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 : ~ 1 e 
 
 FACULTY WORKING 
 PAPER NO. 1123 
 
 
 n Exact and Asymptotic iests of Non-nested Models 
 
 MHii a. Bera 
 Michael McAleer 
 
 College o* Co^mercci and 3usiness Administrai 
 Bureau of fEcorom«c arc Business Researcn 
 University of iijincis. Urbana-Cnampaign 
 
BEBR 
 
 FACULTY WORKING PAPER NO. 1128 
 College of Commerce and Business Administration 
 University of Illinois at Urb ana- Champaign 
 March, 1985 
 
 On Exact and Asymptotic Tests of Non-nested Models 
 
 Anil K. 3era, Assistant Professor 
 Department of Economics 
 
 Michael McAleer 
 Australian National University 
 
 This volume Is bound without ' ' (^3 ' 
 
 which is/are unavailable, 
 
 12-1-at 
 
On Exact and Asymptotic Tests of Non-nested Models 
 
 Anil K. Bera 
 Department of Economics 
 University of Illinois 
 
 Michael McAleer 
 Department of Statistics 
 Australian National University 
 
 Revised: March 1985 
 
 Abstract 
 
 This paper is concerned with alternative exact and asymptotic tests of 
 non-nested regression models. The relations between some tests are 
 developed to indicate how exact test statistics may be calculated from com- 
 puted asymptotic test statistics. The necessary and sufficient conditions 
 for exact tests to achieve maximum power against local alternatives are 
 presented. As an illustration of these conditions, it is shown that exact 
 tests may be applicable even when instrumental variable estimation is used, 
 and that not all exact tests attain maximum local power. The conditions 
 under which tests based on artificial linear regressions are asymptotically 
 equivelant to a Cox test of the null model are also examined. 
 
 Keywords: Asymptotic equivalence, Cox test, finite sample distribution, 
 maximum local power, non-nested regression models. 
 
1 . Introduct ion 
 
 In recent years substantial contributions have been made in developing 
 tests for non-nested regression models; see e.g., Pesaran and Deaton 
 (1978), Davidson and MacKinnon (1981), Fisher and McAleer (1981) and 
 Oastoor (1983). However, the theoretical advances have not led to many 
 empirical applications. The reticence on the part of the econometric pro- 
 fession to use the available tests may stem, in part, from two related fac- 
 tors. First, the relationship between exact and asymptotic tests is not 
 always transparent, and it is sometimes unclear when exact tests may be 
 applied. Since virtually all existing tests have been based upon the 
 modified likelihood ratio of Cox (1961, 1962), they are valid only 
 asymptotically and little is yet known of their small sample proper- 
 ties. As the actual and nominal type I errors for asymptotic tests 
 may differ substantially when the sample size is small, whereas exact 
 tests have known finite sample null distributions, it is important to 
 examine the conditions under which exact tests apply. Second, dif- 
 ferent tests, whether exact or not, may be asymptotically equivalent 
 under the null hypothesis but not under the alternative. Furthermore, 
 they may have different powers against local alternatives and it is 
 not known, in general, which exact tests maximize local power. 
 
 The purpose of this paper is to investigate the issues addressed above. 
 The plan of the paper is as follows. Finite sample relations among alter- 
 native asymptotic tests are given in Section 2. The relations between some 
 exact and asymptotic tests are developed in Section 3, and it is shown how 
 exact test statistics may be calculated from computed asymptotic test sta- 
 tistics. Necessary and sufficient conditions for exact tests to achieve 
 maximum power against local alternatives are presented in Section 4 in an 
 attempt to determine an optimal class of tests procedures. An example is 
 
given to show Chat exact tests are available even when instrumental vari- 
 able estimation is used, and that not ail exact tests achieve maximum local 
 power. The conditions under which tests based on artificial linear 
 regressions are asymptotically equivalent to a Cox test are also examined. 
 Some concluding remarks are given in Section 5. 
 
 2 . Finite Sample Relations Between Alternative Tests 
 
 It is desired to test two non-nested linear regression models, H and 
 H , against each other, where the models are given by 
 
 E x : y = Z Y + u x , -^ ~ N(0, 1^) . (2) 
 
 In equations (1) and (2), y is an n x 1 vector of observations on the 
 dependent variable, X and Z are n x k and n x g matrices of n observations 
 on k and g linearly independent regressors, 8 and y are k x 1 and g x 1 
 vectors of unknown parameters, and u and u are n x 1 vectors of distur- 
 bances. It is assumed that the matrices n X'X and n Z'Z converge to 
 well-defined finite positive definite limits, and n X'Z is non-zero and 
 also converges to a non-zero finite matrix. The two models are non-nested 
 in the sense that at least one column of each regressor matrix cannot be 
 expressed as a linear combination of the columns of the other. The design 
 matrices X and Z are assumed to be fixed in repeated samples, although this 
 
 assumption will be relaxed in Section 4. 
 
 2 9 
 
 Denoting 8' = (3', a Q ) and 6J = (y', o^), the Cox test of H is based 
 
 upon the statistic 
 
 t q = u Q - i x i = n[ P iim n" U q - V'e -e ' (3) 
 
 in which 
 
 2 
 
 I. = - 1/2 n log 2* - 1/2 n log aj - 1/2 n (i = 0,1) 
 
-3- 
 
 is the maximized log-liiceliaood function under B. , plia denotes prob- 
 
 a 
 
 ability limit under H and the second term in (3) is evaluated at 8 =8 . 
 
 The circumflex denotes maximum likelihood estimate, in which case 
 
 ff jj = n~ 1 y'(l-P)y, P = XCX'X)"" X», a^ = n~ y'(I-q)y and Q = Z(Z'Z)~ 1 Z'. The 
 
 Atkinson (1970) variation of the Cox test of H is based upon 
 
 
 
 a a -i a a 
 
 TA Q = (t„ - l 1Q ) - nlpli^n" ( 4() - t 10 )] 9 ' (4) 
 
 in which 
 
 a a a 
 
 £ 1Q = - 1/2 n log 2tt - 1/2 n log a^ Q - 1/2 a(tT l y ' ( I-QP) » ( I-QP)y)/aJ , 
 
 A A A 
 
 2-1 2 
 
 and a, ^ = a. + n y'P(I-Q)Py is a consistent estimate of plim,, a, under H_, 
 10 10 
 
 while the expecatation of QPy equals the expectation of Qy under H . The 
 
 second terms in (3) and (4) are asymptotically equivalent under H . Since, 
 
 2 2-1 9 
 
 in addition, plim Q a. = P lim n ° \q = plim o n y ' ( I_ Q p ) ' ( I_ Q p )y = a J Q > ic 
 
 follows that T and TA are asymptotically equivalent (for further details, 
 see Fisher and McAleer, 1981). The important point to note in the preceding 
 discussion is that both the Cox and Atkinson procedures modify a likeli- 
 hood ratio by subtracting its asymptotic expectation evaluated at con- 
 sistent estimates of the parameters under the null model. Another possible 
 
 modification is given by 
 "0 
 
 T = U " V - nCpliin ^O " *1 )J 3 =V (5) 
 
 where I is the calculated value of the likelihood function under H when 
 
 A A 
 
 8, is replaced by (y , o ), As will be discussed in Section 4 below, we 
 
 A 
 
 consider estimates y of y wnich are linear in y. 
 
 The Cox principle of testing non-nested models was introduced to the 
 econometric literature by Pesaran (1974) for testing linear regression 
 models, and by Pesaran and Deaton (1978) for testing non-linear systems of 
 equations. In this paper, we will refer to the test derived by these 
 
— *f- 
 
 authors as the Cox tesc. The test raay be modified using Atkinson's 
 suggestion of evaluating the entire statistic under the null hypothesis. 
 Denoting a consistent estimate of the asymptotic variance of TA as V , the 
 Atkinson test, namely NA = TA /(V ) , is distributed approximately as 
 
 •J vj u 
 
 N(0,1) in large samples. Fisher and McAleer (1981) show that NA may be 
 
 s u 
 
 written as 
 
 1/2 n(a 2 - o 1 ) + 1/2 y f ( I-P)Q(I-P)y 
 
 •JA = = til 
 
 ■ 
 
 a (y'Pg(I-P)QPy) X ' 
 
 which is equivalent to 
 
 1/2 
 NA Q = -y'PQ(I-P)y/ja (y , PQ(I-P)QPy) X/ ^}. (6) 
 
 Apart from j , NA may be calculated from the artificial regression 
 
 y = X3 + oQPy + u. (7) 
 
 The t-ratio for the ordinary least squares (OLS) estimate of a in (7), 
 
 namely 
 
 t(o) = y'?Q(I-P)y/{a(y , PQ(I-P)QPy) 1/2 }, 
 
 in which j 2 = (n-k-i)" 1 ! y '( I-P)y - (y 'PQ( I-P)y) 2 /(y »?Q( I-P)QPy) } is the 
 
 estimated error variance from (7), is asymptotically equivalent to NA 
 
 2 ? ? 
 
 under H • This result follows from the fact that plim„ o = plim n o~ = err. 
 u u u u 
 
 In the remainder of this section and the next, we will concentrate upon 
 asymptotic approximations to the Cox test based upon artificial regressions 
 such as the one given in (7). 
 
 Three simple asymptotic procedures have been developed by Davidson and 
 MacKinnon (1981) for testing non-nested models. These tests are an 
 approximation to the Cox test and are also very simple to calculate. In 
 the linear case, the large sample J and P tests are identical, while both 
 are preferred to the C test. We shall provide below the finite sample 
 relations between the J and C tests. The sample means of y under H and 
 
o- 
 
 H are given by Py = XS and Qy = Zy , respectively, and the J test is the t- 
 ratio for the OLS estimate of a in the artificial regression 
 
 y = (1 - cOXB + ctQy + u. (8) 
 
 A two-tailed test of a = in (8) is a test of H , whereas a test of H is 
 performed by testing X = in the artificial regression 
 
 y = (1 - X)Zy + XPy + u. (9) 
 
 If Py is substituted for X£ in (8), we have the C test of H , namely 
 the test of a = in 
 
 y = (1 - a)Py + aQy + u, or (I - P)y = a(Q - P)y + u. (10) 
 Davidson and MacKinnon (1981, p. 783) recommend the C test as a preliminary 
 test of H because asymptotically it "...has variance less than unity under 
 H ." The fact that the C test is undersized (i.e., the actual size of the 
 test is less than its nominal size) when both a and the error variance are 
 estimated by maximum likelihood methods is demonstrated below. Denoting 
 the large sample tests of a = (i.e., the t-ratios based upon maximum 
 likelihood estimation) from (8) and (10) as t(a T ) and t(a ), respectively, 
 it follows that 
 
 t (Oj) = Cy»Q(I - P)y) 2 /(o-2(y' Q (i - p)Qy)), 
 
 7 2 U C ) = (y'Q(I - P)y) 2 /(a^(y'(Q - P) 2 y)), 
 where a. = n'^y'U - P)y - (y'Q(I - P)y) 2 /(y'Q(I - P)Qy} 
 
 •J 
 
 ^2 —1 2 ^ 
 
 and a = n {y»(I - P)y - (y'Q(I - P)y) /(y'(Q - P)"y)} are the maximum 
 
 likelihood estimates of the error variances from (8) and (10), respectively. 
 
 2 
 Since y'(Q - P) y - y'Q(I - P)Qy = y'(I - Q)?(I - Q)y is strictly positive 
 
 (even asymptotically) when H is not fitting perfectly, it follows that 
 
 ? 2 
 a > a , and hence 
 
-o- 
 
 t 2 ( aj ) > t 2 (a c ). (II) 
 
 Since Che J test has correct size asymptotically, the C test must be under- 
 sized. 
 
 It should be noted that there are two inherent problems with the C 
 
 A 
 
 test. First, the formula for the variance of a^ is incorrect. Second, 
 the estimated error variance used in calculating the t-ratio for a is 
 
 inappropriate for finite samples. When the two tests are calculated from a 
 
 2 2 
 
 least squares package, the denominators of a and a become (n-k-1) and 
 
 A A 
 
 (n-i), respectively, so that the J and C tests are given by t(a ) and t(a ), 
 The relationships between the tests based upon maximum likelihood and least 
 squares procedures are n t (o ) ■ (n-k-l)t (a ) and n t (a ) ■ (n-l)t (cO, 
 so that the inequality between the J and C tests may be written as 
 
 a a 
 
 (n-1) C 2 (a J ) > (n-k-1) t 2 ( a( J. (12) 
 
 A 
 
 Strictly speaking, there should be (n-k-1) degrees of freedom for t(a ), 
 
 A 
 
 as well as for t(ct ), since k degrees of freedom are lost in estimating 8 
 to obtain (10) from (8). The inequality given by (12) refers solely to 
 the t-ratios obtained from OLS estimation of (8) and (10). In spite of 
 inequality (11), when k is large relative to n and OLS estimation is used, 
 
 A A 
 
 it is clear from (12) that t(a ) may be exceeded by t(a ). 
 
 Under H , Py is distributed independently of (I - P)y, the vector of 
 residuals obtained from least squares estimation of (1). It is noteworthy 
 that Qy is not distributed independently of (I - P)y. Therefore, the J 
 test is not an exact test for H n when there is more than one column in Z 
 that is linearly independent of the columns of X. However, when Z contains 
 only one column that is linearly independent of those of X (e.g., when 
 
 there is one repressor in H, that is not in H ), the J test of H does have 
 
 ° 1 
 
 the exact t distribution in small samples. Following Atkinson (1970), 
 
-7- 
 
 Fisher and McAleer (1981, p. 109) evaluate the entire statistic under H 
 by replacing Qy in (8) with QPy, as in (7), leading to 
 
 y = (1 - a)XS + aQPy + u, (13) 
 
 in which a two-tailed test of a = is a test of H . 
 
 Milliken and Graybill (1970) have shown that if one or more functions 
 of the linear predictions of the linear null model, namely Py, are added to 
 that model, the test statistic of the significance of the additional 
 regressors will have the exact F distribution when the disturbances are 
 normal. In (13), the single additional regressor, QPy, is a function of Py 
 and hence is distributed independently of (I - P)y. Thus, the JA test, 
 which is the t-ratio for the OLS estimate of a, has the exact t distribu- 
 tion with (n-k-1) degrees of freedom under H . If the true spirit of the 
 Cox principle is to be observed, however, the roles of the null and alter- 
 native models should be reversed. The JA test of H , which is the t-ratio 
 for the OLS estimate of X in 
 
 y = (1 - X)Z T + XPQy + u, (14) 
 
 has the t-distribution with (n-g-1) degrees of freedom under H • 
 
 The general results of Milliken and Graybill have been used directly by 
 McAleer (1983) to test a linear null model against several non-nested, non- 
 linear alternatives simultaneously. The resulting test statistic has a 
 central F-distribution under the null but not a non-central F-distribution 
 when the null is false. There is presently very little known about the 
 power of the Milliken and Graybill procedure, and indeed the test may not 
 even be unbiased. Pesaran (1982b) has shown that the JA and J tests have 
 equal power in large samples when H is tested against local alternatives 
 and k > g. However, little is known of power considerations in the 
 interesting case of fixed alternatives, or when k < g. Of course, it is 
 
-3- 
 
 possible for k to exceed g through underspecif ying the alternative, in 
 which case the tests of the null may not even be consistent. For more on 
 this, see McAleer et al . (1982), where it is also shown that overspecifi- 
 cation does not affect the consistency of the tests. 
 
 3. Relations Between Some Exact and Asymptotic Tests 
 
 The numerator of the JA statistic from (13) is given by 
 
 A 
 
 a JA = (y»PQ(l - P)y)/(y , PQ(I - P)QPy), (15) 
 
 A 
 
 and the squared t-ratio from OLS estimation of a r . is 
 
 JA 
 
 A A 
 
 t 2 (a JA ) = (y'PQ(I - P)y) 2 /(o-j A (y'PQ(I - P)QPy)), (16) 
 
 A 
 
 where cj^ = (n-k-l)" 1 } (y » ( I - P)y) - (y'PQ(I - P)y) 2 /(y »PQ(I - P)QPy} 
 is the OLS estimate of the error variance. The exactness of the JA test 
 
 relies upon the independence of QPy and (I - P)y under H so that, given 
 
 *2 2 
 QPy, S (y'PQ(I - P)y) ■ and £ (a ) ■ a~, where E denotes expected value 
 
 2 2 
 
 under H^ . Thus, a T , is an unbiased estimator of a,, and a T . has zero expec- 
 JA JA 
 
 tat ion under H„« 
 
 Transforming both sides of (13) by (I - P) annihilates X and yields the 
 PA test of H , namely the test of a = in 
 
 (I - P)y = a(I - P)QPy + u. (17) 
 
 It should be noted that there are only (n-k) independent observations in 
 
 (17) since the k elements of 8 have already been estimated. Therefore, the 
 
 1/2 1/2 
 t-ratio for a from (17) must be multiplied by (n-k-1) /(n-1) to 
 
 obtain a test statistic which has the exact t distribution with (n-k-1) 
 degrees of freedom. If X8 is replaced by Py in (13), rearrangement yields 
 
 (I - P)y = -a(I - Q)Py + u, (18) 
 
 in which the test of a = is, say, the CA test of H . Note that trans- 
 forming both sides of (18) by (I - P) yields the PA test in (17). 
 
-9- 
 
 It is of interest to obtain the OLS estimate of X in (9) to test H , 
 
 namely 
 
 X J = (y'P(I - Q)y)/(y ? P(I - Q)Py). (19) 
 
 The numerator of X involves the inner product of Py and (I - 0)y so that 
 
 ■J 
 
 the J test of d does not have the exact t distribution, just as it is not 
 exact for H . However, consider testing H by testing the null hypothesis 
 X = 1, for which the squared t-ratio from OLS estimation of X in (9) is 
 
 t 2 (l - \j) = (y'PQ(I - P)y) 2 /(a^ 1 (y'P(I - Q)Py)), (20) 
 
 in which aj x = (n-g-l)" 1 ) (y » ( I - Q)y) - (y'P(I - Q)y) 2 /(y'P(I - Q)Py}. 
 
 Note that, apart from the estimated error variances, the expression for 
 
 t (1 - X ) in (20) is identical to that for t (ot CA ) from (18) » in s P ite of 
 
 the fact that the non-nested models tested in each case are different. 
 
 Thus, if the correct formula (in so far as testing H ) for the variance of 
 
 1 - X T were used, together with a_, as the estimated error variance, it is 
 J JA 
 
 possible for a test of H based upon (9) to be exact. Although it would 
 be a straightforward matter to adjust (20) by the factor 
 
 d = (a^(y'P(I - Q)Py))/(a^ A (y'PQ(I - P)QPy)) 
 to obtain an exact test for H based upon the J test for ft , it is clearly 
 more straightforward to calculate the JA statistic for H n directly from 
 (13). It is interesting to note that simple adjustments for the variance 
 and the estimated error variance can lead to an exact J test of a model 
 that was originally designated as the alternative. Since exactness of a 
 test arises from independence of the additional stochastic regressor and 
 the residual vector of the tested model, a test that is exact for the 
 null cannot be exact for the alternative. Thus, while the JA test of 
 
-10- 
 
 X = in (14) is exact for H , it cannot be modified to give an exact 
 
 test of H^ by any traasf ormation similar to d. 
 
 
 
 Since plini d = (S '(plini n'Vci - Q)X)S)/( S ' ( plira a" X f Q(I - P)QX)3) > 1, 
 where pilau denotes probability limit under ft , it follows that the CA test 
 and the J test of H , when set up initially to test H , will not reject H 
 frequently enough, even though both statistics have zero means under H . 
 
 The reason for this is that the variances of both statistics are incorrect 
 
 2 2 
 
 for testing H , in spite of the fact that a and a (an estimated error 
 
 \J L#r\ J J. 
 
 variance from the J test of H ) are consistent for the 'true' error vari- 
 ance under H . 
 
 4. Exact Tests With Maximum Local Power 
 
 The exact nature of various tests refers to the probability of com- 
 mitting a type I error only. It is clear that substituting any estimator 
 of the form y = VPy for y in 
 
 y = (1 - a)X3 + aZy + u, (21) 
 
 where V is a fixed g x n matrix of rank, g, will lead to an exact test of 
 
 ti since ZVPy is distributed independently of (I - P)y. However, an 
 
 
 
 analysis of the properties of alternative tests would not be complete 
 without a discussion of their relative powers. For cases in which the 
 dimension of the tested model is not exceeded by that of the alternative, 
 Pesaran (1982a) has examined the local power of tests of non-nested models. 
 The conditions under which a test of H based on an artificial linear 
 regression such as (21) has maximum power against local alternatives, in 
 addition to being consistent and asymptotically distributed as unit normal, 
 are stated by Pesaran (1982b). The conditions are presented below for 
 exact tests, assuming that k > g: 
 
-11- 
 
 CONDITION (Gl): lira. (VPu.) = under H.(i = 0,1). 
 
 CONDITION (C2): lira (VX) = D , where D is finite and non-zero. 
 
 CONDITION (C3): lim (VPZ) = D , where D is a positive definite matrix, 
 
 CONDITION (C4): under local alternatives, b lim (VPZ)y = by, in 
 
 J. z 
 
 which b and b ? are constants. 
 
 Conditions (CI) - (C3) are essentially a restatement of those given in 
 Pesaran (1982b), with due consideration for exact tests. The four con- 
 ditions stated are necessary and sufficient for an exact test to be con- 
 sistent, asymptotically N(0,1) under H , and to maximize local power. 
 Condition (C4) arises as a consequence of the Cauchy-Schwarz inequality 
 (see e.g., Rao, 1973, p. 54), and effectively requires that the limit of 
 VPZ be proportional to an identity matrix. 
 
 It should be noted that there are many matrices V which will satisfy 
 the four conditions stated above. However, by relating the test statistic 
 based upon the artificial regression in (21) to a Cox-type test based on 
 equation (5) in a systematic manner, we can restrict the possibilities for 
 V considerably. This may be shown as follows. On the basis of (5), the 
 likelihood function for H evaluated at (y » a ^ ^» where y = VPy, is given 
 
 by 
 
 t = - 1/2 n log 2t - 1/2 n log a* Q - 1/2 n (n y ? ( I-ZVP) ' ( I-ZVP)y)/a 
 
 -1 9 
 
 Since it is required that plim Q n y'(I - ZVP)'(I - ZVP)y = a~ Q , which is 
 
 equivalent to requiring lim n~ 1 X' ( I-ZVP) '( I-ZVP)X = lim n~ X' ( I-QP) ' ( I-QP)X 
 
 = lim n~ X'(I-Q)X, two possible solutions are ZV - Q and ZV = QP. 
 
 However, both of these possibilities simply lead to the JA test considered 
 
 previously, and which is known to be asymptotically equivalent to a Cox 
 
 test. Thus, requiring tests based upon artificial regressions such as (21) 
 
 to be asymptotically equivalent to a Cox test reduces the possibilities 
 
-12- 
 
 for the matrix V. In general, it rules out several forms considered by 
 Pesaran (1982b), such as ZV = QPQ. 
 
 In the remainder of this section we will assume that only the four con- 
 ditions stated hold, with X and Z being stochastic. It is easy to show 
 that exact tests are available for H even when Z is independent of u but 
 not of u . In this case, a set of instruments given by the matrix W of 
 rank h (> g) may be used for consistent estimation under H . If Z is not 
 independent of u , the JA test of H will be exact regardless of whether 
 OLS or instrumental variable (IV) estimation is used for H . This result 
 holds because Z is assumed to be independent of u , even though it is not 
 independent of u . 
 
 In the preceding discussion, we have not assumed that all the columns 
 of X and Z are linearly independent of each other. Therefore, condition 
 (CI) may not be satisfied under H , although conditions (C2) - (C4) are 
 satisfied. For instance, let X and Z be partitioned as [X : Z ? ] and 
 [Z : Z ], respectively, where X and Z are linearly independent. If 
 neither Z nor Z is independent of u , condition (CI) does not hold. 
 However, if Z is independent of u , (CI) is satisfied and hence the power 
 of the JA test of H is maximized whether OLS or IV estimation is used for 
 H . Since V equals (Z f Z)~ Z' and (Z'SZ)~ Z'S for the JA test when OLS and 
 
 IV estimation are used, respectively, in which S = W(W'W) W , then the 
 limit of VPZ will be proportional to an identity matrix. Thus, the JA test 
 of H is not only exact, but in this case also attains maximum power 
 against local alternatives whether OLS or IV estimation is used. 
 
 As a word of warning, however, it should be stressed that some exact 
 tests may not attain maximum local power. The most general form for V is 
 
 V = (Z'S Z) Z'S , where S and S are both projection matrices derived 
 
•13- 
 
 frora sets of instruments. It is clear that setting y in (21) to 
 (Z'S Z) Z ? 5 Py does not alter the exactness of the test of H . Special 
 cases we have already examined are S = S = I and S = S t I. Another 
 possibility is when S t S , in which case (C4) will not be satisfied 
 unless S X = S X = X. Since it has not been assumed that h > k or that the 
 set of instruments for H contains all the columns of X, it follows that an 
 exact test does not necessarily maximize power against local alternatives. 
 
 5. Concluding Remarks and Future Research 
 
 In this paper we have established finite sample relations among some 
 exact and asymptotic tests of non-nested regression models. Necessary and 
 sufficient conditions have been provided for exact tests to attain maximum 
 local power. It was also shown that the JA test satisfies the above con- 
 ditions and is also asymptotically equivalent to a Cox test. While Godfrey 
 and Pesaran (1983) consider the finite sample size and power properties of 
 various exact and asymptotic tests against well-specified alternatives, 
 there are, however, still many issues that remain unanswered, and we men- 
 tion some of those that are currently under investigation. First, little 
 is known about the level of significance and power properties of the JA 
 test when it is not distributed exactly. Second, although Dastoor and 
 McAleer (1984) have obtained some analytical results on the relative powers 
 of paired and joint tests of non-nested models, the small sample properties 
 of these tests are as yet unknown. Third, the robustness of the tests has 
 not been studied in the presence of non-standard error structures and spe- 
 cification errors of various types. There is also a need to develop simple 
 test procedures for non-nested models under a general error specification 
 such as non-normal and serially dependent errors. Finally, since it has 
 been observed in Pesaran (1982a) that a Cox test rejects a 'true' null too 
 
-14- 
 
 frequently, finite sample tests for normality of asymptotic test statistics 
 would be useful. 
 
 Acknowledgement 
 
 The authors would like to thank an anonymous referee for helpful 
 comments on an earlier draft. The financial support of the Research 
 Board and the Bureau of Economic and Business Research of the 
 University of Illinois is gratefully acknowledged. 
 
References 
 
 Atkinson, A. C. (1970), A method for discriminating between models, Journal 
 of the Royal Statistical Society B, 32, 323-344. 
 
 Cox, D. R. (1961), Tests of separate families of hypotheses, Proceedings 
 of the Fourth Berkeley Symposium on Mathematical Statistics and 
 Probability 1 (Berkeley, University of California Press), pp. 105-123. 
 
 Cox, D. R. (1962), Further results on tests of separate families of 
 
 hypotheses, Journal of the Royal Statistical Society B, 24, 406-424. 
 
 Dastoor, N. K. (1983), Some aspects of testing non-nested hypotheses, 
 Journal of Econometrics , 21, 213-228. 
 
 Dastoor, N. K. and M. McAleer (1984), Some comparisons of joint and paired 
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