Szacowanie parametrów modeli autoregresyjnych z uwzględnieniem opóźnień zmiennej objaśniającej

Abstract

Let us assume that a linear relationship exists between a variable Yt and k—1 explanatory variables Xt , Xt—1, . . . , Xt—k and a disturbance term u. If we have a sample of n observations on Y and X's we can write (1) The α, ß0; ß1, . . . , ßk coefficients and the parameters of the u distribution are unknown nad our problem is to obtain estimates of these unknowns. Under the usual assumption about the distribution of u and the independence of X and u there are in principle no new estimation problems in this model. Least-squares will give best linear unbased estimates, if the model has been specified correctly. Several difficulties, however, are likely to arise in practice. First of all one cannot really expect any precise and firm indication from theory of the lenght of lag to be incorporated; rather one hopes to determine the lag from the data by fitting a fairly lag and then examining the significance of the coefficients of various lagged values of X. But this in turn raises two main statistical difficulties; one is that observations are lost due to the lags and the other is that typically the various lagged values of X will be highly intercorrelated leading to very inprecise estimates of the lagged coefficients and great difficulty in making useful inferences about them. These difficulties have lead to the a priori impostion of various assumptions about the form of the weights ß0, . . . , ßk in an attempt to produce a more amenable estimation problem involving fewer than k+1 parameters. The above paper discusses three models for distributed lag analysis that either reduce the number of observations lost due to lagging and/or reduce the number of parameters to be estimated.