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On a simple procedure for testing non-nested regression models
Economics Letters 5 (1980) 0 North-Holland Publishing
45-48 Company
ON A SIMPLE PROCEDURE FOR TESTING NON-NESTED REGRESSION MODELS Russell DAVIDSON and James G. MacKINNON Queen’s University, Kingston, Ont., Canada K7L 3N6 Received
25 April 1980
In this note we discuss some aspects of a procedure nested regression models. It is argued that, although this arbitrariness is inconsequentiaL
which we recently proposed for testing nonthe procedure is arbitrary to some degree,
1. Introduction
Several procedures for non-nested hypothesis testing have been proposed in recent years; see in particular Pesaran (1974) Pesaran and Deaton (1978), and Davidson and MacKinnon (1980). Suppose that one wishes to test the validity of the linear regression model H,:y=xfl+u~,
uo -NO,
4)
(1)
against the evidence provided by an alternative Hi:y=Zy+ui)
Ul
-NO,
4)
>
model (2)
where y is a vector of observations on a dependent variable, X and Z are matrices of observations on independent variables, and p and y are vectors of parameters to be estimated. It is assumed that X does not lie in the space spanned by the columns of Z and vice versa, so that Ho and Hi are indeed non-nested hypotheses. The simplest method for testing the truth of Ho against HI is the J-test of Davidson and MacKinnon (1980). One simply has to estimate the linear regression y=x(3*+cYZ;/+u,
(3)
where T denotes the OLS estimates of y from (2) and /3* = (1 - a) 0. The test statistic is then the t-statistic on OL,which is asymptotically N(O,l) if Ho is true. This assertion is proved in Davidson and MacKinnon (1980). In that paper a simple test procedure is also proposed for the case where Ho and HI are non-linear regression models, and these tests are shown to be asymptotically equivalent, under Ho, to the procedures of Pesaran (1974) and Pesaran and Deaton (1978). 45
R. Davidson, J.C. MacKinnon /Procedure
46
for testing non-nested regression models
It may be thought that this J-test procedure is rather arbitrary. First of all, it is clear that if regression (3) provides a valid test of Ho, then so will any other regression with T replaced by some other estimate of y that is asymptotically uncorrelated with uc. Secondly, it may be thought that the linear artificial nesting used in (3) could be replaced by some other scheme, such as exponential nesting. In this note we shall argue that, while theJ-test is indeed arbitrary, there is no reason to believe that any alternative procedure would perform any better.
2. A family of tests Consider the regression y=xp*+cuh+u,
(4)
where h is any vector whatsoever, stochastic or non-stochastic, subject only to the requirement that it should be asymptotically uncorrelated with U. It is clear that the t-statistic for ar from (4) provides an asymptotically valid test for the truth of Ho. It is easily derived that the numerator of this statistic, which is all we need concern ourselves with here, is simply hTMoV >
(5)
where MO = I - X(X’X)-’ XT. Expression (5) is the most general statistic which is linear iny that could reasonably be used to test the truth of Ho. Consider the apparently more general statistic aTy. We require that, under Ho, plim aTy = 0. Since Ho tells us that y = X/3 + uo, but we do not know p, we can only ensure that this requirement is satisfied by settinga=Moh,forsomeh. We may now prove Theorem 1. Under the compound hypothesis HO, as given in (l), and CO~~OUFUL’ because /3is not specified, the statistic out of the class defined by (5) which yields the greatest power asymptotically against the simple alternative hypothesis HI, with y = yl, is any statistic which is asymptotically proportional to yrZTM~ y. Proofi
If HI is true, then
hTMo y = hTMoZT1 + hTMouI
(6)
Greatest power is achieved when the probability that the right-hand side of (6) is in a neighborhood of zero is minimized. For this we wish to minimize the variance of the ratio of the stochastic to the non-stochastic part of (6) that is, to minimize h TMo h/(h TMoZ~,)’
.
Since expression (7) is homogenous
(7) of degree zero in h, we may normalize h by
R. Davidson, J.C. MacKinnon
/Procedure
for testing non-nested regression models
setting hl’MOh = 1, and then maximize the denominator tion. This yields the first-order condition
41
subject to the normaliza-
where X is the Lagrange multiplier on the normalization constraint. It follows immediately from (8) that for optimum power against Hi, h should be proportional to Zyi. The fact that h does not actually have to equal Zy, is simply a consequence of the well-known result that multiplying any regressor by a constant has no effect on its t-statistic. Q.E.D. Against a compound alternative hypothesis Hi with y unspecified, a result as strong as Theorem 1 is not available. But the step of replacing y by its OLS estimate T is clearly the most straightforward way to proceed, and it is difficult to think of a convincing argument for any alternative procedure. Thus Theorem 1 certainly suggests that the J-test may be expected to have optimal power against a true alternative Hi, among a class of similar tests. It also indicates clearly why the J-test is often able to reject a false hypothesis even when the alternative is also false, although it will of course lack power in that case.
3. Alternative nesting procedures In place of the linear nesting scheme of (3) one might consider the exponential scheme Yt = (xtcl>‘-“G~)”
+ Ut ,
(9)
where subscripts have been added so that we do not have to invent extensions to matrix notation. It would of course be possible to estimate.(g) by non-linear methods, but that possibly difficult task can be avoided by linearizing (9) around /3 = /? and (Y= 0, which yields the linear regression Yt = xts + 4og@t=u> - h%wtL9~ XtB + Ut *
(10)
It is well-known that, if CY= 0, a test of CY= 0 based on (10) will be asymptotically equivalent to one based on (9); see Durbin (1970). Now (10) and (3) are clearly not identical. But the exponential weighting scheme only makes sense if X,p and Z,? are positive for all observations, which can only be guaranteed in advance ify, is always positive and large relative to ut, in which case X,fi/Z,T will tend to be near unity. If so, then a Taylor expansion gives U%GT)
- l%GwN
XtP = m-
XtB ,
so that (10) becomes Yt = XtP +
a3
- xtB> + 4
1
(11)
48
R. Davidson, J.C. MacKinnon /Procedure
for testing non-nested regression models
which must yield identical results to (3), because X,0 is simply a linear combination of the regressors in X,. Thus it seems very likely that (10) and (3) will yield extremely similar results in practice. In view of this, there seems no reason to prefer the more complicated exponential nesting scheme to the simple linear scheme used in the J-test. Other even more complicated schemes, such as a mean of order r (or C.E.S.) formulation, could of course be used, and linear regressions similar to (10) could be derived for them as well. But it seems reasonable to conjecture that, when X,fl/Z,T is near unity, such schemes will also yield results quite similar to (3).
4. Conclusion The J-test is a simple and effective procedure for testing non-nested regression models. Although some features of the test may appear arbitrary, there is no reason to believe that alternative and equally arbitrary tests would perform any better.
References Davidson, Russell and James G. MacKinnon, 1980, Several tests for model specification in the presence of alternative hypotheses, Queen’s University Institute for Economic Research Discussion Paper no. 378’, Econometrica, forthcoming. Durbin, James, 1970, Testing for serial correlation in least-squares regression when some of the regressors are lagged dependent variables, Econometrica 38, 410-421. Pesaran, M.H., 1974, On the general problem of model selection, Review of Economic Studies 41,153-171. Pesaran, M.H. and Angus S. Deaton, 1978, Testing non-nested nonlinear regression models, Econometrica 46, 677-694.
45-48 Company
ON A SIMPLE PROCEDURE FOR TESTING NON-NESTED REGRESSION MODELS Russell DAVIDSON and James G. MacKINNON Queen’s University, Kingston, Ont., Canada K7L 3N6 Received
25 April 1980
In this note we discuss some aspects of a procedure nested regression models. It is argued that, although this arbitrariness is inconsequentiaL
which we recently proposed for testing nonthe procedure is arbitrary to some degree,
1. Introduction
Several procedures for non-nested hypothesis testing have been proposed in recent years; see in particular Pesaran (1974) Pesaran and Deaton (1978), and Davidson and MacKinnon (1980). Suppose that one wishes to test the validity of the linear regression model H,:y=xfl+u~,
uo -NO,
4)
(1)
against the evidence provided by an alternative Hi:y=Zy+ui)
Ul
-NO,
4)
>
model (2)
where y is a vector of observations on a dependent variable, X and Z are matrices of observations on independent variables, and p and y are vectors of parameters to be estimated. It is assumed that X does not lie in the space spanned by the columns of Z and vice versa, so that Ho and Hi are indeed non-nested hypotheses. The simplest method for testing the truth of Ho against HI is the J-test of Davidson and MacKinnon (1980). One simply has to estimate the linear regression y=x(3*+cYZ;/+u,
(3)
where T denotes the OLS estimates of y from (2) and /3* = (1 - a) 0. The test statistic is then the t-statistic on OL,which is asymptotically N(O,l) if Ho is true. This assertion is proved in Davidson and MacKinnon (1980). In that paper a simple test procedure is also proposed for the case where Ho and HI are non-linear regression models, and these tests are shown to be asymptotically equivalent, under Ho, to the procedures of Pesaran (1974) and Pesaran and Deaton (1978). 45
R. Davidson, J.C. MacKinnon /Procedure
46
for testing non-nested regression models
It may be thought that this J-test procedure is rather arbitrary. First of all, it is clear that if regression (3) provides a valid test of Ho, then so will any other regression with T replaced by some other estimate of y that is asymptotically uncorrelated with uc. Secondly, it may be thought that the linear artificial nesting used in (3) could be replaced by some other scheme, such as exponential nesting. In this note we shall argue that, while theJ-test is indeed arbitrary, there is no reason to believe that any alternative procedure would perform any better.
2. A family of tests Consider the regression y=xp*+cuh+u,
(4)
where h is any vector whatsoever, stochastic or non-stochastic, subject only to the requirement that it should be asymptotically uncorrelated with U. It is clear that the t-statistic for ar from (4) provides an asymptotically valid test for the truth of Ho. It is easily derived that the numerator of this statistic, which is all we need concern ourselves with here, is simply hTMoV >
(5)
where MO = I - X(X’X)-’ XT. Expression (5) is the most general statistic which is linear iny that could reasonably be used to test the truth of Ho. Consider the apparently more general statistic aTy. We require that, under Ho, plim aTy = 0. Since Ho tells us that y = X/3 + uo, but we do not know p, we can only ensure that this requirement is satisfied by settinga=Moh,forsomeh. We may now prove Theorem 1. Under the compound hypothesis HO, as given in (l), and CO~~OUFUL’ because /3is not specified, the statistic out of the class defined by (5) which yields the greatest power asymptotically against the simple alternative hypothesis HI, with y = yl, is any statistic which is asymptotically proportional to yrZTM~ y. Proofi
If HI is true, then
hTMo y = hTMoZT1 + hTMouI
(6)
Greatest power is achieved when the probability that the right-hand side of (6) is in a neighborhood of zero is minimized. For this we wish to minimize the variance of the ratio of the stochastic to the non-stochastic part of (6) that is, to minimize h TMo h/(h TMoZ~,)’
.
Since expression (7) is homogenous
(7) of degree zero in h, we may normalize h by
R. Davidson, J.C. MacKinnon
/Procedure
for testing non-nested regression models
setting hl’MOh = 1, and then maximize the denominator tion. This yields the first-order condition
41
subject to the normaliza-
where X is the Lagrange multiplier on the normalization constraint. It follows immediately from (8) that for optimum power against Hi, h should be proportional to Zyi. The fact that h does not actually have to equal Zy, is simply a consequence of the well-known result that multiplying any regressor by a constant has no effect on its t-statistic. Q.E.D. Against a compound alternative hypothesis Hi with y unspecified, a result as strong as Theorem 1 is not available. But the step of replacing y by its OLS estimate T is clearly the most straightforward way to proceed, and it is difficult to think of a convincing argument for any alternative procedure. Thus Theorem 1 certainly suggests that the J-test may be expected to have optimal power against a true alternative Hi, among a class of similar tests. It also indicates clearly why the J-test is often able to reject a false hypothesis even when the alternative is also false, although it will of course lack power in that case.
3. Alternative nesting procedures In place of the linear nesting scheme of (3) one might consider the exponential scheme Yt = (xtcl>‘-“G~)”
+ Ut ,
(9)
where subscripts have been added so that we do not have to invent extensions to matrix notation. It would of course be possible to estimate.(g) by non-linear methods, but that possibly difficult task can be avoided by linearizing (9) around /3 = /? and (Y= 0, which yields the linear regression Yt = xts + 4og@t=u> - h%wtL9~ XtB + Ut *
(10)
It is well-known that, if CY= 0, a test of CY= 0 based on (10) will be asymptotically equivalent to one based on (9); see Durbin (1970). Now (10) and (3) are clearly not identical. But the exponential weighting scheme only makes sense if X,p and Z,? are positive for all observations, which can only be guaranteed in advance ify, is always positive and large relative to ut, in which case X,fi/Z,T will tend to be near unity. If so, then a Taylor expansion gives U%GT)
- l%GwN
XtP = m-
XtB ,
so that (10) becomes Yt = XtP +
a3
- xtB> + 4
1
(11)
48
R. Davidson, J.C. MacKinnon /Procedure
for testing non-nested regression models
which must yield identical results to (3), because X,0 is simply a linear combination of the regressors in X,. Thus it seems very likely that (10) and (3) will yield extremely similar results in practice. In view of this, there seems no reason to prefer the more complicated exponential nesting scheme to the simple linear scheme used in the J-test. Other even more complicated schemes, such as a mean of order r (or C.E.S.) formulation, could of course be used, and linear regressions similar to (10) could be derived for them as well. But it seems reasonable to conjecture that, when X,fl/Z,T is near unity, such schemes will also yield results quite similar to (3).
4. Conclusion The J-test is a simple and effective procedure for testing non-nested regression models. Although some features of the test may appear arbitrary, there is no reason to believe that alternative and equally arbitrary tests would perform any better.
References Davidson, Russell and James G. MacKinnon, 1980, Several tests for model specification in the presence of alternative hypotheses, Queen’s University Institute for Economic Research Discussion Paper no. 378’, Econometrica, forthcoming. Durbin, James, 1970, Testing for serial correlation in least-squares regression when some of the regressors are lagged dependent variables, Econometrica 38, 410-421. Pesaran, M.H., 1974, On the general problem of model selection, Review of Economic Studies 41,153-171. Pesaran, M.H. and Angus S. Deaton, 1978, Testing non-nested nonlinear regression models, Econometrica 46, 677-694.