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Conditional likelihood ratio test with many weak instruments
\address{Department of Economics, London School of Economics, Houghton Street, London, WC2A 2AE, UK.} \email{[email removed]} \address{Graduate School of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8601, Japan.} \email{[email removed]} \address{Department of Economics, London School of Economics, Houghton Street, London, WC2A 2AE, UK.} \email{[email removed]}
Inference in regression models with endogenous structural variables and many, weakly relevant instrumental variables is of great importance in applied research. A canonical example is the paper by Angrist_1991, in which the authors estimate the effect of educational attainment on wages by constructing up to 1,530 instruments for education through interacting quarter, state, and year of birth. A more recent literature that uses many weak instruments employs the “judge design” empirical strategy, which exploits random assignments of cases to judges (e.g., Dahl_2014,Autor_2019). Since each judge is an instrument in these settings, and judges can only process a certain number of cases, the number of instruments typically increases with the sample size. Many weak instrument settings are also found in papers estimating the new Keynesian Phillips curves (see Mavroeidis_2014) or studies that employ the Fama-MacBeth method for pricing assets Fama_1973.
In applications involving many weak instruments, researchers often rely on standard asymptotic approximations when conducting inference. However, asymptotic approximations to the finite sample distributions of conventional estimators and test statistics have been shown to be poor when instruments are weak. The use of many instruments can improve efficiency of estimators or their associated tests, but often causes the usual inference procedures to have poor finite sample properties, and several previous papers have noted this issue. Chao_2005, han2006gmm, and newey2009generalized generalize the many-instrument asymptotic theory to allow for weak instruments or moments. Andrews_2007 show that the Anderson-Rubin (AR), Lagrange multiplier (LM), and conditional likelihood ratio (CLR) tests are robust to many weak instruments, as long as the number of instruments $k$ grows slower than the cube root of the sample size, $n^{1/3}$. For the case where $k$ may be proportional to $n$, hansen2008estimation develop a many-instrument robust standard error and a modification for the LM test, while hausman2012instrumental propose Wald tests with limited information maximum likelihood and Fuller estimators that are robust to heteroskedasticity and many instruments. More recent developments in conducting robust inference with many weak instruments include the jackknife AR tests by Crudu_2021 and Mikusheva_2021, and the jackknife LM test by matsushita2022jackknife.
For the weak (but fixed number of) instruments problem, the seminal work of Moreira_2003 sparked a growing literature on conditional inference. Moreira_2003 introduces a general conditional inference framework for instrumental variable regression models with homoskedastic errors and advocates the CLR test. Andrews_2006 establish a nearly optimal property of the CLR test, while Mills_2014 propose approximately unbiased conditional Wald tests with comparable power to the CLR test. moreira2019optimal extend the conditional inference framework to heteroskedastic and autocorrelated errors.
In this paper, we set out to investigate the finite sample performance of Moreira_2003's (2003) CLR approach when $k$ is relatively large, and may grow proportionally with $n$. Size robustness of the CLR test under $k=o(n^{1/3})$ has already been established by Andrews_2007. However, we show that in a setting with homoskedastic normal errors with unknown variance, the conventional CLR test loses exact similarity and is asymptotically invalid under many weak instrument asymptotics, where $k$ may grow much faster than $n^{1/3}$. We propose a modified version of Moreira_2003's (2003) CLR test, hereafter called the modified CLR (MCLR) test, which is robust to: (i) many instruments, where the number of instruments can grow at the same rate or slower than the sample size and (ii) weak instruments, to the extent that a consistent test may exist Mikusheva_2021. It should be noted that we use the same test statistic as Moreira_2003 (“$LR_{1}$” in his paper), but our proposed test employs a different critical value function which is constructed by representing the likelihood ratio using four statistics, instead of two as in Moreira_2003. Our MCLR test retains asymptotic validity when there are many weak instruments, under the least restrictive condition possible on identification strength Mikusheva_2021. This result holds even when we relax the assumption of normally distributed error terms, as long as we impose an additional moment condition.
The rest of this article is organized as follows. Section (ref) introduces our setup and the likelihood ratio statistic (LR) when error variance is unknown, and discusses its representation by four statistics as well as the properties of those statistics. In Section (ref), we propose our MCLR test by constructing a robust critical value function and establish its asymptotic validity in a many weak instruments setting. We also discuss the lack of validity of the conventional CLR critical value function in our setup. Section (ref) illustrates the usefulness of our proposed method by a simulation study. All proofs are contained in Appendix (ref).
Consider the following instrumental variable regression model:
where $y_{1}=(y_{1i},\ldots,y_{1n})^{\prime}$ is an $n\times1$ vector of dependent variables, $Y_{2}$ is an $n\times l$ matrix of endogenous regressors, $\beta$ is an $l\times1$ vector of unknown structural parameters, $u$ is an $n\times1$ vector of mean-zero disturbances, $Z$ is an $n\times k$ matrix of instruments, $\Pi_{2}$ is a $k\times l$ matrix of unknown parameters, and $V_{2}$ is an $n\times l$ matrix of mean-zero error terms. We assume without loss of generality that there are no exogenous regressors in ((ref)) since one can always partial them out using standard projection methods. The reduced form system can be written as
where $Y=(y_{1},Y_{2})$, $\Pi=(\pi_{1},\Pi_{2})$, and $V=(v_{1},V_{2})$ with $\pi_{1}=\Pi_{2}\beta$ and $v_{1}=V_{2}\beta+u$.
This paper is concerned with testing the null hypothesis $H_{0}:\beta=\beta_{0}$ on the structural parameters, against the alternative $H_{1}:\beta\neq\beta_{0}$, where the coefficients $\Pi_{2}$ are treated as nuisance parameters. In particular, this paper focuses on the situation wherein researchers wish to test such a hypothesis, but only have many weak instruments at their disposal.
To proceed, we impose the following assumptions.
The normality of reduced-form errors in Assumption 1 is useful to motivate our conditional inference approach, which is inspired by the exact similarity of the LR statistic with known $\Omega$. Indeed, Moreira_2003 proves that conditional on the sufficient statistic $\Pi_{2}$ and when errors are normally distributed, the LR statistic with known $\Omega$ has a finite-sample distribution independent of nuisance parameters under $H_{0}$ and its quantiles can be used to construct a similar test (as long as this distribution is continuous). Since we maintain Moreira_2003's conditional inference framework, we begin with normally distributed error terms, although we will show that this assumption can be relaxed for the asymptotic analysis (see Theorem (ref)). Throughout this paper, we focus on the case where $\Omega$ is unknown to researchers. Assumption 2 concerns the instrumental variables. In this paper, we restrict $Z$ to be non-random, which is equivalent to conditioning on $Z$. In order to allow for $k$ to grow proportionally with $n$ as in Assumption 2 (a), we need to impose an additional condition $\frac{\mu^{2}}{\sqrt{k}}\to\infty$, which imposes a lower bound on the strength of the instruments. Indeed, $\frac{\mu^{2}}{\sqrt{k}}\to\infty$ is the least restrictive characterization of weak identification that allows for a consistent test of $H_{0}:\beta=\beta_{0}$ Mikusheva_2021. Chao_2005 also impose this condition to achieve consistency of point estimators under many weak instrument asymptotics. If $k$ grows slower than $n$, as in Assumption 2 (b), there is no requirement on $\mu^{2}$ , i.e., the instruments can be arbitrarily weak.
Note that Wald tests based on many-instrument robust standard errors hansen2008estimation,hausman2012instrumental are asymptotically valid under Assumption 2 (a), but not under Assumption 2 (b). Our MCLR test is asymptotically valid in both cases. Simulation studies in Section (ref) illustrate this distinction numerically. Andrews_2007 show that the conventional CLR test is asymptotically valid under relatively small numbers of instruments, that is when $k^{3}/n\to0$. Assumption 2 allows the number of instruments $k$ to be much larger, and as illustrated in our simulation study, the MCLR test is preferred when $k/n$ is large.
We first introduce some notation. When the variance $\Omega$ of $V$ is known, the LR statistic for testing $H_{0}$ against $H_{1}$ is written as
where $b_{0}=(1,-\beta_{0}^{\prime})^{\prime}$, $P_{Z}=Z(Z^{\prime}Z)^{-1}Z^{\prime}$ is the projection matrix with respect to $Z$, and $\bar{\lambda}$ is the smallest eigenvalue of $\Omega^{-1/2}Y^{\prime}P_{Z}Y\Omega^{-1/2}$ Moreira_2003.
To derive a more convenient expression for $LR_{0}$, note that $Z^{\prime}Y$ is a sufficient statistic for the parameters $(\beta,\Pi)$ under the assumption $V\sim N(0,\Omega)$ with known $\Omega$. This implies that $Z^{\prime}YD$ is also a sufficient statistic, for any nonsingular matrix $D$. So, we set $D=(b_{0},\Omega^{-1}A_{0})$ and obtain the partition $Z^{\prime}YD=[S:T]$, where \[ S=Z^{\prime}Yb_{0},\qquad T=Z^{\prime}Y\Omega^{-1}A_{0}. \] This is a convenient partitioning because $S$ and $T$ are independent and only $T$ depends on the nuisance parameters $\Pi$. Indeed, under the null hypothesis, $T$ alone is a sufficient statistic for $\Pi$.
By using standardized versions of $S$ and $T$: \[ \bar{S}=(Z^{\prime}Z)^{-1/2}Z^{\prime}Yb_{0}(b_{0}^{\prime}\Omega b_{0})^{-1/2},\qquad\bar{T}=(Z^{\prime}Z)^{-1/2}Z^{\prime}Y\Omega^{-1}A_{0}(A_{0}^{\prime}\Omega^{-1}A_{0})^{-1/2}, \] the LR statistic $LR_{0}$ can be alternatively written as
where $\bar{\lambda}$ is the smallest eigenvalue of $(\bar{S},\bar{T})^{\prime}(\bar{S},\bar{T})$. See Proposition 1 in Moreira_2003 for a proof. If $\Omega$ is known, we can apply the conventional CLR test by Moreira_2003 based on $LR_{0}$, even with many weak instruments. Since this paper focuses on the case of unknown $\Omega$ as stated in Assumption 1, the conventional test is infeasible. Its feasible counterpart, obtained by plugging in a consistent estimator of $\Omega$, turns out to be invalid under the many weak instrument asymptotics (see, Remark (ref) below).
We now introduce our test statistic of interest, for the case of unknown $\Omega$. The error variance matrix $\Omega$ can be estimated by
where $M_{Z}=I_{n}-P_{Z}$. By replacing $\Omega$ in ((ref)) with the estimator $\hat{\Omega}$, the LR statistic for testing $H_{0}$ with unknown $\Omega$ is written as
where $\hat{\lambda}$ is the smallest eigenvalue of $\frac{1}{n-k}\hat{\Omega}^{-1/2}Y^{\prime}P_{Z}Y\hat{\Omega}^{-1/2}$.
To obtain an analogous expression to ((ref)) for $LR_{1}$, we introduce two more objects: \[ \tilde{S}=M_{Z}Yb_{0}(b_{0}^{\prime}\Omega b_{0})^{-1/2},\qquad\tilde{T}=M_{Z}Y\Omega^{-1}A_{0}(A_{0}^{\prime}\Omega^{-1}A_{0})^{-1/2}. \] Based on this notation, we obtain the following representation of the $LR_{1}$ statistic.
This proposition says that the LR statistic $LR_{1}$ depends on six objects, instead of three as for $LR_{0}=\psi_{0}(\bar{S}^{\prime}\bar{S},\bar{S}^{\prime}\bar{T},\bar{T}^{\prime}\bar{T})$ in ((ref)). In order to develop our conditional inference method based on $LR_{1}$, we first establish the following properties of those six objects.
Based on the test statistic $LR_{1}$ and its properties presented in the last section, we now develop our conditional inference method. To begin with, recall that $\bar{T}$ is a sufficient statistic for $\Pi$, and consider the critical value function for given $\bar{T}=t$:
\[ c_{1,\alpha}(t)=(1-\alpha)\text{-th quantile of }\psi_{1,n,k}(\mathcal{S}^{\prime}\mathcal{S},\mathcal{S}^{\prime}t,t^{\prime}t,\mathcal{W}_{1},\mathcal{W}_{2},\mathcal{W}_{3}), \] where $\psi_{1,n,k}$ is defined in Proposition (ref), and $\mathcal{S}\sim N(0,I_{k})$ and $\left(
\right)\sim Wishart(n-k,I_{l+1},0)$ are independent. Propositions \ref{prop:1} and \ref{prop:2} directly imply the following property of $c_{1,\alpha}(t)$.
This theorem says that if $\bar{T}$ is observable, the LR test using $\frac{LR_{1}}{n-k}$ with the critical value $c_{1,\alpha}(\bar{T})$ is exactly similar. However, since $\bar{T}$ is unobservable for the case of unknown $\Omega$, the test based on ((ref)) is infeasible. To develop a feasible version, we estimate $\bar{T}$ by \[ \hat{T}=(Z^{\prime}Z)^{-1/2}Z^{\prime}Y\hat{\Omega}^{-1}A_{0}(A_{0}^{\prime}\hat{\Omega}^{-1}A_{0})^{-1/2}, \] where $\hat{\Omega}$ is as defined in ((ref)). Based on this estimator, our proposed rejection rule is defined as:
The next theorem is the main result of our paper, and it establishes asymptotic validity of the MCLR test in ((ref)).
This theorem is derived under the normality assumption (Assumption 1). For non-normal errors, as long as $k/n\to0$, we can also establish asymptotic validity of the MCLR test by requiring an additional moment condition. Let $P_{ii}$ be the $(i,i)$-th element of $P_{Z}$.
Specifically, as long as the number of instruments $k$ grows slower than the sample size $n$, and $\frac{1}{k}\sum_{i=1}^{n}P_{ii}^{2}\to0$, our MCLR test is asymptotically valid even when reduced form errors are non-normal. Note, Assumption 2 (b) is still more general than $k=o(n^{1/3})$, which Andrews_2007 require for asymptotic validity of the CLR test with non-normal errors.
In this section, we compare the critical value function $c_{1,\alpha}(t)$ of the MCLR test with $c_{0,\alpha}(t)$ of the conventional CLR test (Section (ref)), and use Monte Carlo simulations to evaluate the finite sample performance of our MCLR test relative to existing competitors.
The critical value function $c_{1,\alpha}(t)$ of our MCLR test does not have a closed form, as is the case with Moreira_2003's (2003) critical value function $c_{0,\alpha}(t)$. Panel A of Table (ref) presents the critical value function of the MCLR test for the 5% significance level. Critical values are shown for $n=100$, using different values of $\tau=\bar{T}^{\prime}\bar{T}$ and calculated using 10,000 Monte Carlo replications. Although $c_{1,\alpha}(t)$ takes $\bar{T}$ as input, we choose to focus on $\tau$ for simulations, which is indicative of identification strength. This also aids comparison with Moreira_2003, whose critical value function only depends on $\tau$.
When $k=1$, the critical value function of the MCLR test is a constant equal to 3.93 for all values of $\tau$, with the slight variation in the final row being due to simulation error. Just as is the case for the CLR test, the critical value function of the MCLR test for any given $k$ has an approximately exponential shape. Figure (ref) illustrates this with a plot of the critical value function of our MCLR test when $k=4$. When instruments are weak (i.e., $\tau$ is small), the critical values are larger. When $\tau$ is larger, such that instruments are stronger, the test behaves as if it were unconditional and the critical values are relatively stable around 3.93.
For comparison, in Panel B of Table (ref), we present the critical value function $c_{0,\alpha}(t)$ of the conventional CLR test. As shown in Theorem (ref), this test runs into size problems when there are many weak instruments. This is evident in the critical value function - once the number of instruments exceeds a tenth of the sample size, the critical values of the CLR test lie everywhere below those of our MCLR test. This suggests that the conventional CLR test would lead to over-rejection of the null hypothesis $H_{0}:\beta=\beta_{0}$ when the number of instruments is large.
This subsection conducts a simulation study based on Design I of Staiger_1997 with $l=1$, and $\beta_{0}=0$. $Z$ comprises of: a constant $Z_{1}$, $Z_{2}$ drawn from $N(0,1)$, $Z_{2}^{2}$, $Z_{2}^{3}$ and $k-4$ elements independently drawn from $N(0,1)$. Once drawn, $Z$ is then held fixed across simulations. To vary the strength of instruments, we use a population version of Stock_2005's (2005) pre-test for weak instruments. We use three different values of $\Pi_{2}$ so that the population first-stage F-statistic $\delta^{2}=\Pi_{2}^{\prime}Z^{\prime}Z\Pi_{2}/\omega_{22}$ takes the values 2 (very weak instruments), 10 (weak instruments), and 30 (strong instruments), for different values of $k$. The rows of $(u,V_{2})$ are i.i.d. normal random vectors with unit variances and correlation $\rho$, which measures the degree of endogeneity of $Y_{2}$ in ((ref)). The number of Monte Carlo replications is 10,000 for the size analysis and 2,500 for the power analysis.
Table (ref) investigates the size properties of five tests for $n=100$ and $H_{0}:\beta=\beta_{0}$: (i) the t-test with the heteroskedasticity robust limited information maximum likelihood estimator by hausman2012instrumental (H-LIML), (ii) the conditional likelihood ratio test by Moreira_2003 (CLR), (iii) the modified Lagrange multiplier test by hansen2008estimation (mKLM), (iv) the jackknifed version of the Anderson--Rubin (AR) test proposed by Mikusheva_2021 (J-AR) and our proposed modified CLR test (MCLR). Firstly, note that the size distortions of H-LIML are large, except when $\delta^{2}$ is large relative to $k$. The degree of endogeneity of $Y_{2}$ also seems to matter; when $\rho=0.2$, the t-test tends to under-reject the null hypothesis, while when $\rho=0.6$, the null is over-rejected. The distortions of the test are most severe when $\delta^{2}$ is small relative to $k$, and $k$ is large. As a result, we do not investigate power of this test.
The CLR test attains roughly the correct size when $k=5$, even when identification is weak and the degree of endogeneity is high. However, size distortions can be observed when $k/n\ge0.1$. Surprisingly, this is not visibly exacerbated when $\delta^{2}$ is low, suggesting that it is the existence of many instruments that has the more severe empirical consequences on the conventional CLR test. Overall, even when the CLR test experiences little size distortion, it always has empirical rejection frequency farther from $5\%$ than our proposed MCLR test.
The mKLM test works well. Although it tends to under-reject when $\delta^{2}$ is relatively small, its size distortions never exceed $1\%$. Similarly, the J-AR test appears relatively robust to many weak instruments, although it tends to over-reject the null in all cases.
Compared to the other tests that we consider, the rejection frequencies of our proposed MCLR test are on average closest to the nominal level. As our theory in Section (ref) suggests, the MCLR test is robust to weak instruments, many instruments, and many weak instruments.
Figure (ref) presents calibrated power curves for the MCLR, J-AR and mKLM tests for $H_{0}:\beta=\beta_{0}$, under the alternative hypotheses $H_{1}:\beta=\beta_{0}+\Delta$. These power curves are plotted with respect to the $5\%$ significance level, i.e., the critical values for these three tests are given by the $95$-th percentiles of these test statistics under $H_{0}$, computed via 10,000 Monte Carlo replications. Each curve is plotted for $n=200$, $l=1$, and $\beta_{0}=0$. We present four different cases, with different values of $\delta^{2}/k$ , holding $\rho=0.2$. As we move from left to right, and top to bottom, the figures show the cases of $\delta^{2}/k=1/3$, $1/2$, $1$, and $2$. Our MCLR test is uniformly more powerful at all values of $\delta^{2}/k$, although is more pronounced for low $\delta^{2}/k$. The mKLM test experiences spurious declines in power under alternative hypotheses that are further away from the null, with consistently low power when $\delta^{2}/k=1/3$. When $\delta^{2}/k$ is high, i.e., identification is strong and/or the number of instruments is small, the J-AR has similar, although everywhere lower power, than our M-CLR test. While we do not present theoretical results on power, this result suggests that the MCLR test shares the superior power properties of the conventional CLR test, which has near optimal power with small $k$ Andrews_2006.
In this paper, we propose a modification of Moreira_2003's (2003) conditional likelihood ratio (CLR) test, namely the MCLR test. We prove that in instrumental variable regression models with unknown error variance and many weak instruments, the MCLR test is asymptotically valid under many weak instrument asymptotics, unlike the CLR test. This is true even when the number of instruments grows proportionally to the sample size, and identification is as weak as possible to the extent that a consistent test may exist Mikusheva_2021.
Our simulation study indicates that the MCLR test has superior size properties to the CLR test and is more powerful than competing tests that are robust to many weak instruments, including the modified Lagrange multiplier test by hansen2008estimation and the jackknife Anderson--Rubin test by Mikusheva_2021.
The size and power results presented in our theorems and simulation study lead us to recommend the MCLR test for general use in scenarios when instruments are many and potentially weak. This is based on the fact that the MCLR test is asymptotically valid with little size distortion in finite samples with many weak instruments, while also retaining the favorable power properties of the CLR test.