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Further evidence on asymptotic tests for homogeneity and symmetry in large demand systems
101
Economics Letters 8 (1981) 101-105 North-Holland Publishing Company
FURTHER EVIDENCE ON ASYMPTOTIC TESTS FOR HOMOGENEITY AND SYMMETRY IN LARGE DEMAND SYSTEMS A.K. BERA, Austrulim Received
R.P. BYRON
Nutrmul
and C.M. JARQUE
Unicersltr~, Cunherru. ACT .?6OO,Austrdru
2 July 1981
The asymptotically equivalent Lagrange multiplier. likelihood ratio and Wald tests are compared when testing for homogeneity and symmetry. The need for size correction becomes apparent as does the superior performance of the LM test under H,,.
Laitinen (1978) and Meisner (1979) set the cat among the pigeons with their studies pinpointing the considerable bias towards rejection of the asymptotic X2 Wald (W) test when applied to large demand systems with relatively few observations. Laitinen identified the problem using only homogeneity restrictions, and because of the within equation nature of the homogeneity constraint, was able to derive conclusive analytical results. Meisner considered symmetry restrictions and using Monte Carlo methods demonstrated that the problem with the Wald test of the form b’R’{ R[ S @ ( XX) ~ ’ ]R’}- ‘Rb is that the sample estimate of the variance -covariance matrix S must be used. If the population value Z replaces S in the formula the test no longer yields excessive type I errors. The problem for the econometrician is to find a test which gives the desired significance level and which has reasonable power. Both authors confined themselves to the Wald test. Here we also examine, in the same context, the performance of the likelihood ratio (LR) and Lagrange multiplier (LM) tests. The three test statistics considered may be written in the form [e.g., see Berndt and Savin (1977, p. 1270)] (1) 0165-1765/81/0000-0000/$02.75
0 1981 North-Holland
LR=Tln{J~//j~l}
=Tln{(!?‘oT(/(&!?)},
LM=Ttr~-‘(~-_)==ttr(~~)~‘(~~--_~),
(2) (3)
where the usual regression notation is employed, T denotes the sample size, and tilde and hat denote constrained and unconstrained estimates respectively. Given the Savin (1976) inequality, i.e. W > LR > LM, one might anticipate the LM test would be less prone to the rejection bias discovered by Laitinen and Meisner. This is found to be the case below, although the basic problem remains; as the number of equations increase relative to the number of observations, all tests become biased in type I error. It is essentially a small sample problem and in the present state of the art we are confined to large sample tests. Even in a single equation context, as Evans and Savin (1980) point out, the asymptotic x2 Wald test will be biased. However, if the test is properly size corrected the bias disappears and, in fact the three tests will tend to have identical distributions under H,, and H,. Unfortunately, as will become clear, the appropriate size corrections are not yet obvious in the context of systems of equations with across equation restrictions. Our first step was to apply the three tests to the 14 equation, 31 observation Rotterdam Model used by the authors. Our results, which were based on Barten’s (1966) original Dutch data, were as given in table 1. ’ The differences between the values of the three test statistics are much larger than we anticipated, with the LA4 test bordering on acceptance of Ho. Size correction would reduce all the calculated statistics. ’ Next, to compare the LA4 and LR tests with the Wald test. When testing for homogeneity, the results of Evans and Savin (1980) hold and therefore all tests will have identical power under H, Furthermore, given that the finite sample distribution of the Wald test is known to be Hotelling’s 5 * [see Laitinen (1978)] there is no need to use asymptotic critical values. We now consider testing for (i) symmetry and (ii) homogeneity and symmetry. We carried out Monte Carlo experiments analogous to those of Laitinen and Meisner, using 500 replications in each
’ The LR and LM tests were based on eqs. (2) and (3) after convergence, iterating on 2. Convergence was achieved after 4-5 iterations. 2 In table I the value of W for homogeneity is 393.4. This differs from that of Laitinen (1978, p. 188).267.85, because here W is computed with a factor T rather than T-k, and a different data set is used.
A.K.
Table I Computed
Homogeneity Symmetry Homogeneity a q denotes
Bern et al. / Asymptotic
test statistics
on 14 sector Rotterdam
and symmetry the number
tests for homogemty
rend syrnntet~~
103
Model.
W
I-R
LM
qn
x2 0.95 (4)
393.4 354.0 775.9
81.1 173.7 245.5
28.7 102.6 121.7
13 78 91
22.3 99.6 114.2
of restrictions.
case. The first pass, under H,, was used to calculate type I error and test. In a distributional closeness to a x2 via the Kolmogorov-Smirnov second pass the population parameters, calculated using the previous authors’ guidelines, were perturbed, and the power of each test, based on the previous empirical critical value was calculated. 3 The results are fairly obvious and need little discussion. The LM test did best under H, as judged by type I error and the Kolmogorov-Smirnov test; however, it tended to be inferior in terms of power. The LR test was generally in the middle. All tests were unsatisfactory as n, the number of equations, increased. 4 The need for size correction is brought out by the final table which gives, for each test, the ratio of the empirical 5 percent critical value to the asymptotic critical x2 value. A size correction of T/(Tk), for example, such as Meisner used for the Wald test, is reasonable when n = 5 but quite inadequate when n = 14 [e.g., with T = 31, the values of T/( T - k) for n = 5, 8, 11 and 14 are 1.24, 1.49, 1.63 and 1.93 respectively]. Similarly, corrections suggested by Anderson (1958, p. 208) for the LR test are inappropriate for cross-equation restrictions and, when adapted in a common-sense way, by averaging the total number of restrictions per equation, prove inadequate in application.’ Given the success of Evans and Savin (1980) using Edgeworth expansions to To perturb the parameter values, we added a quantity cz, =&,(I +2/((1I),l+j)) ton,,, defined in Laitinen (1978, footnote 3). where n is the number of equations. We used ~~0.09 for ‘symmetry’, and c=O.Ol for ‘homogeneity and symmetry’. These choices were made to get powers within a ‘convenient illustrative range’ for all the three test statistics. Although type I errors can be compared as PI increases, power cannot. There appeared no way of standardising the deviation of H, from H,, in terms of perturbations on the population parameters. Consequently, power can only be compared across rows. One striking result is that the ratio corresponding to the LM test does not appear to change as n (and hence 4) increases (see last column in table 3).
104
A. K. Beru et ul. / Asymptotic
Table 2 Comparison
tests for homogeneit_v and symmetry
of power. Symmetry
Symmetry
and homogeneity
LR
LM
W
LR
LM
T_vpe I error a n= 5 19.8 n= 8 58.8 n=Il 97.8 ,I= 14 100.0
14.8 38.0 83.0 100.0
9.8 12.4 22.4 46.6
25.4 70.8 99.6 100.0
16.2 46.8 90.4 100.0
7.8 12.2 19.6 48.8
n= 5 tl= 8 n=ll n= I4
Power b 49.0 77.0 100.0 98.0
50.4 82.0 100.0 95.6
50.6 81.8 95.4 77.4
43.4 54.6 84.8 46.2
39.0 35.2 35.8 11.2
n= 5 II= 8 n=Il ?I= 14
Kolmogoroo-Smirnoo 0.238 0.591 0.932 0.990
W
45.0 65.2 96.2 6 I .O
test stutistrts ’ 0.198 0.167 0.47 I 0.297 0.827 0.469 0.96 I 0.648
0.351 0.713 0.972 0.992
0.282 0.54 I 0.865 0.989
0.197 0.288 0.471 0.669
a Percentage of rejections of Ha: AP=O using asymptotic 5 percent x2 level. b Percentage of rejections of He under H, : RP#O using empirical 5 percent level ’ Based on 100 replications.
Table 3 Comparison
of critical (I
values (CV). Empirical
CV
W
LR
LM
W
LR
LM
12.6 32.7 61.7 99.6
18.8 66.5 188.3 565.4
16.0 47.3 107.2 211.0
13.9 36.0 71.0 113.4
1.49 2.03 3.05 5.68
1.27 I .45 1.74 2.12
I.1 I I.1 I I.15 1.14
29.8 94.6 251.8 1008.5
23.8 63.0 130.4 258.4
19.4 45.x 82.0 128.8
I .63 2.29 3.44 8.83
1.30 1.53 1.78 2.26
I .06 I.1 I 1.12 I.13
Symmetty n= 5 n= 8 n=ll n=14
6 21 45 78
Sytimetty n= 5 n=8 n=ll n=l4
nnd homogeneity 10 18.3 28 41.3 55 73.3 91 114.3
a Ratio is defined
Ratios ’
Asymptotic cv
as empirical
CV over asymptotic
CV.
A.K.
Bercr et ul. / Asymptottc
tests for homogeneit_v urd symmetcv
105
effectively adjust large sample tests there is reason to hope that analytical work in the near future will achieve a similar result in the present context. If so, despite the small sample nature of most empirical demand analysis, it should be possible to reliably test hypotheses.
References Anderson, Theodore W., 1958, An introduction to multivariate statistical analysis (Wiley. New York). Batten. Anton P., 1966, Het verbruik door gezinshuishoudingen in Nederland 1921- 1939 en l948- 1962, Netherlands School of Economics, Report 6604. Berndt, Ernst R. and N. Eugene Savin, 1977. Conflict among criteria for testing hypothesis in the multivariate linear regression model, Econometrica 45, 1263-1277. Evans, G.B.A. and N. Eugene Savin, 1980, Conflict among the criteria revisited: The W, LR. and LM tests, Econometrics Society, World Congress (Aix-en-Provence). Laitinen, Kenneth, 1978, Why is demand homogeneity so often rejected? Economics Letters I, 187-191. Meisner, James F., 1979. The sad fate of the asymptotic Slutsky symmetry test for large systems, Economics Letters 2, 231-233. Savin, N. Eugene, 1976, Conflict among testing procedures in a linear regression model with autoregressive disturbances, Econometrica 44, 13033 I3 15.
Economics Letters 8 (1981) 101-105 North-Holland Publishing Company
FURTHER EVIDENCE ON ASYMPTOTIC TESTS FOR HOMOGENEITY AND SYMMETRY IN LARGE DEMAND SYSTEMS A.K. BERA, Austrulim Received
R.P. BYRON
Nutrmul
and C.M. JARQUE
Unicersltr~, Cunherru. ACT .?6OO,Austrdru
2 July 1981
The asymptotically equivalent Lagrange multiplier. likelihood ratio and Wald tests are compared when testing for homogeneity and symmetry. The need for size correction becomes apparent as does the superior performance of the LM test under H,,.
Laitinen (1978) and Meisner (1979) set the cat among the pigeons with their studies pinpointing the considerable bias towards rejection of the asymptotic X2 Wald (W) test when applied to large demand systems with relatively few observations. Laitinen identified the problem using only homogeneity restrictions, and because of the within equation nature of the homogeneity constraint, was able to derive conclusive analytical results. Meisner considered symmetry restrictions and using Monte Carlo methods demonstrated that the problem with the Wald test of the form b’R’{ R[ S @ ( XX) ~ ’ ]R’}- ‘Rb is that the sample estimate of the variance -covariance matrix S must be used. If the population value Z replaces S in the formula the test no longer yields excessive type I errors. The problem for the econometrician is to find a test which gives the desired significance level and which has reasonable power. Both authors confined themselves to the Wald test. Here we also examine, in the same context, the performance of the likelihood ratio (LR) and Lagrange multiplier (LM) tests. The three test statistics considered may be written in the form [e.g., see Berndt and Savin (1977, p. 1270)] (1) 0165-1765/81/0000-0000/$02.75
0 1981 North-Holland
LR=Tln{J~//j~l}
=Tln{(!?‘oT(/(&!?)},
LM=Ttr~-‘(~-_)==ttr(~~)~‘(~~--_~),
(2) (3)
where the usual regression notation is employed, T denotes the sample size, and tilde and hat denote constrained and unconstrained estimates respectively. Given the Savin (1976) inequality, i.e. W > LR > LM, one might anticipate the LM test would be less prone to the rejection bias discovered by Laitinen and Meisner. This is found to be the case below, although the basic problem remains; as the number of equations increase relative to the number of observations, all tests become biased in type I error. It is essentially a small sample problem and in the present state of the art we are confined to large sample tests. Even in a single equation context, as Evans and Savin (1980) point out, the asymptotic x2 Wald test will be biased. However, if the test is properly size corrected the bias disappears and, in fact the three tests will tend to have identical distributions under H,, and H,. Unfortunately, as will become clear, the appropriate size corrections are not yet obvious in the context of systems of equations with across equation restrictions. Our first step was to apply the three tests to the 14 equation, 31 observation Rotterdam Model used by the authors. Our results, which were based on Barten’s (1966) original Dutch data, were as given in table 1. ’ The differences between the values of the three test statistics are much larger than we anticipated, with the LA4 test bordering on acceptance of Ho. Size correction would reduce all the calculated statistics. ’ Next, to compare the LA4 and LR tests with the Wald test. When testing for homogeneity, the results of Evans and Savin (1980) hold and therefore all tests will have identical power under H, Furthermore, given that the finite sample distribution of the Wald test is known to be Hotelling’s 5 * [see Laitinen (1978)] there is no need to use asymptotic critical values. We now consider testing for (i) symmetry and (ii) homogeneity and symmetry. We carried out Monte Carlo experiments analogous to those of Laitinen and Meisner, using 500 replications in each
’ The LR and LM tests were based on eqs. (2) and (3) after convergence, iterating on 2. Convergence was achieved after 4-5 iterations. 2 In table I the value of W for homogeneity is 393.4. This differs from that of Laitinen (1978, p. 188).267.85, because here W is computed with a factor T rather than T-k, and a different data set is used.
A.K.
Table I Computed
Homogeneity Symmetry Homogeneity a q denotes
Bern et al. / Asymptotic
test statistics
on 14 sector Rotterdam
and symmetry the number
tests for homogemty
rend syrnntet~~
103
Model.
W
I-R
LM
qn
x2 0.95 (4)
393.4 354.0 775.9
81.1 173.7 245.5
28.7 102.6 121.7
13 78 91
22.3 99.6 114.2
of restrictions.
case. The first pass, under H,, was used to calculate type I error and test. In a distributional closeness to a x2 via the Kolmogorov-Smirnov second pass the population parameters, calculated using the previous authors’ guidelines, were perturbed, and the power of each test, based on the previous empirical critical value was calculated. 3 The results are fairly obvious and need little discussion. The LM test did best under H, as judged by type I error and the Kolmogorov-Smirnov test; however, it tended to be inferior in terms of power. The LR test was generally in the middle. All tests were unsatisfactory as n, the number of equations, increased. 4 The need for size correction is brought out by the final table which gives, for each test, the ratio of the empirical 5 percent critical value to the asymptotic critical x2 value. A size correction of T/(Tk), for example, such as Meisner used for the Wald test, is reasonable when n = 5 but quite inadequate when n = 14 [e.g., with T = 31, the values of T/( T - k) for n = 5, 8, 11 and 14 are 1.24, 1.49, 1.63 and 1.93 respectively]. Similarly, corrections suggested by Anderson (1958, p. 208) for the LR test are inappropriate for cross-equation restrictions and, when adapted in a common-sense way, by averaging the total number of restrictions per equation, prove inadequate in application.’ Given the success of Evans and Savin (1980) using Edgeworth expansions to To perturb the parameter values, we added a quantity cz, =&,(I +2/((1I),l+j)) ton,,, defined in Laitinen (1978, footnote 3). where n is the number of equations. We used ~~0.09 for ‘symmetry’, and c=O.Ol for ‘homogeneity and symmetry’. These choices were made to get powers within a ‘convenient illustrative range’ for all the three test statistics. Although type I errors can be compared as PI increases, power cannot. There appeared no way of standardising the deviation of H, from H,, in terms of perturbations on the population parameters. Consequently, power can only be compared across rows. One striking result is that the ratio corresponding to the LM test does not appear to change as n (and hence 4) increases (see last column in table 3).
104
A. K. Beru et ul. / Asymptotic
Table 2 Comparison
tests for homogeneit_v and symmetry
of power. Symmetry
Symmetry
and homogeneity
LR
LM
W
LR
LM
T_vpe I error a n= 5 19.8 n= 8 58.8 n=Il 97.8 ,I= 14 100.0
14.8 38.0 83.0 100.0
9.8 12.4 22.4 46.6
25.4 70.8 99.6 100.0
16.2 46.8 90.4 100.0
7.8 12.2 19.6 48.8
n= 5 tl= 8 n=ll n= I4
Power b 49.0 77.0 100.0 98.0
50.4 82.0 100.0 95.6
50.6 81.8 95.4 77.4
43.4 54.6 84.8 46.2
39.0 35.2 35.8 11.2
n= 5 II= 8 n=Il ?I= 14
Kolmogoroo-Smirnoo 0.238 0.591 0.932 0.990
W
45.0 65.2 96.2 6 I .O
test stutistrts ’ 0.198 0.167 0.47 I 0.297 0.827 0.469 0.96 I 0.648
0.351 0.713 0.972 0.992
0.282 0.54 I 0.865 0.989
0.197 0.288 0.471 0.669
a Percentage of rejections of Ha: AP=O using asymptotic 5 percent x2 level. b Percentage of rejections of He under H, : RP#O using empirical 5 percent level ’ Based on 100 replications.
Table 3 Comparison
of critical (I
values (CV). Empirical
CV
W
LR
LM
W
LR
LM
12.6 32.7 61.7 99.6
18.8 66.5 188.3 565.4
16.0 47.3 107.2 211.0
13.9 36.0 71.0 113.4
1.49 2.03 3.05 5.68
1.27 I .45 1.74 2.12
I.1 I I.1 I I.15 1.14
29.8 94.6 251.8 1008.5
23.8 63.0 130.4 258.4
19.4 45.x 82.0 128.8
I .63 2.29 3.44 8.83
1.30 1.53 1.78 2.26
I .06 I.1 I 1.12 I.13
Symmetty n= 5 n= 8 n=ll n=14
6 21 45 78
Sytimetty n= 5 n=8 n=ll n=l4
nnd homogeneity 10 18.3 28 41.3 55 73.3 91 114.3
a Ratio is defined
Ratios ’
Asymptotic cv
as empirical
CV over asymptotic
CV.
A.K.
Bercr et ul. / Asymptottc
tests for homogeneit_v urd symmetcv
105
effectively adjust large sample tests there is reason to hope that analytical work in the near future will achieve a similar result in the present context. If so, despite the small sample nature of most empirical demand analysis, it should be possible to reliably test hypotheses.
References Anderson, Theodore W., 1958, An introduction to multivariate statistical analysis (Wiley. New York). Batten. Anton P., 1966, Het verbruik door gezinshuishoudingen in Nederland 1921- 1939 en l948- 1962, Netherlands School of Economics, Report 6604. Berndt, Ernst R. and N. Eugene Savin, 1977. Conflict among criteria for testing hypothesis in the multivariate linear regression model, Econometrica 45, 1263-1277. Evans, G.B.A. and N. Eugene Savin, 1980, Conflict among the criteria revisited: The W, LR. and LM tests, Econometrics Society, World Congress (Aix-en-Provence). Laitinen, Kenneth, 1978, Why is demand homogeneity so often rejected? Economics Letters I, 187-191. Meisner, James F., 1979. The sad fate of the asymptotic Slutsky symmetry test for large systems, Economics Letters 2, 231-233. Savin, N. Eugene, 1976, Conflict among testing procedures in a linear regression model with autoregressive disturbances, Econometrica 44, 13033 I3 15.