Uniform Validity of the Subset Anderson-Rubin Test under Heteroskedasticity and Nonlinearity

Uniform Validity of the Subset Anderson-Rubin Test under Heteroskedasticity and Nonlinearity

JEL classification codes: C11, C12, C22, C32, C44.

\noindentKeywords: uniformly valid inference, lack of identification, local projection, impulse response.

Introduction

\global\long\global\long\global\long\global\long\global\long\global\long\global\long\global\long\global\long\global\long We analyze Anderson-Rubin (AR) statistics based on the continuous updating estimator (CUE) for parametric non-linear moment conditions and under a null hypothesis that restricts only a subset of the parameters. We extend results about the uniform validity of these statistics by Guggenberger, Kleibergen, Mavroeidis and Chen GuggenbergerKleibergenMavroeidisChen2012, GKMC12, who considered the case of linear models in iid settings and under homoskedasticity. Our CUE framework is non-linear, allows for heteroskedasticity, general dependence and heterogeneity. In time series environments stationarity is not assumed but we do rule out trending behavior including the presence of unit roots. We make no parametric assumptions about the data generating process. The null hypothesis only imposes a set of possibly non-linear moment restrictions that are parameterized by a finite dimensional parameter. Under our assumptions, estimated parameters may be partially or completely unidentified and there may not be a valid limiting distribution for estimated parameters even in situations where the parameters are identified. The setting also includes classical weak and non-identified scenarios where the limiting distribution of parameter estimates is non-standard and/or depends on nuisance parameters that cannot be estimated. This means that Wald type inference is potentially highly unreliable. Our inference procedure using the Anderson-Rubin statistic is simple to implement and only requires conventional chi-square based critical values. Our conditions center around objects that the researcher observes and controls. These include the data used in the analysis and the moment conditions that are being specified for inference. On the other handwe treat the unobserved data generating process non-parametrically.

Our theory contributes to the literature on inference when parameters are unidentified, partially or weakly identified. Phillips1989 is an early investigation of inference in partially identified models. The weak instrument literature to which Dufour1997, Staiger1997, Stock2000, Kleibergen2002, Kleibergen2005 and Moreira2003 made important early contributions focuses on inference that is robust to a lack of identification. Dufour1997 and Staiger1997 are the first to note that the Anderson-Rubin (AR) statistic, introduced by Anderson1949, can be used to construct valid confidence intervals in weakly identified settings. Kleibergen2002 develops a modified LM statistic for linear instrumental variables models under homoskedasticity that reduces to the AR statistic in the just identified case. Stock2000 propose a generalized version of the AR statistic based on the CUE. Stock2000 and Dufour2005 consider subvector tests in settings where only parameters that are part of the null hypothesis may be underidentified. Kleibergen2005 proposes the K-statistic which is based on the first order conditions of CUE to account for conditional heteroskedasticity and serial correlation. Of these papers, Stock2000 consider moment conditions and test statistics that are similar to our setting. Their theoretical analysis is limited to weak instrument asymptotic sequences which are not sufficient to establish uniform validity. Kleibergen2005 considers subvector tests for a general set of moment conditions as we are but imposes strong identification of the parameters not subject to the null hypothesis. His asymptotic analysis includes cases where the parameters restricted under the null are unidentified or weakly identified but is pointwise rather than uniform. As GKMC12, we do not impose any restrictions on the degree of identification of either set of parameters and provide a uniform asymptotic analysis. There is a large subsequent literature on inference robust to identification failure including Andrews, Moreira and Stock (2006)\nocite{Andrews2006}, Chaudhuri2011, Andrews2012a, Andrews2016, Andrews2019, Andrews, Marmer and Yu Andrews2019a, Andrews, Cheng and Guggenberger Andrews2020 and Guggenberger, Kleibergen and Mavroeidis Guggenberger2024.

Our theoretical results are closest to Guggenberger, Kleibergen, Mavroeidis and Chen GuggenbergerKleibergenMavroeidisChen2012, GKMC12, who show that the AR statistic applied to a subset of the parameters in linear instrumental variables models has uniformly valid size over a class of distributions that include completely unidentified scenarios. Their results are valid for iid data with homoskedastic errors. Guggenberger, Kleibergen and Mavroeidis Guggenberger2019, GKM19, develop a more powerful subvector test under homoskedasticity but require numerically computed critical values. Andrews2017a proposes a two step subvector test that is robust to heteroskedasticity but requires a data dependent significance level in the second step. Guggenberger, Kleibergen and Mavroeidis Guggenberger2024, GKM24, extend GKM19 to allow for conditional heteroskedasticity by introducing a model selection procedure that selects between the AR statistic for linear models analyzed in GMK19 and the AR/AR test of Andrews2017a. There does not seem to exist theoretical results that show the uniform validity of a simple one step subvector test robust to non-identification of the untested parameters for general heteroskedasticity, temporal dependence and non-stationarity. We show that a test based on the Anderson-Rubin testing principle formulated for the CUE is uniformly valid. The AR test is based on conventional critical values for the chi-square distribution and thus easy to implement. Our assumptions are high level and do not assume a specific model, nor identification of any of the estimated or tested parameters. Andrews2019 consider tests under general identification failure and heteroskedasticity but only consider subvector tests for the case when the parameters not subject to the null are strongly identified. Of the four types of regularity conditions that Andrews and Guggenberger (2019,p. 1709) allow to fail, we account for two failures, namely that the moment functions have multiple solutions and that the Jacobian of the moment functions are column rank deficient. On the other hand, and unlike Andrews2019, we do maintain that the moment functions have full rank covariance matrices and that parameters under the null are in the interior of the parameter space. We also consider valid confidence intervals that are based on inverting the AR statistic. Inverting the AR statistic to construct a confidence interval dates back to Anderson1949 and was more recently investigated in the context of weakly identified or unidentified models by Dufour1997, Staiger1997 and Stock2000.

We illustrate the assumptions of our theory using examples of simultaneous equations models with heteroskedasticity and temporal dependence as well as a time series application with inference for impulse response analysis using the local projections proposed by Jorda2005. To the extent that the robust testing literature considered time series settings, stationarity has been maintained to the best of our knowledge. We allow for general forms of non-stationarity for processes with well defined second moments. While this rules out trending forms of non-stationarity such as unit roots and non-stationary fractionally integrated processes we do allow for time-changing relationships between outcomes, policy variables and controls.

The paper is organized as follows. Section (ref) defines the inference problem and the test statistic we consider. Section (ref) discusses high level assumptions and presents the main theoretical result establishing uniform validity. The section also discusses the main steps in our proof strategy. Section (ref) presents two examples that illustrate the high level assumptions we impose. Section (ref) presents Monte Carlo evidence using the New Keynesian Phillips curve model considered by Kleibergen2009. Proofs as well as some Lemmas are contained in the Appendix.

Test Statistics

Consider a probability space $\left(\Omega,\mathcal{F},P\right)$ on which a vector valued double array\footnote{A double array does not restrict the index $t$ for a given $n,$ while a triangular array imposes the restriction $t\leq n$.} of random variables $\chi_{n,t}\in\mathbb{R}^{d_{\chi}}$ is defined and where $n$ is an integer value representing sample size and $t$ is the observation index. We do not assume a parametric data generating process but instead consider sequences of induced probability measures $P_{\chi_{n,t}}$. We are interested in testing a null hypothesis that imposes constraints on $P_{\chi_{n,t}}$ that can be expressed in terms of moment conditions and that are parametrized by finite dimensional parameters $\beta$ and $\gamma$. More specifically, the null hypothesis imposes the constraint $H_{0}:\beta=\beta_{n,0}$ while $\gamma$ remains unconstrained under the null. Constraints on moments are expressed using moment functions $g\left(\chi_{n,t};\beta,\gamma\right)$ where $g\left(.\right):\mathbb{R}^{d_{\chi}}\times\mathbb{R}^{d_{\beta}+d_{\gamma}}\rightarrow\mathbb{R}^{d}$. For ease of notation we define $g_{t}\left(\beta,\gamma\right)\equiv g\left(\chi_{n,t};\beta,\gamma\right)$ and suppress the dependence on $\chi_{n,t}$. We do assume that the moment condition

holds under $H_{0}$ for at least one value $\gamma_{n,0}$. In a strongly identified setting $\beta_{n,0}$ and $\gamma_{n,0}$ are the unique solutions to (ref). However, we do allow for partially, weakly and unidentified cases where the values $\beta_{n,0}$ and $\gamma_{n,0}$ may not be the only parameter values that satisfy (ref).

Our framework allows for general dependence patters, including time series data. We distinguish between outcome variables $y_{n,t}$, exogenous or endogenous variables $x_{n,t}$ associated with the parameters $\beta$ in linear models, additional endogenous, exogenous or predetermined variables $w_{n,t}$ associated with auxiliary parameters $\gamma$ and instruments $z_{n,t}$ that contain exogenous components of $x_{nt}$ and $w_{n,t}$. When moment conditions are based on non-linear functions the association between parameters $\beta$, $\gamma$ and specific covariates is often less strong and is not required for our results. In linear models where all variables in $x_{n,t}$ and $w_{n,t}$ are exogenous $z_{n,t}$ may be composed only of these variables. The vector $\chi_{n,t}$ contains all the distinct random variables contained in $y_{n,t},x_{n,t},w_{n,t}$ and $z_{n,t}$. We define $X_{n,t}:=\left(x_{n,t}',w_{n,t}'\right)'$ with $X_{n,t}\in\mathbb{R}^{d_{x}}$. We assume that we observe a sample $\chi_{n,t}$ for $t=1,...,n.$

The empirical analog to the moment condition is based on sample averages

When evaluated at $\theta_{n,0}=\left(\beta_{n,0}',\gamma_{n,0}'\right)'$ we use the shorthand notation $\hat{g}_{n,0}=\hat{g}_{n}\left(\beta_{n,0},\gamma_{n,0}\right).$ The moment conditions (ref) may be general scores of $Z$-estimators or may be moment functions for a GMM estimator with instruments $z_{t}$ such that

and where $q\left(.\right):\mathbb{R}^{d_{x}+1}\times\mathbb{R}^{d_{\beta}+d_{\gamma}}\rightarrow\mathbb{R}^{d_{q}}$ is a $d_{q}$-dimensional vector of functions. The total number of moment restrictions is $d=d_{q}d_{z}$.

Test statistics for $H_{0}$ are based on the CUE criterion function evaluated under $H_{0}$. The CUE was proposed by Sargan1958 and analyzed by Hansen, Heaton and Yaron HansenHeatonYaron1996 and more recently Donald2000 and Newey2004. Stock2000 were the first to consider the CUE criterion function for hypothesis testing in a weakly identified setting involving possibly non-linear moment restrictions.

The CUE criterion function depends on an estimate of the variance of $\hat{g}_{n}\left(\beta,\gamma\right)$. Following the literature on heteroskedasticity and autocorrelation robust standard errors (HAC), see Newey and West (1987, 1994)Newey1987, Newey2004, Andrews1991, Andrews1992, and in particular Jong2000, we define the matrix \[ \Omega_{n}\left(\beta,\gamma\right)=n^{-1}\sum_{t=1}^{n}\sum_{s=1}^{n}E\left[g_{t}\left(\beta,\gamma\right)g_{s}\left(\beta,\gamma\right)'\right]. \] In addition, when evaluated at a sequence of pseudo-true values $\beta_{n,0}$ and $\gamma_{n,0}$ the covariance matrix $\Omega_{n,0}$ is defined as \[ \Omega_{n,0}=\Omega_{n}\left(\beta_{n,0},\gamma_{n,0}\right)=\operatorname{Var}\left(n^{-1/2}\sum_{t=1}^{n}g_{t}\left(\beta_{n,0},\gamma_{n,0}\right)\right) \] where the last equality holds because the moment conditions $n^{-1/2}\sum_{t=1}^{n}E\left[g_{t}\left(\beta_{n,0},\gamma_{n,0}\right)\right]=0$ are satisfied for $\beta_{n,0}$ and $\gamma_{n,0}$.

The matrix $\Omega_{n,0}$ can be replaced by a sample analog following constructions in the HAC literature, in particular De Jong and Davidson (2000) who explicitly account for triangular arrays that are central to our theoretical discussion. To define the estimators of the GMM weight matrix and long run variances, we adopt the following definition of the class of kernel functions $k\left(.\right)$ with bandwidth sequence $a_{n}$ from Jong2000, see also Andrews1991.

Then

The continuous updating estimator for $\theta=\left(\beta',\gamma'\right)'$ is now given by \[ \hat{\theta}_{CUE}=\left(\hat{\beta}'_{CUE},\hat{\gamma}'_{CUE}\right)'=\textrm{argmin}_{\beta,\gamma}n^{-1}Q_{n}\left(\beta,\gamma\right) \] where $Q_{n}\left(\beta,\gamma\right)=n\hat{g}_{n}\left(\beta,\gamma\right)'\hat{\Omega}_{n}^{-1}\left(\beta,\gamma\right)\hat{g}_{n}\left(\beta,\gamma\right).$ The subset AR statistic for the null hypothesis $H_{0}$ based on the CUE is given as

Let $d_{\gamma}$ be the dimension of $\gamma$ and assume that $d-d_{\gamma}>0.$ Under regularity conditions, and when $\gamma$ is strongly identified it is well known from Stock2000, Newey2004 and Kleibergen2005 that $\textrm{AR}\left(\beta_{n,0}\right)$ has a limiting $\chi_{d-d_{\gamma}}^{2}$distribution. When the AR statistic is formulated based on the LIML estimator for linear models and iid observations with homoskedastic errors GKMC12 show that (ref) is bounded by a random variable with a limiting $\chi_{d-d_{\gamma}}^{2}$ distribution uniformly over sequences of data distributions where $\beta$ and $\gamma$ may not be identified. A nominal size $\alpha$ test of $H_{0}$ then is based on rejecting the null when \[ \textrm{AR}_{\textrm{C}}\left(\beta_{n,0}\right)>c_{1-\alpha,\chi_{d-d_{\gamma}}^{2}} \] where $c_{1-\alpha,\chi_{d-d_{\gamma}}^{2}}$ is the $1-\alpha$ quantile of the $\chi_{d-d_{\gamma}}^{2}$ distribution. Based on Anderson and Rubin (1949) we can construct a confidence region with coverage rate of at least $1-\alpha$ by inverting the AR statistic \[ \textrm{CI}=\left\{ \beta\in\mathbb{R}^{d_{\beta}}|\mathit{\textrm{AR}_{\textrm{C}}}\left(\beta\right)\leq c_{1-\alpha,\chi_{d-d_{\gamma}}^{2}}\right\} . \] The confidence set CI is the set of all values $\beta$ for which the AR statistic does not reject $H_{0}$. AR based confidence sets were analyzed in the context of weakly identified or non-identified linear IV methods by Dufour1997, and for weakly identified instrumental variables by Staiger1997 and Stock2000. GKMC12 are the first to show uniform validity of the subset AR test over all sequences of linear models, including weakly identified, partially identified and unidentified cases.

A key ingredient to uniform validity of the AR statistic in more general settings than linear models with homoskedastic errors is that the statistic is based on the CUE. The CUE implicitly orthogonalizes the moment functions $\hat{g}_{n}\left(\theta\right)$ from the influence of estimating nuisance parameters $\gamma$. Orthogonalization lies at the core of the $C\left(\alpha\right)$ test of Neyman1959 and was shown in Donald2000 to be a feature of the first order conditions of the CUE. Let $\hat{\gamma}_{n}=\hat{\gamma}_{n}\left(\beta_{n,0}\right)=\operatorname{argmin}_{\gamma}Q_{n}\left(\beta_{n,0},\gamma\right).$ If $\hat{\gamma}_{n}$ is not unique then any value for $\gamma$ that minimizes $Q_{n}\left(\beta_{n,0},\gamma\right)$ can be chosen. We show that the only effect of estimating $\gamma$ by $\hat{\gamma}_{n}$ on the asymptotic distribution $\textrm{AR}_{\textrm{C}}\left(\beta_{n,0}\right)=Q_{n}\left(\beta_{n,0},\hat{\gamma}_{n}\right)$ is a reduction in the degrees of freedom of the resulting $\chi^{2}$ distribution. This result does not hold when first stage estimators for $\gamma$ such as two stage least squares or GMM are plugged into $Q_{n}\left(\beta_{n,0},\gamma\right).$

Theoretical Results

We start by stating detailed regularity conditions we impose. Let $\left(\Omega,\mathcal{F\mathit{,P}}\right)$ be a probability space. Let $\chi_{n}=\left\{ \chi_{n,t}:t=1,2,...;n=1,2,...\right\} $ be a double array of random vectors of dimension $d_{\chi}$. Let $y_{n,t}$ and $x_{n,t}$ be double arrays of random variables with elements from $\chi_{n,t}.$ Let $w_{n,t}$ be a $d_{w}\times1$ vector and $z_{n,t}$ a $d_{z}\times1$ vector of elements from $\chi_{n,t}$. Assume that $x_{n,t}$ and $\text{\ensuremath{w_{n,t}} }$ contain different elements, while $z_{n,t}$ may contain $x_{n,t}$ and $\text{\ensuremath{w_{n,t}} }$ as well as additional variables. The exact configuration of $z_{n,t}$ depends on whether $x_{n,t}$ and $\text{\ensuremath{w_{n,t}} }$ are believed to be endogenous or exogenous. Let $\lambda_{min}\left(A\right)$ denote the smallest eigenvalue of the matrix $A$, let $\left\Vert A\right\Vert _{2}=\operatorname{tr}\left(A'A\right)^{1/2}$ be the Frobenius norm and $\left\Vert A\right\Vert =\sup_{\left\Vert x\right\Vert =1}\left\Vert Ax\right\Vert $ be the operator norm where $\left\Vert .\right\Vert $ applied to a vector in $\mathbb{R}^{k}$ is the usual Euclidean norm. Let $\chi_{n}:\Omega\rightarrow\mathbb{R}^{\infty}$ and denote by $\chi_{n}$$\rightsquigarrow$$\chi$ weak convergence of $\chi_{n}$ to a random element $\chi$ in $\mathbb{R}^{\infty}$, vanderVaartWellner1996 (1996, Definition 1.3.3).

Let $\mathscr{X}$ be a collection of measurable processes $\chi$. Each $\chi\in\mathscr{X}$ induces a probability distribution $P_{\chi}$. We consider sequences of $\chi_{n}=\left(\chi_{n,1},\chi_{n,2},...,\chi_{n,n},...\right)$ of processes $\chi_{n}\in\mathscr{X}$ that contain subsequences that converge to a limiting random element $\chi$ in $\mathbb{R}^{\infty}$. This is formalized in Assumption (ref).

\phantom{.}

Assumption (ref) and Prohorov's theorem (see vanderVaartWellner1996, 1996, Theorem 1.3.9) imply that for every subsequence $n_{k}$ there is a further subsequence of $n_{k}$ denoted by $n_{k_{l}}$ such that $\chi_{n_{k_{l}}}$$\rightsquigarrow$$\chi$ and where the limit $\chi$ may depend on the particular subsequence. In our notation we suppress the dependence of limits $\chi$ on the subsequence. For ease of notation we omit references to specific subsequences and refer to converging subsubsequences $n_{k_{l}}$ simply as converging subsequences $n_{k}$. Assumption (ref) implies that all finite dimensional distributions of $\chi_{n_{k}}$ converge along converging subsequences. To the extent that moments of $\chi_{n}$ exist, moments related to finite dimensional distributions also converge along converging subsequences. Assumption (ref) neither implies that ordinary limits (as opposed to subsequential limits) exist, nor does it rule out the existence of such ordinary limits. See Andrews2020 for a related assumption that makes the dependence of the limit $\chi$ on the subsequence explicit.

Assumption (ref) is a nonparametric generalization of semiparametric assumptions in GKMC12 and Andrews2020 and is similar to assumptions imposed in Andrews2019. These papers define classes of data generating processes and sequences taking values in these model classes that converge to limiting data generating processes as the sample size increases. Special cases of such classes can be found in the weak instrument literature that follows the approach of Staiger1997 and where only particular sequences leading to so called weak instrument limits are considered. In order to analyze uniformity, the setting in GKMC12 is significantly more general as it accounts for completely unidentified, partially identified as well as fully identified data generating processes.

We do not assume that $\chi_{n}$ is generated by a particular parametric or semiparametric model. We only assume that under the null hypothesis a set of finite dimensional moment restrictions parameterized by a set of finite dimensional parameters hold. The null hypothesis is specified in terms of restrictions on a subset of the parameter vector. We do not specifically consider the case where the null hypothesis restricts the entire parameter vector since in that scenario the AR statistic trivially has uniformly correct size under our assumptions and results. The next assumption postulates that the moment conditions (ref) hold along parameter sequences under the null hypothesis.

In applications the null hypothesis is usually formulated for a fixed constant $\beta_{0}$ such that $H_{0}:\beta=\beta_{0}$. Our theory covers a fixed null as a special case. Without assuming homogeneity or stationarity $\gamma_{n,0}$ typically varies with $n$ even if $\beta_{n,0}=\beta_{0}$ is fixed. Note that $\theta_{n,0}$ is not necessarily unique under our assumptions and in general there are infinitely many solutions to the moment condition. Our theory is not intended to directly test for violations of Assumption (ref) which imposes both functional form restrictions summarized by $g_{t}\left(.\right)$ as well as a restriction on the parameter $\beta_{0}.$ In other words, and in line with existing theory, a test of $H_{0}$ is interpreted as testing $\beta=\beta_{0}$ while taking for granted that the moment condition in (ref) is satisfied for some $\theta.$ However, since our theoretical analysis is only conducted under the null we do not take a stand on whether (ref) holds for some $\theta$ under the alternative.

\phantom{.}

The full rank assumption for $\Omega\left(\theta\right)$ corresponds to similar assumptions in Stock2000 and Kleibergen2005. The assumption has been relaxed by Andrews2019 for a modified version of the AR statistic. The convergence of $\hat{\Omega}_{n_{k}}\left(\theta\right)\rightarrow_{p}\Omega\left(\theta\right)$ was shown for triangular near epoch dependent arrays defined on mixing sequences by Jong2000. The limits in Assumption (ref) may depend on the specific subsequence. Additional regularity conditions may guarantee that convergence obtains for $n_{k}=n$, a scenario that is a special case under our assumptions.

When $g_{t}\left(.\right)$ is of the form (ref) and the data are stationary a necessary condition for Assumption (ref) is that for all $a\in\mathbb{R}^{d_{z}}$, $\left\Vert a\right\Vert =1$ and all $n\geq1$ the variables $z_{n,t}$ are not colinear in the sense that

To see this, assume that this condition does not hold. Then, for some $a\in\mathbb{R}^{d_{z}}$ it follows that $z_{n,t}'a=0\text{ a.s.}$ for all $n$ and $t.$ But this implies that $g_{t}\left(\beta,\gamma\right)\left(I_{q}\otimes a\right)=0\text{ a.s.}$ for all $n$ and $t$ which in turn implies that Assumption (ref) cannot hold. In addition, (ref) implies that $\lambda_{min}\left(E\left[z_{n,t}z_{n,t}'\right]\right)>0$ and $\lambda_{min}\left(n^{-1}\sum_{t=1}^{n}E\left[z_{n,t}z_{n,t}'\right]\right)>0.$

Using the pseudo true parameter value $\theta_{n,0}$ we now define the pseudo residual

where $\varepsilon_{n,t}$ is a $d\times1$ vector of random variables and where by Assumption (ref) it follows $n^{-1/2}\sum_{t=1}^{n}E\left[\varepsilon_{n,t}\right]=0$. Also define $d\times d_{\gamma}$ dimensional matrix of random functions $g_{t}^{\gamma}\left(\beta,\gamma\right)=\partial g_{t}\left(\beta,\gamma\right)/\partial\gamma'$ and let

and

The behavior of the derivative $g_{t}^{\gamma}\left(\beta,\gamma\right)$ determines whether $\gamma_{n,0}$ is identified. Stock2000 consider moment functions that additively separate into a component that only depends on $\gamma$ and assume that the derivative of the moment function with respect to that component is full rank. Similarly, Kleibergen2005 imposes a full column rank assumption on a term equivalent to $\hat{g}_{n,\gamma}$ for the subset version of his test. Instead, we allow for the derivatives $\hat{g}_{n,\gamma}$ to be of reduced column rank in the limit. We Impose the following assumptions on $\hat{g}_{n,\gamma}.$

Define the $d_{\gamma}\times d_{\gamma}$ non-stochastic matrix $\Gamma=g_{\gamma}'\Omega_{0}^{-1}g_{\gamma}$ with eigenvalues $\Delta_{1}\geq\Delta_{2}...\geq\Delta_{d_{\gamma}}\geq0$. Assumption (ref) does not restrict the column rank of $g_{\gamma}$. Thus, we allow the rank $r$ of $\Gamma$ to take values $0\leq r\leq d_{\gamma}$ and where in general $\Delta_{j}$ and $r$ depend on the subsequence $n_{k}.$

Define $\Pi{}_{n,\gamma}=n^{-1}\sum_{t=1}^{n}E\left[g_{t}^{\gamma}\left(\beta_{n,0},\gamma_{n,0}\right)'\right]$ where $\Pi_{n,\gamma}$ is a $d\times d_{\gamma}$ dimensional matrix. Define $V_{n,t}$ as a $d\times d_{\gamma}$ dimensional matrix of pseudo-residuals that can be written as

such that $\sum_{t=1}^{n}E\left[V_{n,t}\right]=0$ by construction. As in GKMC12 no other restrictions are imposed on $\Pi_{n,\gamma}$ so that scenarios where $\Pi_{n,\gamma}$ is rank deficient, or even $\Pi_{n,\gamma}=0$ are allowed. The properties of $\Pi_{n,\gamma}$ depend on the data distribution $P_{\chi}$ . Rank deficiency of $\Pi_{n,\gamma}$ arises when $\gamma$ is not identified, partially or weakly identified.

The analysis of the CUE also involves derivatives of $\hat{\Omega}_{n_{k}}\left(\theta\right)$ given by

where

To guarantee convergence of $\hat{\Lambda}_{n}\left(\gamma\right)$ define the matrices

and

and impose the following restrictions on $\hat{\Lambda}_{z,n,0}$ and $\hat{\Lambda}_{V,n,0}$.

\phantom{.}We now consider the process \[ \omega_{n,t}=\left(\varepsilon_{n,t}',\operatorname{vec}\left(V_{n,t}\right)'\right)' \] where $\omega_{n,t}$ is a $d\left(1+d_{\gamma}\right)$ vector of double arrays of random variables. Let $S_{n}=\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\omega_{n,t}.$ Note that $E\left[S_{n}\right]=0$ by (ref), (ref) and Assumption (ref), such that, without loss of generality, $\omega_{n,t}$ can be assumed to be mean zero. The next assumption imposes that $\nu_{n,t}$ has a non-degenerate limiting distribution along all converging subsequences.

Assumptions (ref), (ref), (ref) and (ref) are high level assumptions. They could be formulated more directly by imposing restrictions on the moments and the distribution of $\chi_{n}$. Such assumptions then could be used to invoke laws of large numbers and central limit theorems implying Assumptions (ref)-(ref). We do not follow this approach here because low level assumptions tend to be more specific to a particular testing context and obscure the fundamental requirements for our results. However, we consider two specific examples in Section (ref) were we discuss a set of low level assumptions that can be imposed to guarantee that Assumptions (ref), (ref), (ref) and (ref) hold.

Our main theoretical result is now stated. Recall the definition of the AR statistic $\textrm{AR}_{\textrm{C}}\left(\beta_{n,0}\right)$ in (ref). The asymptotic size of the subset AR test is defined as

where $P_{\chi_{n}}$ is the induced probability measure of the process $\chi_{n}$ on the underlying probability space and where each $\chi_{n}$ implies a corresponding sequences of parameters $\theta_{n,0}$ under $H_{0}$. We prove the following result which extends Theorem 1 of GKMC12 to the case of the AR statistic based on the CUE rather than LIML and that allows for nonlinear moment conditions and non-stationary or heterogenous processes with conditional heteroskedasticity.

The challenge in proving Theorem (ref) under the maintained assumptions of this paper is that the CUE $\hat{\gamma}_{n}$ of $\gamma$ is not necessarily unique and may not converge to a well defined limit. Then, conventional approximation arguments for the first order conditions of the CUE fail. The proof of Theorem (ref) is based on a novel approach of regularizing the first order condition. We then perturb the pseudo true value $\gamma_{n,0}$ toward a value $\tilde{\gamma}_{n,0}$ in the direction of the regularized first order conditions. This is done in a way that keeps $\tilde{\gamma}_{n,0}$ close enough to $\gamma_{n,0}$ for the central limit theorem of Assumption (ref) to apply while at the same time approximately satisfying the first order conditions. An immediate consequence of the definition of $\textrm{AR}_{\textrm{C}}\left(\beta_{n,0}\right)$ is that all sequences $\gamma_{n}$ produce upper bounds because $\textrm{argmin}_{\gamma}Q\left(\beta_{n,0},\gamma\right)\leq Q\left(\beta_{n,0},\gamma_{n}\right)$ for any sequence $\gamma_{n}$. While $\gamma_{n,0}$ leads to one such upper bound, the bound based on $\gamma_{n,0}$ is too large in the sense that $Q\left(\beta_{n,0},\gamma_{n,0}\right)$ has a limiting $\chi_{d}^{2}$ rather than a $\chi_{d-d_{\gamma}}^{2}$ distribution. This occurs because $\gamma_{n,0}$ does not satisfy the first order conditions well enough even in an asymptotic sense. The perturbed point $\tilde{\gamma}_{n,0}$ is constructed in such a way that it approximately solves the CUE moment conditions evaluated at the infeasible sequence $\theta_{n,0}$ under $H_{0}:\beta=\beta_{n,0}$ and that as a result $Q\left(\beta_{n,0},\tilde{\gamma}_{n,0}\right)$ has the desired $\chi_{d-d_{\gamma}}^{2}$ distribution in the limit irrespective of whether $\gamma$ is identified or not. We stress that $\tilde{\gamma}_{n,0}$ is a construct exclusively used in the proofs and not needed to compute the test statistic $\textrm{AR}_{\textrm{C}}\left(\beta_{n,0}\right)$ itself.

Donald2000 show that the CUE moment conditions remove the highest order bias term of a conventional GMM estimator by showing that the CUE moment conditions are centered at zero. Their insight implies an asymptotic orthogonality condition between the limiting process of the moment function $\hat{g}_{n}$ and the residual of the projection of $\hat{g}_{n,\gamma}$ onto $\text{\ensuremath{\hat{g}_{n,0}}.}$ The same insight underlies Kleibergen2005 but only for the case where $\gamma$ is strongly identified. To show uniform validity, the asymptotic orthogonality needs to be established for all converging subsequences irrespective of whether $\gamma$ is identified or not. This then allows to characterize the upper bound of the $\textrm{AR}_{\textrm{C}}\left(\beta_{n,0}\right)$ statistic in terms of a $\chi_{d-d_{\gamma}}^{2}$ limiting distribution along all converging subsequences. The limiting $\chi_{d-d_{\gamma}}^{2}$ distribution is obtained by representing the AR-statistic asymptotically as the sum of squares of the residuals of a projection of the moment conditions $\hat{g}_{n}$ onto the column space spanned by the residualized $\hat{g}_{n,\gamma}$. The rank of this projection residual is $d-d_{\gamma}$ irrespective of whether $\gamma$ is identified or not.

We now explain the construction of the sequence $\tilde{\gamma}_{n,0}$ and the proof strategy behind Theorem (ref) in more detail. Use (ref) to define $\hat{g}_{n}\left(\gamma\right)=\hat{g}_{n}\left(\beta_{n,0},\gamma\right)$, $\hat{g}_{n,0}=\hat{g}_{n}\left(\gamma_{n,0}\right)$ and $\tilde{g}_{n,0}=\hat{g}_{n}\left(\tilde{\gamma}_{n,0}\right)$. Recall the definition of $\hat{g}_{n,\gamma}$ in (ref) and use (ref) to define $\tilde{g}_{n,\gamma}=\hat{g}_{n,\gamma}\left(\tilde{\gamma}_{n,0}\right)$. Also recall the definition of $\hat{\Lambda}_{n}\left(\gamma\right)$ in (ref) where $\hat{\Lambda}_{n}\left(\gamma\right)$ has dimension $d^{2}\times d_{\gamma}$. Now construct the matrix

and consider the empirical moment conditions

To gain some intuition for the moment condition (ref) we focus on the case where $g_{t}\left(.\right)$ is of the form (ref) with $d_{q}=1$ such that (ref) can be written as \[ 0=\left(\hat{g}_{n,\gamma}-\hat{g}_{n,0}'\hat{\Omega}_{n,0}^{-1}\hat{\Lambda}_{n,0}\right)'\hat{\Omega}_{n,0}^{-1}\hat{g}_{n}\left(\gamma\right). \] As argued in Donald2000, $\hat{g}_{n,0}'\hat{\Omega}_{n,0}^{-1}\hat{\Lambda}_{n,0}$ is the projection of $\hat{g}_{n,\gamma}$ onto $\hat{g}_{n,0}$ in the case of independently distributed data. The term $\hat{g}_{n,\gamma}-\hat{g}_{n,0}'\hat{\Omega}_{n,0}^{-1}\hat{\Lambda}_{n,0}$ then is the projection residual, which by construction is orthogonal to $\hat{g}_{n,0}$. As shown by Donald2000, the orthogonality holds exactly in the iid setting and in expectation in a time series framework. The use of long run variance-covariance matrices in the time series case ensures that this interpretation remains valid in the limit in the more general setting of this paper. By constructing our perturbed moment vector $\tilde{g}_{n,0}$ to be close to $\hat{g}_{n,0}$ and approximately orthogonal to $\hat{A}_{n}$ we guarantee that the moment vector $\hat{g}_{n,0}$ is stochastically independent of the column space spanned by $\hat{A}_{n}$ at least in the limit.

The construction of $\tilde{\gamma}_{n,0}$ depends on the particular subsequence $n_{k}$ such that $\tilde{\gamma}_{n,0}$ is only defined for $n_{k}$ and may differ for different sequences $n_{k}$ and $n_{k'}$. We start with the mean value expansion of $\hat{g}_{n}\left(\gamma\right)$ around $\gamma_{n,0}$,

with $\left\Vert \check{\gamma}_{n,0}-\gamma_{n,0}\right\Vert \leq\left\Vert \gamma-\gamma_{n,0}\right\Vert $ which upon substitution into the moment conditions (ref) leads to

If $\hat{A}_{n}$ and $\hat{g}_{n,\gamma}$ were full column rank at least for large enough samples, then one could solve (ref) for $\gamma.$ Under the sequences considered in this paper, both matrices may have reduced ranks for finite $n$ and in the limit. Solutions based on the Moore-Penrose inverse have delicate convergence properties, see for example Wedin1973. In fact, such solutions often do not converge because the operator norm of the MP inverse becomes unbounded as eigenvalues of its argument tend towards zero, see Stewart1977 (1977, Theorem 3.1). To stabilize these solutions we adopt a regularization scheme called truncated singular value decomposition (TSVD), see the numerical analysis literature Hanson1971 (1971), Varah1973 (1973) and Hansen1987.

Let the singular value decomposition (SVD), which in this case coincides with the spectral representation of $\hat{\Gamma}_{n}=\hat{g}_{n,\gamma}'\hat{\Omega}_{n,0}^{-1}\hat{g}_{n,\gamma}$ be equal to $\hat{\Gamma}_{n}=\hat{R}_{n}\hat{\Delta}_{n}\hat{R}'_{n}$ and where $\hat{\Delta}_{n}$ is a diagonal matrix of the eigenvalues $\hat{\Delta}_{1,n}\geq....\geq\hat{\Delta}_{d_{\gamma},n}\geq0$ of $\hat{\Gamma}_{n}$. By Assumptions (ref) and (ref) and for a converging subsequence $n_{k}$, $\hat{\Gamma}_{n_{k}}\rightarrow_{p}\Gamma$ such that by Theorem (ref), $\hat{\Delta}_{j,n_{k}}\rightarrow_{p}\Delta_{j}\geq0$. Assume that $\Delta_{j}>0$ for $j\leq r$ and some $0\leq r\leq d_{\gamma}$. Fix $\varepsilon$ arbitrary with $\Delta_{r}>\varepsilon>0$. For each converging subsequence $n_{k}$ define $\mathring{\Delta}_{n_{k}}$ as the matrix with diagonal elements $\text{\ensuremath{\mathring{\Delta}_{n_{k},j}}}$ such that for all $l\in\left\{ 1,...,d_{\gamma}\right\} ,$ \[ \ensuremath{\mathring{\Delta}_{n_{k},l}}=

. \] The TSVD of $\hat{\Gamma}_{n_{k}}$ is now given by

We construct an infeasible solution $\tilde{\gamma}_{n_{k},0}$ to (ref) with two properties. The solution $\tilde{\gamma}_{n,0}$ is a perturbation of $\gamma_{n,0}$ small enough to converge to $\gamma_{n,0}$ but such that it also satisfies (ref) with sufficient accuracy in large samples. The purpose of the construction is to evaluate the CUE criterion function at a point that allows for a limiting distribution and where the moment conditions of the CUE, and thus orthogonality between $\hat{A}_{n_{k}}$ and $\tilde{g}_{n_{k},0}$ hold approximately. The parameter $\tilde{\gamma}_{n_{k},0}$ is defined as a perturbation of the parameters $\gamma_{n_{k},0}$ in the direction of the first order conditions for the CUE estimator by setting

Note that the MP-inverse of $\mathring{\Gamma}_{n_{k}}$ denoted by $\mathring{\Gamma}_{n_{k}}^{+}$ is continuous and therefore converges along converging subsequences. Continuity of $\mathring{\Gamma}_{n_{k}}^{+}$ holds because the eigenvalues of $\mathring{\Gamma}_{n_{k}}$ are bounded away from zero by $\varepsilon$ due to regularization. This is in contrast to the MP-inverse of $\hat{\Gamma}_{n_{k}}$ which may not be continuous and thus not converge along converging subsequences. The following properties of $\tilde{\gamma}_{n_{k},0}$ can now be established.

A proof of Lemma (ref) is given in Appendix (ref). Let $\tilde{\Omega}_{n,0}=\hat{\Omega}_{n}\left(\beta_{n,0},\tilde{\gamma}_{n,0}\right)$. Using Lemma (ref) and denoting by $P_{\hat{A}_{n}}$ the projection onto the column space spanned by $\hat{A}_{n}$ and letting $M_{\hat{A}_{n}}=I-P_{\hat{A}_{n}}$, the proof of Theorem (ref) then proceeds to show in Lemma (ref) that \[ \textrm{AR}_{\textrm{C}}\left(\beta_{n,0}\right)\leq n\tilde{g}_{n,0}'\tilde{\Omega}_{n,0}^{-1/2}M_{\hat{A}_{n}}\tilde{\Omega}_{n,0}^{-1/2}\tilde{g}_{n,0}+\varpi_{n}, \] where along converging subsequences $n_{k}$ \[ \varpi_{n_{k}}=n_{k}\tilde{g}_{n_{k}}'\tilde{\Omega}_{n_{k}}^{-1/2}P_{\hat{A}_{n_{k}}}\tilde{\Omega}_{n_{k}}^{-1/2}\tilde{g}_{n_{k}}=o_{p}\left(1\right). \] This result holds because $\tilde{g}_{n,0}=\hat{g}_{n}\left(\tilde{\gamma}_{n,0}\right)$ approximately satisfies the orthogonality condition in Lemma (ref)(ii). With the help of Lemma $\ref{lem:gamma-conv}$ and (ref) we show in Lemma (ref) that along converging subsequences \[ n_{k}\tilde{g}_{n_{k},0}'\tilde{\Omega}_{n_{k},0}^{-1/2}M_{\hat{A}_{n_{k}}}\tilde{\Omega}_{n_{k},0}^{-1/2}\tilde{g}_{n_{k},0}=n_{k}\hat{g}_{n_{k},0}'\hat{\Omega}_{n_{k},0}^{-1/2}M_{\hat{A}_{n_{k}}}\hat{\Omega}_{n_{k},0}^{-1/2}\hat{g}_{n_{k},0}+o_{p}\left(1\right). \] By Assumption (ref) $\hat{g}_{n_{k},0}$ converges to a Gaussian limit variable $\omega_{\varepsilon}$. We show in Lemma (ref) that along converging subsequences $P_{\hat{A}_{n_{k}}}\rightarrow_{d}P_{A}$ where $P_{A}$ has constant rank $d_{\gamma}$ independent of the subsequential limit and where $P_{A}$ is a possibly random matrix with elements that are stochastically independent of $\omega_{\varepsilon}.$ The last result is essentially due to Lemma (ref)(ii) which is at the heart of establishing asymptotic uncorrelatedness between the elements of $\hat{A}_{n_{k}}$ and $\hat{g}_{n_{k},0}$ along all converging subsequences. Combining these results, we establish in Lemma (ref) that $\textrm{AR}_{\textrm{C}}\left(\beta_{n,0}\right)$ is bounded above by a random variable that is asymptotically $\chi_{d-d_{\gamma}}^{2}$ along all converging subsequences.

Examples

Our general framework in Section (ref) and our theoretical results in Section (ref) do not assume that the data are generated by a parametric model. Nevertheless, parametric models are often the basis for empirical work and serve as useful benchmarks that allow more precise interpretations of the parameters $\theta.$ In this section we consider two specific examples of such parametric models and discuss how they fit into our general framework.

Linear Instrumental Variables

Consider as an example a simple univariate static model where $g_{t}\left(.\right)$ is of the form (ref) with $d_{q}=1$ and $q\left(.\right)$ is linear. The data generating process is given by

and

where $z_{n,t},\varepsilon_{n,t}$ and $V_{n,t}=\left(V_{x,t}',V_{w,t}'\right)'$ are the basic inputs with $z_{n,t}$ observed and $\varepsilon_{n,t},V_{n,t}$ unobserved. We assume that $\left(z_{n,t},\varepsilon_{n,t},V_{n,t}'\right)$ are independent across $t$ and where $w_{n,t}$ has $d_{w}$ elements and $z_{n,t}$ has $d_{z}>d_{w}$ elements and $x_{n,t}$ is univariate. The parameter vector $\pi_{x,n}$ and the $d_{z}\times d_{w}$ matrix $\pi_{w,n}$ control instrument strength. The model is the same as in GKMC12 but without the assumption of homoskedasticity of $\varepsilon_{n,t}$ and $V_{n,t}$, nor do we assume that the random variables are identically distributed over $t.$ At the expense of a slightly more restrictive example, one could assume that $\left(z_{n,t},\varepsilon_{n,t},V_{n,t}'\right)$ and $\gamma_{n}$ do not depend on $n.$ This scenario would still contain weak instrument limits as considered by Staiger and Stock (1997) by allowing $\text{\ensuremath{\pi_{x,n}}}$ and $\text{\ensuremath{\pi_{w,n}}}$ to be in a $1/\sqrt{n}$ neighborhood local to zero, but would also include non-identified scenarios where $\text{\ensuremath{\pi_{x,n}}}=0$ and/or $\pi_{w,n}=0.$ In the more general setting considered here and in GKMC12 one can allow for $\operatorname{Cov}\left(\varepsilon_{n,t},V_{n,t}\right)$ to depend on $n$, and thus add additional dimensions along which inference may be challenging.

If $E\left[z_{n,t}\left(\varepsilon_{n,t},V_{n,t}'\right)\right]=0$ then the moment conditions $n^{-1}\sum_{t=1}^{n}E\left[\left(y_{n,t}-X'_{n,t}\theta\right)z_{n,t}\right]=0$ have at least one solution $\theta_{n}=\left(\beta,\gamma_{n}'\right)'$. If $n^{-1}\sum_{t=1}^{n}E\left[X_{n,t}z_{n,t}'\right]$ is of reduced column rank then the model is not identified and the average moment condition may have multiple solutions.

The CUE for this model is formed from moment functions

and can be evaluated at $H_{0}:\beta=\beta_{0}$ by setting $\hat{g}_{n}\left(\beta_{0},\gamma\right)=0$. The weight matrix $\hat{\Omega}_{n}$ is exploiting independence and is given by

The criterion function $Q_{n}\left(\beta,\gamma\right)$ and the test statistic $\textrm{AR}_{\textrm{C}}$ then are formulated in the same way as in Section (ref) using the expressions for $\hat{g}_{n}$ and $\hat{\Omega}_{n}$ given in ((ref)) and ((ref)). The derivatives $E\left[g_{n,\gamma}\right]=n^{-1}\sum_{t=1}^{n}E\left[z_{n,t}w_{n,t}'\right]=n^{-1}\sum_{t=1}^{n}E\left[z_{n,t}z_{n,t}'\right]\pi_{n,w}$ are reduced column rank if $\pi_{n,w}$ is of reduced column rank.

Assumption (ref) is satisfied in the following way. Assume $\eta_{n,t}=\left(z_{n,t}',\varepsilon_{n,t},V_{n,x,t},V_{n,w,t}'\right)'$ is a double array of independent random variables, where in addition $z_{n,t}$ is independent of $\varepsilon_{n,t}$, $V_{n,x,t}$ and $V_{n,w,t}$. Assume that $y_{n,t}$, $x_{n,t}$ and $w_{n,t}$ are generated by (ref), (ref) and (ref) respectively for parameters $\theta,$ $\pi_{n,x}$ and $\pi_{n,w}$ in some compact parameter space $\Theta.$ When the DGP is linear as in (ref)-(ref) then Assumptions (ref), (ref) and (ref) can be replaced with the following assumption.

Assumption (ref) holds under the following sufficient low level assumptions. Let $\eta_{n,t,j}$ be the $j$-th element of $\eta_{n,t}$. Assume $\sup_{t}E\left[\left|\eta_{n,t,j}\right|^{2+\delta}\right]<\infty$ for some $0<\delta<1$ and all $j$ and all $n\geq1$. This implies that the elements $\chi_{n,t,j}$ of $\chi_{n,t}$ also have bounded $2+\delta$ moments. Also assume that there exist subsequences $n_{k}$ such that $n_{k}^{-1}\sum_{t=1}^{n_{k}}E\left[\chi_{n_{k},t,j}\chi_{n_{k},t,l}\right]$ converges to some constant $c$ that may depend on the subsequence $n_{k}$. If $\chi_{n,t}$ does not depend on $n$ Theorem 1 in Chow1997 (1997, p.124) implies that Assumption (ref) holds for all such converging subsequences. A strong law for double arrays of independent heterogenous $\chi_{n,t}$ is given in Teicher1985. Assumption (ref) holds if for $\varepsilon_{n,t}\left(\theta\right)=y_{n,t}-x_{n,t}\beta-w'_{n,t}\gamma$ it follows that $\inf_{\theta}\det\left(E\left[\varepsilon_{n,t}\left(\theta\right)^{2}z_{n,t}z_{n,t}^{'}\right]\right)>0.$ Sufficient conditions are that $\sup_{\theta}\Pr\left(\varepsilon_{n,t}\left(\theta\right)=0\right)<1$ which holds if $y_{n,t}$ is not co-linear with $x_{n,t}$ and $w_{n,t}$ and $\sup_{a}\Pr\left(z_{n,t}'a=0\right)<1$. Assumption (ref) holds if for any $\ell\in\mathbb{R}^{\left(2+d_{w}\right)d_{z}}$ and $\nu_{n,t}=\left(\varepsilon_{n,t},V_{n,x,t},V_{n,w,t}'\right)'\otimes z_{n,t}$ it follows that $E\left[\ell'\nu_{n,t}\right]=0$, $\sigma_{n,t}^{2}=E\left[\left(\ell'\nu_{n,t}\right)^{2}\right]>0$, $c_{n}^{2}=\sum_{t=1}^{n}\sigma_{n,t}^{2}$ and $\lim{}_{k\rightarrow\infty}\sum_{t=1}^{n_{k}}E\left[\left(\ell'\nu_{n_{k},t}/c_{n_{k}}\right)^{2}1\left\{ \left|\ell'\nu_{n_{k},t}\right|>\varepsilon c_{n_{k}}\right\} \right]\rightarrow0$ for every converging subsequence $n_{k}$ by Theorem 9.6.1 of Dudley1989.

Local Projections

We consider time series data with a non-parametric data generating process and dependence structure. The parameter of interest is the local projection of Jorda2005 for a univariate outcome variable $y_{n,t+h}$ at horizons $h\in\left\{ 0,...,H\right\} $ onto a policy or intervention variable $x_{n,t}$ and additional controls $w_{n,t}$ under heterogeneity and nonstationarity. To simplify the discussion we assume that $x_{n,t}$ is univariate. We assume the researcher is specifying a linear model under $H_{0}$ which is formulated as a restriction on the conditional mean function $E\left[y_{n,t+h}|x_{n,t},w_{n,t}\right]=x_{n,t}\beta_{t}\left(h\right)+w'_{n,t}\gamma_{t}\left(h\right)$ and where $\beta_{t}\left(h\right)$ for $h\in\left\{ 0,...,H\right\} $ are the parameters of interest. As in Section (ref) we only assume that a linear moment restriction holds under $H_{0}$ but we do not assume that $y_{n,t}$ is necessarily generated by a linear model. The conditioning variables $w_{n,t}$ summarize information $\mathcal{I}_{t-1}$ up to time $t-1$, which may also contain instrumental variables necessary to identify structural parameters or to give the impulse coefficient a causal interpretation.

When $x_{n,t}$ depends on unobserved components that are potentially correlated with unobserved factors affecting the outcome variable, identification is achieved with an instrumental variable $z_{n,t}$. In addition, in settings without a recursive structure some of the variables in $w_{n,t}$ may also be endogenous. Note that in the case of selection on observables, $z_{n,t}$ may be identical to $x_{n,t}$ and $w_{n,t}$.

Let $d=H+1,$ $Y_{n,t}=\left(y_{n,t},....,y_{n,t+H}\right)$, $\beta_{n}=\left(\beta_{n}\left(0\right),....,\beta_{n}\left(H\right)\right)'$ and $\gamma_{n}=\left(\gamma_{n}\left(0\right)',....,\gamma_{n}\left(H\right)'\right)'$. Let $X_{n,t}=\left(x_{n,t},w_{n,t}'\right)'$,

and $g_{t}\left(\beta,\gamma\right)=q\left(Y_{n,t},X_{n,t},\beta,\gamma\right)\otimes z_{n,t}.$ If the underlying process generating the observed data is weakly stationary $\beta_{t}\left(h\right)$ does not depend on $t$. In the absence of stationarity, we define $\beta_{n,0},\gamma_{n,0}$ as the solution to

for any fixed $n\geq1$. It is then possible to define pseudo residuals $\varepsilon_{n,t}:=Y_{t}-\left(I_{d}\otimes x_{n,t}\right)\beta_{n,0}-\left(I_{d}\otimes w'_{n,t}\right)\gamma_{n,0}$ which inherit the property $\sum_{t=1}^{n}E\left[\varepsilon_{n,t}\left(h\right)z_{n,t}\right]=0$ from (ref) under $H_{0}$. With $\hat{g}_{n}\left(\beta,\gamma\right)$ as defined in (ref) and $\hat{\Omega}_{n}\left(\beta,\gamma\right)$ as defined in (ref) the statistic $\textrm{A\ensuremath{\textrm{R}_{C}}}\left(\beta_{n,0}\right)$ is defined in (ref).

Assumption (ref) is satisfied for many weakly dependent processes. Assumption (ref) can be satisfied by imposing the conditions of Jong2000 which include L$_{2}$ near-epoch dependent triangluar arrays. Assumptions (ref) and (ref) can be replaced by Assumption (ref) due to linearity of (ref). Assumption (ref) holds for $L_{1}$-mixingale arrays and $L_{1}$-near epoch dependent arrays by results for weak laws of large numbers in Andrews1988. De Jong (1996)Jong1996 proves strong laws for $L_{q}$-mixingale arrays for $q\geq1.$ The multivariate central limit theorem in Assumption (ref) can be established with the help of the Cramer-Wold theorem and by verifying regularity conditions for $\lambda'\Sigma_{n}^{-1/2}\nu_{n,t}$ for all $\lambda$ with $\left\Vert \lambda\right\Vert =1$. The assumption that $E\left[\text{\ensuremath{\left\Vert S_{n}\right\Vert ^{2}}}\right]<\infty$ is not sufficient in itself to guarantee a CLT without additional restrictions on dependence. Low level conditions for univariate double array processes can be found in McLeish1977 who imposes a weak dependence condition in the form of a mixingale assumption. McLeish's theory allows for non-stationarity, while trending behavior, including unit roots are ruled out by the assumptions on dependence. Wooldridge1988 prove a related result under near epoch dependence, which implies a mixingale property (see Proposition 2.9 of Wooldridge1988, 1988). They also prove a multivariate version, directly leading to the condition in Assumption (ref). For non-stationary triangular arrays instead of double arrays, Jong1997 proves a central limit theorem directly for heterogenous $L_{2}$-mixingale arrays. The result is obtained by showing that a blocking scheme can be used to obtain a martingale approximation for which a martingale CLT can be applied. Jong1997 and Davidson2000 establish a functional CLT that is based on Jong1997. Rio1997 establishes a CLT extending Lindeberg's CLT for iid random variables to double arrays of strongly mixing random variables. An alternative approach is explored by Neumann2013 and Merlevede, Peligrad and Utev Merlevede2019 who impose a maximal correlation measure to control dependence. When $\nu_{n,t}$ is a martingale difference array, McLeish1974, see also HallHeyde1980, provides a central limit theorem.

Implementation

Practitioners are often more interested in confidence intervals for a single parameter than in confidence sets for the entire parameter vector. One can obtain such confidence intervals from our confidence set via the projection method, see e.g., Dufour1997 and Dufour2005.

When $\beta$ is a scalar ($d_{\beta}=1$), which is often the case in empirical applications of LP-IV, a $100(1-\alpha)$% confidence interval for $\beta$ is given by $[\underline{\beta}_{1-\alpha},\overline{\beta}_{1-\alpha}]$, where $\underline{\beta}_{1-\alpha}=\min\left\{ \beta|AR_{c}(\beta)\leq c_{1-\alpha}\right\} $, $\overline{\beta}_{1-\alpha}=\max\left\{ \beta|AR_{c}(\beta)\leq c_{1-\alpha}\right\} $, and $c_{1-\alpha}$ is the $1-\alpha$ quantile of the $\chi^{2}$ distribution with $d-d_{\gamma}$ degrees of freedom. If the confidence set is convex, this confidence interval has exact $(1-\alpha)$ asymptotic coverage probability. Otherwise, it is conservative.

When $d_{\beta}>1$, let $\tilde{c}_{1-\alpha}$ be the $1-\alpha$ quantile of the $\chi^{2}$ distribution with $d-d_{\beta}-d_{\gamma}+1$ degrees of freedom. A $100(1-\alpha)$% confidence interval for $\beta_{1}$ (without of loss of generality) is given by $[\underline{\beta}_{1,1-\alpha},\overline{\beta}_{1,1-\alpha}]$, where for some $\beta_{2:d_{\beta}}$

and $\beta_{2:d_{\beta}}$ denotes the $(d_{\beta}-1)\times1$ subvector of $\beta$ excluding $\beta_{1}$. In practice, the condition, $AR_{c}(\beta_{1},\beta_{2:d_{\beta}})\leq\tilde{c}_{1-\alpha}$, can be checked by $\min_{\beta_{2:d_{\beta}},\gamma}Q_{n}(\beta,\beta_{2:d_{\beta}},\gamma)\leq\tilde{c}_{-1-\alpha}$ for a given value of $\beta_{1}$, where the minimum on the left-hand side is achieved by numerical optimization methods. For example, to obtain $\underline{\beta}_{1,1-\alpha}$ {[}$\overline{\beta}_{1,1-\alpha}${]}, one minimizes $\beta_{1}$ {[}resp. -$\beta_{1}${]} subject to $\min_{\beta_{2:d_{\beta}},\gamma}Q_{n}([\beta_{1}\;\beta_{2:d_{\beta}}],\gamma)\leq\tilde{c}_{1-\alpha}$. In MATLAB, one can use fminunc to evaluate the constraint function and use fmincon to minimize $\beta_{1}$ {[}$-\beta_{1}${]}.

When $d_{\beta}>1$, these confidence intervals are conservative in that the asymptotic coverage probability is greater than or equal to $1-\alpha$.

Monte Carlo Experiments

The data generating process is based on the bivariate VAR model of inflation ($\pi_{t}$) and output gap ($x_{t}$) in equations (4) and (7) of Kleibergen2009:

where $\rho_{1}=\rho(1-\rho_{2})=0.9(1-\rho_{2})$, $\alpha_{0}=\lambda/[1-\gamma_{f}(\rho_{1}+\gamma_{f}\rho_{2})]$, $\alpha_{1}=\lambda\gamma_{f}\rho_{2}/[1-\gamma_{f}(\rho_{1}+\gamma_{f}\rho_{2})]$, $\lambda=\rho_{\eta\nu}\frac{\sigma_{\eta}}{\sigma_{\nu}}\left(1-\gamma_{f}\left(\rho_{1}+\gamma_{f}\rho_{2}\right)\right)$. We introduce conditional heteroskedasticity to allow the conditional variance of the moment conditions to depend on the instruments: $\eta_{t}=\exp(0.5h_{1t})\epsilon_{1t}/\kappa$, $\nu_{t}=\exp(0.5h_{2t})\epsilon_{2t}/\kappa$, $\epsilon_{1t}$ and $\epsilon_{2t}$ are iid $N(0,1)$ with correlation $\rho_{\eta\nu}$, , $\kappa$ is set so that $\textrm{Var}(\eta_{t})=\sigma_{\eta}^{2}=1$ and $\textrm{Var}(\nu_{t})=\sigma_{\nu}^{2}=1$, and $h_{1t}$ and $h_{2t}$ are two independent AR(1) processes:

and $\xi_{1t}$ and $\xi_{2t}$ are independent iid $N(0,0.2)$ random variables that are independent of $\epsilon_{1t}$ and $\epsilon_{2t}$.

We estimate the Phillips curve,

using the moment conditions,

We consider two choices of instruments: (i) $z_{t}=[\pi_{t-1}\;x_{t-1}\;\pi_{t-2}\;x_{t-2}'\;\pi_{t-3}\;x_{t-3}]'$ as in Kleibergen2009 and (ii) $z_{t}=[x_{t}\;x_{t-1}]$ which represents a just identified specification.

It follows from equation 6 and other equations in Kleibergen2009 that the population reduced-form equations for the RHS endogenous variables can be written as

There are at least two cases of identification failures. First, when $\rho_{2}=0$, the projection coefficient matrix simplifies to

Thus, $\gamma_{f}$ and $\lambda$ are only partially identified (i.e., their linear combination is identified). Second, when $\gamma_{\eta\nu}=0$, the matrix simplifies to

In this case, $\gamma_{f}$ is identified but $\lambda$ is not.

We consider $\rho_{2}\in\{-0.05,-0.65,-0.99\}$ and $\textrm{Corr}(\eta_{t},\nu_{t})\in\{0.2,0.99\}$. The number of Monte Carlo simulations is set to 5,000. For HAC estimation, the lag truncation parameter is set to zero, as the moment conditions are serially uncorrelated.

Tables (ref) and (ref) report the finite-sample size of the $t$ test, the subset KLM test and the subset $AR_{c}$ test when testing $\gamma_{f}=0.5$ at the 5% significance level. Table 1 shows that, when the three lags of $\pi_{t}$ and $x_{t}$ are used as instruments, the conventional $t$ test suffers from size distortions especially when the instruments are weak and the degree of endogeneity is high ($\rho_{2}=-0.05$ and $\rho_{\epsilon\eta}=0.99$) However, when the parameters are just identified, the size distortions of the conventional $t$ test are much smaller, as shown in Table 2. In either case, the $AR_{c}$ and $KLM$ tests do not suffer from size distortions as expected.

Figures (ref)(a)-(f) and Figures (ref)(a)-(f) show the power curve of these three tests when $z_{t}=[\pi_{t-1}\;x_{t-1}\;\pi_{t-2}\;x_{t-2}\;\pi_{t-3}\;x_{t-3}]'$ and $z_{t}=[x_{t-1}\;x_{t-2}]'$, respectively. The sample size is set to 1,000. When the instruments are weak ($\rho_{2}=-0.05$ or $\rho_{\epsilon\eta}=0.0$), the $\textrm{AR}_{c}$ and KLM tests have correct size, although they do not have any power, as shown in Figures (ref)(a) and (d) and Figures (ref)(a) and (d). \footnote{Interestingly, when $\rho_{\epsilon\eta}=0.0$, the $AR_{c}$ and KLM tests do not have power when testing $\gamma_{f}=0.5$.} As the instruments become stronger, the tests become more powerful, as demonstrated in Figures (ref)(b),(c) and (d),(f) and Figures (ref) (b),(c) and (d),(f). When the parameters are overidentified and are very strongly identified, the KLM test is more powerful than the $\textrm{AR}_{c}$ test, seeFigures (ref)(c) and (f). However, when the parameter is mildly strongly identified, the $\textrm{AR}_{c}$ test is more powerful than the KLM test when the true parameter value is close to the null value, see Figures Figures (ref)(b) and (d). When the parameters are just identified, their power curves are virtually identical, as seen in Figures Figures (ref)(a) through (f). These results are consistent with Moreira2009 who shows that in the linear Gaussian case the just identified $\textrm{AR}_{c}$ based tests as well as score tests are uniformly most powerful for certain testing problems.

Conclusions

We obtain a theoretical upper bound for the Anderson Rubin subset statistic based on the continuous updating estimator for general parametric non-linear moment restrictions with heteroskedasticity, dependence and heterogeneity. The upper bound is shown to be valid for all converging subsequences of data distributions irrespective of whether nuisance parameters are identified or not. The upper bound is sharp in the sense that the $\textrm{AR}_{c}$ attains it when nuisance parameters are identified. We establish that the upper bound has a limiting $\chi_{d-d_{\gamma}}^{2}$ distribution along all converging subsequences. These results imply that the $\textrm{AR}_{c}$ statistic has uniformly correct size and can be used to construct confidence intervals that are uniformly valid.

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