Advanced Econometrics Takeshi Amemiya

Three Error Components Models

6.6.1 Three error components models are defined by

= л. i= 1, 2,. . . ,N, (6.6.1)

t= 1,2,. . . ,Г, and

ий = а + Л, + €й, (6.6.2)

where Ці and A, are the cross-section-specific and time-specific components mentioned earlier. Assume that the sequence {pi), (A,), and {ей} are i. i.d. random variables with zero mean and are mutually independent with the variances a *, a, and a, respectively. In addition, assume that хй is a Af-vector of known constants, the first element of which is 1 for all i and t.

We shall write (6.6.1) and (6.6.2) in matrix notation by defining several symbols. Define y, u, €, and X as matrices of size NTX 1, NTX 1, NTX 1, and NTXK, respectively, such that their [(г — 1)Г + r]th rows are yit, ии, €й, and хй, respectively. Also define ц = (цх, ц2, • . - ,Цц)’, A = (А,, Aj,. . . , Ar)',L = Ідт ® 1Г, where lris a Г-vector of ones, and 1 = 1^® Ir. Then we can write (6.6.1) and (6.6.2) as

y = X0 + u (6.6.3)

and

u = 1/i + IA + e. (6.6.4)

The covariance matrix (2 = Emi' can be written as (2 = ffjjA + oВ +

where A = LL' and В = IT. Its inverse is given by

П-1 = ~ (Int ~ У. А - y2В + 7з3), (6.6.6)

where у, = al(a + Га?)-1,

У2 = ^К^ + ^я)_1.

Уз = ЬУ№*1 + Таї + + Таї + МаЬ~

and J is an ATX NT njatrix consisting entirely of ones.

In this model the LS estimator of ft is unbiased and generally consistent if both N and T go to °°, but if Q is known, GLS provides a more efficient estimator. Later we shall consider FGLS using certain estimates of the var­iances, but first we shall show that we can obtain an estimator of ft that has the same asymptotic distribution as GLS (as both A and T go to ») but that does not require the estimation of the variances.

To define this estimator, it is useful to separate the first element of fi (the intercept) from its remaining elements. We shall partition fi = (/?0, /?{)' and X = (1, X,).® We shall call this estimator the transformation estimator because it is based on the following transformation, which eliminates the cross-sec­tion - and time-specific components from the errors. Define the NTX NT matrix

It is easy to show that Q is a projection matrix of rank NT —N—T+ 1 that is orthogonal to 1, L, and I. Let H be an NTX (NT — N— T + 1) matrix, the columns of which are the characteristic vectors of Q corresponding to the unit roots. Premultiplying (6.6.3) by H' yields

H'y = H'X1j91+H4 (6.6.8)

which is Model 1 because fH'ee'H = аЦ. The transformation estimator of fii, denoted fiQi, is defined as LS applied to (6.6.8):

fiQ^WQXir'X'iQy - (6.6.9)

The transformation estimator can be interpreted as the LS estimator of, treating Ці, fi2,. . . , fiN and A,, Aj, . . . , Ar as if they were unknown re­gression parameters. Then formula (6.6.9) is merely a special case of the general formula for expressing a subset of the LS estimates given in (1.2.14). This interpretation explains why the estimator is sometimes called the fixed-
effects estimator (since ffs and A’s are treated as fixed effects rather than as random variables) or the dummy-variable regression. Still another name for die estimator is the covariance estimator.

To compare (6.6.9) with the corresponding GLS estimator, we need to derive the corresponding subset of GLS PG. We have

fiGl = [XfO-'X, - Х,1£1-,іа/0“11)",ГП“,Х,]"1 (6.6.10)

X [XJO^y - XjQ-,l(rft_Il)-,l, fl_iy]

= [x;(i - Уіа - 72b + удаг'хіа - у, a - у2в + y4j)u,

where yA = (NTafel — a*)/NTo2 + To2^{o + Noj). Note the similarity be­tween Q and I — — y2B + y4J. The asymptotic equivalence between pQl

and fiai essentially follows from this similarity. If both N and T go to «(it does not matter how they go to °°), it is straightforward to prove that under reason­able assumptions on X and u, VAT0?Q1 — px) and VAT {flGl — fi{) converge to N[0, lim AT(XJQX,)-1]. A proof of the special case of l'X, =0 has been given by Wallace and Hussain (1969).

The GLS estimator of P0 is given by

Similarly, we can define the transformation estimator of Д, by

ъ _Vy - I'xjot NT

We have

and similarly for /Jqo-A>-Note that

/-1 t-1 <-l 1-І

where the probabilistic orders of the three terms in the right-hand side of (6.6.14) are TjN, NfT, and '{NT, respectively. Because the probabilistic order of ГХ,( Ди - Pi) or ТХ,(Д}і - /?,) is -/NT, it does not affect the asymp­totic distribution of Pqo or Ду. Hence, these two estimators have the same asymptotic distribution. To derive their asymptotic distribution, we must

specify whether N от T goes to00 faster. If TV grows faster than T, 'TT (Poo — Д) and JT(Pqo — Po) have the same limit distribution as whereas if Г

grows faster than N, - JN(Pgo ~ Po) and - IN(— Д) have the same limit distribution as 2JL, njJN.

Because of the form of fl_I given in (6.6.6), FGLS can be calculated if we can estimate the three variances a, a*, and a. Several estimators of the variances have been suggested in the literature. (A Monte Carlo comparison of several estimators of the variances and the resulting FGLS has been done by Baltagi, 1981.) Amemiya (1971) proved the asymptotic normality of the fol­lowing so-called analysis-of-variance estimators of the variances:

and

A

where u = у — xpQ. Amemiya also proved that they are asymptotically more efficient than the estimates obtained by using у — X0 for fi, where P is the LS estimator.

These estimates of the variances, or any other estimates with the respective probabilistic order of (NT)~in, N~l/2, and T~l/2, can be inserted into the right-hand side of (6.6.6) for calculating FGLS. Fuller and Battese (1974) proved that under general conditions FGLS and GLS have the same asymp­totic distribution.