Inference on local causality and tests of non-causality in time series

Abstract

The study of the causal relationships in a stochastic process $(Y_{t},Z_{t})_{t\in\mathbb{Z}}$ is a subject of a particular interest in finance and economy. A widely-used approach is to consider the notion of Granger causality, which in the case of first order Markovian processes is based on the joint distribution function of ${{(Y_{t+1},Z_{t})}}$ given ${{Y_{t}}}$. The measures of Granger causality proposed so far are global in the sense that if the relationship between ${{Y_{t+1}}}$ and ${{Z_{t}}}$ changes with the value taken by ${{Y_{t}}}$, this may not be captured. To circumvent this limitation, this paper proposes local causality measures based on the conditional copula of ${{(Y_{t+1},Z_{t})}}$ given ${{Y_{t}}}=x$. Exploiting results by [5] on the asymptotic behavior of two kernel-based conditional copula estimators under $\alpha$-mixing, the asymptotic normality of nonparametric estimators of these local measures is deduced and asymptotically valid confidence intervals are built; tests of local non-causality are also developed. The suitability of the proposed methods is investigated with simulations and their usefulness is illustrated on the time series of Standard & Poor’s 500 prices and trading volumes.