3 - Unit roots

Summary

Introduction

In the previous chapter we discussed the econometric problems that might occur when we run a regression with different orders of integrated variables. To avoid this problem, the first thing we have to do is to identify the correct order of the integration of each variable. In the context of ARIMA modeling, this identification is equivalent to determining the parameter d in the ARIMA(p, d, q) model. The Box-Jenkins approach discussed in the previous chapter suggested the use of visual inspection of correlograms for determining the parameter d. The recent development of unit root tests is nothing but the use of formal statistical tests in place of the visual inspection of the correlogram.

The idea that the parameter d is equal to the number of unit roots led Dickey and Fuller to replace the subjective inspection of the sample autocorrelation function with a formal test of the unit root null hypothesis. This test, known as the standard Dickey-Fuller (DF) test, is based on independently and identically distributed (iid) errors. For a wide class of errors which allows some heterogeneity and serial correlations in errors, two approaches have been proposed to modify the standard DF test. One is the parametric approach which suggests the augmented Dickey-Fuller (ADF) test. The nonparametric approach leads to the Phillips-Perron tests. The DF tests were applied to many US macroeconomic time series by Nelson and Plosser (1982). The evidence presented by them showed that most macroeconomic variables are well described by the ARIMA models with one unit root.