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A note on a fixed effect model with arbitrary interpersonal covariance
Journal of Econometrics 22 (1983) 391-393. North-Holland
A NOTE ON A FIXED EFFECT MODEL WITH ARBITRARY INTERPERSONAL COVARIANCE Peter SCHMIDT Michigan State University, East Lansing, MI 48824, USA Received July 1982, final version received December 1982
1. Introduction and summary In this note, we consider a model with fixed individual effects. Usually in such a model the doubly-indexed error term is taken to be i.i.d., and hence uncorrelated across both individuals and time. Here we allow it to have arbitrary correlation across individuals (at a given point in time) by assuming a SUR error structure. The result which is obtained is that the within transformation still works, in the sense that SUR estimation with individual dummies is identical to SUR estimation on the within-transformed data. This result is not true for an arbitrary covariance structure, but it holds for the SUR error structure as well as for the i.i.d. case.
2. Statement of the model The model is of the form Yit =XitP+Ui+W~ty
i=l ,..., N,
t=l,...,
i?
(1)
Here i indexes individuals and t indexes time. The ui are treated as fixed. The model can be written in matrix form as y=xp+u+w,
(2)
with the NT observations in the usual order (T observations for individual 1, etc.). The simplest and most common assumption about the wtt is that they are i.i.d. N(0, o2), so that cov(w) = a’I,r. Instead, here we assume that the 0304-4076/83/$3.00 0 1983, Elsevier Science Publishers B.V. (North-Holland)
P. Schmidt, A jxed
392
effect model
vectors
w,=(w1t,. . * w,,)’ 2
are i.i.d. as N(0, z). This amounts to treating (1) as a set of N seemingly unrelated regressions, with identical coefficients but different intercepts. This is a feasible and plausible specification for cases in which N is small but T is large. (This is the opposite of the commonly assumed panel case in which N is large but T is small. It is nevertheless a case sometimes encountered in empirical work.) When N is small, it is obvious to model the individual effects as fixed, since consistent estimation of the variance of random individual effects requires large N. Furthermore, when N is small and T is large, it is feasible to estimate the N x N covariance matrix ,J5in unrestricted form. This seems preferable to accounting for correlations across individuals by introducing random time effects, since it is more general; these correlations need not be the same for all possible sets of individuals. There is some similarity between this model and the model of Kiefer (1980), who allowed for an arbitrary intertemporal covariance matrix. However, the difficulties encountered in Kiefer’s model do not arise in this model, since with N fixed there is no ‘incidental parameters problem’ [see also Chamberlain (1980)]. The result presented below does not hold in Kiefer’s model. However, by the symmetry of the model it does hold for the case of arbitrary intertemporal covariance, with large N and small rl: if we have fixed time effects rather than fixed individual effects.
3. The result We rewrite the model (2) as
y=xp+zy+w,
(3)
where y=(u,,u,,..., uN)‘, and Z=Z, @ i, is a matrix of dummies (i, being a T x 1 vector of ones). The covariance matrix of w is SzE CO I,. The SUR (GLS) estimate, assuming known C, is
[IB
B = [(X, Z)‘G?_‘(X, Z)] -
yx, Z)'K'y,
which, for the parameters p, is the same as
B= (X'QJ)
- 'X'Q,y,
(5)
P. Schmidt, A fixed
eflect model
393
with Q,=52-‘-0-‘Z(2’52-‘Z)-‘Z’SZ-‘.
(6)
This is a general result, true for any Z and !2. However, with Q=E 0 1, and Z = I, @ i,, it simplifies considerably:
Q*=C-'QA,
(7)
where A = IT-(l/T)i& is the idempotent T x T matrix which converts a T x 1 vector to deviations from means. Thus we can also write Q+, as Q, = (I, 0 A)@ - 10 I,)(I,
0 A),
(8)
where I, @ A is the idempotent NT x NT matrix which performs the within transformation. Inserting this in (5), it is apparent that the SUR estimator with dummies, j?, can also be calculated by applying the within transformation to the data, and then applying the usual SUR estimator, still treating 52 as C 0 I,. 4. Concluding remarks This result may seem very unsurprising. If so, consider that it is not a general result. In (3), the Z variables can always be eliminated by transforming by M= I- Z(Z’Z)-‘Z’, and then GLS using Sz can be applied to the transformed data. This would yield ~=W’-&+~W’X’Q,,Y>
(9)
where Q,, = M!2- ‘M. For the special case considered here (Z=I,@ iT, and so the two procedures are Q=C@Z,), it happens that Q,,=Q, identical. But this is not so for general Z and 51. It is not even so in the somewhat similar case of arbitrary intertemporal covariance (Z = I, 0 iT, Sz = I, Q C) considered by Kiefer (1980, p. 199).
References Chamberlain, G., 1980, Analysis of covariance with qualitative data, Review of Economic Studies 47, 225-238. Kiefer, N.M., 1980, Estimation of fixed effect models for time series of cross-sectioks with arbitrary intertemporal covariance, Journal of Econometrics 14, 195-202.
A NOTE ON A FIXED EFFECT MODEL WITH ARBITRARY INTERPERSONAL COVARIANCE Peter SCHMIDT Michigan State University, East Lansing, MI 48824, USA Received July 1982, final version received December 1982
1. Introduction and summary In this note, we consider a model with fixed individual effects. Usually in such a model the doubly-indexed error term is taken to be i.i.d., and hence uncorrelated across both individuals and time. Here we allow it to have arbitrary correlation across individuals (at a given point in time) by assuming a SUR error structure. The result which is obtained is that the within transformation still works, in the sense that SUR estimation with individual dummies is identical to SUR estimation on the within-transformed data. This result is not true for an arbitrary covariance structure, but it holds for the SUR error structure as well as for the i.i.d. case.
2. Statement of the model The model is of the form Yit =XitP+Ui+W~ty
i=l ,..., N,
t=l,...,
i?
(1)
Here i indexes individuals and t indexes time. The ui are treated as fixed. The model can be written in matrix form as y=xp+u+w,
(2)
with the NT observations in the usual order (T observations for individual 1, etc.). The simplest and most common assumption about the wtt is that they are i.i.d. N(0, o2), so that cov(w) = a’I,r. Instead, here we assume that the 0304-4076/83/$3.00 0 1983, Elsevier Science Publishers B.V. (North-Holland)
P. Schmidt, A jxed
392
effect model
vectors
w,=(w1t,. . * w,,)’ 2
are i.i.d. as N(0, z). This amounts to treating (1) as a set of N seemingly unrelated regressions, with identical coefficients but different intercepts. This is a feasible and plausible specification for cases in which N is small but T is large. (This is the opposite of the commonly assumed panel case in which N is large but T is small. It is nevertheless a case sometimes encountered in empirical work.) When N is small, it is obvious to model the individual effects as fixed, since consistent estimation of the variance of random individual effects requires large N. Furthermore, when N is small and T is large, it is feasible to estimate the N x N covariance matrix ,J5in unrestricted form. This seems preferable to accounting for correlations across individuals by introducing random time effects, since it is more general; these correlations need not be the same for all possible sets of individuals. There is some similarity between this model and the model of Kiefer (1980), who allowed for an arbitrary intertemporal covariance matrix. However, the difficulties encountered in Kiefer’s model do not arise in this model, since with N fixed there is no ‘incidental parameters problem’ [see also Chamberlain (1980)]. The result presented below does not hold in Kiefer’s model. However, by the symmetry of the model it does hold for the case of arbitrary intertemporal covariance, with large N and small rl: if we have fixed time effects rather than fixed individual effects.
3. The result We rewrite the model (2) as
y=xp+zy+w,
(3)
where y=(u,,u,,..., uN)‘, and Z=Z, @ i, is a matrix of dummies (i, being a T x 1 vector of ones). The covariance matrix of w is SzE CO I,. The SUR (GLS) estimate, assuming known C, is
[IB
B = [(X, Z)‘G?_‘(X, Z)] -
yx, Z)'K'y,
which, for the parameters p, is the same as
B= (X'QJ)
- 'X'Q,y,
(5)
P. Schmidt, A fixed
eflect model
393
with Q,=52-‘-0-‘Z(2’52-‘Z)-‘Z’SZ-‘.
(6)
This is a general result, true for any Z and !2. However, with Q=E 0 1, and Z = I, @ i,, it simplifies considerably:
Q*=C-'QA,
(7)
where A = IT-(l/T)i& is the idempotent T x T matrix which converts a T x 1 vector to deviations from means. Thus we can also write Q+, as Q, = (I, 0 A)@ - 10 I,)(I,
0 A),
(8)
where I, @ A is the idempotent NT x NT matrix which performs the within transformation. Inserting this in (5), it is apparent that the SUR estimator with dummies, j?, can also be calculated by applying the within transformation to the data, and then applying the usual SUR estimator, still treating 52 as C 0 I,. 4. Concluding remarks This result may seem very unsurprising. If so, consider that it is not a general result. In (3), the Z variables can always be eliminated by transforming by M= I- Z(Z’Z)-‘Z’, and then GLS using Sz can be applied to the transformed data. This would yield ~=W’-&+~W’X’Q,,Y>
(9)
where Q,, = M!2- ‘M. For the special case considered here (Z=I,@ iT, and so the two procedures are Q=C@Z,), it happens that Q,,=Q, identical. But this is not so for general Z and 51. It is not even so in the somewhat similar case of arbitrary intertemporal covariance (Z = I, 0 iT, Sz = I, Q C) considered by Kiefer (1980, p. 199).
References Chamberlain, G., 1980, Analysis of covariance with qualitative data, Review of Economic Studies 47, 225-238. Kiefer, N.M., 1980, Estimation of fixed effect models for time series of cross-sectioks with arbitrary intertemporal covariance, Journal of Econometrics 14, 195-202.