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A Hausman test with trending data
Economics Letters 19 (1985) 323-325 North-Holland
A HAUSMAN Walter
323
TEST WITH TRENDING DATA
KtiMER
Vienna Institute for Advanced Studies and Scientific Research, A 1060 Vienna, Austrra
Received
3 April 1985
A simple Hausman test for errors in variables asymptotic null distribution and the consistency
is investigated when there is a trend in the data. It is shown of the test remain intact in this non-standard situation.
that both the
1. Introduction Let /‘? and fi be two estimates, based on a sample of size T, of the parameter vector from some econometric model. For concreteness and simplicity, take this to be the standard linear regression 1 model y = Xp + U, and let ,6 be the OLS and fl some IV estimate. If, under correct specification,
where 4 = fi - 8, a consistent asymptotic specification test can then be based provided plim b # plim a under the respective alternative. The test statistic m=
Tq’P(a)-4,
on this difference,
(2)
where c(Q)- is a consistent estimate for some generalized inverse of P’(g), has an asymptotic x2 distribution under H,, with degrees of freedom equal to the rank of V(g), and will tend to infinity in probability otherwise [see, e.g., Hausman and Taylor (1981)]. Now consider the case of trending data, which by (1) are ruled out in the standard analysis. As shown in, e.g., Krgmer (1984,1985b) for the errors-in-variables and simultaneous equation alternatives, trending data require some second thoughts concerning the asymptotic normality of the estimates, and can induce identical limiting distribution for fi and B, after proper normalization, both under H, and the alternative. This raises the following questions on the large sample asymptotics of the Hausman test: (i) will the test statistic under H,, still have an asymptotic x2 distribution when there is a trend in the data. and (ii) can we hope that the Hausman test will remain consistent in the face of an identical limiting distribution, under the alternative, of B and j?? The following section addresses these issues for a simple Hausman test for errors in variables, with the reassuring result that both questions can be answered in the affirmative. 0165-1765/85/$3.30
0 1985. Elsevier Science Publishers
B.V. (North-Holland)
2. Errors in variables
To avoid unrelated complications to the single regressor model vi =
px,-t u,,
that arise from the possible
singularity
t = l,....T.
where, under H,,, u, is iid(0, u,‘) and E( x,u,) = 0 Vt. s. The alternative error. i.e., that the .v,‘s are generated by )‘,
=
Pz, = c,,
of V’(q), I confine
t=l
,
, T,
myself
(3)
is that x, is measured
with
(4)
where z, = x, - 11, is unobservable and the measurement errors u, are iid(O, a,:). with E( u,~,) = 0. A convenient IV estimate p is based on lagged values of x,. i.e.. ,&’= C.u,~ ,.v,/~.x,~ ,x,. If 0 < m,, = plim Z’Z/T 7-L-E
< cc,
(5)
normal under and 0 < C= plim._,Cz,z,~ ,/T < 00, both fi and b are consistent and asymptotically and therefore, by the fundamental Hausman lemma. H,,, with V(b) = u,‘/m,,, V( fi) = u,fm,=/C’, V( Lj) = V( fi) - V(b). Under the alternative, plim j = p. whereas plim fi = /3 - fiu,?/( m,, + a,‘) f ,8. so clearly plim 4 f 0 and m 4 00 [provided only that l/p(q) remains O,,(l) also under the alternative, which is not a very strong requirement]. Now consider the case where there is a trend in z,. For simplicity, let z, = t. Then
Z’Z=
ft’=T(T+l)(2T+l)/h=O(T’),
(6)
1=1
so (5) clearly can no longer hold. By Theorem 2 in Kramer (1985b), both 4 T’(fi - p) and now have an asymptotic N(0, 30,:) distribution, whether there is measurement error or 4 T3(&P) not. It is easily seen that this does not affect the asymptotic xf,, null distribution of the test statistic, provided V( Lj) is estimated as usual by
(7) where 6’ is an arbitrary consistent estimate of I$. This is trivially so when the u,‘s are normal (since the term in braces is just the variance of 4) and can be inferred from Theorem 1 in Kramer (1985a) otherwise. More generally, write Lj as 4 = Wy with a (K x T) matrix W. By Theorem 1 in Kramer (1985a), a sufficient condition for (i to be asymptotically normal is that the maximum diagonal element of W( W’W)- W’ tends to zero in probability as T --) co. This will always hold in standard cases, but also for many types of trended data. Exceptions are exponential types of trend, where one can show by simple counterexamples (take for instance z, = 2’) that Lj will no longer be asymptotically normal, after proper normalization, unless the ~4,‘s themselves are normal. In such cases there is no hope to establish an asymptotic x2 distribution for the test statistic m.
After several lines of straightforward written as P(4)=6’37-‘(T+
l)(T-
l)/Cr”(Ct(r-
but tedious algebra,
l))?,
(7) can in the present
simple case also be
(8)
which is 0,,(T~m4) and thus of smaller order than 6’(p) and 6’(B), which are both O,,(T -‘). This implies that in genera1 V( 0) f V(p) - V(B) with trending data, when the same normalizing factors are used. Now consider the power of the test under the alternative. If C-J,! > 0, it is easily checked that 4 = -
&q&x”
+ O,,(T-‘),
where T’Cu,u,/Cx,? z - 3pu,? > 0. Since k( 4) is still as in (8). plus terms which are O,,( T-4). we thus have m 3 cc and a consistent test. This is so despite the asymptotic equivalence of the two estimators, and follows from the rapid convergence to zero of P(G), which outpaces the squared difference in the estimates.
References Hausman, J.A. and W.E. Taylor. 1981. A generalized specification test, Economics Letters 8. 239-245. KrBmer, W., 1984. On the consequences of trend for simultaneous equation estimation. Economics Letters 14 23-30 KrBmer. W., 1985a. Asymptotic normality of ordinary least squares estimates in the linear regrewon model. Method5 of Operations Research, forthcoming. Krtimer. W., 1985b. Asymptotic distributions of some estimators when there are errors in the varlsbles and trending data (in German), Metrika. forthcoming.
A HAUSMAN Walter
323
TEST WITH TRENDING DATA
KtiMER
Vienna Institute for Advanced Studies and Scientific Research, A 1060 Vienna, Austrra
Received
3 April 1985
A simple Hausman test for errors in variables asymptotic null distribution and the consistency
is investigated when there is a trend in the data. It is shown of the test remain intact in this non-standard situation.
that both the
1. Introduction Let /‘? and fi be two estimates, based on a sample of size T, of the parameter vector from some econometric model. For concreteness and simplicity, take this to be the standard linear regression 1 model y = Xp + U, and let ,6 be the OLS and fl some IV estimate. If, under correct specification,
where 4 = fi - 8, a consistent asymptotic specification test can then be based provided plim b # plim a under the respective alternative. The test statistic m=
Tq’P(a)-4,
on this difference,
(2)
where c(Q)- is a consistent estimate for some generalized inverse of P’(g), has an asymptotic x2 distribution under H,, with degrees of freedom equal to the rank of V(g), and will tend to infinity in probability otherwise [see, e.g., Hausman and Taylor (1981)]. Now consider the case of trending data, which by (1) are ruled out in the standard analysis. As shown in, e.g., Krgmer (1984,1985b) for the errors-in-variables and simultaneous equation alternatives, trending data require some second thoughts concerning the asymptotic normality of the estimates, and can induce identical limiting distribution for fi and B, after proper normalization, both under H, and the alternative. This raises the following questions on the large sample asymptotics of the Hausman test: (i) will the test statistic under H,, still have an asymptotic x2 distribution when there is a trend in the data. and (ii) can we hope that the Hausman test will remain consistent in the face of an identical limiting distribution, under the alternative, of B and j?? The following section addresses these issues for a simple Hausman test for errors in variables, with the reassuring result that both questions can be answered in the affirmative. 0165-1765/85/$3.30
0 1985. Elsevier Science Publishers
B.V. (North-Holland)
2. Errors in variables
To avoid unrelated complications to the single regressor model vi =
px,-t u,,
that arise from the possible
singularity
t = l,....T.
where, under H,,, u, is iid(0, u,‘) and E( x,u,) = 0 Vt. s. The alternative error. i.e., that the .v,‘s are generated by )‘,
=
Pz, = c,,
of V’(q), I confine
t=l
,
, T,
myself
(3)
is that x, is measured
with
(4)
where z, = x, - 11, is unobservable and the measurement errors u, are iid(O, a,:). with E( u,~,) = 0. A convenient IV estimate p is based on lagged values of x,. i.e.. ,&’= C.u,~ ,.v,/~.x,~ ,x,. If 0 < m,, = plim Z’Z/T 7-L-E
< cc,
(5)
normal under and 0 < C= plim._,Cz,z,~ ,/T < 00, both fi and b are consistent and asymptotically and therefore, by the fundamental Hausman lemma. H,,, with V(b) = u,‘/m,,, V( fi) = u,fm,=/C’, V( Lj) = V( fi) - V(b). Under the alternative, plim j = p. whereas plim fi = /3 - fiu,?/( m,, + a,‘) f ,8. so clearly plim 4 f 0 and m 4 00 [provided only that l/p(q) remains O,,(l) also under the alternative, which is not a very strong requirement]. Now consider the case where there is a trend in z,. For simplicity, let z, = t. Then
Z’Z=
ft’=T(T+l)(2T+l)/h=O(T’),
(6)
1=1
so (5) clearly can no longer hold. By Theorem 2 in Kramer (1985b), both 4 T’(fi - p) and now have an asymptotic N(0, 30,:) distribution, whether there is measurement error or 4 T3(&P) not. It is easily seen that this does not affect the asymptotic xf,, null distribution of the test statistic, provided V( Lj) is estimated as usual by
(7) where 6’ is an arbitrary consistent estimate of I$. This is trivially so when the u,‘s are normal (since the term in braces is just the variance of 4) and can be inferred from Theorem 1 in Kramer (1985a) otherwise. More generally, write Lj as 4 = Wy with a (K x T) matrix W. By Theorem 1 in Kramer (1985a), a sufficient condition for (i to be asymptotically normal is that the maximum diagonal element of W( W’W)- W’ tends to zero in probability as T --) co. This will always hold in standard cases, but also for many types of trended data. Exceptions are exponential types of trend, where one can show by simple counterexamples (take for instance z, = 2’) that Lj will no longer be asymptotically normal, after proper normalization, unless the ~4,‘s themselves are normal. In such cases there is no hope to establish an asymptotic x2 distribution for the test statistic m.
After several lines of straightforward written as P(4)=6’37-‘(T+
l)(T-
l)/Cr”(Ct(r-
but tedious algebra,
l))?,
(7) can in the present
simple case also be
(8)
which is 0,,(T~m4) and thus of smaller order than 6’(p) and 6’(B), which are both O,,(T -‘). This implies that in genera1 V( 0) f V(p) - V(B) with trending data, when the same normalizing factors are used. Now consider the power of the test under the alternative. If C-J,! > 0, it is easily checked that 4 = -
&q&x”
+ O,,(T-‘),
where T’Cu,u,/Cx,? z - 3pu,? > 0. Since k( 4) is still as in (8). plus terms which are O,,( T-4). we thus have m 3 cc and a consistent test. This is so despite the asymptotic equivalence of the two estimators, and follows from the rapid convergence to zero of P(G), which outpaces the squared difference in the estimates.
References Hausman, J.A. and W.E. Taylor. 1981. A generalized specification test, Economics Letters 8. 239-245. KrBmer, W., 1984. On the consequences of trend for simultaneous equation estimation. Economics Letters 14 23-30 KrBmer. W., 1985a. Asymptotic normality of ordinary least squares estimates in the linear regrewon model. Method5 of Operations Research, forthcoming. Krtimer. W., 1985b. Asymptotic distributions of some estimators when there are errors in the varlsbles and trending data (in German), Metrika. forthcoming.