A Consistent Characteristic-Function-Based Test for Conditional Independence

By Liangjun Su, Halbert White. 2003

170 citations recorded by the source corpus.

This paper proposes a nonparametric test of conditional independence based on the notion that two conditional distributions are equal if and only if the corresponding conditional characteristic functions are equal. We use the functional delta method to expand the test statistic around the population truth and establish asymptotic normality under $\beta -$mixing conditions. We show that the test is consistent and has power against local alternatives at distance $n^{-1/2}h_{1}^{-(d_{1}+d_{3})/4}.$ The cases for which not all random variables of interest are\ continuously valued or observable are also treated, and we show that the test is nuisance-parameter free. Simulation results suggest that the test has better finite sample performance than the Hellinger metric test of Su and White (2002) in detecting nonlinear Granger causality in the mean. Applications to exchange rates and to stock prices and trading volumes indicate that our test can reveal some interesting nonlinear causal relations that the traditional linear Granger causality test fails to detect.

Subfields: Finance, Statistics and Probability

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Citations in the corpus

Kernel-based Conditional Independence Test and Application in Causal Discovery (2011, background): “For example, Su and White (2008) deﬁne the test statistic as some distance be-tween the estimated conditional densities p ( X | Y, Z ) and p ( X | Y ), and Su and White (2007) exploit the difference between the characteristic functions of these conditional densities.”
Testing Conditional Independence Via Empirical Likelihood (2014, methodology): “Following Su and White (2010), we define the mixing coefficients: αn (j) = sup1≤l≤n−j{P (A ∩B) −P (A)P (B)|A ∈ σ (Wnt : 1 ≤ t ≤ l) , B ∈ σ (Wnt : l + j ≤ t ≤ n)} if j ≤ n− 1, and αn (j) = 0 if j ≥ n.”
Valuing travel time variability: Characteristics of the travel time distribution on an urban road (2012, background): “Su and White (2007)) would be applicable to check this formally.”
A Nonparametric Test for Granger Causality in Distribution With Application to Financial Contagion (2014, methodology): “Testing procedures for Granger causality in the whole distribution in a time series context are developed only in Su and White (2007, 2008, 2012, 2014), Bouezmarni, Rombouts, and Taamouti (2012), and Taamouti, Bouezmarni, and El Ghouch (2014).”
Estimation and Forecasting of Dynamic Conditional Covariance: A Semiparametric Multivariate Model (2009, methodology): “A similar idea was used in Lavergne and Vuong (2000), Su and White (2007), and Su and Ullah (2009).”
Strongly consistent nonparametric tests of conditional independence (2012, methodology): “Under density assumption Su and White (2008a, 2008b), used Hellinger metric and characteristic functions, respectively, relaxing the independence assumption on the sample.”
Granger Causality and Dynamic Structural Systems (2010, methodology): “…consistent against arbitrary alternatives are readily available (e.g., Linton and Gozalo, 1997; Fernandes and Flores, 2001; Delgado and Gonzalez-Manteiga, 2001; and Su and White, 2007a, 2007b, 2008), but given the large number of covariates in most applications, these are often not practical.”
On Azadkia–Chatterjee’s conditional dependence coefficient (2021, background): “…made to provide nonparametric solutions, and notable examples include Linton and Gozalo (1996) (on conditional cumulative distribution functions); Su and White (2007, 2008, 2014) (on conditional characteristic functions, conditional probability density functions, and smoothed empirical…”
The conditional permutation test (2018, background): “There is also a large literature on extending measures of marginal independence to the conditional setting, including partial distance covariance [24]; conditional mutual information [15]; characteristic functions [21]; Hellinger distances [22]; and smoothed empirical likelihoods [23].”
Minimax optimal conditional independence testing (2020, background): “Wang and Hong [38] develop a new test based on characteristic functions, which achieves faster rates against certain local alternatives in comparison to the test developed by Su and White [32].”
Gaussian Processes for Independence Tests with Non-iid Data in Causal Inference (2015, methodology): “In the case of conditional independence, several tests have been proposed, including a test based on characteristic functions [Su and White 2007], the Normalized Conditional Cross-Covariance Operator (NOCCO) [Fukumizu et al. 2007], Kernel-based Conditional Independence (KCI) [Zhang et al. 2011], a…”
Approximate Kernel-Based Conditional Independence Tests for Fast Non-Parametric Causal Discovery (2017, methodology): “A second method involves measuring the distance between estimates of the conditional densities f ( X | Y, Z ) and f ( X | Z ), or their associated characteristic functions, by observing that f ( X | Y, Z ) = f ( X | Z ) when X ⊥⊥ Y | Z (Su and White, 2007, 2008).”
The conditional permutation test for independence while controlling for confounders (2018, background): “…independence to the conditional setting, including partial distance covariance (Székely and Rizzo, 2014), conditional mutual information (Runge, 2018), characteristic functions (Su and White, 2007), Hellinger distances (Su and White, 2008) and smoothed empirical likelihoods (Su and White, 2014).”
Causality in quantiles and dynamic stock return–volume relations (2009, result): “This agrees with the finding of Su and White (2007) which is based on a test at the distribution level. where T is the sample size and q ≥ 1; this model will be referred to as a lag-q model.”
Statistical Dependence: Beyond Pearson’s ρ (2018, background): “A number of authors have considered such an approach; both in the iid and time series case, see e.g. Rosenblatt (1975), Robinson (1991), Skaug and Tjøstheim (1993b; 1996), Granger, Maasoumi, and Racine (2004), Hong and White (2005), Su and White (2007) and Berrett and Samworth (2017).”
Testing conditional independence using maximal nonlinear conditional correlation (2010, background): “l independence has a long history. However, there are relatively few results on nonparametric tests when the vectors X, Y and Zare continuous. Some examples of such tests can be found in Su and White [12, 13], where they also proposed conditional independence tests based on a weighted Hellinger dist”