Testing the presence of polynomial way trend in the conditional variance of stationaryu macroeconomic and financial time series

Abstract

In this paper, we consider stationary first order autoregressive models assuming that the errors that generate our process exhibits polynomial trend in their variance. We organize this work into three major sections: theory, Monte Carlo simulations and empirical work. In the first part, basic theoretical concepts of first order autoregressive models are presented. Limiting results, statistical inference and forecasting using the ordinary least squares estimator in AR (1) regressions are discussed. We continue with a discussion on heteroskedasticity. We describe it in details by defining it, presenting the generated difficulties and suggesting existing methods for dealing with this violation of the classical linear regression assumptions. Next, we focus on our hypotheses for the conditional variance of the error. The exact required properties of the error driving the model are stated and discuss the limiting behavior of our series under the existence of polynomial trend in the second moment of the AR (1) errors. Then we make some preliminary comments on how these theoretical results is expected to affect the discussed concepts (statistical inference, forecasting and limiting results) of AR(1) models. In the next section we present the results of Monte Carlo simulations. We conducted these experiments in order to demonstrate the effect of polynomial trend in the variance of the AR(1) disturbances in the least squares test statistic and compare alternative methods of correcting the suggested kind of heteroskedasticity. The last section of this paper demonstrates empirical results. For the purposes of this work, we created a database which consists of a variety of macroeconomic and financial data. These time series are tested for the presence of a polynomial way trend in the variance of the fitted residuals by an AR(1) regression. We aim in locating whether real data exhibits or not such a behavior in their disturbances. Afterwards, we conclude with a discussion on how our results changes existing results on the efficiency of the tested markets.