Nonparametric Tests for Conditional Independence
(Dissertation Summary)
Liangjun Su

As David (1979) puts it, independence and conditional independence form the basis of probability theory and are equally fundamental in the theory of statistical inference. Applications of conditional independence are ubiquitous in econometrics. In a seminal paper, Granger (1980) introduces the concept of Granger non-causality at the distribution level, which is a particular case of conditional independence. Angrist (1997) shows that the problem of sample selection bias can be cast in terms of conditional independence. Further, conditional independence is often used as an identification condition in econometrics (e.g., Imbens and Newey, 2001; Lechner and Miquel, 2002).

Despite its wide use, few tests have been developed to test for conditional independence. Two exceptions are Linton and Gozalo (1997) who propose two nonparametric tests for conditional independence based on empirical distribution functions and Fernandes and Flores (1999) who employ a generalized entropy measure to test for conditional independence. Nevertheless, Monte Carlo simulations suggest that these tests have poor power against conditional dependence in a variety of data generating processes. My dissertation develops some more powerful nonparametric tests for the null hypothesis of conditional independence. Hence, I resolve a long-standing challenge in econometrics by providing asymptotic theory for powerful nonparametric tests of conditional independence.

In "A Nonparametric Hellinger Metric Test for Conditional Independence'' I propose a nonparametric test of conditional independence based on the weighted Hellinger distance between f(y|x,z) and f(y|x), where f(y|x,z) (resp. f(y|x)) is the conditional density of random variable Y given random variable (X,Z) (resp. X). Under the null hypothesis (Z does not carry extra information about Y once X is known), the distance is identically zero whereas under the alternative it is nonzero. The unnormalized test statistic takes values between 0 and 1 and serves as a first measure of the strength of conditional dependence. The normalized test statistic is asymptotically normal under regularity conditions that are standard in the literature. Due to the "curse of dimensionality'', however, the test is less satisfactory when the dimension of (X,Y,Z) is large.

To lessen the adverse effect of the dimension of (X,Y,Z) on the power of the test, I next look at an equivalent characterization of conditional distributions: conditional characteristic functions. In "Consistent Characteristic-Function-Based Test for Conditional Independence'' I explore tests based on the notion that two conditional distributions are equal if and only if their corresponding conditional characteristic functions are equal and propose a new test for conditional independence. The test is less severely subject to the adverse effect of the dimension of (X,Y,Z)  on the power than the Hellinger metric test (because the dimension of Y does not affect the convergence rate of the new test statistic), and at the same time it maintains the good properties of the Hellinger metric test: (1) the normalized test statistic is asymptotically normal under some regularity conditions; and (2) the test has power against deviations from the null. Simulation results suggest that this new test complements the Hellinger metric test when the dimension of  (X,Y,Z) is small, and it is also powerful when the dimension of (X,Y,Z) is relatively large.

"Testing Conditional Independence via Empirical Likelihood'' [Job Market Paper] I examine a class of empirical-likelihood-based tests for the null of conditional independence. Motivated by the optimality of the parametric likelihood ratio test, I look at its nonparametric analogue: the empirical likelihood test. I extend the applicability of empirical likelihood from testing a finite number of moment or conditional moment restrictions (e.g., Tripathi and Kitamura, 2002) to testing an infinite collection of conditional moment restrictions. Writing the null hypothesis in terms of conditional-distribution-based moment restrictions and employing the idea of ``smoothed'' empirical likelihood, I construct an intuitively appealing test statistic and show that it is asymptotically normal under the null. I also derive its asymptotic distribution under a sequence of local alternatives. Although this test statistic has intuitive appeal, it delivers poor power in small samples because of the discrete nature of the indicator functions used in forming the sample analogue of the moment restrictions. Thus I build on Su and White (2003) and use a class of smoother moment conditions to construct a new empirical-likelihood-based test. Then I show that in large samples both tests are weakly optimal in that they attain maximum average local power with respect to different spaces of functions for the local alternatives. Simulations suggest that the smoother-moment-conditions-based test outperforms all previous tests in small samples. I apply this latter test to a number of economic and financial time series and find that the test reveals some interesting nonlinear Granger causal relations that the traditional linear Granger causality test fails to detect. Further, the approach outlined in the paper easily extends to testing many other distribution-based hypotheses such as conditional goodness-of-fit, conditional homogeneity and conditional symmetry.