Lecture 7 Regression Models with Serially Correleated Disturbances 1. <a href="/keyword/asymptotic-normality/" >asymptotic normality</a> of the <a href="/keyword/ols-estimator/" >ols estimator</a> with Serially Correlated Errors Proposition 2.1 in Hayashi uses the conditions that {xt, t} is a jointly stationary and ergodic process and {xt t} is an m.d.s. in order to establish the <a href="/keyword/asymptotic-normality/" >asymptotic normality</a> of the <a href="/keyword/ols-estimator/" >ols estimator</a> . The m.d.s. restriction rules out serial correlation in the s. It turns out that the m.d.s. restriction can be replaced with the weaker condition that {xt t} is a mixingale and the <a href="/keyword/ols-estimator/" >ols estimator</a> will be consistent and asymptotically normal. The mixingale condition is essentially the condition that the sequence behaves asymptotically like an m.d.s. in the sense that E(xt+s t+s t, t-1, ,xt,xt-1, ) goes to 0 as s goes to . (See White, 2001, for more on this.) This assumption does allow for serial correlation in the disturbances. However, if the disturbances are serially correlated, then the orthogonality condition (A.3) will, for example, rule out lagged dependent variables among the regressors: yt = 0 + 1yt-1 + t and t = t-1 + ut , ut ~ i.i.d. In this case, yt-1 will correlated with t because both are determined by t-1. 2. The GLS and FGLS Estimator In the linear regression model with strictly exogenous regressors and heteroskedastic or serially correlated, normally-distributed disturbances, the GLS estimator is the BUE, MLE, and efficient estimator of the s. What are the large sample properties of the GLS estimator in time series regressions? If we assume that the regressors are predetermined but not necessarily strictly exogenous, then in general the GLS estimator will be inconsistent! The idea Suppose that we have a linear model with a single regressor, which is predetermined, so that E( t xt, xt-1, ) = 0 for all t. Let var( 1, , T) = 2 T Suppose, that T is known for each T. Let CTCT = T-1 Now, recall that GLS amounts to transforming the original model to: ~ ~ = ~ + y t xt t where ~ = c y + c y + ... + c y yt t1 1 t2 2 tT T ~ = c x + c x + ... + c x xt t1 1 t2 2 tT T ~ t = ct1 1 + ct 2 2 + ... + ctT T then fitting the model by OLS. But note that unless x is strictly exogenous, E ( ~t t ) 0 x~ and the GLS estimator will be inconsistent! Note: The GLS will be consistent and asymptotically superior to OLS in the important special case where the disturbances follow a finite-order autoregressive process. However, unless the regressors are strictly exogenous, the FGLS estimator will be inconsistent regardless of the form of the error process. We will return to this case in the next section of the course. 3. The Linear Trend Model A commonly encountered time series regression model is the linear trend model y t = 0 + 1 t + t where t is a zero-mean, covariance stationary process. That is, yt is the sum of a deterministic linear trend and a stationary process. In this case, we say that yt is a trend stationary process. This is a very simple and appealing way to think about many trending economic time series, like the log of real GDP. In this case, it is natural to consider estimating the linear trend: the estimated trend function, + t , provides us with an estimate of the long-run path of yt the residuals, = y t , provide us with an estimate of the short-run or cyclical behavior of yt 0 1 t t 0 1 we can use the residuals in regression models that require stationary variables If t is an i.i.d. N(0, 2) process, then for any fixed sample size T, this model satifies all of the assumptions of the classical normal linear regression model: linearity, strictly exogenous regressors, no multicollinearity, spherical normal disturbances. OLS is the BUE and MLE of the s; the simple OLS t and F statistics can be applied in the usual way. If t is i.i.d. (0, 2), the <a href="/keyword/ols-estimator/" >ols estimator</a> is consistent, asymptotically normal, and asymptotically efficient. The simple OLS t and F statistics can be applied in the usual way, though the justification will be asymptotic (since the disturbances are not assumed to be normal). (More on this shortly.) Suppose that t is a ser...