# Joint Presence of Lagged Endogenous Variables and Serial Correlation

In this last subsection we shall depart from Model 6 and consider briefly a problem that arises when X in (6.1.1) contains lagged values of y. An example of such a model is the geometric distributed-lag model (5.6.4), where the errors are serially correlated. This is an important problem because this situation often occurs in practice and many of the results that have been obtained up to this point in Section 6.3 are invalidated in the presence of lagged endogenous variables.

Consider model (5.6.4) and suppose {u,} follow AR(1), u, = pu,-x 4- e„ where {€,) are i. i.d. with Ее, = 0 and Ve, = a1. The LS estimators of A and a are clearly inconsistent because plimr_« T-12 Ф 0. The GLS estimators

of A and a based on the true value of p possess not only consistency but also all the good asymptotic properties because the transformation R, defined in

(5.2.10) essentially reduces the model to the one with independent errors. However, it is interesting to note that FGLS, although still consistent, does not have the same asymptotic distribution as GLS, as we shall show.

Write (5.6.4) in vector notation as

y^Ay., + ax + ussZy + u, (6.3.27)

where y_ і = (y0, Уі> ■ • • > Ут-1)’- Suppose, in defining FGLS, we use a con­sistent estimator of p, denoted p, such that ‘/Tip —p) converges to a non­degenerate normal variable. Then the FGLS estimator of y, denoted yF, is given by

% = (Z’ftift. Z)-‘Z’ftiR. y, (6.3.28)

where ft, is derived from R, by replacingp with p. The asymptotic distribution of % can be derived from the following equations:

~У) = (7’_1Z, R,1R1Z)_,7’"1/2Z, R’IR1u (6.3.29)

= (plim r-‘Z’RJRjZ)-1 X [r-‘^Z’RIRjU + (T~l/2Z’ftjfeju – r-^z’R^u)].

Let wT be the first element of the two-dimensional vector (r_1/2Z’R{RjU — 7’-i/22’rjri|1). Then we have

wT= ‘Ifip2 ~P2)j.^ УА+1 (6.3.30)

1 1-1

_ 1 /г-1 Г—2

-p)~[ 2 w+ 2 w+z)

1 r—1 1—0 /

Thus we conclude that the asymptotic distribution of FGLS differs from that of GLS and depends on the asymptotic distribution of p (see Amemiya and Fuller, 1967, for further discussion). Amemiya and Fuller showed how to obtain an asymptotically efficient feasible estimator. Such an estimator is not as efficient as GLS.

The theory of the test of independence discussed in the preceding subsec­tion must also be modified under the present model. If X contained lagged dependent variables, y,_,, y,_2,. . . , we will still have Eq. (6.3.17) formally, but {Q will no longer be independent normal because H will be a random matrix correlated with u. Therefore the Durbin-Watson bounds will no longer be valid.

Even the asymptotic distribution of d under the null hypothesis of indepen­dence is different in this case from the case in which X is purely nonstochastic. The asymptotic distribution ofrfis determined by the asymptotic distribution of p because of (6.3.14). When X is nonstochastic, we have fT(p — p)—> N(0, 1 — pi2) by the results of Section 6.3.2. But, if X contains the lagged dependent variables, the asymptotic distribution of p will be different. This can be seen by looking at the formula for ff(p — p) in Eq. (6.3.4). The third term, for example, of the right-hand side of (6.3.5), which is 7’-,/2S£.2(/? — Д)’х, м,_, does not converge to 0 in probability because x, and u(_, are correlated. Therefore the conclusion obtained there does not hold.

We consider the asymptotic distribution of p in the simplest case

yt = Ctyt-X + U„ U,= рЩ— + e„ (6.3.31)

where |a|, p < 1, {et) are i. i.d. with Ее, = 0 and Eej = a2, and both {j>,} and {u,} are stationary. Define

£ й’-А

Р = Ч——– > (6-3.32)

where й, = у, — ay,-i and a is the least squares estimator. Consider the limit distribution of •JTp under the assumption p = 0. Because the denominator times T~l converges to a2 in probability, we have asymptotically

№JJ3>

Therefore the asymptotic variance (denoted AV) of ‘ТГр is given by

AV(Vfa – ji Y У f І [ц-л – (1 – c?)y,-iuJ } (6.3.34)

■ ? ?[E (I +“ – “,,!£ (І *-■*)’ (-2 r-2 / J

= «2(1 – Г"1).

Hence, assuming that the asymptotic normality holds, we have

yfTp —»N(0, a2). (6.3.35)

Durbin (1970) obtained the following more general result: Even if higher – order lagged values of y, and purely exogenous variables are contained among the regressors, we have under the assumption p = 0

Vfp ^N[0, 1 – AV(VTo,)], (6.3.36)

where a, is the least squares estimate of the coefficient on y,_,. He proposed

that the test of the independence be based on the asymptotic normality above.3