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The size bias of White's information matrix test
Economics Letters North-Holland
24 (1987) 63-67
63
THE SIZE BIAS OF WHITE’S INFORMATION Larry W. TAYLOR
*
Unioersi
Lxhrgh
Received Accepted
MATRIX TEST
PA 18015,
USA
16 December 7986 24 March 1987
A recently substantial
proposed diagnostic test for maximum-likelihood estimators is examined. and this makes the test impractical to use in its current form.
It is found
that
the size bias is very
Recently White (1982) proposed a diagnostic testing procedure based upon the asymptotic equivalence of the Hessian and outer product forms of Fisher’s information matrix. Lancaster (1984) and Chesher (1983) derived a simple Lagrange multiplier form of the test by altering its covariance matrix. The purpose of this paper is to show that the size properties of the simplified version are very poor. Monte Carlo methods are used for both the linear and tobit frameworks to illustrate the size problems. The information matrix (IM) test is very general and applicable to any maximum-likelihood problem. Of particular interest is the widely used linear model
c,-N(0, a’),
yn=x,J+~,,
(1)
where X, is a 1 X K vector of exogenous variables and p a K concave log-likelihood function for the linear model is
E
ln[ f( y,; f9)] = - p
ln(2a)
- r
In a* -
n=l
5
(”
~~~“*
vector of coefficients.
X 1
The globally
(2)
.
n=l
Under general regularity conditions, consistent estimates of the parameters can be obtained by maximizing (2) with respect to ,B and a*. Another population model among empiricists is the censored normal model, popularly referred to as tobit. For the tobit framework, consider the following relationship: y,=x,P+e
n,
= 0,
if
(x,P+e,)>O,
if
(xnP+c,)lO,
(3)
* I would like to thank Cornelis Los for helpful comments that I received at the 1986 American Econometric Society. All shortcomings of the paper, however, should be credited to me.
016%1765/87/$3.50
0 1987, Elsevier Science Publishers
B.V. (North-Holland)
Summer
Meetings
of the
with s,,, J;,, and E,, defined is
5 In[f(J;,; ,I= I
0)] =C
as in (1). Olsen’s (1978) globally
ln[F(-x,,P)]
+C
0
I
concave
log-likelihood
function
ln(h)-1/2C(hl;,-x,,p). 1
for tobit
(4)
C,, is the summation over all observations where ??,I= 0, C, is the summation over all observations where );, > 0, F( -x,,/3) is the cumulative standard normal distribution function evaluated at -x/3, and h = l/a. Maximum-likelihood estimation can again be used to get consistent estimates of /3 and 0’. Of course, if the likelihood function is misspecified in either the linear framework or for tobit. then consistency of the estimators is no longer guaranteed. Such misspecifications include omitted variables, misspecified functional form, simultaneity bias, and mismeasured regressors. For the tobit model, a mismeasured dependent variable will also result in estimator inconsistency [cf. Stapleton and Young (1984)]. The IM test is designed to detect this inconsistency. The IM test can become quite involved computationally and is cumbersome due to the third derivatives of the log-likelihood function required to form the test statistic. Fortunately, the Lancaster and Chesher (henceforth LC) version is much easier to compute and is asymptotically equivalent to White’s, To define the LC test. let
d,O;,;
~)]/ae,)
0) = (a ln[f(J,,:
i=l.....
p.
j=i
,....
p.
.(a
ln[f(.r;,:
fl)]/ae,)
+ (a? ln[f(.,;,:
~)l/a~,~~,).
l=l,...,q.
where QI Y, p-
= an N x q matrix = an N x p matrix =
(f,.
whose rows are of the form d,( y,,; 6); I = 1,. whose rows are of the form a ln[f(?;,; e)]/a0,;
, q; .j = 1,.
, p;
I$);
= column of N ones; SSE = sum of the squared The LC test statistic is i
nz = N(l
- SSE/N)
errors
for the regression
= N - SSEd$
of i on p.
(5)
which is a very simple alternative to the original version as presented in White (1982. p. 11). Finite-sample properties of the LC version are now presented. Simulation experiments indicate that asymptotic theory for the LC test is not sufficient to explain finite-sample behavior concerning its size. The test tends to reject a correct null hypothesis too often. The model used for the simulation experiments is I;,=I.O+~.OX,,,+~.OX~,~+~~
with
c,,-N(O,l).
(6)
L. W. Taylor / The sue bras of White’s mformation
Given
model (6) the symmetric Paran
F,,
F,,
63
F,4
F-1
F23
G4
Ku
I::.
matrix
has the form:
,ter vector
PO PI P2
F74
F44
information
mrrtrix test
1
a
2
1
Several combinations of ‘indicators’ [cf. White (1982, p. 9)] could be chosen when computing White’s test. Such combinations of indicators correspond to using different combinations of the d,(_r,,; 0)‘s for the LC test. For illustrative purposes, four combinations were chosen for this paper; those corresponding to row 4, rows 3 and 4, rows 2 through 4 and, finally, rows 1 through 4 but excluding the indicator associated with F,,. As was suggested by White, the intercept indicator is identically equal to zero for both the linear and tobit models and thus useless in a statistical setting. For the LC test, the size problems are so severe that a slight modification of the test was . made to improve the finite-sample properties. Instead of using all of the derivatives for the Y2 vector described above, only the first derivatives corresponding to the rows were used [i.e., the derivative [Cl In f( y,; e)]/aa * was used for the ‘ Row 4’ test, [ i3 In f( y,,; B)]/ap, and [ 8 In f( y,:,,; b)]/aa’ were used for the ‘Rows 3 and 4’ test, etcetera]. Dropping some of the elements in the gradient vector will cause the test to have a smaller size. Unfortunately, this modification did not greatly improve the size properties of the test. An evaluation of the size properties of the LC test is now given. Size properties were evaluated by using significance levels of 5 and 10 percent with samples ranging from N = 20 to N = 8000 for the linear model and from N = 40 to N = 1000 for tobit. Referring to table 1, it is clear that the actual size of the LC test is much larger than the nominal size for the smaller samples. For example, using 1000 Monte Carlo replications for the linear model. ‘Rows l-4’ rejected the null hypothesis of correct specification 752 times for a sample size of N = 20. 810 for N = 50, 346 for N = 500, 220 for N = 1000 and 85 for N = 8000, as opposed to a nominal expected value of 50 at the 5% level of significance. ‘Rows 2-4’ performs better but has an unacceptable number of rejections even for rather large sample sizes. The results for the tobit model concur that the current form of the LC test is unacceptable. It does seem a bit surprising that the LC test appears to have better size properties for the tobit model than for the linear model. Further research is needed to discover the reason for this particular phenomenon and, in general, why the test behaves so poorly. A brief examination of White’s original version of the IM test follows next and indicates that there are again size problems. An anonymous referee has pointed out that White’s original test and the modified LC test will not be asymptotically equivalent except in special circumstances. This is due to the omission of some of the gradient vector elements when computing the modified LC test. The purpose for including all of the elements of the gradient vector is to compensate for the presence of estimated parameters in the indicators. Without this correction, asymptotically the test will generally have a true size smaller than the nominal size (e.g., a test based on the 5% level of significance will have a true size of less than 5%). Of course, in light of the Monte Carlo results, the fact that the modified LC test will not reject often enough in very large samples may not be of much consolation for the typical empiricist who deals with relatively small data sets. For example, the ‘Rows l-4’ version of the test has the correct asymptotic properties since it uses all of the gradient elements - but this test rejects entirely too often for even N = 8000. There is at least one situation for which the modified LC test will be asymptotically equivalent to White’s original version.
66
L. W. Taylor / The sire bias of White’s informatmn
Table 1 Size evaluation of the LC test. ’ Model: Y=l.O+l.O X, +l.O X, +Normal
mnrrix test
(0, 1).
Model data
N
Row 4
Rows 3 and 4
Rows 2-4
Rows l-4
10%
5%
10%
5%
10%
5%
10%
5%
OLS/Normal
20 50 100 250 500 1000 2000 3000 4000 6000 8000
0.385 0.293 0.224 0.176 0.166 0.147 0.121 0.098 0.092 0.102 0.100
0.312 0.243 0.173 0.115 0.106 0.098 0.070 0.052 0.055 0.052 0.060
0.653 0.529 0.416 0.294 0.228 0.183 0.142 0.133 0.133 0.116 0.120
0.528 0.450 0.326 0.209 0.159 0.119 0.088 0.073 0.076 0.072 0.066
0.824 0.727 0.590 0.429 0.358 0.247 0.181 0.161 0.153 0.144 0.153
0.711 0.631 0.517 0.327 0.260 0.174 0.110 0.107 0.084 0.087 0.069
0.908 0.879 0.736 0.556 0.448 0.300 0.209 0.184 0.165 0.153 0.142
0.752 0.810 0.653 0.449 0.346 0.220 0.144 0.120 0.106 0.104 0.085
Tobit/Normal
40 100 250 500 1000
0.219 0.169 0.117 0.108 0.093
0.135 0.106 0.075 0.059 0.054
0.398 0.281 0.244 0.184 0.133
0.266 0.203 0.154 0.123 0.081
0.669 0.523 0.434 0.317 0.196
0.537 0.425 0.331 0.230 0.134
0.870 0.767 0.592 0.486 0.291
0.773 0.666 0.511 0.369 0.212
Tobit/Uniform
40 100 250 500 1000
0.271 0.172 0.137 0.121 0.122
0.196 0.128 0.087 0.083 0.068
0.484 0.337 0.241 0.204 0.179
0.392 0.267 0.175 0,142 0.101
0.628 0.495 0.328 0.278 0.239
0.508 0.388 0.249 0.208 0.164
0.761 0.615 0.435 0.352 0.267
0.644 0.520 0.342 0.259 0.199
a 1000 replications were performed for each test. The values reported are the percentage of rejections. Standard errors are calculated as the square root of ~(1 - p)/lOOO, where p is the percentage of rejections from the table. The ‘Row 4’ test has 1 degree of freedom (d.f.), the ‘Rows 3 and 4’ test has 3 d.f., the ‘Rows 2-4’ test has 6 d.f., and the ‘Rows l-4’ test has 9 d.f. For Normal data X, and X, were id Normal (0,l). For Uniform data X, was still Normal (0, 1) but X, was Uniform (0, 20). The regressors were fixed in repeated samples (i.e., a new set of c’s were generated for each replication, but X,, and X,, were fixed). Different seeds for the random number generator yield similar results to those reported. An IBM mainframe, the IMSL statistical package and a double-precision FORTRAN program were used to investigate the LC test.
Consider combination ‘Row 4’ of the modified LC test in the linear regression framework. This test will have the correct asymptotic size because the estimation of /I does not affect the asymptotic distribution of the indicator and because the maximum-likelihood estimator (MLE) for p is uncorrelated with the MLE for u2. One thousand Monte Carlo replications for White’s original test using the ‘Row 4’ combination were performed to be compared with the LC version. At the 10 percent level of significance, there were 264 rejections for N = 20, 248 for N = 50, 207 for N = 100, 163 for N = 250,164 for N = 500 and 144 for N = 1000 as opposed to an expected 100 rejections. At the 5 percent level of significance, there were 210 rejections for N = 20, 214 for N = 50, 160 for N = 100, 112 for N = 250, 104 for N = 500 and 98 for N = 1000. These results are similar to those obtained for the LC test and indicate that there are also size problems for White’s original test. The Monte Carlo results then indicate that some type of finite sample alterations could be very valuable for the IM test. Corrections are indeed necessary before any formal evaluation of the power properties of the test can be undertaken. Such was the attempt in this paper when the LC test was modified. Another approach which may also hold promise is to provide finite-sample corrected critical values for the IM test without actually trying to alter the test statistic directly. Phillips’ (1983)
L. W. Taylor / The size bias of White’s information
extended rational values. Preliminary
approximation procedure is being results indicate that this approach
matrix test
considered in order to provide may work very well.
61
those
critical
References Chesher, A., 1983, The information matrix test: Simplified calculation via a score test interpretation. Economics Leters 13. 45-48. Lancaster, T., 1984. The covariance matrix of the information matrix test, Econometrica 52. 1051-1054. Olsen, R.J., 1978, Note on the uniqueness of the maximum likelihood estimator for the Tobit model, Econometrica 5. 1211-1215. Phillips, P.C.B.. 1983, ERAS: A new approach to small sample theory, Econometrica 51. 1505-1525. Stapleton, D.C. and D.J. Young, 1984, Censored normal regression with measurement error on the dependent variable, Econometrica 52, 737-760. White, H., 1982. Maximum likelihood estimation of misspecified models, Econometrica 50, l-25.
24 (1987) 63-67
63
THE SIZE BIAS OF WHITE’S INFORMATION Larry W. TAYLOR
*
Unioersi
Lxhrgh
Received Accepted
MATRIX TEST
PA 18015,
USA
16 December 7986 24 March 1987
A recently substantial
proposed diagnostic test for maximum-likelihood estimators is examined. and this makes the test impractical to use in its current form.
It is found
that
the size bias is very
Recently White (1982) proposed a diagnostic testing procedure based upon the asymptotic equivalence of the Hessian and outer product forms of Fisher’s information matrix. Lancaster (1984) and Chesher (1983) derived a simple Lagrange multiplier form of the test by altering its covariance matrix. The purpose of this paper is to show that the size properties of the simplified version are very poor. Monte Carlo methods are used for both the linear and tobit frameworks to illustrate the size problems. The information matrix (IM) test is very general and applicable to any maximum-likelihood problem. Of particular interest is the widely used linear model
c,-N(0, a’),
yn=x,J+~,,
(1)
where X, is a 1 X K vector of exogenous variables and p a K concave log-likelihood function for the linear model is
E
ln[ f( y,; f9)] = - p
ln(2a)
- r
In a* -
n=l
5
(”
~~~“*
vector of coefficients.
X 1
The globally
(2)
.
n=l
Under general regularity conditions, consistent estimates of the parameters can be obtained by maximizing (2) with respect to ,B and a*. Another population model among empiricists is the censored normal model, popularly referred to as tobit. For the tobit framework, consider the following relationship: y,=x,P+e
n,
= 0,
if
(x,P+e,)>O,
if
(xnP+c,)lO,
(3)
* I would like to thank Cornelis Los for helpful comments that I received at the 1986 American Econometric Society. All shortcomings of the paper, however, should be credited to me.
016%1765/87/$3.50
0 1987, Elsevier Science Publishers
B.V. (North-Holland)
Summer
Meetings
of the
with s,,, J;,, and E,, defined is
5 In[f(J;,; ,I= I
0)] =C
as in (1). Olsen’s (1978) globally
ln[F(-x,,P)]
+C
0
I
concave
log-likelihood
function
ln(h)-1/2C(hl;,-x,,p). 1
for tobit
(4)
C,, is the summation over all observations where ??,I= 0, C, is the summation over all observations where );, > 0, F( -x,,/3) is the cumulative standard normal distribution function evaluated at -x/3, and h = l/a. Maximum-likelihood estimation can again be used to get consistent estimates of /3 and 0’. Of course, if the likelihood function is misspecified in either the linear framework or for tobit. then consistency of the estimators is no longer guaranteed. Such misspecifications include omitted variables, misspecified functional form, simultaneity bias, and mismeasured regressors. For the tobit model, a mismeasured dependent variable will also result in estimator inconsistency [cf. Stapleton and Young (1984)]. The IM test is designed to detect this inconsistency. The IM test can become quite involved computationally and is cumbersome due to the third derivatives of the log-likelihood function required to form the test statistic. Fortunately, the Lancaster and Chesher (henceforth LC) version is much easier to compute and is asymptotically equivalent to White’s, To define the LC test. let
d,O;,;
~)]/ae,)
0) = (a ln[f(J,,:
i=l.....
p.
j=i
,....
p.
.(a
ln[f(.r;,:
fl)]/ae,)
+ (a? ln[f(.,;,:
~)l/a~,~~,).
l=l,...,q.
where QI Y, p-
= an N x q matrix = an N x p matrix =
(f,.
whose rows are of the form d,( y,,; 6); I = 1,. whose rows are of the form a ln[f(?;,; e)]/a0,;
, q; .j = 1,.
, p;
I$);
= column of N ones; SSE = sum of the squared The LC test statistic is i
nz = N(l
- SSE/N)
errors
for the regression
= N - SSEd$
of i on p.
(5)
which is a very simple alternative to the original version as presented in White (1982. p. 11). Finite-sample properties of the LC version are now presented. Simulation experiments indicate that asymptotic theory for the LC test is not sufficient to explain finite-sample behavior concerning its size. The test tends to reject a correct null hypothesis too often. The model used for the simulation experiments is I;,=I.O+~.OX,,,+~.OX~,~+~~
with
c,,-N(O,l).
(6)
L. W. Taylor / The sue bras of White’s mformation
Given
model (6) the symmetric Paran
F,,
F,,
63
F,4
F-1
F23
G4
Ku
I::.
matrix
has the form:
,ter vector
PO PI P2
F74
F44
information
mrrtrix test
1
a
2
1
Several combinations of ‘indicators’ [cf. White (1982, p. 9)] could be chosen when computing White’s test. Such combinations of indicators correspond to using different combinations of the d,(_r,,; 0)‘s for the LC test. For illustrative purposes, four combinations were chosen for this paper; those corresponding to row 4, rows 3 and 4, rows 2 through 4 and, finally, rows 1 through 4 but excluding the indicator associated with F,,. As was suggested by White, the intercept indicator is identically equal to zero for both the linear and tobit models and thus useless in a statistical setting. For the LC test, the size problems are so severe that a slight modification of the test was . made to improve the finite-sample properties. Instead of using all of the derivatives for the Y2 vector described above, only the first derivatives corresponding to the rows were used [i.e., the derivative [Cl In f( y,; e)]/aa * was used for the ‘ Row 4’ test, [ i3 In f( y,,; B)]/ap, and [ 8 In f( y,:,,; b)]/aa’ were used for the ‘Rows 3 and 4’ test, etcetera]. Dropping some of the elements in the gradient vector will cause the test to have a smaller size. Unfortunately, this modification did not greatly improve the size properties of the test. An evaluation of the size properties of the LC test is now given. Size properties were evaluated by using significance levels of 5 and 10 percent with samples ranging from N = 20 to N = 8000 for the linear model and from N = 40 to N = 1000 for tobit. Referring to table 1, it is clear that the actual size of the LC test is much larger than the nominal size for the smaller samples. For example, using 1000 Monte Carlo replications for the linear model. ‘Rows l-4’ rejected the null hypothesis of correct specification 752 times for a sample size of N = 20. 810 for N = 50, 346 for N = 500, 220 for N = 1000 and 85 for N = 8000, as opposed to a nominal expected value of 50 at the 5% level of significance. ‘Rows 2-4’ performs better but has an unacceptable number of rejections even for rather large sample sizes. The results for the tobit model concur that the current form of the LC test is unacceptable. It does seem a bit surprising that the LC test appears to have better size properties for the tobit model than for the linear model. Further research is needed to discover the reason for this particular phenomenon and, in general, why the test behaves so poorly. A brief examination of White’s original version of the IM test follows next and indicates that there are again size problems. An anonymous referee has pointed out that White’s original test and the modified LC test will not be asymptotically equivalent except in special circumstances. This is due to the omission of some of the gradient vector elements when computing the modified LC test. The purpose for including all of the elements of the gradient vector is to compensate for the presence of estimated parameters in the indicators. Without this correction, asymptotically the test will generally have a true size smaller than the nominal size (e.g., a test based on the 5% level of significance will have a true size of less than 5%). Of course, in light of the Monte Carlo results, the fact that the modified LC test will not reject often enough in very large samples may not be of much consolation for the typical empiricist who deals with relatively small data sets. For example, the ‘Rows l-4’ version of the test has the correct asymptotic properties since it uses all of the gradient elements - but this test rejects entirely too often for even N = 8000. There is at least one situation for which the modified LC test will be asymptotically equivalent to White’s original version.
66
L. W. Taylor / The sire bias of White’s informatmn
Table 1 Size evaluation of the LC test. ’ Model: Y=l.O+l.O X, +l.O X, +Normal
mnrrix test
(0, 1).
Model data
N
Row 4
Rows 3 and 4
Rows 2-4
Rows l-4
10%
5%
10%
5%
10%
5%
10%
5%
OLS/Normal
20 50 100 250 500 1000 2000 3000 4000 6000 8000
0.385 0.293 0.224 0.176 0.166 0.147 0.121 0.098 0.092 0.102 0.100
0.312 0.243 0.173 0.115 0.106 0.098 0.070 0.052 0.055 0.052 0.060
0.653 0.529 0.416 0.294 0.228 0.183 0.142 0.133 0.133 0.116 0.120
0.528 0.450 0.326 0.209 0.159 0.119 0.088 0.073 0.076 0.072 0.066
0.824 0.727 0.590 0.429 0.358 0.247 0.181 0.161 0.153 0.144 0.153
0.711 0.631 0.517 0.327 0.260 0.174 0.110 0.107 0.084 0.087 0.069
0.908 0.879 0.736 0.556 0.448 0.300 0.209 0.184 0.165 0.153 0.142
0.752 0.810 0.653 0.449 0.346 0.220 0.144 0.120 0.106 0.104 0.085
Tobit/Normal
40 100 250 500 1000
0.219 0.169 0.117 0.108 0.093
0.135 0.106 0.075 0.059 0.054
0.398 0.281 0.244 0.184 0.133
0.266 0.203 0.154 0.123 0.081
0.669 0.523 0.434 0.317 0.196
0.537 0.425 0.331 0.230 0.134
0.870 0.767 0.592 0.486 0.291
0.773 0.666 0.511 0.369 0.212
Tobit/Uniform
40 100 250 500 1000
0.271 0.172 0.137 0.121 0.122
0.196 0.128 0.087 0.083 0.068
0.484 0.337 0.241 0.204 0.179
0.392 0.267 0.175 0,142 0.101
0.628 0.495 0.328 0.278 0.239
0.508 0.388 0.249 0.208 0.164
0.761 0.615 0.435 0.352 0.267
0.644 0.520 0.342 0.259 0.199
a 1000 replications were performed for each test. The values reported are the percentage of rejections. Standard errors are calculated as the square root of ~(1 - p)/lOOO, where p is the percentage of rejections from the table. The ‘Row 4’ test has 1 degree of freedom (d.f.), the ‘Rows 3 and 4’ test has 3 d.f., the ‘Rows 2-4’ test has 6 d.f., and the ‘Rows l-4’ test has 9 d.f. For Normal data X, and X, were id Normal (0,l). For Uniform data X, was still Normal (0, 1) but X, was Uniform (0, 20). The regressors were fixed in repeated samples (i.e., a new set of c’s were generated for each replication, but X,, and X,, were fixed). Different seeds for the random number generator yield similar results to those reported. An IBM mainframe, the IMSL statistical package and a double-precision FORTRAN program were used to investigate the LC test.
Consider combination ‘Row 4’ of the modified LC test in the linear regression framework. This test will have the correct asymptotic size because the estimation of /I does not affect the asymptotic distribution of the indicator and because the maximum-likelihood estimator (MLE) for p is uncorrelated with the MLE for u2. One thousand Monte Carlo replications for White’s original test using the ‘Row 4’ combination were performed to be compared with the LC version. At the 10 percent level of significance, there were 264 rejections for N = 20, 248 for N = 50, 207 for N = 100, 163 for N = 250,164 for N = 500 and 144 for N = 1000 as opposed to an expected 100 rejections. At the 5 percent level of significance, there were 210 rejections for N = 20, 214 for N = 50, 160 for N = 100, 112 for N = 250, 104 for N = 500 and 98 for N = 1000. These results are similar to those obtained for the LC test and indicate that there are also size problems for White’s original test. The Monte Carlo results then indicate that some type of finite sample alterations could be very valuable for the IM test. Corrections are indeed necessary before any formal evaluation of the power properties of the test can be undertaken. Such was the attempt in this paper when the LC test was modified. Another approach which may also hold promise is to provide finite-sample corrected critical values for the IM test without actually trying to alter the test statistic directly. Phillips’ (1983)
L. W. Taylor / The size bias of White’s information
extended rational values. Preliminary
approximation procedure is being results indicate that this approach
matrix test
considered in order to provide may work very well.
61
those
critical
References Chesher, A., 1983, The information matrix test: Simplified calculation via a score test interpretation. Economics Leters 13. 45-48. Lancaster, T., 1984. The covariance matrix of the information matrix test, Econometrica 52. 1051-1054. Olsen, R.J., 1978, Note on the uniqueness of the maximum likelihood estimator for the Tobit model, Econometrica 5. 1211-1215. Phillips, P.C.B.. 1983, ERAS: A new approach to small sample theory, Econometrica 51. 1505-1525. Stapleton, D.C. and D.J. Young, 1984, Censored normal regression with measurement error on the dependent variable, Econometrica 52, 737-760. White, H., 1982. Maximum likelihood estimation of misspecified models, Econometrica 50, l-25.