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Optimal Linear Instrumental Variables Approximations
The Ordinary Least Squares (OLS) estimator has an appealing nonparametric interpretation---it provides the optimal linear approximation (in a mean-square error sense) to the true regression function. That is, the OLS estimand is a meaningful and easily interpretable parameter under misspecification of the linear model. Unfortunately, except in special circumstances (such as with random assignment), this parameter does not have a causal interpretation. Commonly used estimands based on Instrumental Variables (IV) do have a causal interpretation (see, e.g., LATE), but they do not share with OLS the appealing nonparametric interpretation (see, e.g., AIG_2000). The main goal of our paper is to fill this gap and to propose an IV estimand that has the same nonparametric interpretation as OLS, but under endogeneity.
The parameter of interest is thus the vector of slopes in the optimal linear approximation of the structural regression function. We call this parameter the Optimal Linear IV Approximation (OLIVA). We investigate regular identification of the OLIVA, i.e. identification with a finite efficiency bound, based on the results in Severini_Tripathi_2012. The main contribution of our paper is to show that a necessary condition for regular identification of the OLIVA is also sufficient for existence of an IV estimand in a linear structural regression. That is, we show that, under a minimal condition for standard inference on the OLIVA, it is possible to obtain an IV estimator for it.
The identification result is constructive and leads to a Two-Step IV (TSIV) estimation strategy. The necessary condition for regular identification is a conditional moment restriction that is used to estimate a suitable instrument in a first step. The second step is simply a standard linear IV estimator with the estimated instrument from the first step. The situation is somewhat analogous to optimal IV (see, e.g., Robinson and Newey_1990), but more difficult due to the lack of identification of the first step and the first step problem being statistically harder than a nonparametric regression problem. To select an instrument among potentially many candidates, we use Tikhonov regularization, combined with a sieve approach to obtain a Penalized Sieve Minimum Distance (PSMD) first step estimator (cf. Chen_Pouzo_2012 ). The instrument choice based on Tikhonov is statistically and empirically justified. Statistically, a Tikhonov instrument exhibits a certain sufficiency property explained below. Empirically, the resulting PSMD estimator can be computed with standard regression routines. The TSIV estimator is shown to be asymptotically normal and to perform favorably in simulations when compared to alternative estimators, being competitive with the oracle IV under linearity of the structural model, while robustifying it otherwise.
An important application of our approach is to a Hausman test for exogeneity that is robust to misspecification of the linear model. This robustness comes from our TSIV being nonparametrically comparable to OLS under exogeneity. The robust Hausman test is a standard t-test in an augmented regression that does not require any correction for standard errors for its validity, as we show below. Lochner_Moretti_2015 consider a different exogeneity test comparing the classical IV estimator with a weighted OLS estimator when the endogenous variable is discrete. In contrast, our test compares the standard OLS with our TSIV estimator--more in the spirit of the original Hausman 's exogeneity test--while allowing for general endogenous variables (continuous, discrete or mixed). Monte Carlo simulations confirm the robustness of the proposed Hausman test, and the inability of the standard Hausman test to control the empirical size under misspecification of the linear model.
Our paper contributes to two different strands of the literature. The first strand is the nonparametric IV literature; see, e.g., Newey_Powell_2003 , Ai_Chen_2003, HH_2005, BCK_2007, H_2007, H_2011, DFFR_2011, Santos_2012, Chetverikov_Wilhem_2017, and Freyberger, among others. Severini_Tripathi_2006, Severini_Tripathi_2012 discuss identification and efficiency of linear functionals of the structural function without assuming completeness. Their results on regular identification are adapted to the OLIVA below. Santos_2011 establishes regular asymptotic normality for weighted integrals of the structural function in nonparametric IV, also allowing for lack of nonparametric identification of the structural function. BH_2007 develop a nonparametric test of exogeneity under the maintained assumption of nonparametric identification$.$ The OLIVA functional was not considered in Severini_Tripathi_2006, Severini_Tripathi_2012 or Santos_2011, and the semiparametric robust Hausman test complements the nonparametric test of BH_2007.
Our paper is also related to the Causal IV literature that interprets IV nonparametrically as a Local Average Treatment Effect (LATE); see LATE. A forerunner of our paper is Abadie_2003. He defines the Complier Causal Response Function and its best linear approximation in the presence of covariates. He also develops two-step inference for the linear approximation coefficients when the endogenous variable is binary. Within this binary case, we show that the OLIVA's slope parameter is the IV estimand resulting from using the propensity score as instrument, a recommended IV estimator in the literature (see LATE and pg. 623 in Wooldridge2). Our asymptotic results for the binary endogenous case can thus be viewed as extensions of existing methods (such as, e.g., Theorem 3 in LATE) to a nonparametrically estimated propensity score.
The main theoretical contributions of this paper are thus the interpretation of the regular identification of the OLIVA as existence of an IV estimand, the asymptotic normality of a TSIV estimator, and the robust Hausman test. The identification, estimation and exogeneity test of this paper are all robust to the lack of the identification of the structural function (i.e. lack of completeness) and lack of identification of the first step instrument. Furthermore, the proposed methods are also robust to misspecification of linear model, sharing the nonparametric robustness of OLS, but in a setting with endogenous regressors.
We illustrate the utility of our methods with an empirical application estimating the elasticity of intertemporal substitution (EIS) with quarterly US data, revisiting previous work by Yogo. If the structural relationship between consumption growth and interest rates is linear, then the TSIV and standard IV estimands should be the same. In contrast, we obtain a TSIV estimate much larger than the standard IV estimate, with a similar level of precision, thereby suggesting that nonlinearities matter in this application. The TSIV and OLS estimates are rather close, and the robust Hausman test fails to reject the null hypothesis of exogeneity of real interest rates.
The rest of the paper is organized as follows. Section (ref) defines formally the parameter of interest and its regular identification. Section (ref) proposes a PSMD first step and establishes the asymptotic normality of the TSIV. Section (ref) derives the asymptotic properties of the robust Hausman test for exogeneity. The finite sample performance of the TSIV and the robust Hausman test is investigated in Section (ref). Finally, Section (ref) reports the results of our empirical application to the EIS. Appendix A presents notation, assumptions and some preliminary results that are needed for the main proofs in Appendix B. A Supplemental Appendix contains further simulation results, including extensive sensitivity analysis.
Let the dependent variable $Y$ be related to the $p-$dimensional vector $X$ through the equation
where $E[\left. \varepsilon\right\vert Z]=0$ almost surely (a.s), for a $q-$dimensional vector of instruments $Z.$
The OLIVA parameter $\beta$ solves, for $g$ satisfying ((ref)),
where henceforth $A^{\prime}$ denotes the transpose of $A$. Note that $X$ may (and in general, will) contain an intercept. For extensions to weighted least squares versions of ((ref)) see Section (ref).
If $E[XX^{\prime}]$ is positive definite, then
When $X$ is exogenous, i.e. $E[\left. \varepsilon\right\vert X]=0$ a.s., the function $g(\cdot)$ is the regression function $E[\left. Y\right\vert X=\cdot]$ and $\beta$ is identified and consistently estimated by OLS under mild conditions. In many economic applications, however, $X$ is endogenous, i.e. $E[\left. \varepsilon\right\vert X]\neq0$, and identification and estimation of ((ref)) becomes a more difficult issue than in the exogenous case, albeit less difficult than identification and estimation of the structural function $g$ in ((ref)). Of course, if $g$ is linear $g(x)=\gamma_{0}^{\prime}x,$ then $\beta=\gamma_{0}.$
We first investigate regular identification of $\beta$ in ((ref) )-((ref)). The terminology of regular identification is proposed in KhanTamer, and refers to identification with a finite efficiency bound. Regular identification of a parameter is desirable because it means possibility of standard inference (see Chamberlain86). It will be shown below that a necessary condition for regular identification of $\beta$ is
for an square integrable $h(\cdot);$ see Lemma (ref), which builds on Severini_Tripathi_2012. We show that condition ((ref)) is also sufficient for existence of an IV estimand identifying $\beta.$ That is, we show that ((ref)) implies that $\beta$ is identified from a linear structural regression
The IV estimand uses the unknown, possibly not unique, transformation $h(\cdot)$ of $Z$ as instruments. We propose below a Two-Step IV (TSIV) estimator that first estimates the instruments from ((ref)) and then applies IV with the estimated instruments. The proposed IV estimator has the same nonparametric interpretation as OLS, but under endogeneity.
If the nonparametric structural function $g$ is identified, then $\beta$ is of course identified (from (ref)). Conditions for point identification and consistent estimation of $g$ are given in the references above on the nonparametric IV literature. Likewise, asymptotic normality for continuous functionals of a point-identified $g$ has been analyzed in Ai_Chen_2003 , Ai_Chen_2007, CFR_hb, CFR_2014, Chen_Pouzo_2015 and BreunigJohannes_2016, among others.
Nonparametric identification of $g$ is, however, not necessary for identification of the OLIVA; see Severini_Tripathi_2006, Severini_Tripathi_2012. It is indeed desirable to obtain identification of $\beta$ without requiring completeness assumptions, which are known to be impossible to test (cf. Canay_Santos_Shaikh). In this paper we focus on regular identification of the OLIVA without assuming completeness, i.e. without assuming identification of $g$.
Section (ref) below shows the necessity of the conditional moment restriction ((ref)) for regular identification of the OLIVA. When regular identification of the OLIVA does not hold, but the OLIVA is identified, we expect our estimator to provide a good approximation to the OLIVA. This follows because (i) under irregular identification of the OLIVA, the first step instrument approximately solves the first step conditional moment, and (ii) small errors in the first step equation lead to small errors in the second step limit.\footnote{We thank Andres Santos for making this point to us.} Inference under irregular identification is known to be less stable, see Chamberlain86, and it is beyond the scope of this paper. See Babii_Florens for recent advances in this direction, and Escanciano_Li_Bounds for partial identification results.
We observe a random vector $W=(Y,X^{\prime},Z^{\prime})^{\prime}$ satisfying ((ref)), or equivalently,
where $T^{\ast}$ denotes the adjoint operator of the operator $T,$ with $Th(x)=E[\left. h(Z)\right\vert X=x]$ a.s. Let $\mathcal{G}$ denote the parameter space for $g.$ Assume $g\in\mathcal{G}\subseteq L_{2}(X)$ and $r\in L_{2}(Z),$ where henceforth, for a generic random variable $V,$ $L_{2}(V)$ denotes the space of (measurable) square integrable functions of $V,$ i.e. $f\in L_{2}(V)$ if $\left\Vert f\right\Vert ^{2}:=E\left[ \left\vert f(V)\right\vert ^{2}\right] <\infty,$ and where $\left\vert A\right\vert =trace\left( A^{\prime}A\right) ^{1/2}$ is the Euclidean norm.\footnote{When $f$ is vector-valued, by $f(V)\in L_{2}(V)$ we mean that its components are all in $L_{2}(V).$}
The next result, which follows from an application of Lemma 4.1 in Severini_Tripathi_2012, provides a necessary condition for regular identification of the OLIVA. Define $g_{0}:=\arg\min_{g:r=T^{\ast}g}\left\Vert g\right\Vert ,$ and note that correct specification of the model guarantees that $g_{0}$ is uniquely defined; see EHN. Define $\xi=Y-g_{0}(X)$, $\Omega(z)=E[\left. \xi^{2}\right\vert Z=z],$ and let $\mathcal{S}_{Z}$ denote the support of $Z.$ For future reference, define the range of the operator $T$ as $\mathcal{R}(T):=\{f\in L_{2}(X):\exists s\in L_{2} (Z),Ts=f\},$ and for a subspace $V,$ let $V^{\perp}$ and $\overline{V}$ denote, respectively, its orthogonal complement and its closure.
\noindentAssumption 1: ((ref)) holds, $g\in\mathcal{G}\subseteq L_{2}(X),$ $r\in L_{2}(Z),$ and $E[XX^{\prime}]$ is finite and positive definite$.$
\noindentAssumption 2: $0<\inf_{z\in\mathcal{S}_{Z}}\Omega(z)\leq \sup_{z\in\mathcal{S}_{Z}}\Omega(z)<\infty$ and $T$ is compact.
\noindentAssumption 3: There exists $h(\cdot)\in L_{2}(Z)$ such that ((ref)) holds.
The proof of Lemma (ref) and other results in the text are gathered in Appendix B. Assumptions 1 and 2 are taken from Severini_Tripathi_2012 and are standard in the literature. Given the necessity of Assumption 3 and its importance for our results it is useful to provide some discussion on it. The first observation is that although sufficient conditions for Assumption 3 to hold can be obtained for parametric settings, such as those in the Monte Carlo section, it is hard to give primitive sufficient conditions in nonparametric settings. The second observation is that Assumption 3 may hold when $L_{2}-$completeness of $X$ given $Z$ fails and $g$ is thus not identified (see Newey_Powell_2003 for discussion of $L_{2}-$completeness). To illustrate this point, we consider the empirically relevant case where $X$ is continuous and $Z$ is discrete. It is well known that in this case $g$ is not identified. In contrast, Assumption 3 may hold and, importantly, it is testable. To see this, let $\{z_{1} ,...,z_{J}\}$ denote the support of $Z,$ with $J<\infty,$ and note that any function $h$ can be identified with a $J\times p$ matrix through the representation \[ h(z)=\sum_{j=1}^{J}h(z_{j})1(z=z_{j}), \] where $1(A)$ is the indicator function of the event $A.$ Assumption 3 is then simply the conditional moment restriction with a finite number of parameters $\theta=(h(z_{1}),...,h(z_{J}))\in\mathbb{R}^{p\times J}$ given by
where $\mathbf{1}=(1(Z=z_{1}),...,1(Z=z_{J}))^{\prime}.$ To deal with the potential lack of identification of $h$ (i.e. of $\theta)$ we use the minimum norm estimator described below, which is consistent for a population analog $h_{0}(Z)=\theta_{0}\mathbf{1}$. Furthermore, the estimator of $\theta_{0}$ can be shown to be asymptotically normal. Thus, standard tools from nonparametric regression testing can be used to test for ((ref)); see, e.g., Bierens and Escanciano_2006. Whether the nonparametric conditional moment restriction in Assumption 3 is testable more generally (i.e. with continuous $Z)$ is a delicate issue, see ChenSantos, and it will be investigated elsewhere.
When Assumption 3 does not hold two possibilities may arise: (i) $\beta$ is identified, but it has infinite efficiency bound, and (ii) $\beta$ is not identified. When $\beta$ is identified and Assumption 3 fails, $X$ belongs to the boundary of the range of $T$ (i.e. $X\in\overline{\mathcal{R}(T)} \setminus\mathcal{R}(T),$ see Severini_Tripathi_2012), and thus our IV estimand can be made arbitrarily close to $\beta$. As we explain below in Remark (ref), even when Assumption 3 does not hold, our estimator has a well-defined IV estimand as its limit, provided a mild condition is satisfied.
The main observation of this paper is that the necessary condition for regular identification of $\beta$ is also sufficient for existence of an IV estimand. This follows because by the law of iterated expectations, Assumption 3 and $E[\left. \varepsilon\right\vert Z]=0$ a.s.,
which is the IV estimand using $h(Z)$ as instruments for $X.$ We note that to obtain this IV representation in ((ref)) a weaker exogeneity than $E[\left. \varepsilon\right\vert Z]=0$ suffices, namely $E[\varepsilon h(Z)]=0.$ We maintain the \textquotedblleft strict\textquotedblright \ exogeneity $E[\left. \varepsilon\right\vert Z]=0$ because it is often used in the literature and simplifies some of our subsequent asymptotic results, although see Remark (ref). The following result summarizes this finding and shows that, although there are potentially many solutions to ((ref)), the corresponding $\beta$ is unique.
Proposition (ref) suggests a TSIV estimation method where, first, an $h$ is estimated from ((ref)) and then, an IV estimator is considered using the estimated $h$ as instrument. To describe the estimator, let $\{W_{i}\equiv(Y_{i},X_{i}^{\prime},Z_{i}^{\prime})^{\prime}\}_{i=1}^{n}$ be an independent and identically distributed $(iid)$ sample of size $n$ satisfying ((ref)). The TSIV estimator follows the steps:
For ease of exposition, we consider first the case where $X$ and $Z$ have no overlapping components (i.e. no included exogenous or controls) and both are continuous. We also analyze below the case of control variables and discrete variables.
As argued in pg. 130 of Santos_2012 identification of $h$ in ((ref)) is problematic, as in most instances instruments posses a variation that is unrelated to the endogenous regressor (i.e., there exists a function $\psi(z)$ such that $E[\left. \psi(Z)\right\vert X]=0$ a.s.). To deal with the problem of lack of uniqueness of $h,$ we consider a Tikhonov-type estimator. This approach is commonly used in the literature estimating $g$, see HH_2005, CFR_hb, FJV_2011, Chen_Pouzo_2012 and Gagliardini_Scaillet_2012, among others. Chen_Pouzo_2012 propose a PSMD estimator of $g$ and show the $L_{2} -$consistency of a solution identified via a strict convex penalty. These authors also obtain rates in Banach norms under point identification. Our first-step estimator $\hat{h}_{n}$ is a PSMD estimator of the form considered in Chen_Pouzo_2012 when identification is achieved with an $L_{2} $-penalty. As it turns out, the Tikhonov-type or $L_{2}$-penalty estimator is well motivated in our setting, as we explain below. It implies that our instrument satisfies a certain sufficiency property.
Defining $m(X;h):=E[h(Z)-X|X]$, we estimate the unique $h_{0}$ satisfying $h_{0}=\lim_{\lambda\downarrow0}h_{0}(\lambda),$ where \[ h_{0}(\lambda)=\arg\min\{||m(\cdot;h)||^{2}+\lambda||h||^{2}:h\in L_{2}(Z)\}, \] and $\lambda>0.$ Assumption 3 guarantees the existence and uniqueness of $h_{0},$ see EHN. The sufficiency property mentioned above is that for any distinct solution $h_{1}$ of ((ref)), $h_{1}\neq h_{0},$ it holds that in the first stage regression
$\alpha_{1}$ must be zero, as shown in the next result. We note that $V$ is simply a least squares (i.e. reduced form) error term.
This result states that after controlling for $h_{0}(Z)$ in the first stage regression, any other distinct solution to ((ref)) is irrelevant in the first stage. It is in this precise sense that we say $h_{0}(Z)$ is sufficient. We note, however, that this property does not imply that $h_{0}$ is better than any other solution to ((ref)) in terms of leading to a more efficient estimation of $\beta$. For efficiency considerations see Severini_Tripathi_2012.
Having motivated the Tikhonov-type instrument, we introduce now its PSMD estimator. Let $E_{n}[g(W)]$ denote the sample mean operator, i.e. $E_{n}[g(W)]=n^{-1}\sum_{i}^{n}g(W_{i}),$ let $||g||_{n}=\left( E_{n}[\left\vert g(W)\right\vert ^{2}]\right) ^{1/2}$ be the empirical $L_{2}$ norm, and let $\hat{E}[\left. h(Z)\right\vert X]$ be a series-based estimator for the conditional mean $E[\left. h(Z)\right\vert X],$ which is given as follows. Consider a vector of approximating functions \[ p^{K_{n}}(x)=(p_{1}(x),...,p_{K_{n}}(x))^{\prime}, \] having the property that a linear combination can approximate $E[\left. h(Z)\right\vert X=x]$ well. Then, \[ \hat{E}[\left. h(Z)\right\vert X=x]={p^{K_{n}}}^{\prime}(x)(P^{\prime} P)^{-1}\sum_{i=1}^{n}p^{K_{n}}(X_{i})h(Z_{i}), \] where $P=[p^{K_{n}}(X_{1}),...,p^{K_{n}}(X_{n})]^{\prime}$ and $K_{n} \rightarrow\infty$ as $n\rightarrow\infty$.
Let $\mathcal{H}\subseteq L_{2}(Z)$ denote the parameter space for $h.$ Then, define the estimator
where $\mathcal{H}_{n}\subset\mathcal{H}\subseteq L_{2}(Z)$ is a linear sieve parameter space whose complexity grows with sample size$,$ $\hat{m} (X_{i};h)=\hat{E}(h(Z)-X|X_{i})$, and $\lambda_{n}$ is a sequence of positive numbers satisfying that $\lambda_{n}\downarrow0$ as $n\uparrow\infty,$ and some further conditions given in the Appendix A. In our implementation $\mathcal{H}_{n}$ is the finite dimensional linear sieve given by
where $q^{J_{n}}(z)=(q_{1}(z),...,q_{J_{n}}(z))^{\prime}$ is a vector containing a linear sieve basis, with $J_{n}\rightarrow\infty$ as $n\rightarrow\infty$.
To better understand the first step estimator and how it can be computed by standard methods consider the approximation \[ X=E[\left. h(Z)\right\vert X]\approx E[\left. a^{\prime}q^{J_{n} }(Z)\right\vert X]=a^{\prime}E[\left. q^{J_{n}}(Z)\right\vert X], \] which suggests a two step procedure for obtaining $\hat{h}_{n}:$ (i) first compute the fitted values $\hat{q}(X)=\hat{E}[\left. q^{J_{n}}(Z)\right\vert X]$ by OLS of $q^{J_{n}}(Z)$ on $p^{K_{n}}(X);$ and then (ii) run Ridge regression $X$ on $\hat{q}(X).$ Indeed, if we define $D_{n}=E_{n}[\hat {q}(X)X^{\prime}]$, $Q_{2n}=E_{n}[q^{J_{n}}(Z)q^{J_{n}}(Z)^{\prime}],$ and \[ \hat{A}_{\lambda_{n}}=E_{n}[\hat{q}(X)\hat{q}(X)^{\prime}]+\lambda_{n}Q_{2n}. \] Then, the closed form solution to ((ref)) is given by
This estimator can be easily implemented by an OLS and a standard Ridge regression steps: (i) standardize $q^{J_{n}}$ so that $Q_{2n}$ becomes the identity (simply multiply the original $q^{J_{n}}$ by $Q_{2n}^{-1/2});$ (ii) run OLS $q^{J_{n}}(Z)$ on $p^{K_{n}}(X)$ and keep fitted values $\hat{q}(X);$ (iii) run standard Ridge regression of $X$ on $\hat{q}(X);$ the slope coefficient in the last regression is $D_{n}^{\prime}\hat{A}_{\lambda_{n} }^{-1}.$ Section (ref) further discusses implementation of the estimation of $h_{0}$ in a more general setting with additional exogenous variables.
An alternative minimum norm approach requires choosing two sequences of positive numbers $a_{n}$ and $b_{n}$ and solving the program \[ \tilde{h}_{n}:=\arg\min\{||h||_{n}^{2}:h\in\mathcal{H}_{n},||\hat {m}(X;h)||_{n}^{2}\leq b_{n}/a_{n}\}. \] This is the approach used in Santos_2011 for different functionals than the OLIVA. We prefer our implementation, since we only need one tuning parameter rather than two, and data driven methods for choosing $\lambda_{n}$ are readily available; see Section (ref).
This section establishes the consistency and asymptotic normality of $\hat{\beta},$ and the consistency of its asymptotic variance, which is useful for inference. Recall $W=(Y,X^{\prime},Z^{\prime})^{\prime}$ and define
with the short notation $m_{0}=m(W,\beta,h_{0},g_{0}).$ The second term in ((ref)) accounts for the asymptotic impact of estimating the instrument $h_{0}.$ When the minimum norm structural function $g_{0}$ is linear, like with a binary treatment, this second term is zero and there will be no impact from estimating $h_{0}$ on inference. Thus, we can interpret this second term in $m$ as accounting for a \textquotedblleft nonlinearity bias\textquotedblright\ in inference of the IV estimator.
To estimate the asymptotic variance of $\hat{\beta}$ is useful to estimate $g_{0}$. We introduce a Tikhonov-type estimator that is the dual of $\hat {h}_{n}$. Let $\hat{g}_{n}(\cdot)$ denote a PSMD estimator of $g_{0}$ given by
with $G_{n}=E_{n}[\hat{p}(Z)Y],$ $\hat{p}(Z)=\hat{E}[\left. p^{K_{n} }(X)\right\vert Z],$ $\hat{E}[\left. g(X)\right\vert Z=z]={q^{J_{n}}} ^{\prime}(z)(Q^{\prime}Q)^{-1}\sum_{i=1}^{n}q^{J_{n}}(Z_{i})g(X_{i})$, $Q=[q^{J_{n}}(Z_{1}),...,q^{J_{n}}(Z_{n})]^{\prime},$ $P_{2n}=E_{n}[p^{K_{n} }(X)p^{K_{n}}(X)^{\prime}],$ and $\hat{B}_{\lambda_{n}}=E_{n}[\hat{p} (Z)\hat{p}(Z)^{\prime}]+\lambda_{n}P_{2n}.$ For ease of presentation, we use the same notation for the tuning parameters in $\hat{h}_{n}$ and $\hat{g} _{n},$ although of course we will use different tuning parameters $K_{n}$ and $J_{n}$ for estimating $\hat{h}_{n}$ or $\hat{g}_{n},$ see Section (ref) for issues of implementation.
The assumptions in Theorem (ref) are standard in the literature of two-step semiparametric estimators. Theorem (ref) can be then used to construct confidence regions for $\beta$ and testing hypotheses about $\beta$ following standard procedures. The proof of Theorem (ref) relies on new $L_{2}-$rates of convergence for $\hat{h}_{n}$ and $\hat{g}_{n}$ under partial identification of $h$ and $g$ (note that the rates in Chen_Pouzo_2012 are given under point identification and Santos_2011 obtained related rates but for a weak norm).
We provide now a partial effects interpretation for subvectors of the OLIVA parameter $\beta$ that are analogous to OLS. Define $X=(X_{1}^{\prime} ,X_{2}^{\prime})^{\prime}$ and partition $\beta$ accordingly as $\beta =(\beta_{1}^{\prime},\beta_{2}^{\prime})^{\prime}.$ Suppose we are only interested in $\beta_{2}.$ From standard OLS theory, we obtain \[ \beta_{2}=E[V_{2}V_{2}^{\prime}]^{-1}E[V_{2}g(X)], \] where $V_{2}$ is the OLS error from the regression of $X_{2}$ on $X_{1}.$ This result could be used to obtain an estimator of $\beta_{2}$ that does not compute an estimator for $\beta_{1}$ and that reduces the dimensionality of the problem of estimating $h$ (from the dimension of the original $X$ to the dimension of $X_{2}$)$,$ since now we can use the condition \[ E[\left. h(Z)\right\vert V_{2}]=V_{2}\text{ a.s.} \] This method might be particularly useful when the dimension of $X_{1}$ is large and $g$ has a partly linear structure
since then $\beta_{2}=E[V_{2}V_{2}^{\prime}]^{-1}E[V_{2}g_{2}(X_{2})]$ can be interpreted as providing a best linear approximation to $g_{2}(X_{2})$ with a linear function of $V_{2}$, i.e. \[ \beta_{2}=\arg\min_{b_{2}}E[\left( g_{2}(X_{2})-b_{2}^{\prime}V_{2}\right) ^{2}]. \] In this discussion, $X_{1}$ could be variables that are of secondary interest.
Suppose now that there are exogenous variables included in the structural equation $g$. This means $X$ and $Z$ have common components. Specifically, with some abuse of notation, define $X=(X_{1}^{\prime},X_{2}^{\prime} )^{\prime}$ and $Z=(Z_{1}^{\prime},Z_{2}^{\prime})^{\prime}$ where $X_{1}=Z_{1}$ denote the overlapping components of $X$ and $Z,$ with dimension $p_{1}=q_{1}.$ This is a very common situation in applications, where exogenous controls are often used. In this setting a solution of $E[\left. h(Z)\right\vert X]=X$ a.s. has the form $h(Z)=(Z_{1}^{\prime},h_{2}^{\prime }(Z))^{\prime},$ where
Following the arguments of the general case, we could obtain an estimator given by $\hat{h}_{n}=(Z_{1}^{\prime},\hat{h}_{2n}^{\prime})^{\prime},$ where
and $D_{2n}:=E_{n}[\hat{q}(X)X_{2}^{\prime}].$ This setting also covers the case of an intercept with no other common components, where $X_{1}=Z_{1}=1$ and $q_{1}=1.$ The asymptotic normality for $\hat{\beta}$ continues to hold, with no changes in the asymptotic distribution.
If the dimension of $X_{1}$ is high and the sample size is moderate, the method above may not perform well due to the curse of dimensionality. Equation ((ref)) implies
so that nonparametric estimation of $h_{20}$ only involves functions $p^{K_{n}}(X_{2})$ and $q^{J_{n}}(Z).$ Equation ((ref)) is still necessary for regular identification. Summarizing, for implementing our methods with moderate or high dimensional controls $X_{1}$ we recommend our general algorithm above with bases $\{p^{K_{n}}(X_{2})\}$ and $\left\{ q^{J_{n} }(Z)\right\} ,$ which is consistent with the specification in ((ref)). Further details on implementation are provided in Section (ref).
Simplifications occur when some variables are discrete. When the endogenous variable $X$ is discrete we do not need $K_{n}\rightarrow\infty,$ and we can choose $p^{K_{n}}$ as a saturated basis. Consider first the important case of a binary endogenous variable $X=(1,X_{2})$ with $X_{2}\in\left\{ 0,1\right\} $. Define the propensity score $\pi(z):=\Pr\left( \left. X_{2}=1\right\vert Z=z\right) .$ We show below that under the mild assumption that $\pi(z)$ is not constant, Assumption 3 holds. Furthermore, the minimum norm solution $h_{0}$ is simply
where $\alpha=\bar{\pi}\left( 1-\gamma\right) ,$ $\gamma=\bar{\pi} (1-\bar{\pi})/var(\pi(Z))$ and $\bar{\pi}=\Pr\left( X_{2}=1\right) $. An implication of this representation is that the slope of the OLIVA is
i.e., the LATE estimand $\alpha_{\pi}^{IV}$ using the propensity score as instrument, which was suggested in LATE. Thus, the OLIVA\ in the binary endogenous case coincides with an important IV estimand recommended in the literature. We summarize our findings in the following result. The proof can be found in the Appendix.
This result implies that for the binary endogenous case estimating $h_{0},$ and then $\beta_{0},$ simply requires estimating nonparametrically the propensity score.
More generally, if $X$ has $d$ points of support, say $\left\{ x_{1} ,...,x_{d}\right\} ,$ then we can set $K_{n}=d$ and $p_{k}(x)=1(x=x_{k}),$ $k=1,..,K_{n},$ in our general algorithm. Define the unconditional probabilities $\Pr\left( X=x_{j}\right) =\pi_{j},$ $j=1,...,d.$ Then, Assumption 3 boils down to existence of $h$ satisfying the linear equalities, for $k=1,...,d,$
Using Theorem 2, pg. 65, in Luenberger_Book, we can find a closed form solution for $h_{0}$ as follows. Define the generalized propensity scores $\pi_{j}(z):=\Pr\left( \left. X=x_{j}\right\vert Z=z\right) $ and the random vector $\Pi\equiv\Pi(Z)=(\pi_{1}(Z),...,\pi_{d}(Z))^{\prime}.\ $If $E[\Pi\Pi^{\prime}]$ is positive definite, then the minimum norm solution to ((ref)) is given by $h_{0}(z)=\gamma^{\prime}\Pi(z)$ where $\gamma=\left( E[\Pi\Pi^{\prime}]\right) ^{-1}S$ and $S=(\pi_{1}x_{1},...,\pi_{d} x_{d})^{\prime}.$ Thus, for discrete endogenous variables our nonparametric algorithm with $K_{n}=d$ and $p_{k}(x)=1(x=x_{k}),$ $k=1,..,K_{n},$ is a semiparametric method where $h_{0}(z)=\gamma^{\prime}\Pi(z)$ is estimated by estimating the conditional probabilities $\Pi(z)$ by $\hat{\Pi}(z)=(\hat{\pi }_{1}(Z),...,\hat{\pi}_{d}(Z)),$ where $\hat{\pi}_{k}(z)=\hat{E}[\left. p_{k}(X)\right\vert Z=z].$ In estimating $\gamma,$ if the sample analog of $E[\Pi\Pi^{\prime}]$ is positive definite, then there is no need to choose $\lambda$ for estimating $h_{0}.$ If this matrix is not invertible, we can apply the Tikhonov-type estimator, as proposed above.
Similarly, when $Z$ is discrete we do not need $J_{n}$ diverging to infinity. As before, we can choose a linear sieve $\mathcal{H}_{n}$ that is saturated and $q^{J_{n}}(Z)$ could be a saturated basis for it. Specifically, if $Z$ takes $J$ discrete values, $\{z_{1},...,z_{J}\}$, we can take $q_{j} (z)=1(z=z_{j}),$ $j=1,...,J_{n}\equiv J.$
Summarizing, all the different cases (with or without controls, nonparametric or semiparametric structural functions, discrete or continuous variables) can be implemented with the same algorithm but with different definitions of the approximation bases $\{p^{K_{n}}(X),q^{J_{n}}(Z)\}.$ In all these cases, the formulas for the asymptotic variance of $\hat{\beta}$ remain the same. The following section provides further details on implementation.
To enhance the practical applicability of our method we summarize its implementation in what we think is the most useful case in empirical applications: estimation in the presence of a vector of controls entering linearly in the models for $g$ and $h$. Since the vector of controls can be high dimensional, we do not think of the linearity of the controls as a strong assumption. As before, we split $X=(X_{1}^{\prime},X_{2}^{\prime})^{\prime}$ and $Z=(Z_{1}^{\prime},Z_{2}^{\prime})^{\prime},$ where $X_{1}=Z_{1}$ denote the vector of exogeneous controls (containing an intercept), with dimension $p_{1}=q_{1}.$ The endogenous variable of interest $X_{2}$ has dimension $p_{2}=p-p_{1},$ and the instrument $Z_{2}$ has dimension $q_{2}=q-q_{1}.$ Following the discussion above, for implementation one has to choose bases $\{p^{K_{n}}(x),q^{J_{n}}(z)\}$ and the tuning parameters $\{J_{n} ,K_{n},\lambda_{n}\}.$ Using these imputs, we estimate an instrument $\hat {h}_{n}=(Z_{1}^{\prime},\hat{h}_{2n}^{\prime})^{\prime},$ where $\hat{h}_{2n}$ estimates a minimum norm solution $h_{20}$ of \[ E[\left. h_{2}(Z)\right\vert X_{2}]=X_{2}\text{ a.s.} \] An appealing feature of sieve estimation is that additional semiparametric restrictions can be imposed on $h_{2}$ simply by restricting the terms in the basis $\{q^{J_{n}}(z)\}.$ These include additivity or exclusion restrictions, among others. For example, one restriction that we impose in this section is that $h_{2}$ is linear in $Z_{1},$ i.e. $h_{2}(Z)=a_{1}^{\prime}Z_{1} +h_{20}(Z_{2}).$ This is, of course, not necessary for regular identification, but it ameliorates the curse of dimensionality, specially when $Z_{1}$ is high dimensional, and it may lead to better finite sample performance (by reducing variance).
As explained in the previous section, the implementation varies according to the nature of the endogenous variable $X_{2}$ and the instrument $Z_{2}$ (whether continuous or discrete). In the continuous case we need to choose $\{J_{n},K_{n},\lambda_{n}\}$ for estimating $h_{0}.$ We can make these choices simultaneously by Generalized Cross-validation (cf. Wahba_Book, GCV henceforth). To simplify the computations we implement GCV by setting first $J_{n}=q_{1}+j_{n}$ for fixed value $j_{n}$ in a small grid (e.g. $j_{n}\in\{4,5,6,7\}),$ then setting $K_{n}=p_{1}+\left\lfloor cj_{n} \right\rfloor ,$ for a grid of values for $c$ in $[1,3],$ where $\left\lfloor x\right\rfloor $ is the floor function, and then minimizing in $\tau =\{j_{n},c,\lambda\}$ the GCV criteria $GCV_{n}(\tau)$ given below in ((ref)) over the grid values.
Details are given as follows. Let $H_{n}$ denote the $n\times p$ matrix with rows $\hat{h}_{n}(Z_{i})$ $i=1,...,n.$ Let $\mathbf{X}=[X_{1},...,X_{n} ]^{\prime}$ and denote by $\mathbf{X}_{1}\equiv\mathbf{Z}_{1}$ and $\mathbf{X}_{2},$ respectively, the corresponding $n\times p_{1}$ and $n\times p_{2}$ design matrices based on the partition $X=(X_{1}^{\prime},X_{2} ^{\prime})^{\prime}$. Construct the $n\times J_{n}$ matrix $Q=[\mathbf{Z}_{1}$ $Q_{2}],$ $J_{n}=q_{1}+j_{n},$ $Q_{2}=[q^{j_{n}}(Z_{21}),...,q^{j_{n}} (Z_{2n})]^{\prime}$ ($Q_{2}$ excludes an intercept), and similarly the $n\times K_{n}$ matrix $P=[\mathbf{X}_{1}$ $P_{2}],$ $K_{n}=p_{1}+k_{n},$ $k_{n}=\left\lfloor cj_{n}\right\rfloor ,$ $P_{2}=[p^{k_{n}}(X_{21} ),...,p^{k_{n}}(X_{2n})]^{\prime}$ ($P_{2}$ excludes an intercept), and their corresponding projection matrices $\Pi_{P}$ and $\Pi_{Q},$ where $\Pi _{A}=A(A^{\prime}A)^{-1}A^{\prime}$ for a generic matrix $A.$ Denote also $I_{d}$ as the $d\times d$ identity matrix$.$ Then, to provide an expression for $H_{n}$ we construct \[ H_{2n}=Q\hat{A}_{\lambda_{n}}^{-1}Q^{\prime}\Pi_{P}\mathbf{X}_{2}, \] where \[ \hat{A}_{\lambda_{n}}=Q^{\prime}(\Pi_{P}+\lambda_{n}I_{n})Q. \] Finally, \[ H_{n}=[\mathbf{Z}_{1}\text{ }H_{2n}] \] and
where $\mathbf{Y}=[Y_{1},...,Y_{n}]^{\prime}.$ This provides a matrix formula implementation for our estimator.
To give the GCV criteria define $L_{\tau}=\mathbf{X}\left( H_{n}^{\prime }\mathbf{X}\right) ^{-1}H_{n}^{\prime},$ $\hat{Y}_{\tau}=L_{\tau} \mathbf{Y}=(\hat{Y}_{\tau1},...,\hat{Y}_{\tau n})^{\prime}$ and $v_{\tau }=trace(L_{\tau}).$ Then, the GCV criteria for estimating $\hat{\beta}$ is
To estimate $g_{0}$ in the presence of a high dimensional vector of controls we follow the specification in ((ref)). The $n\times1$ vector $G_{n}$ of fitted values $\hat{g}_{n}(X_{i}),$ $i=1,...,n,$ is given by
where \[ \hat{B}_{\lambda_{n}}=P^{\prime}(\Pi_{Q}+\lambda_{n}I_{n})P. \] Since $G_{n}$ is linear in $\mathbf{Y},$ we can easily set another GCV method for selecting $\{J_{n},K_{n},\lambda_{n}\}$ for $\hat{g}_{n}$ (simply replaced $L_{\tau}$ above by $L_{\tau}=P\hat{B}_{\lambda_{n}}^{-1}P^{\prime}\Pi_{Q}).$ See also CFF_2017$.$
The following algorithm summarizes the main steps for implementation\footnote{Matlab and R code to implement the TSIV and related inferences are available from the authors upon request.}:
For continuous variables we recommend using B-splines as sieve basis. If $Z_{2}$ is discrete, with support $\{z_{21},...,z_{2j_{2}}\}$, we set $J_{n}=q_{1}+j_{2}-1$ and $q_{j}(z_{2})=1(z_{2}=z_{2j}),$ $j=2,...,j_{2},$ in the algorithm above. Similarly, if $X_{2}$ is discrete, with support $\{x_{21},...,x_{2k_{2}}\}$, we set $K_{n}=p_{1}+k_{2}-1,$ and $p_{k} (x_{2})=1(x_{2}=x_{2k}),$ $k=2,...,k_{2}.$ In this discussion, we exclude the first element in the indicators because the intercept is part of the exogenous controls.
Our previous discussion can be extended to weighted least squares criteria. That is, suppose that the OLIVA is now defined as
for an square integrable $h(\cdot);$ and under this condition and if $E[XX^{\prime}w(X)]$ is positive definite, then it follows that
The estimation proceeds as in our basic case (where $w=1)$. If $w$ is unknown, we can estimate $w$ nonparametrically and use the plugging estimator with the estimated $w$ to solve for $h$ in ((ref)). Our estimator will be consistent and asymptotically normal under regularity conditions, as in the basic case. It remains to study if estimation of $w$ changes the asymptotic variance of the resulting estimator of $\beta_{w}$. This issue is, however, beyond the scope of this paper and is left for future research.
Applied researchers are concerned about the presence of endogeneity, and they traditionally use tools such as the Hausman's exogeneity test for its measurement. This test, however, is uninformative under misspecification; see Lochner_Moretti_2015. The reason for this lack of robustness is that in these cases OLS and IV estimate different objects under exogeneity, with the estimand of standard IV depending on the instrument itself. As an important by-product of our analysis, we robustify the classic Hausman test of exogeneity against nonparametric misspecification of the linear regression model.
The classical Hausman test of exogeneity (cf. Hausman) compares OLS with IV. If we use the TSIV as the IV estimator, we obtain a robust version of the classical Hausman test, robust to the misspecification of the linear model. For implementation purposes it is convenient to use a regression-based test (see Wooldridge, pg. 481). We illustrate the idea in the case of one potentially endogenous variable $X_{2}$ and several exogenous variables $X_{1}$, with $X_{1}$ including an intercept.
In the model \[ Y=\beta_{1}^{\prime}X_{1}+\beta_{2}X_{2}+U,\qquad E[Uh(Z)]=0,\text{ }h(Z)=(X_{1}^{\prime},h_{2}(Z))^{\prime}, \] the variable $X_{2}$ is exogenous if $Cov(X_{2},U)=0.$ If we write the first-stage as \[ X_{2}=\alpha_{1}^{\prime}X_{1}+\alpha_{2}h_{2}(Z)+V,\qquad E[Vh(Z)]=0, \] then weak exogeneity of $X_{2}$ is equivalent to $Cov(V,U)=0.$ This in turn is equivalent to $\rho=0$ in the least squares regression \[ U=\rho V+\xi. \] A simple way to run a test for $\rho=0$ is to consider the augmented regression \[ Y=\beta^{\prime}X+\rho V+\xi, \] estimated by OLS and use a standard $t-test$ for $\rho=0.$
Since $V$ is unobservable, we first need to obtain residuals from a regression of the endogenous variable $X_{2}$ on $X_{1}$ and $\hat{h}_{2n}(Z),$ say $\hat{V}.$ Then, run the regression of $Y$ on $X$ and $\hat{V}$. The new Hausman test is a standard two-sided t-test for the coefficient of $\hat{V},$ or its Wald version in the multivariate endogenous case. Denote the t-test statistic by $t_{n}.$ The benefit of this regression approach is that under some regularity conditions given in Appendix A no correction is necessary in the OLS standard errors because $\hat{V}$ is estimated. Denote $S=(X,V)^{\prime},$ and consider the following mild assumption.
\noindentAssumption 4: The matrix $E[SS^{\prime}]$ is finite and non-singular.
{The proof of Theorem (ref) is involved and requires stronger conditions than that of }Theorem (ref). In particular, for obtaining the result that standard OLS theory applies under the null hypothesis we have used a conditional exogeneity assumption between $U$ and $Z,$ $E[\left. U\right\vert Z]=0$ a.s. Simulations below show that, at least for the models considered, this assumption leads to a robust Hausman test that is able to control the empirical size. We note that under the null of exogeneity we do not require the model to be linear in the sense of $E[\left. U\right\vert X]=0$ a.s.
This section studies the finite sample performance of the proposed methods. Consider the following Data Generating Process (DGP): \[ \left\{
\right. \qquad\left(
\right) \sim N\left( \left(
\right) ,\left(
\right) \right) , \] where $H_{j}(x)$ is the $j-th$ Hermite polynomial, with the first four given by $H_{0}(x)=1,$ $H_{1}(x)=x,$ $H_{2}(x)=x^{2}-1$ and $H_{3}(x)=x^{3}-3x;$ $V=X-E[\left. X\right\vert Z],$ $\zeta$ is a standard normal, drawn independently of $X$ and $D,\ $and $s$ is a monotone function given below. The DGP is indexed by $p$ and the function $s.$ To generate $V$ note \[ E[\left. X\right\vert Z]=E[\left. E[\left. X\right\vert D]\right\vert Z]=\gamma E[\left. D\right\vert Z]=\gamma s^{-1}(Z), \] where $s^{-1}$ is the inverse of $s.$ Thus, by construction $Z$ is exogenous, $E[\left. \varepsilon\right\vert Z]=0,$ while $X$ is endogenous because $E[\left. \varepsilon\right\vert X]=\rho X,$ with $\rho=\rho_{\varepsilon }(1-\gamma^{2})$, $\rho_{\varepsilon}>0$ and $-1<\gamma<1.$
The structural function $g$ is given by \[ g(x)=\sum_{j=1}^{p}H_{j}(X), \] and is therefore linear for $p=1$, but nonlinear for $p>1.$ It follows from the orthogonality of Hermite polynomials that the true value for OLIVA is $\beta=1\ $and that $g$ is identified if $\gamma\neq0$ (since $Var(E[\left. g(X)\right\vert Z])=\sum_{j=1}^{\infty}g_{j}^{2}\gamma^{2j},$ where $g_{j}=E[g(X)H_{j}(X)]$ is the $j-th$ Hermite coefficient, and thus, $E[\left. g(X)\right\vert Z]=0\Longrightarrow g=0$).
Note also that the OLIVA is regularly identified, because $h(Z)=s^{-1} (Z)/\gamma$ solves \[ E[\left. h(Z)\right\vert X]=X. \] We consider three different DGPs, corresponding to different values of $p$ and functional forms for $s$:
Several values for the parameters $(\gamma,\rho)$ will be considered: $\gamma\in\{0.4,0.8\}$ and $\rho\in\{0,0.3,0.9\}$. We will compare the TSIV with OLS and standard IV (using instrument $Z).$ For DGP1, $h(Z)=\gamma^{-1}Z$ and hence the standard IV estimator with instrument $Z$ is a consistent estimator for the OLIVA. Indeed, the standard IV can be seen as an oracle (infeasible version of our TSIV) under DGP1, where $h$ is known rather than estimated. This allows us to see the effect of estimating $h_{0}$ on inferences. For DGP2 and DGP3, IV is not consistent for the OLIVA. The number of Monte Carlo replications is $5000$. The sample sizes considered are $n=100,$ $500$ and $1000$.
Tables 1-3 report the Bias and MSE for OLS, IV and the TSIV for DGP1-DGP3, respectively. Our estimator is implemented with B-splines, following the GCV described in Section (ref), where to simplify the computations we set $J_{n}=6$ and $K_{n}=2J_{n},$ and optimize only in $\lambda$ for each simulated data. A similar strategy was followed in BCK_2007. Likewise, we have followed a simple rule for selecting $\{J_{n},K_{n},\lambda_{n}\}$ for $\hat{g}_{n}$: switch the values of $J_{n}$ and $K_{n}$ used for $\hat{h}_{n}$ to compute $\hat{g}_{n}$ (so now $J_{n}=2K_{n}),$ and use same value of $\lambda_{n}$ for $\hat{g}_{n}$ as for estimating $\hat{h}_{n},$ which seems to work well in our simulations. Remarkably, for DGP1 in Table 1 our TSIV implemented with GCV performs comparably or even better than IV (which does not estimate $h$ and uses the true $h)$. Thus, our estimator seems to have an oracle property, performing as well as the method that uses the correct specification of the model. As expected, OLS is best under exogeneity, but it leads to large biases under endogeneity. For the nonlinear models DGP2 and DGP3, IV deteriorates because the linear model is misspecified. Our TSIV performs well, with a MSE that converges to zero as $n$ increases. Increasing $\gamma$ makes the instrument stronger, thereby reducing the MSE of IV estimates, while for a fixed $\gamma,$ increasing the level of endogeneity increases the MSE.
We have done extensive sensitivity analysis on the performance of the TSIV estimator. Simulations in the Supplemental Appendix report the sensitivity of the estimator to different choices of tuning parameters, $J_{n}$, $K_{n}$ and $\lambda_{n}$. From these results, we see that the TSIV estimator is not sensitive to the choice of these parameters, within the wide ranges for which we have experimented. This is consistent with the regular identification, which means that the estimator should be robust to local perturbations of the tuning parameters. Likewise, unreported simulations with other DGPs confirm the overall good performance of the proposed TSIV under different scenarios.
Table 4 provides the results for coverage of confidence intervals based on the asymptotic normality of the TSIV using the GCV-computed $\lambda_{n}$, along with that using $0.7\lambda_{n}$ and $0.9\lambda_{n}$. The coverage is very stable for the three choices of $\lambda_{n}$ considered. The performance in DGP1 and DGP2 is fairly good, while in DGP3 it noticeably improves when the sample size increases.
We now turn to the Hausman test. Practitioners often use the Hausman test to empirically evaluate the presence of endogeneity. As mentioned above, the standard Hausman test is not robust to misspefication of the linear model, because in that case OLS and IV estimate different parameters (Lochner_Moretti_2015). We confirm this by simulating data from DGP1-DGP3 and reporting rejection frequencies for the standard Hausman test for $\gamma\in\{0.4,0.8\}$. Table 5 contains the results. For DGP1, the rejection frequencies for $\rho=0$ are close to the nominal level of 5% across the different sample sizes, confirming the validity of the test under correct specification. However, for DGP2 and DGP3 we observe large size distortions for the standard Hausman test, as large as 85%. This shows that the standard Hausman test is unreliable under misspecification of the linear model. In contrast, the proposed robust tests is able to control type-I error uniformly across the three DGPs. We also report size-corrected empirical rejections under the alternative. For the linear model, the standard Hausman test has (slightly) larger power than the robust test, while for the nonlinear model DGP2, the robust test has much larger power. For DGP3, the robust Hausman test outperforms the standard test for low values of $\rho,$ while for large values of $\rho$ they have comparable powers. In all cases we observe an empirical power that increases with the sample size and the level endogeneity, suggesting consistency against these alternatives. Despite these simulation results and others in the Supplemental Appendix, we stress that standard and robust Hausman tests should be viewed as complements rather than substitutes, given that they work under different set of assumptions.
We also report in the Supplemental Appendix further simulation results for cases where $Z$ is discrete and $X$ is continuous. For these DGPs $g$ is not identified, although Assumption 3 is satisfied. These additional simulation results provide further evidence of the excellent finite sample performance of the TSIV and the robust Hausman test relative to their standard IV counterparts.
Overall, these simulations confirm the robustness of the proposed methods to misspecification of the linear IV model and their adaptive behaviour when correct specification holds. Furthermore, the TSIV estimator does not seem to be too sensitive to the choice of tuning parameters. Finally, the proposed Hausman test is indeed robust to the misspecification of the linear model, which makes it a reliable tool for economic applications. These finite sample robustness results confirm the claims made for the TSIV estimator as a nonparametric analog to OLS under endogeneity.
In its log-linearized version, the Consumption-based Capital Asset Pricing Model (CCAPM) leads to the equation
where $\psi$ is the elasticity of intertemporal substitution (EIS), $\Delta c_{t+1}$ is the growth rate of consumption (the first difference in log real consumption per capita), $r_{t+1}$ is the real interest rate at time $t+1$, $\alpha$ is a constant and $Z_{t}$ is a vector of variables in the agent's information set at time $t$. The parameters $\beta_{0}=(\alpha,\psi)^{\prime}$ can be estimated from ((ref)) by several estimation strategies; see, e.g., HansenSingleton_1983. Yogo, using data from Campbell, applied Two-Step Least Squares (TSLS), among other methods, to obtain estimates of $\psi$ across different countries, arguing that in most cases the TSLS is subject to weak identification. Here we focus on quarterly US interest rate data, for which there is empirical evidence suggesting identification (the first-stage F statistic is 15.5). The data set is available at Motohiro Yogo's web page. A full description of the data is given in Campbell.\footnote{It should be possible to extend our asymptotic results above to strictly stationary and ergodic time series data, although doing so is beyond the scope of this paper. Following much of the literature, including Yogo, we compute standard errors assuming that the influence functions of the reported estimators are uncorrelated.}
Following Yogo, we use as instruments $Z_{t}=(r_{t-1},\pi_{t-1},\Delta c_{t-1},dp_{t-1}),$ where $r_{t}$ is the nominal interest rate, $\pi_{t}$ is inflation, and $dp_{t}$ is the log dividend-price ratio. The sample size is $n=206.$ The TSLS point estimate of $\psi$ is 0.06, with a standard error of 0.09. We compare the TSLS with the proposed TSIV. To deal with the curse of dimensionality, we estimate $h_{0}(Z)$ with an additive nonparametric model, $h_{0}(Z_{t})=h_{01}(Z_{t1})+\cdots+h_{04}(Z_{t4})$. Specifically, we follow the implementation in our Monte Carlo and use B-splines with $4,5$ or $6$ knots for each instrument, leading to $J_{n}=12,15,18$, respectively, $K_{n}=2J_{n}$ and GCV for choosing $\lambda$. The matrix $Q=[q^{J_{n}} (Z_{1}),...,q^{J_{n}}(Z_{n})]^{\prime}$ simply concatenates the corresponding matrix for each instrument. For estimating $g_{0}$ for the TSIV's standard errors we choose the same $J_{n}$ and $K_{n}$ as before, and compute $\lambda_{n}$ by GCV. Further details on implementation are given in Section (ref) for the case with an intercept (so $X_{1}=Z_{1}\equiv1).$
Not surprisingly, our TSLS coincides with that of Yogo. The TSIV is relatively much larger than the TSLS, and closer to the OLS, while the standard errors of both IV methods are similar in magnitude. These results are robust to the choice of $J_{n}$ and $K_{n}$ (we have experimented with $K_{n}=cJ_{n}$ for $c$ between 1 and 3 and obtain qualitatively the same conclusions). If we apply our robust Hausman test of exogeneity we obtain very large p-values. Again, this result is robust to the choice of $J_{n}$ and $K_{n}.$ In contrast, the standard Hausman test leads to a p-value of 0.154.
We reach several conclusions from these results. First, the difference between the TSLS and the TSIV suggests that nonlinearities might be important in this application (indeed, the plot of the estimated $\hat{g}_{n},$ which is not reported here for the sake of space, reveals a marked nonlinear estimate). Second, once one accounts for the misspecification uncertainty, the null hypothesis of exogeneity cannot be rejected, thereby suggesting that for the purpose of estimating a log-linearized version of the Euler equation, endogeneity bias may be a second-order concern.