Asymptotic Properties of Multiperiod Control Rules in the Linear Regression Model*

IK A~ULTIPERIODCONTROL PROBLEMS with unkilown parameters. curreilt decisioiis aKect iiot only current performance, but also the alnoulit of information that is obtained about the uiiknown paraineters. The purpose of this study is to investigate such aspects of m~~l t ipe r iod control in a siinple linear regression model with one unlino\vn parameter, where t!le illdependent variable is set at certain levels in order to bring the dependent variable to soine desired level. The approach uses the methods and criteria of statistical estimation theory (such as stroiig consisteiicy and eliiciency) to jiivestigate the properties of various control rules. This approacl~ sceins particularly useful in coiitrol problems of this type wliere estimation of unknown parameters plays an important role. Previous i:zvestigatioiis of this type of illultiperiod control problem (Aoki [2]), Zellner [9],aild Prescott [ 5 ] ) have been from a Bayesian point of view. By specifying a loss fiu~~ction, prior distributioris on the paraineters, and a distribution for the random disturbance term, a Bayes co~ltrol rule call be calculated, in principle, with the methods of dy~ialllic programming. However, as these studies have shown, ca1culatio.11 or even characterization of Bayes co~itrol rules has proved quite difficult. The approach of this study is 11011-Bayesian. Thc methods and results should c o m p l e m e ~ ~ t the usual Bayeslail viewpoint in eventually leading to reasoiiabie decisioiis in practical probleins. In Section 2 the nlodel is introduced and two coiltrol rules are defined. In Sectio:~3 we prove that these control rilles converge with probability 1 to tlie value \vI:icli would be used if the u~ilinown parameter were kliown with certainty. Iil Section 4 we derive tlie asyinptotic distribution of tlie coiitrol ruies aiitl parameter estimates, and in Section 5 we show that these coiitrol rules lead to parameter estilnates whicli have as small an asynlptotic variance as any other control rule in a fairly wide class. 111 particular this nieaiis that control rules which are designed for experimentation d o not give parameter estimates which are any better asy~ilptotically than tile inore simple control rulcs of this paper.