Wald tests of common factor restrictions

Economics Letters North-Holland

WALD

Allan

TESTS

Received

OF COMMON

W. GREGORY

Unrcwrsiy

of

203

22 (1986) 203-208

FACTOR

and Michael

RESTRICTIONS

R. VEALL

*

Western Onturro. Landon. Ont., Cunudu N6A 3K7

14 July 1986

In this letter using Monte Carlo analysis, we investigate the sensitivity of Wald tests to alternative common factor restrictions. For there examples considered, there is no one form of the restrictions different parameter settings.

formulations that performs

of non-linear well over the

1. Introduction

The value of the Wald test statistic for two algebraically equivalent non-linear restrictions will not be equal in finite samples. While this point has been made, for example by Burguete, Gallant and Souza (1982, p. 165), its importance in econometric applications has only recently been documented [Gregory and Veal1 (1985 and 1986), Lafontaine and White (1986)]. Monte Carlo evidence in Gregory and Veal1 (1985 and 1986) suggests that the multiplicative form of the non-linear restriction yields empirical test sizes that are reasonably close to the asymptotic theory for quite small samples. Therefore, principally on the basis of this, it was concluded that non-linear restrictions should probably be specified in multiplicative form. While in a general sense we continue to hold this belief, the purpose of this letter is to demonstrate that there are occasions for which no one form of the restriction (including the multiplicative form) performs well over all the different parameter settings. Using Monte Carlo analysis we investigate alternative formulations of Wald tests of common factor restrictions and show that the small sample properties of the multiplicative form deteriorate dramatically as the serial correlation parameter approaches the unit root. This emphasizes the need for further research in finding the best way to specify non-linear restrictions for Wald tests.

2. Wald tests of common

factor restrictions

This easiest way to investigate alternative formulations of Wald tests of common tions as analyzed by Hendry and Mizon (1978) is to consider a very simple dynamic the unrestricted form y, = PY,-1 + P1xt + * The authors acknowledge

I%%1 +

(1)

01.

would like to thank G. Fisher, CR. Rao, M. Sampson, R. Tihshirani and A. Ullah for helpful financial support from the Social Sciences and Humanities Research Council of Canada.

01651765/86/$3.50

factor restricexample with

B 1986. Elsevier Science Publishers

B.V. (North-Holland)

discussions

and

where 1p 1 < 1, and uI is an independently and identically distributed error term with a mean of zero and a constant variance. Eq. (1) may be written more compactly using the lag operator L as

(2) If & is equal to -&p,

then

(l-pL)y,=P,(l-pL.)x,+u,,

yt =

or

(3)

PlX, + uI)

(4)

where U, = u/(1 - pL). This error term may be expressed as a simple first-order stationary autoregressive process u, = put-1 + cr. Thus, in terms of eq. (2) the polynomials in the lag operator share a common root and this root is the serial correlation coefficient in the autoregressive process for the static model (4). Therefore expressing the model as (4) reduces the parameter space by one and gives rise to one testable non-linear factor restrictions by Wald tests.

restriction.

Hendry

and Mizon

suggest testing

the common

Although the usual way in which to test this restriction is in the form (i) &p + & = 0, we might also consider three other possible forms (ii) /3t + &/p = 0, (iii) p + &/pl = 0, and (iv) plp/& + 1 = 0. The general expression

for the Wald test is

(5) where h are the unrestricted parameters to be tested, g(X) = 0 is the restriction which could be in forms (i)-(iv) or others, G( fi) = ag(X)/aXr evaluated at fi, a consistent estimate of X, V(i) is the estimated

variance-covariance

matrix

of fi and q is the dimension

of g, the number

of restrictions.

As can be verified easily using (5) each of forms (i)-(iv) would yield a different Wald statistic in finite samples. Moreover infinitely many other parameterizations equivalent to forms (i)-(iv) can be calculated, each with its own corresponding test statistic. ’ All such Wald tests are asymptotically equivalent. Note also that tests based on the likelihood ratio (LR) or the score or Lagrangian multiplier (LM) principles are invariant to such reparameterizations. To investigate the small sample properties of Wald tests based on different parameterizations of the null hypothesis in first-order mixed autoregressive models, we conduct a simple Monte Carlo experiment. For each repetition, an x vector is obtained for each of 1000 replications using a first-order autoregressive process with a lag parameter value of 0.75, and a random standard normal deviate generator. 2 A y vector is generated from eq. (1) for both the case where the null hypothesis (i)-(iv) is assumed to be true and where it is assumed to be false. Different values are assigned for PI and p, and u, is again drawn from a standard normal generator. The four forms of the Wald test are then calculated

using the ordinary

As the asymptotic heuristic

distribution

interpretation

there is no difficulty restriction

severe problems ’

Random

normal

the University observations

pp. 115-11611,

with the existence

approached

estimates

in Gregory

of gradient

the coefficients

G in that cast.

and Veal1 (1985)

zero, the approximate

we found in our simple

violation

of the required

of Western were set.

were generated Ontario.

using IMSL

All variances

subroutine

were

of (ii)-(k) that as the

of derivatives

this does not affect as implemented

Processes

of rejections of Wald (1943)

or the

‘true’

denominator

[Wald (1943.

on the CDC 20 periods

and

p = & = 0 or p. 463)j

the test’s asymptotic

were started

for

must be non-zero

the trivial cases where either

example

continuity

GGNML

set at unity.

[see pp. 445-446

in the denominators

(This excepts

with the size of the Wald test in ratio form. Of course deviates

of eq. (1) and the number

of the Wald test is derived under the null hypothesis

of Silvey (1975,

p1 = & = 0.) However,

least squares

Cyber

of a led to

Justification. 170/835

before

at

the first

60 71 56 44 47

20 30 50 100 500

20 30 50 100 500

p, = 0.5 p=o.5 p2 = 0.25

p, = 1.0 p = 0.5 p2 = 0.5

p, = 0.1 p = 0.9 pz = 0.09

51 59 43 50 51 92 92 85 87 51 104 106 90 79 61

20 14 11 10 11

47 31 29 24 13

52 43 31 26 9

42 34 25 25 18

68 68 48 57 52

114 104 94 89 57

131 123 96 80 61

114 93 103 92 66

20 30 50 100 500

20 30 50 100 500

20 30 50 100 500

p, = 0.5 p = 0.9 pr = 0.45

p, =I.0 p = 0.9 pz = 0.9 30 25 20 21 17

27 33 23 21 9

26 24 23 22 12

56 54 71 80 65

63 69 73 71 61

47 70 70 19 56

11 2 6 18 49

5 8 6 14 38

5 5 9 18 56

0.05

x2

13 11 13 20 18

14 16 18 18 9

12 19 11 16 12

1 0 0 0 5

1 1 0 1 5

1 0 0 0 5

0.01

wofP,+(Pz/P)=O

38 47 64 71 65

45 54 66 68 60

36 54 59 75 56

8 2 4 18 46

5 6 5 12 38

0

4 3 5 17 55

7 6 10 16 17

6 10 14 18 9

4 10 7 14 12

0 0 0 0 5

1 1 0 0 5

4

0

0

0

0.01

0.05

F

79 72 91 84 62

49 57 51 55 58

18 11 4 7 19

52 58 44 59 47

32 43 33 34 42

22 15 11 14 24

0.05

X2

30 19 18 24 17

12 16 9 11 10

5 3 0 0 0

18 15 12 14 10

11 12 6 9 8

9 8 1 1 3

0.01

w,ofP+(P,/P,)=O”

60 59 84 81 61

33 44 46 51 56

12 10 3 5 19

39 48 41 59 47

28 36 32 33 42

16 14 9 13 23

0.05

F

11 15 14 22 17

6 6 8 9 10

1 3 0 0 0

13 8 10 11 10

7 9 5 7 8

5 4 1 0 2

0.01

a For I+‘,, in the case where PI = 0.1, p = 0.9 and pz = 0.09, and with 2000 observations, the row would read 41, 10, 41 and 10.

88 14 90 86 64

10 10 8 8 11

44 56 46 40 46

12 17 9 10 11 8 9 7 9 11

12 12 8 10 7

0.01

53 49 55 53 61

23 16 9 10 9

0.01

0.05

70 63 62 51 61

0.05

20 30 50 100 500

x2

p1 = 0.1 p = 0.5 pz = 0.05

F

wofP,P+&=O

N

Case

Table 1 Number of rejections using Wald statistics when common factor restrictions arc true

16 72 63 60 55

17s 146 86 58 38

231 166 140 109 41

61 53 38 32 24

198 188 125 89 52

341 279 249 189 79

0.05

X2

43 30 24 11 11

115 91 49 30 6

65 61 60 57 55

162 138 84 57 38

199 151 135 106 41

47 48 34 30 24

28 12 12 7 1 138 105 82 155 17

185 175 116 88 50

322 267 241 188 79

0.05

F

123 116 71 42 17

259 202 165 108 36

0.01

W,of(PP,/P,)+l=O

29 23 18 10 11

91 71 46 28 6

112 95 70 54 17

12 11 9 7 1

90 102 63 38 17

227 188 150 102 35

0.01

286

46X

X26

1000

490

658

932

1000

30

50

100

500

p = 0.9

p* = 0.7

792

loo0

892

1000

30

50

100

500

p = 0.9

pz = 0.2 786

X8X 1000

431

662 1000

120 219

314 429

x12 1000

928

432

231

1000

647

457

124

281

24X 972

513 996

7x1 1000

1000

412

650 8X5

187

111

385

266

803 1000

926 loo0

194 403

642

93

210 42X

68 964

6X

103 361

8

27

53

996

2 0

15

28

266

178

101

966

994

976

995

8 73

102 269

529

71

1

389

2

14 21

56

0.05

26

263

179

0.01

X2

w,ofP,+(Pz/P)=O

57

12

7

are f&c.

1

9

1000

762

879 1000

370

632

63 137

220 345

7x5 1Ow

922

623 1000

141 374

380

57

172

56

345

264

1000

74x

281

830

1000

887

544

235

155

1000

27

1000

725

241

59

43

Y18

15

17

50

94

965 1000

466

763

658

514

16

26

83

151

223

1000

Y4

23

7x

953

31

3

1

1

997

734

982

554

96

467

361

329

621

1

11

25

61

1000

916

657

535

404

917

14

14

45

x5

1000

963

754

640

48X

15

25

12 0

79

141

204

997

731

540

445

54

103

156

9Y3

628

427

345

240

W,of(PB,/P*)+l=O~

296

46

the row would read 926, 669, 926. 668.

1000

X90

558

78 102

196 288

497 1000

840

324

1000

124

38 115

135

994

265

52

1X

14

997

561

359

32

9x

86 163

152

954

33

3

995

281

68

4

1

3

18

52 964

121 316 9X2

27

996

44 67

997

569

2x1

18X

101

0.05

KofP+(&/P,)=O

0

16

964

60

6

1

0

994

371

F

factor restrictions

a For w,. in the case where /j’, = 0.5. p = 0.5 and & = 0.1. and with 2000 observations.

464

672

20

p,=o.5

18X

185

322

20

2X6

972

359

273

519

996

100

500

463

126

286

50

p,=o.5

78

&=O.l

50

137

30

200

20

p = 0.5

995

500

p, = 0.5

976

535

117

2x7

216

50

100

&=0.4

50

7x

131

30

201

20

10x

0.05

0.05

0.01

F

x2

W,ofp,p+p,=o

using Wald statistics when common

p = 0.5

N

of rejections

p,=os

Case

Number

Table 2

612

1

8

41 21

1000

494 631 90x

355

11 0

91 44

141

* 2 3 3 % 3 $ r: $ $ z c %

$ B k 5 5

2 =z \

S 3

409 615 992

Z-G ._3 196 307

1

A. W. Gregory,

M. R. Veull /

W&J tests of comn~on juctor

207

restrrctiom

each test at the 5 and 1 percent levels of confidence are registered using both the x2 and F distributions. Each experiment is done for sample sizes of N = 20, 30, 50, 100 and 500 observations. Since this kind of test generally would use standard regression output, the estimated variances are obtained

by

the least

squares

formula

with

the degrees

of freedom

adjustment

N - 3 in the

denominator. 3 Table 1 contains a selection of our Monte Carlo results in which the null hypothesis is true. The 95 percent confidence intervals for the expected number of rejections for 1000 repetitions at the 5 percent and 1 percent levels are [36, 651 and [4, 161, respectively. empirical sizes consistently within these intervals and compared across Wald tests are much more substantial.

For example,

However, none of the tests provides to our earlier work, the differences

using the chi-square

5 percent

level at the

N = 20 level, the number of rejections for the same data varies from 5 to 347. Moreover, for some parameter settings, important differences persist to N = 500. Unlike the results in our earlier work, there is no one form that can perform well over all the different parameter settings. The small sample performance of each formulation depends heavily upon the particular parameter values. For instance, the multiplicative form tends to reject a true null hypothesis too often as the serial correlation coefficient approaches the unit root. This tendency for the product form to overreject remains when comparisons are made against the F distribution and even when N is fairly large. Also, for certain configurations of the parameters, some ratio forms of the test are biased towards the null hypothesis. That is, they fail to reject a true null hypothesis the number of times consistent with asymptotic theory. However, same ratio form can now reject the null too frequently. Even in cases where the empirical

sizes appear

similar,

for another

set of parameter

values, the

there may indeed be substantial

conflicts

between tests. For example, for & = 1.0, p = 0.5, & = 0.5, N = 20 and a significance level of 5 percent, the results for W, (68 rejections) and W, (61 rejections) seem similar. However, this conceals that there are 38 cases where W, rejects but W, does not reject and 31 cases where the opposite occurs for a total of 69 conflicts. Hence we see a ‘conflict percentage’ in excess of the test size. Turning to table 2, we observe that the various forms of the Wald test are markedly different in their ability to detect false restrictions. However, examining tables 1 and 2, we can see that these differences are largely attributable to differences in test size, thus making power comparisons difficult. For this experiment, there does not seem to be any one form that can provide test results over a wide range of parameter values.

reasonable

3. Conclusion Although asymptotically the form of the non-linear restriction is irrelevant, the Monte Carlo evidence for testing common factor restrictions suggests that for finite samples, persistent and important differences can occur merely through rearrangement of the null hypothesis. Unfortunately in this experiment and contrary to our earlier work, there is no one formulation of the restriction which dominates (in terms of empirical test size) the others over all the parameter continue to favour the multiplicative form, applied researchers employing Wald

settings. While we tests of non-linear

restrictions should at least make explicit the form in which the restrictions are tested and perhaps should also consider the effects on test conclusions with alternative formulations. Finally these results emphasize the need for an analytical resolution to the problem of Wald test sensitivity. 3 We did some variance the results.

calculations

using N in the denominator

but found that this did not change

the qualitative

nature

of

208

A. W. Gregory, M. R. Veirll / Weld tests

of common factorrestrictions

References Burguete, J.F.. A.R. Gallant and G. Souza, 1982, On unification of the asymptotic theory of nonlinear econometric models, Econometric Review 1, 151-190. Gregory, A.W. and M.R. Veall, 1985, On formulating Wald tests of nonlinear restrictions, Econometrica 53. 1465-1468. Gregory, A.W. and M.R. Veall, 1986, Wald tests of the rational expectations hypothesis, Unpublished manuscript (University of Western Ontario, London). Hendry, D.F. and G.E. Mizon, 1978, Serial correlation as a convenient simplification. not as a nuisance: A comment on a study of the demand for money by the Bank of England, Economic Journal 88, 549-563. Lafontaine, F. and K.J. White, 1986, Obtaining any Wald statistic you want, Economics Letters 21, 35-40. Silvey, SD., 1975, Statistical inference (Chapman and Hall, London). Wald. A., 1943, Tests of statistical hypotheses concerning several parameters when the number of observations is large, Transactions of American Mathematical Society 54, 426-482.