Predictive Model With Highly Persistent Predictors:Theory And Applications Of Robust Inference

Testing predictability of asset returns has been studied for recent three decades as a cornerstone research topic in economics and finance.It not only attracts attention.from financial practitioners as it is a key component to evaluate mutual fund man-agers' performance,examine the validity of asset pricing models,and improve asset allocation efficiency,but also has important implications in finance theoretical re-search such as the efficient market hypothesis(Fama,1965[1],1970[2]).There are two major directions in the area of testing asset return predictability.The first one is to check whether the return as a time series is a white noise process,or random walk process,or martingale difference sequence,while the second stream of the literature is to expand predictive regressions to include the lag of other economic and finan-cial variables.Nevertheless,as pointed out by Fama(1991),the past realized returns are noisy measures of expected return so that the test based on them lacks of power.Meanwhile,it is restrictive to use historical returns only as predictors since investors could observe other information.Therefore,many recent literature and this paper focus on the second direction.However,a series of recent studies find that the statistical inference for predic-tive regressions,or more specifically,the limiting distribution of t-statistics,crucially relies on the time series properties of the regressors,i.e,their persistence containing nuisance parameter which could not be estimated consistently.Empirical analyses conclude evidently that most of predictors widely used in the literature are highly persistent with autoregressive roots extremely close to unity,which contains the s-tationary and the nonstationary cases.Ignoring such persistence and the embedded endogeneity induced by the contemporary correlation between the innovations of predictors and stock returns may lead to an over-rejection of the null hypothesis for conventional test(Torous et al.2004[4];Campbell and Yogo,2006[5]).Similarly,Lee(2016)[6]revealed these problems still existed in predictive quantile regression.To offer the valid test statistics in mean predictive regression,this dissertation first introduces an weighted estimator of the slope coefficient based on variable addition approach(Breitung and Demetrescu,2015[7])and constructs the consistent test statis-tics with all kinds of persistence.Second,this dissertation expands the linear projec-tion method(Cai and Wang,2014[8])to the predictive quantile regression framework to construct the consistent test statistics with nonstationary predictors.Finally,this dissertation expands this new approach to construct the weighted estimator based on variable approach in first part to predictive quantile regression and applies a different additional variable in this procedure.Based on this weighted estimator,this disser-tation constructs the consistent test statistics in predictive quantile regression.The main part of this dissertation contains three parts as follows.Firstly,this dissertation expands the variable addition approach to construct the new test statistics which achieve the local power under the optimal rate,i.e.,(?)with stationary predictors and T with nonstationary predictors.Moreover,it is valid in multivariate mean predictive regression and the cases some predictors are sta-tionary while some predictors are nonstationary.Variable addition approach is to decompose predictor into two new variables which replace the original predictor in the predictive regression.The estimators of coefficients of these two new variables both converges to the coefficient of the original predictor.Unlike constructing the test statistics only by the estimator of the slope of the additional variable as Breitung and Demetrescu(2015)[7],i.e.ignoring the constraint that both estimators converge the same coefficient,we construct a new estimator which is the weighted sum of the estimators of coefficients of these two new variables.Then the new test statistics based on this estimator achieves the local power under the optimal rate when the predictor is nonstationary.However,this test statistics is invalid when the predictor is stationary.Therefore,we construct a new test statistic which the weighted sum of the previous test statistics and the traditional t test statistics.Monte Carlo simulation demonstrate the good finite sample performance of this new test statistics in terms of siz.e and power under all persistence.Secondly,to test the predictability of the risk of stock return,this dissertation expands the linear projection method to the predictive quantile regression such that the pivotal test statistics achieve the local power under the optimal rate T with non?stationary predictors.Linear projection method is to decompose the innovation of the predictive regression into two parts.One part associate with the innovation of predictor while the other part as the new innovation of predictive regression is uncor-related with the predictor.As a result,the embedded endogeneity is dismissed by the linear projection method such that the new estimator follows the mixture normal dis-tribution.Monte Carlo simulation demonstrates the good finite sample performance of this new test statistics in terms of size and power with non-stationary predictors.Finally,since the linear projection method in predictive quantile regression is invalid with stationary predictors,we expand the new method introduced in the first part to the predictive quantile regression with the additional variable different from that in first part.Still,this method in predictive quantile regression is always valid and achieve the local power under the optimal rate,i.e.,(?)with stationary predictors and T with nonstationary predictors.Monte Carlo simulation demonstrate the good finite sample performance of this new test statistics in terms of size and power under all persistence.