Notes on Asymptotic Estimation Theory for Integrated and Cointegrated Processes

1 Preliminaries Let fX t g be a discrete-time stochastic process. Throughout, we will assume that any process fX t g is indexed by the integers or some subset of the integers (for example, the positive integers). In practice, we observe a nite length sample (or time series) X 1 ; ; X n from fX t g. (In this presentation, we will distinguish between a stochastic process fX t g and a nite time series X 1 ; ; X n ; this terminology is not particularly standard and one often uses time series to mean stochastic process.) A stochastic process fX t g is said to be strictly stationary if for each nite collection (t 1 ; ; t m) of integers and each integer s, (X t 1 ; ; X tm) and (X t 1 +s ; ; X tm+s) have the same joint probability distribution. fX t g is second-order stationary if E(X t) = ((nite), E(X 2 t) < 1 and E(X t X t+s) depends only on jsj. (Hence Cov(X t ; X t+s) depends only on jsj.) Note that strict stationarity implies second-order stationarity whenever E(X 2 t) is nite. To simplify exposition, we will use \stationary" to mean \strictly stationary"; any process which is not (strictly) stationary is said to be non-stationary. A simple measure of the dependence structure of a stationary process fX t g is the function r(s) = Cov(X t ; X t+s) which is called the autocovariance function of fX t g. A scaled version of r(s), (s) = r(s)=r(0) is called the autocorrelation function of fX t g. Estimates of (s) turn out to be of some use in the identiication of time series models. EXAMPLE 1. Let fX t g be a sequence of independent, identically distributed random variables. It is trivial to verify that fX t g is stationary. Such stochastic processes are called white noise processes and are the basis of many stationary and non-stationary models in time series analysis. 3 EXAMPLE 2. Deene the process fX t g by the diierence equation where f" t g is white noise and " t is independent of X t?1 ; X t?2 ;. Such a process is called an autoregressive moving average (ARMA) process. It can be shown that the fX t g is stationary if the parameters 1 ; ; p satisfy the …