Sankhya: The Indian Journal of Statistics

1997, Volume 59, Series B, Pt. 2, 127-141

MODEL MISSPECIFICATION PROBLEMS FOR NONSTATIONARY TIME SERIES

By

DONG WAN SHIN, *Ewha Womans University, Seoul,*

And

SAHADEB SARKAR, *Oklahama State University, Stillwater*

SUMMARY. The ordinary least squares estimation in regression models with integrated regression and integrated errors is considered. It is shown that the asymptotics derived under this general framework can be used to unify various theoretical and empirical results available on model misspcification problems. The theory has been used to analyse how the asymptotic behaviour of the ordinary least squares estimator, and the *t,* DW and *R ^{2}* statistics is affected in misspecified nonstationary time series models. Such models include spurious regression, cointegrated regression, misspecified polynomial regression and misspecified autoregression. In many situations some regression statistics diverge as the sample size goes to infinity, implying misleading significance of the test statistics. It is concluded that a small DW statistics value may be taken as an indication of possible misspecification. Using the theory developed , we have discussed tests of cointegration and methods to decide between an autoregressive model and a polynomial regression model for a better fit. We have also illustrated how to guard against underspecifying the multiplicity of autoregressive unit roots in the Divkey-Fuller unit root tests. Applications have been illustrated through several real and simulated example datasets.

*AMS (1991) subject classification.* 62M10, 62F05, 62F10, 62F11

*Key words and phrases*. Brownian motion, cointregation, integrated error, integrated regressors, least squares estimation, nonstationarity, spurious regression.