Finite sample sign-based procedures in linear and non-linear statistical models: with applications to Granger causality analysis

Abstract

This thesis consists of three essays on hypothesis testing and Granger causality analysis. The two main topics under consideration are: (1) exact point-optimal sign-based inference in linear and non-linear predictive regressions with a financial application; and (2) sign-based measures of causality in the Granger sense with an economics application.

These essays can be regarded as an extension to the sign-based procedures proposed by Dufour and Taamouti (2010a). The distinction is that in our study the predictors are stochastic and the signs may exhibit serial dependence. As a consequence, the task of obtaining feasible test statistics and measures of Granger causality is more challenging. Therefore, in each essay we either impose an assumption on the sign process or propose tools with which the entire dependence structure of the signs can feasibly be modeled. The three essays are summarized below.

In the first chapter, we acknowledge that the predictors of stock returns (e.g. dividend-price ratio, earnings-price ratio, etc.) are often persistent, with innovations that are highly correlated with the disturbances of the predictive regressions. This generally leads to invalid inference using the conventional T-test.
Therefore, we propose point-optimal sign-based tests in the context of linear and non-linear models that are valid in the presence of stochastic regressors. In order to obtain feasible test statistics, we impose an assumption on the dependence structure of the signs; namely, we assume that the signs follow a finite order Markov process. The proposed tests are exact,distribution-free, and robust against heteroskedasticity of unknown form. Furthermore, they may be inverted to build confidence regions for the parameters of the regression function. Point-optimal sign-based tests depend on the alternative hypothesis, which in practice is unknown. Therefore, a problem exists: that of finding an alternative which maximizes the power. To choose the alternative, we adopt the adaptive approach based on the split-sample technique suggested by Dufour and Taamouti (2010a). We present a Monte Carlo study to assess the performance of the proposed "quasi"-point-optimal sign test by comparing its size and power to those of certain existing tests that are intended to be robust against heteroskedasticity. The results show that our procedures outperform the other tests. Finally, we consider an empirical application to illustrate the usefulness of the proposed tests for testing the predictability of stock returns.

In the second chapter, we relax the assumption imposed earlier on the dependence structure of the signs. We had provided a caveat that to obtain feasible test statistics, the Markovian assumption must be imposed on the signs. In this essay, we extend the flexibility of the exact point-optimal sign-based tests proposed in the first chapter, by considering the entire dependence structure of the signs and building feasible test statistics based on pair copula constructions of the sign process. In a Monte Carlo study, we compare the performance of the proposed "quasi"-point-optimal sign tests based on pair copula constructions by comparing its size and power to those of certain existing tests that are intended to be robust against heteroskedasticity. The simulation results maintain the superiority of our procedures to existing popular tests.


In the third chapter, we propose sign-based measures of Granger causality based on the Kullback-Leibler distance that quantify the degree of causalities. Furthermore, we show that by using bound-type procedures, Granger non-causality tests between random variables can be developed as a byproduct of the sign-based measures. The tests are exact, distribution-free and robust against heteroskedasticity of unknown form. Additionally, as in the first chapter, we impose a Markovian assumption on the sign process to obtain feasible measures and tests of causality. To estimate the sign-based measures, we suggest the use of vector autoregressive sieve bootstrap to reduce the bias and obtain bias-corrected estimators. Furthermore, we discuss the validity of the bootstrap technique. A Monte Carlo simulation study reveals that the bootstrap bias-corrected estimator of the causality measures produce the desired outcome. Furthermore, the tests of Granger non-causality based on the signs perform well in terms of size control and power. Finally, an empirical application is considered to illustrate the practical relevance of the sign-based causality measures and tests.