restrictions and can be extended to account for aggregate shocks. Finally, we apply our methodology to test for rational expectations about future earnings. While individuals tend to be right on average about their future earnings, our test strongly rejects rational expectations. Keywords. Rational expectations, test, subjective expectations, data combination. JEL classification. C12, D84, E24. Xavier D’Haultfoeuille: [email protected] Christophe Gaillac: [email protected] Arnaud Maurel: [email protected] This paper is based on portions of our working paper D’Haultfœuille, Gaillac, and Maurel (2018b). We thank three anonymous referees, Peter Arcidiacono, Levon Barseghyan, Federico Bugni, Pierre Cahuc, Zhuoli Chen, Tim Christensen, Valentina Corradi, Christian Gourieroux, Nathael Gozlan, Gregory Jolivet, Max Kasy, Jia Li, Matt Masten, Magne Mogstad, Andrew Patton, Aureo de Paula, Mirko Wiederholt, Basit Zafar, Yichong Zhang, and participants of various seminars and conferences for useful comments and suggestions. The Federal Reserve Bank of New York disclaims any responsibility or legal liability for this analysis and interpretation of the Survey of Consumer Expectations (SCE) data used in this paper. The SCE data are available without charge at http://www.newyorkfed.org/microeconomics/sce and may be used subject to license terms posted there. Xavier D’Haultfoeuille thanks the hospitality of PSE where part of this research was conducted. Christophe Gaillac acknowledges financial support from the grants ERC POEMH 337665 and ANR-17-EURE-0010. ©2021 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://qeconomics.org.https://doi.org/10.3982/QE1724 818 D’Haultfoeuille, Gaillac, and Maurel Quantitative Economics 12 (2021) 1. Introduction How individuals form their beliefs about uncertain future outcomes is critical to understanding decision making. Despite longstanding critiques (see, among many others, Pesaran (1987), Manski (2004)), rational expectations remain by far the most popular framework to describe belief formation (Muth (1961)). This theory states that agents have expectations that do not systematically differ from the realized outcomes, and efficiently process all private information to form these expectations. Rational expectations (RE) are a key building block in many macro and microeconomic models, and in particular in most of the dynamic microeconomic models that have been estimated over the last two decades (see, e.g., Aguirregabiria and Mira (2010), Blundell (2017), for recent surveys). In this paper, we build a new test of RE. Our test only requires having access to the marginal distributions of subjective beliefs and realizations, and, as such, can be applied quite broadly. In particular, this test can be used in a data combination context, where individual realizations and subjective beliefs are observed in two different datasets that cannot be matched. Such situations are common in practice (see, e.g., Delavande (2008), Arcidiacono, Hotz, and Kang (2012), Arcidiacono, Hotz, Maurel, and Romano (2014), Stinebrickner and Stinebrickner (2014a), Gennaioli, Ma, and Shleifer (2016), Kuchler and Zafar (2019), Boneva and Rauh (2018), Biroli, Boneva, Raja, and Rauh (forthcoming)). Besides, even in surveys for which an explicit aim is to measure subjective expectations, such as the Michigan Survey of Consumers or the Survey of Consumer Expectations of the New York Fed, expectations and realizations can typically only be matched for a subset of the respondents. And of course, regardless of attrition, whenever one seeks to measure long or medium-term outcomes, matching beliefs with realizations does require waiting for a long period of time before the data can be made available to researchers.1 The tests of RE implemented so far in this context only use specific implications of the RE hypothesis. In contrast, we develop a test that exploits all possible implications of RE. Using the key insight that we can rationalize RE if and only if the distribution of realizations is a mean-preserving spread of the distribution of beliefs, we show that rationalizing RE is equivalent to satisfying one moment equality and (infinitely) many moment inequalities.2As a consequence, if these moment conditions hold, RE cannot be refuted, given the data at our disposal. By exhausting all relevant implications of RE, our test is able to detect much more violations of rational expectations than existing tests. To develop a statistical test of RE rationalization, we build on the recent literature on inference based on moment inequalities, and more specifically, on Andrews and Shi (2017). By applying their results to our context, we show that our test controls size asymptotically and is consistent over fixed alternatives. We also provide conditions under which the test is not conservative. 1Situations where realizations can be perfectly predicted beforehand, such as in school choice settings where assignments are a known function of observed inputs, are notable exceptions. 2Interestingly, the equivalence on which we rely, which is based on Strassen’s theorem (Strassen (1965)), is also used in the microeconomic risk theory literature; see, in particular, Rothschild and Stiglitz (1970). Quantitative Economics 12 (2021) Rationalizing rational expectations 819 We then consider several extensions to our baseline test. First, we show that by using a set of covariates that are common to both data sets, we can increase our ability to detect violations of RE. Another important issue is that of unanticipated aggregate shocks. Even if individuals have rational expectations, the mean of observed outcomes may differ from the mean of individual beliefs simply because of aggregate shocks. We show that our test can be easily adapted to account for such shocks. Finally, we prove that our test is robust to measurement errors in the following sense. If individuals have rational expectations but both beliefs and outcomes are measured with (classical) errors, then we can still rationalize RE with such data provided that the amount of measurement errors on beliefs does not exceed the amount of intervening transitory shocks plus the measurement errors on the realized outcomes. In that specific sense, imperfect data quality does not jeopardize the validity of our test. In particular, this allows for elicited beliefs to be noisier than realized outcomes. This provides a rationale for our test even in cases where realizations and beliefs are observed in the same data set, since a direct test based on a regression of the outcome on the beliefs (see, e.g., Lovell (1986))is,atleastatthepopulationlevel,notrobusttoanyamountof measurement errors on the subjective beliefs. We apply our framework to test for rational expectations about future earnings. To do so, we combine elicited beliefs about future earnings with realized earnings, using data from the Labor Market module of the Survey of Consumer Expectations (SCE, New York Fed), and test whether household heads form rational expectations on their annual labor earnings. While a naive test of equality of means between earnings beliefs and realizations shows that earnings expectations are realistic in the sense of not being significantly biased, thus not rejecting the rational expectations hypothesis, our test does reject rational expectations at the 1% level. Taken together, our findings illustrate the practical importance of incorporating the additional restrictions of rational expectations that are embedded in our test. The results of our test also indicate that the RE hypothesis is more credible for certain subpopulations than others. For instance, we reject RE for individuals without a college degree, who exhibit substantial deviations from RE. On the other hand, we fail to reject the hypothesis that college-educated workers have rational expectations on their future earnings. By developing a test of rational expectations in a setting where realizations and subjective beliefs are observed in two different data sets, we bring together the literature on data combination (see, e.g., Cross and Manski (2002), Molinari and Peski (2006), Fan, Sherman, and Shum (2014), Buchinsky, Li, and Liao (forthcoming), and Ridder and Moffitt (2007) for a survey), and the literature on testing for rational expectations in a microenvironment (see, e.g., Lovell (1986), Gourieroux and Pradel (1986), Ivaldi (1992), for seminal contributions). On the empirical side, we contribute to a rapidly growing literature on the use of subjective expectations data in economics (see, e.g., Manski (2004), Delavande (2008), Van der Klaauw and Wolpin (2008), Van der Klaauw (2012), Arcidiacono et al. (2014), de Paula, Shapira, and Todd (2014), Stinebrickner and Stinebrickner (2014b), Wiswall and Zafar (2015)). In this paper, we show how to incorporate all of the relevant information from subjective beliefs combined with realized data to test for rational expectations. 820 D’Haultfoeuille, Gaillac, and Maurel Quantitative Economics 12 (2021) The remainder of the paper is organized as follows. In Section 2,wepresentthegeneral set-up and the main theoretical equivalences underlying our RE test. In Section 3, we introduce the corresponding statistical tests and study their asymptotic properties. Section 4illustrates the finite sample properties of our tests through Monte Carlo simulations. Section 5applies our framework to expectations about future earnings. Finally, Section 6concludes. The Appendix gathers the proofs of the equivalence results. We consider in the Online Supplementary Material (D’Haultfoeuille, Gaillac, and Maurel (2021)) various theoretical extensions, additional simulation results, additional material on the application, and all the remaining proofs. Finally, the companion R package RationalExp, described in the user guide (D’Haultfœuille, Gaillac, and Maurel (2018a)), performs the test of RE. 2. Set-up and characterizations 2.1 Set-up We assume that the researcher has access to a first data set containing the individual outcome variable of interest, which we denote by Y. She also observes, through a second data set drawn from the same population, the elicited individual expectation on Y, denoted by ψ. The two data sets, however, cannot be matched. We focus on situations where the researcher has access to elicited beliefs about mean outcomes, as opposed to probabilistic expectations about the full distribution of outcomes. The type of subjective expectations data we consider in the paper has been collected in various contexts, and used in a number of prior studies (see, among others, Delavande (2008), Zafar (2011b), Arcidiacono, Hotz, and Kang (2012), Arcidiacono et al. (2014), Hoffman and Burks (2020)). Formally, ψ=E[Y|I],whereIdenotes the σ-algebra corresponding to the agent’s information set and E[·|I]is the subjective expectation operator (i.e., for any U,E[U|I] is a I-measurable random variable). We are interested in testing the rational expectations (RE) hypothesis ψ=E[Y|I],whereE[· | I]is the conditional expectation operator generated by the true data generating process. Importantly, we remain agnostic throughout most of our analysis on the information set I. Our setting is also compatible with heterogeneity in the information different agents use to form their expectations. To see this, let (U1Um)denote mvariables that agents may or may not observe when they form their expectations, and let Dk=1if Ukis observed, 0otherwise. Then, if Iis the information set generated by (D1U1DmUm), agents will use different subsets of the (Uk)k=1m (i.e., different pieces of information) depending on the values of the (Dk)k=1m. Our setup encompasses a wide variety of situations, where individuals have private information and form their beliefs based on their information set. This includes various contexts where individuals form their expectations about future outcomes, including education, labor market as well as health outcomes. By remaining agnostic on the information set, our analysis complements several studies, which primarily focus on testing for different information sets, while maintaining the rational expectations assumption (see Cunha and Heckman (2007), for a survey). Quantitative Economics 12 (2021) Rationalizing rational expectations 821 It is easy to see that the RE hypothesis imposes restrictions on the joint distribution of realizations Yand beliefs ψ. In this data combination context, the relevant question of interest is then whether one can rationalize RE, in the sense that there exists a triplet (YψI)such that (i) the pair of random variables (Yψ)are compatible with the marginal distributions of Yand ψ;and(ii) ψcorrespond to the rational expectations of Y, given the information set I,thatis,E(Y|I)=ψ.Hence,weconsiderthetestof the following hypothesis: H0:there exists a pair of random variables Yψand a sigma-algebra Isuch that σψ⊂IY∼Yψ∼ψ,andEY|I=ψ where ∼denotes equality in distribution. Rationalizing RE does not mean that the true realizations Y, beliefs ψ, and information set Iare such that E[Y|I]=ψ.Instead,it means that there exists a triplet (YψI)consistent with the data and such that E[Y| I]=ψ. In other words, a violation of H0implies that RE does not hold, in the sense that the true realizations, beliefs, and information set do not satisfy RE (E[Y|I]=ψ). The converse, however, is not true. 2.2 Equivalences 2.2.1 Main equivalence Let Fψand FYdenote the cumulative distribution functions (cdf) of ψand Y,x+=max(0x), and define (y) =y −∞ FY(t) −Fψ(t)dt Throughout most of our analysis, we impose the following regularity conditions on the distributions of realized outcomes (Y) and subjective beliefs (ψ): Assumption 1. E(|Y|)<∞and E(|ψ|)<∞. The following preliminary result will be useful subsequently. Lemma 1. Suppose that Assumption 1holds.Then H0holds if and only if there exists a pair of random variables (Yψ)such that Y∼Y,ψ∼ψ,and E[Y|ψ]=ψ. Lemma 1states that in order to test for H0, we can focus on the constraints on the joint distribution of Yand ψ, and ignore those related to the information set. This is intuitive given that we impose no restrictions on this set. Our main result is Theorem 1 below. It states that rationalizing RE (i.e., H0) is equivalent to a continuum of moment inequalities, and one moment equality. Theorem 1. Suppose that Assumption 1holds.The following statements are equivalent: (i) H0holds; (ii) (FYmean-preserving spread of Fψ)(y) ≥0for all y∈Rand E[Y]=E[ψ]; (iii) E[(y −Y)+−(y −ψ)+]≥0for all y∈Rand E[Y]=E[ψ]. 822 D’Haultfoeuille, Gaillac, and Maurel Quantitative Economics 12 (2021) The implication (i) ⇒(iii) and the equivalence between (ii) and (iii) are simple to establish. The key part of the result is to prove that (iii) implies (i). To show this, we first use Lemma 1, which states that H0is equivalent to the existence of (Yψ)such that Y∼Y,ψ∼ψand E[Y|ψ]=ψ. Then the result essentially follows from Strassen’s theorem (Strassen (1965, Theorem 8)). It is interesting to note that Theorem 1is related to the theory of risk in microeconomic theory. In particular, using the terminology of Rothschild and Stiglitz (1970), (ii) states that realizations (Y) are more risky than beliefs (ψ). The main value of Theorem 1, from a statistical point of view, is to transform H0into the set of moment inequality (and equality) restrictions given by (iii). We show in Section 3how to build a statistical test of these conditions. Comparison with alternative approaches We now compare our approach with alternative ones that have been proposed in the literature. In the following discussion, as in this whole section, we reason at the population level and thus ignore statistical uncertainty. Accordingly, the “tests” we consider here are formally deterministic, and we compare them in terms of data generating processes violating the null hypothesis associated with each of them. Our approach can clearly detect many more violations of rational expectations than the “naive” approach based solely on the equality E(Y ) =E(ψ). It also detects more violations than the approach based on the restrictions E(Y ) =E(ψ) and V(Y ) ≥V(ψ) (approach based on the variance), which has been considered in particular in the macroeconomic literature on the accuracy and rationality of forecasts (see, e.g., Patton and Timmermann (2012)). On the other hand, and as expected since it relies on the joint distribution of (Yψ), the “direct” approach for testing RE, based on E(Y |ψ) =ψ,can detect more violations of rational expectations than ours. To better understand the differences between these four different approaches (“naive,” variance, “direct,” and ours), it is helpful to consider important particular cases. Of course, if ψ=E[Y|I], individuals are rational and none of the four approaches leads to reject RE. Next, consider departures from rational expectations of the form ψ=E[Y|I]+η,withηindependent of E[Y|I].IfE(η) = 0, subjective beliefs are biased, and individuals are on average either overpessimistic or overoptimistic. It follows that E(Y) =E(ψ), implying that all four approaches lead to reject RE. More interestingly, if E(η) =0, individuals’ expectations are right on average, and the naive approach does not lead to reject RE. However, it is easy to show that, as long as deviations from RE are heterogeneous in the population (V(η) > 0), the direct approach always leads to a rejection. In this setting, our approach constitutes a middle ground, in which rejection of RE depends on the degree of dispersion of the deviations from RE (η) relative to the uncertainty shocks (ε=Y−E(Y |I)). In other words and intuitively, we reject RE whenever departures from RE dominate the uncertainty shocks affecting the outcome. Formally, and using similar arguments as in Proposition 4in Section 2.2.4,one can show that if εis independent of E[Y|I], we reject H0as long as the distribution of the uncertainty shocks stochastically dominates at the second order the distribution of the deviations from RE. Quantitative Economics 12 (2021) Rationalizing rational expectations 823 Specifically, if ε∼N(0σ2 ε)and η∼N(0σ2 η), we reject RE if and only if σ2 η>σ2 ε.In such a case, our approach boils down to the variance approach mentioned above: we reject whenever V(ψ) > V(Y). But interestingly, if the discrepancy (η) between beliefs and RE is not normally distributed, we can reject H0even if V(ψ) ≤V(Y). Suppose for instance that ε∼N(01)and η=a−1{U≤01}+1{U≥09}U∼U[01]and a>0 In other words, 80% of individuals are rational, 10% are overpessimistic and form expectations equal to E[Y|I]−a, whereas 10% are overoptimistic and expect E[Y|I]+a. Then one can show that our approach leads to reject RE when a≥1755, while for a=1755,V(η) ≃0616 <V(ε) =1. Binary outcome Our equivalence result does not require the outcome Yto be continuously distributed. In the particular case where Yis binary, our test reduces to the naive test of E(Y) =E(ψ). Indeed, when Yis a binary outcome and ψ∈[01], one can easily show that as long as E(Y) =E(ψ), the inequalities E[(y −Y)+−(y −ψ)+]≥0automatically hold for all y∈R.Thisappliestoexpectationsaboutbinaryevents,suchas,for example, being employed or not at a given date. Interpretation of the boundary condition To shed further light on our test and on the interpretation of H0, it is instructive to derive the distributions of Y|ψthat correspond to the boundary condition ((y) =0). The proposition below shows that, in the presence of rational expectations, agents whose beliefs ψlies at the boundary of H0have perfect foresight, that is, ψ=E[Y|I]=Y.3 Proposition 1. Suppose that (Yψ) satisfies RE,u→F−1 Y|ψ(τ |u) is continuous for all τ∈ (01),and (y0)=0for some y0in the interior of the support of ψ.Then the distribution of Yconditional on ψ=y0is degenerate:P(Y =y0|ψ=y0)=1. 2.2.2 Equivalence with covariates In practice, we may observe additional variables X∈ RdXin both data sets. Assuming that Xis in the agent’s information set, we modify H0 as follows:4 H0X:there exists a pair of random variables Yψand a sigma-algebra Isuch that σψX⊂IY|X∼Y|Xψ|X∼ψ|X,andEY|I=ψ Adding covariates increases the number of restrictions that are implied by the rational expectation hypothesis, thus improving our ability to detect violations of rational expectations. Proposition 2below formalizes this idea and shows that H0Xcan be expressed as a continuum of conditional moment inequalities, and one conditional moment equality. 3For any cdf F,weletF−1denote its quantile function, namely F−1(τ) =inf{x:F(x)≥τ}. 4See complementary work by Gutknecht, Hoderlein, and Peters (2018), who use subjective expectations data to relax the rational expectations assumption, and propose a method allowing to test whether specific covariates are included in the agents’ information sets. 824 D’Haultfoeuille, Gaillac, and Maurel Quantitative Economics 12 (2021) Proposition 2. Suppose that Assumption 1holds.The following two statements are equivalent: (i) H0Xholds; (ii) Almost surely,E[(y −Y)+−(y −ψ)+|X]≥0for all y∈Rand E[Y−ψ|X]=0. Moreover,if H0Xholds,H 0holds as well. 2.2.3 Equivalence with unpredictable aggregate shocks Oftentimes, the outcome variable is affected not only by individual-specific shocks, but also by aggregate shocks. We denote by Cthe random variable corresponding to the aggregate shocks. The issue, in this case, is that we observe a single realization of C(c, say), along with the outcome variable conditional on that realization C=c. In other words, we only identify FY|C=c rather than FY, as the latter would require to integrate over the distribution of all possible aggregate shocks. Moreover, the restriction E[Y|C=cψ]=ψis generally violated, even though the rational expectations hypothesis holds. It follows that one cannot directly apply our previous results by simply replacing FYby FY|C=c.Insuchacase,one has to make additional assumptions on how the aggregate shocks affect the outcome. To illustrate our approach, let us consider the example of individual income. Suppose that the logarithm of income of individual iat period t, denoted by Yit, satisfies a Restricted Income Profile model: Yit =αi+βt+εit where βtcapture aggregate (macroeconomic) shocks, εit follows a zero-mean random walk, and αi,(βt)tand (εit)tare assumed to be mutually independent. Let Iit−1denote individual i’s information set at time t−1, and suppose that Iit−1= σ(αi(βt−k)k≥1(εit−k)k≥1). If individuals form rational expectations on their future outcomes, their beliefs in period t−1about their future log-income in period tare given by ψit =E[Yit |Iit−1]=αi+Eβt|(βt−k)k≥1+εit−1 Thus, Yit =ψit +Ct+εit −εit−1,withCt=βt−E[βt|(βt−k)k≥1]. The corresponding conditional expectation is given by E[Yit |Iit−1Ct=ct]=ψit +ct=ψit To get closer to our initial set-up, we now drop indexes iand tand maintain the conditioning on the aggregate shocks C=cimplicit. Under these conventions, rationalizing RE does not correspond to E[Y|I]=ψ, but instead to E[Y|I]=c0+ψfor some c0∈R. A similar reasoning applies to multiplicative instead of additive aggregate shocks. In such a case, the null takes the form E[Y|I]=c0ψ,forsomec0>0.Inthesetwoexamples, c0is identifiable: by c0=E(Y) −E(ψ) in the additive case, by c0=E(Y)/E(ψ) in the multiplicative case. Moreover, there exists in both cases a known function q(yc) such that E(q(Y c0)) =E(ψ), namely q(yc) =y−cand q(yc) =y/c for additive and multiplicative shocks, respectively. Quantitative Economics 12 (2021) Rationalizing rational expectations 831 regularity conditions on q(··), which hold in particular for the leading examples of additive and multiplicative shocks (q(yc) =y−cand q(yc) =y/c). We refer the reader to Appendix 1.1 for a detailed discussion of this extension. 4. Monte Carlo simulations In the following, we study the finite sample performances of the test without covariates through Monte Carlo simulations. The finite sample performances of the version of our test that accounts for covariates are reported and discussed in Appendix 5. We suppose that the outcome Yis given by Y=ρψ +ε with ρ∈[01],ψ∼N(01)and ε=ζ−1{U≤01}+1{U≥09} where ζ,U,andψare mutually independent, ζ∼N(201),andU∼U[01].Inthis setup, E(Y |ψ) =ρψ and expectations are rational if and only if ρ=1. But since we observe Yand ψin two different data sets, there are values of ρ= 1for which our test is not consistent. More precisely, we can show that the test is consistent if and only if ρ≤ρ∗≃0616. Besides, given this data generating process, the naive test E(Y ) =E(ψ) is not consistent for any ρ, while the RE test based on variances is only able to detect a subset of violations of RE that correspond to ρ<0445. To compute our test, we need to choose the tuning parameters b0,κ,,andη(see Section 3for definitions). As mentioned in Section 3,weset=005 and η=10−6,following Andrews and Shi (2017). Andrews and Shi (2013) show that there exists in practice a large range of admissible values for the other tuning parameters parameters. Regarding b0and κ, we follow Beare and Shi (2019, Section 4.2) and compute, for a grid of candidate parameters, the rejection rate under the null and under one alternative (namely, ρ=05), through Monte Carlo simulations. Then we set (b0κ) so as to maximize the power subject to the constraint that the rejection rate under the null is below the nominal size 005. That way, we obtain b0=03and κ=0001. The parameter phas a distinct effect, in that its choice does not affect size, at least asymptotically. Rather, this parameter selects to what extent the test aims power at the equality constraint E(Y −ψ) =0 versus the inequalities E[(y −Y)+−(y −ψ)+]≥0(y∈R). Setting pto 005 leads to slightly higher power in our DGP, but values of pin [0031]provide similar finite sample performances, with power always greater than 90% of the maximal power. Results reported in Figure 1show the power curves of the test ϕαfor five different sample sizes (nY=nψ=n∈{400;800;1200;1600;3200}) as a function of the parameter ρ, using 800 simulations for each value of ρ.Weuse500 bootstrap simulations to compute the critical values of the test. Several remarks are in order. First, as expected, under the alternative (i.e., for values of ρ≤ρ∗=0616), rejection frequencies increase with the sample size n. In particular, for the largest sample size n=3200, our test always results in rejection of the RE hypothesis 832 D’Haultfoeuille, Gaillac, and Maurel Quantitative Economics 12 (2021) Figure 1. Power curves. Notes: The vertical line at ρ≃0616 corresponds to the theoretical limit for the rejection of the null hypothesis using our test. The dotted horizontal line corresponds to the 5% level. for values of ρas large as 045. Second, in this setting, our test is conservative in the sense that rejection frequencies under the null are smaller than α=005, for all sample sizes. This should not necessarily come as a surprise since the test proposed by AS has been shown to be conservative in alternative finite-sample settings (see, e.g., Table 1, p. 22 in AS for the case of first-order stochastic dominance tests). However, for the version of our test that accounts for covariates and for the data generating process considered in Section S5 of the Appendix in the Online Supplementary Material, rejection frequencies under the null are very close to the nominal level. 5. Application to earnings expectations 5.1 Data Using the tests developed in Section 3, we now investigate whether household heads form rational expectations on their future earnings. We use for this purpose data from the Survey of Consumer Expectations (SCE), a monthly household survey that has been conducted by the Federal Reserve Bank of New York since 2012 (see Armantier, Topa, Van der Klaauw, and Zafar (2017), for a detailed description of the survey, and Kuchler and Zafar (2019); Conlon, Philossoph, Wiswall, and Zafar (2018); Fuster, Kaplan, and Zafar (forthcoming) for recent articles using the SCE). The SCE is conducted with the primary goal of eliciting consumer expectations about inflation, household finance, labor market, as well as housing market. It is a rotating internet-based panel of about 1200 household heads, in which respondents participate for up to 12 months.9Each month, the panel consists of about 180 entrants, and 1100 repeated respondents. While entrants are overall fairly similar to the repeated respondents, they are slightly older and also have slightly lower incomes (see Table 1 in Armantier et al. (2017)). Of particular interest for this paper is the supplementary module on labor market expectations. This module is repeated every 4months since March 2014. Since March 9Each survey takes on average about 15 minutes to complete, and respondents are paid $15 per survey completed. Quantitative Economics 12 (2021) Rationalizing rational expectations 833 Table 1. Descriptive statistics of the SCE sample. Mean Std. Dev. Male 053 050 White 074 043 College degree 049 050 Low numeracy 033 047 Tenure ≤6months 017 038 Age 458130 ψ(Earnings beliefs) $50,592 $40,889 Y(Realized earnings) $52,354 $38,634 2015, respondents are asked the following question about labor market earnings expectations (ψ)overthenext4months: “What do you believe your annual earnings will be in 4months?” Implicit throughout the rest of our analysis is the assumption that these elicited beliefs correspond to the mean of the subjective beliefs distribution.10 In this module, respondents are also asked about current job outcomes, including their current annual earnings (Y), through the following question: “How much do you make before taxes and other deductions at your [main/current] job, on an annual basis?” Specifically, we use for our baseline test the elicited earnings expectations (ψ), which are available for two cross-sectional samples of household heads who were working either full-time or part-time at the time of the survey, and responded to the labor market module in March 2015 and July 2015, respectively. We combine this data with current earnings (Y) declared in July 2015 and November 2015 by the respondents who are working full-time or part-time at the time of the survey.11 This leaves us with a final sample of 2993 observations, which is composed of 1565 earnings expectation observations, and 1428 realized earnings observations. 51% (1536) of these observations correspond to the subsample of respondents who are reinterviewed at least once. We refer to Table 1for additional details on our sample. 5.2 Implementation of the test We summarize how we implemented the test in practice, either on the overall sample or on each subsample corresponding to the binary covariates in Table 1. For each case, we start by winsorizing the distribution of realized earnings (Y) and earnings beliefs 10This assumption, while often made in the subjective expectations literature, is apriorirestrictive. In this application, for the vast majority of the subgroups of the population, the mean of ψcannot be statistically distinguished from the one of Y(see Table 2below). This provides empirical support for this assumption. 11Throughout our analysis (with the exception of the number of observations reported in Table 2), we use the monthly survey weights of the SCE in order to obtain an estimation sample that is representative of the population of U.S. household heads. See Armantier et al. (2017) for more details on the construction of these weights. We also Winsorize the top 5percentile of the distributions of realized earnings and earnings beliefs. 834 D’Haultfoeuille, Gaillac, and Maurel Quantitative Economics 12 (2021) (ψ)atthe95% level.12 Then we perform the test without covariates, where we allow for multiplicative aggregate shock, and thus test H0S,withq(y;c) =y/c.13 Then we use the function test of our companion R package RationalExp.14 We choose the same values for the tuning parameters b0=03and κ=0001 as in the Monte Carlo simulations in Section 4.Wealsosetp=005,=005,andη=10−6. Following Andrews and Shi (2017), the interval  Yis approximated by a grid of length 100 from mini=1n  Yito maxi=1n  Yi. Finally, we use 5000 bootstrap simulations to compute the critical values of the test. 5.3 Are earnings expectations rational? In Table 2, we report the results from the naive test of RE (E(Y) =E(ψ)), and our preferred test (“Full RE”), where we allow for multiplicative aggregate shocks. We implement the tests both on the overall population and on separate subgroups. The latter approach allows us not only to identify which groups fail to rationalize RE, but also, and importantly, to account for the possibility that aggregate shocks may in fact differ across subgroups. Several remarks are in order. First, using our test, we reject for the whole population, at any standard level, the hypothesis that agents form rational expectations over their Table 2. Tests of RE on annual earnings. p-Value Number of Obs. E(Y −ψ)/E(Y) Naive RE Variance RE Full RE ψY All 0034 023 071 <0001 1565 1428 Women 0059 013 062 <0001 730 649 Men 0025 048 058 0210 835 779 White 0032 031 067 0015 1200 1097 Minorities 0046 043 060 <0005 365 331 College degree −0001 096 050 0129 1106 1053 No college degree 0093 004 057 0015 459 375 High numeracy 0033 028 062 0013 1158 1070 Low numeracy 0055 027 058 0022 407 358 Tenure ≤6months 0105 024 063 0001 271 180 Tenure >6months 0007 081 065 0074 1294 1248 Note: “Naive RE” denotes the naive RE test of equality of means between Yand ψ. “Variance RE” denotes the variance RE test where the null hypothesis is the variance of Ybeing greater or equal than the variance of ψ, once we account for aggregate, multiplicative shocks. “Full RE” denotes the test without covariates, where we test H0Swith q(yc) =y/c.Weuse5000 bootstrap simulations to compute the critical values of the Full RE test. Distributions of realized earnings (Y) and earnings beliefs (ψ)are both Winsorized at the 95% quantile. 12We show in Table SI of the Online Supplementary Material that our results are robust to other levels of Winsorization. 13In our application, the parameter cis estimated using survey weights from the SCE. 14See Section 3 in our user’s guide (D’Haultfœuille, Gaillac, and Maurel (2018a)) for details on this function. Quantitative Economics 12 (2021) Rationalizing rational expectations 835 future earnings. Second, we also reject RE (at the 5% level) when we apply our test separately for whites (non-Hispanics) and minorities, as well as low versus high numeracy test scores.15 Third, the results from our test point to beliefs formation being heterogeneous across schooling (college degree vs. no college degree) and tenure (more or less than 6months spent in current job) levels. In particular, we cannot rule out that the beliefs about future earnings of individuals with more schooling experience correspond to rational expectations with respect to some information set. Similarly, while we reject RE at any standard level for the subgroup of workers who have accumulated less than 6 months of experience in their current job, we can only marginally reject at the 10% level RE for those who have been in their current job for a longer period of time. As such, these findings complement some of the recent evidence from the economics of education and labor economics literatures that individuals have more accurate beliefs about their ability as they progress through their schooling and work careers (see, e.g., Stinebrickner and Stinebrickner (2012), Arcidiacono, Aucejo, Maurel, and Ransom (2016)). Fourth, using the naive test of equality of means between earnings beliefs and realizations, one would instead generally not reject the null at any standard levels. The one exception is the subgroup of workers without a college degree, for whom the naive test yields rejection of RE at the 5% level. But, as discussed before, one cannot rule out that such a rejection is due to aggregate shocks. Even though individuals in the overall sample form expectations over their earnings in the near future that are realistic, in the sense of not being significantly biased, the result from our preferred test shows that earnings expectations are nonetheless not rational. Taken together, these findings highlight the importance of incorporating the additional restrictions of rational expectations that are embedded in our test, using the distributions of subjective beliefs and realized outcomes to detect violations of rational expectations. That the variance test of RE never rejects the null at any standard levels indicates that it is important in practice to go beyond the first moments, and exploit instead the full distributions of beliefs and outcomes to detect departures from rational expectations. These results also suggest that, in order to rationalize the realized and expected earnings data, one should consider alternative models of expectation formation that primarily differ from RE in their third, or higher-order moments. The results of the direct test of RE on the subsample of individuals who are followed over four months are reported in Table 3below. While these results generally paint a similar picture to the results of our test, there are some differences. In particular, the direct test rejects RE at the 5% level for men and at 1% for individuals with tenure greater than 6months, whereas we do not reject RE for the former group and only marginally so, at the 10% level, for the latter. The direct test also rejects with less power than our test for certain groups (low numeracy, tenure lower than 6months, and minorities). This lower power may seem surprising given that the direct test can exploit the joint distribution 15Respondents’ numeracy is evaluated in the SCE through five questions involving computation of sales, interests on savings, chance of winning lottery, of getting a disease and being affected by a viral infection. Respondents are then partitioned into two categories: “High numeracy” (4or 5correct answers), and “low numeracy” (3or fewer correct answers). 836 D’Haultfoeuille, Gaillac, and Maurel Quantitative Economics 12 (2021) of (Yψ), but is simply due to the important reduction in sample size when focusing on the subsample of individuals who are followed over 4months results. There are also important issues associated with the direct test, which generally warrant caution when interpreting the results from this test. Most importantly, as already discussed in Section 2.2.4, the direct test is not robust to measurement errors on the subjective beliefs ψ. As shown in Proposition S3 in the Appendix in the Online Supplementary Material, it is however possible to derive a restriction on βunder RE. Specifically, if ξψis positively correlated with ε+ξY, we have, under RE, β≥1−1 1+λ(2) where λis a lower bound on the signal-to-noise ratio V(ψ)/V(ξψ).Table3also reports the results of tests combining (2) with the restrictions on the marginal distributions used in our full RE test. Adding the restriction (2) does not change the results for values of signal-to-noise ratio between 5and 20 (i.e., for noise-to-signal ratios between 5% and 20%). Overall, using the subsample of linked data (Y ψ) through this additional restriction does not add much to our test, at least once we account for possible measurement errors on the elicited beliefs. Another significant concern with the direct test, and more generally, the use of linked data on (Y ψ), is that attrition may be endogenous. We discuss this issue in more details in Appendix 6.2. Table 3. Direct test, our test, and combined test of RE on annual earnings. p-Value Combined Test Direct Test Full RE Bound on Signal/Noise λ:520 βImplied Bound on β:0833 0952 Number of Obs. ψY(ψY) All 0954 0001 <0001 <0001 <0001 1565 1428 768 Women 0956 0002 <0001 <0001 <0001 730 649 356 Men 0960 0021 0210 0276 0276 835 779 412 White 0963 0004 0021 0019 0010 1200 1097 596 Minorities 0928 0010 0006 0007 0005 365 331 172 College degree 0974 0060 0130 0182 0182 1106 1053 560 No college degree 0954 0044 0013 0017 0017 459 375 208 High numeracy 0959 0001 0012 0016 0016 1158 1070 573 Low numeracy 0954 0094 0022 0030 0030 407 358 195 Tenure ≤6months 0942 0015 0001 0002 0001 271 180 98 Tenure >6months 0956 0001 0091 0094 0094 1294 1248 670 Note: “Direct test” denotes the direct test of RE when (ψY ) is observed. βis the coefficient of the regression of Yon ψin that case. “Full RE” denotes the test without covariates, where we test H0Swith q(y c) =y/c.Weuse5000 bootstrap simulations to compute the critical values of the Full RE test. “Combined RE test” denotes the test without covariates, where we test H0S with q(yc) =y/c, which is the “Full RE” test, combined with the additional restriction β≥1−1/(1+λ), where λis an a priori bound on the signal-to-noise ratio. Distributions of realized earnings (Y) and earnings beliefs (ψ) are both Winsorized at the 95% quantile. Quantitative Economics 12 (2021) Rationalizing rational expectations 837 Coming back to our test, the rejection of RE for the overall population but also for most of the subpopulations are, in view of Proposition 4, unlikely to be due to data quality issues. In that sense, these results may be seen as robust evidence against the RE hypothesis for individual earnings, at least in this context. As a result, conclusions of behavioral models based on the assumption that agents form rational expectations about their future earnings may be misleading. Exploring this important question requires one to go beyond testing though, by quantifying the extent to which model predictions are actually sensitive to the violations from rational expectations that have been detected with our test. We investigate this issue in D’Haultfœuille, Gaillac, and Maurel (2018b)in the context of a life-cycle consumption model. 6. Conclusion In this paper, we develop a new test of rational expectations that can be used in a broad range of empirical settings. In particular, our test only requires having access to the marginal distributions of realizations and subjective beliefs. As such, it can be applied in frequent cases where realizations and beliefs are observed in two separate data sets, or only observed for a selected subpopulation. By bypassing the need to link beliefs to future realizations, our approach also enables to test for rational expectations without having to wait until the outcomes of interest are realized and made available to researchers. We establish that whether one can rationalize rational expectations is equivalent to the distribution of realizations being a mean-preserving spread of the distribution of beliefs, a condition which can be tested using recent tools from the moment inequalities literature. We show that our test can easily accommodate covariates and aggregate shocks, and, importantly for practical purpose, is robust to some degree of measurement errors on the elicited beliefs. We apply our method to test for rational expectations about future earnings, using data from the Survey of Consumer Expectations. While individuals tend to be right on average about their future earnings, our test strongly rejects rational expectations. Beyond testing, in this application as in any other situations where rational expectations are violated, a natural next step is to evaluate the deviations from rational expectations that one can rationalize from the available data. In the context of structural analysis, a central question then becomes to which extent the main predictions of the model are sensitive to those departures from rational expectations. We explore this important issue and propose in D’Haultfœuille, Gaillac, and Maurel (2018b) a tractable sensitivity analysis framework on the assumed form of expectations. Appendix A: Proofs of the equivalence results A.1 Proof of Lemma 1 Under H0,thereexistY,ψand Isuch that Y∼Y,ψ∼ψ,σ(ψ)⊂Iand E(Y|I)= ψ. Then, by the law of iterated expectations, EY|ψ=EEY|I|ψ=Eψ|ψ=ψ 838 D’Haultfoeuille, Gaillac, and Maurel Quantitative Economics 12 (2021) Conversely, if there exists (Yψ)such that Y∼Y,ψ∼ψand E[Y|ψ]=ψ,letI= σ(ψ).Thenψ=E[Y|ψ]=E[Y|I]and H0holds. A.2 Proof of Theorem 1 (i) ⇔(iii). By Strassen’s theorem (Strassen (1965, Theorem 8)), the existence of (Yψ) with margins equal to FYand FψandsuchthatE[Y|ψ]=ψis equivalent to fdF ψ≤ fdF Yfor every convex function f. By, for example, Proposition 2.3 in Gozlan, Roberto, Samson, Shu, and Tetali (2018), this is, in turn, equivalent to (iii). (ii) ⇔(iii). By Fubini–Tonelli’s theorem, y −∞ FY(t)dt =E[y −∞ 1{t≥Y}dt]=E[(y − Y)+]. The same holds for ψ.Hence,(y) ≥0for all y∈Ris equivalent to E[(y −Y)+]≥ E[(y −ψ)+]for all y∈R. The result follows. A.3 Proof of Proposition 1 First, by Jensen’s inequality, we obtain E(y0−Y)+|ψ≥y0−E(Y |ψ)+=(y0−ψ)+ Moreover, (y0)=0implies that E((y0−Y)+)=E((y0−ψ)+). Hence, almost surely, we have E(y0−Y)+|ψ=(y0−ψ)+ Equality in the Jensen’s inequality implies that the function is affine on the support of the random variable. Therefore, for almost all u,weeitherhaveSupp(Y |ψ=u) ⊂[y0∞)or Supp(Y |ψ=u) ⊂(−∞y0]. Because E[Y|ψ]=ψ,Supp(Y |ψ=u) ⊂[y0∞)for almost all u>y 0and Supp(Y |ψ=u) ⊂(−∞y0]for almost all u<y 0. Then, for all τ∈(01), F−1 Y|ψ(τ |u) ≥y0for almost all u≥y0and F−1 Y|ψ(τ |u) ≤y0for almost all u≤y0.Thus,for all τ∈(01), by continuity of F−1 Y|ψ(τ |·),F−1 Y|ψ(τ |y0)=y0. This implies that Y|ψ=y0is degenerate. A.4 Proof of Proposition 2 We first prove that H0Xis equivalent to the existence of (Yψ)such that DY+ (1−D)ψ= Y,D⊥⊥ (Yψ)|Xand E(Y|ψX) =ψ. First, under H0X,thereexists (YψI)such that DY+(1−D)ψ= Y,D⊥⊥ (Yψ)|X,σ(ψX) ⊂Iand E(Y|I)=ψ.Then EY|ψX=EEY|I|ψX=Eψ|ψX=ψ Conversely, if there exists (Yψ)such that DY+(1−D)ψ= Y,D⊥⊥(Yψ)|Xand E(Y|ψX)=ψ,letI=σ(Xψ).Thenψ=E(Y|ψX)=E(Y|I)and H0Xholds. The proposition then follows as Theorem 1. Quantitative Economics 12 (2021) Rationalizing rational expectations 839 A.5 Proof of Proposition 4 For all y,ξ→E[(y −ψ−ξ)+]is decreasing and convex. 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