Many estimators have been proposed for regressioni models with lagged dependent variables and general stationary disturbances, with Hannan [1965] and Amemiya and Fuller [1967] as the initial contributors. Generalizations to higher order, simultaneous equation, and differential equation representations have been introduced by Dhrymes [1970], Espasa [1977], Espasa alid Sargaln [1977] and Robinson [1976]. All of these estimators are asymptotically inaximum likelihood. However, because they employ spectral metlhods, they have often been criticized for reliance onl asymptotic criteria as it is widely believed that the finite sample performance is inferior to the asymptotic approximations. Potential improvement could result from using exact maximum likelihood estimators. In the time domain, exact maximum likelihood estimators have been claimed to be better than asymptotic approximations. Newbold [1974], Galbraith and Galbraith [1974], Wallis and Prothero [1976] and Osborn [1976] have argued for maximizing the exact likelihood functioni of various ARMA models. In this paper, the exact maximum likelihood estimators are found foi dynamic regression models under two sets of assumptions onl initial values. Althouglh they are asymptotically equivalent, the estimators differ for finite samples. It is also shown that Hannan's likelihood function is exact if one of these assumptions is taken. It is sometimes possible to see that this assumption is completely untenable in the data. One would therefore expect the finite sample performance of this estimator to be inferior to its asymptotic properties. These results are then applied to the band spectrum regression problem where there are presumably economic argUrments for excludinlg some frequenicy bands from regressioin. This may be to test whether the regressioll is stable across frequencies, (Engle [1978, 1974], Espasa and Sargani [1977]), or becaulse there are measurement errors at some frequencies, (Engle anid Foley [1975]) or because there is seasonality in certain bands (Hylleberg [1975]). The estimation technique has not previously been available for extending these to dynamic regressions with stationary errors. Engle [1980] solves the testing and estimation problem for