In this section we consider the pxh order autoregressive model


(13.2.1) yt = X Pjjt-j + et, t = p+l, p+2, . . . , T,

i= і

where {є(} are i. i.d. with Eet = 0 and Vet = ct2, and (уі, у2> • • • ,Ур) are independent of (є^+ь єр+2,. . . , £r). This model differs from (13.1.26) only in that the {yt} are observable, whereas the {ut} in the earlier equation are not. We can write (13.2.1) in matrix notation as (13.2.2) у = Yp + є by defining

У = (Ур+ъУр+2, ■ ■ ■ ,Ут)’, E = (Є^+1, Єр+2, ■ • • , Єт)’,

Р — (Pb Р2> • • • > Рр)’,

Ур Ур-1 Уі

Ур+1 Ур У2

Ут-і Ут-2 Ут-р

Although the model superficially resembles (12.1.3), it is not a classical regression model because Y cannot be regarded as a nonstochastic matrix.

The LS estimator p = (Y’Y) *Y’y is generally biased but is consistent with the asymptotic distribution

(13.2.3) p ~ iV[p, a2(Y’Y)-1].

Since Theorem 12.2.4 implies that (3 ~ 1V[0, cr2(X’X) ’], the above result shows that even though (13.2.2) is not a classical regression model, we can asymptotically treat it as if it were. Note that (13.1.29), obtained earlier, is a special case of (13.2.3).

It is useful to generalize (13.2.2) by including the independent variables on the right-hand side as

(13.2.4) у = Yp + Zy + e,

where Z is a known nonstochastic matrix. Essentially the same asymptotic results hold for this model as for (13.2.2), although the results are more difficult to prove. That is, we can asymptotically treat (13.2.4) as if it were a classical regression model with the combined regressors X = (Y, Z). Economists call this model the distributed-lag model See a survey of the topic by Griliches (1967).

We now consider a simple special case of (13.2.4),

(13.2.5) yt = py(_! +7^ + є,.

This model can be equivalently written as


(13.2.6) yt = y^PiZt-j + wt,


where wt = pwt- + £;. The transformation from (13.2.5) to (13.2.6) is called the inversion of the autoregressive process. The reverse transformation is the inversion of the moving-average process. A similar transformation is possible for a higher-order process. The term “distributed lag” describes the manner in which the coefficients on z,_7 in (13.2.6) are distributed over j. This particular lag distribution is referred to as the geometric lag, or the Koyck lag as it originated in the work of Koyck (1954).

The estimation of p and у in (13.2.5) presents a special problem if {є,} are serially correlated. In this case, plim T’1 ^=2^-іЄ( ф 0, and therefore the LS estimators of p and у are not consistent.

In general, this problem arises whenever plim T_1X’u A 0 in the regres­sion model у = X0 + u. We shall encounter another such example in

Section 13.3. In such a case we can consistently estimate P by the instru­mental variables (TV) estimator defined by

(13.2.7) p/у = (S’X)_1S’y,

where S is a known nonstochastic matrix of the same dimension asX, such that plim T! S’X is a nonsingular matrix. It should be noted that the nonstochasticness of S assures plim T au = 0 under fairly general as­sumptions on u in spite of the fact that the u are serially correlated. Then, under general conditions, we have

(13.2.8) fW ~ N[p, (S’Xr^’SSfX’S)-1],

where X = £uu’. The asymptotic variance-covariance matrix above sug­gests that, loosely speaking, the more S is correlated with X, the better.

To return to the specific model (13.2.5), the above consideration sug­gests that z, and zt- constitute a reasonable set of instrumental variables. For a more efficient set of instrumental variables, see Amemiya and Fuller (1967).

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