- Email: [email protected]
Distributional specification tests against semiparametric alternatives
Journal
of Econometrics
47 (1991) 175-194.
North-Holland
Distributional specification tests against semiparametric alternatives* Simon Peters Unir~ersityof Bristol, Bristol BS8 IW,
Richard
U.K.
J. Smith
Unkersity of Cambridge, Cambridge CB3 900,
U.K.
Score or Lagrange multiplier versions of a Hausman test for distributional specification are presented for limited dependent variable models. These tests are based on the first derivatives of semiparametric criterion functions associated with robust estimation of such models and only require the computation of the maximum likelihood estimator. Various regression forms of the statistic are also presented. Monte Carlo results for the Tobit model indicate that such tests may be efficacious in detecting situations in which the maximum likelihood estimator is seriously biased due to incorrect distributional specification.
1. Introduction
This paper considers distributional specification tests for parametric models estimated by the method of maximum likelihood. As is now widely appreciated for limited dependent variable models, failure of the distributional assumption is usually fatal for the consistency property of the maximum likelihood estimator (MLE). Recently, therefore, there has been an upsurge of interest in robust estimation techniques for such models. In particular, such methods typically specify only part of the generating process for the data, leaving the remainder to be dealt with in a nonparametric way. Such methods are termed semiparumetric; for example, see inter alia Powell (1984,1986) for two approaches to the semiparametric estimation of a Tobit model. For some *We are grateful for the helpful comments of the Editors, two anonymous referees, Halbert White, and other participants at the ESRC Econometric Study Group Conference, 1989, University of Warwick, where an earlier version of this paper was presented. Research for this paper was supported partially by the ESRC under grant B 0023 2150.
0304-4076/91/$03.500
l991-Elsevier
Science
Publishers
B.V. (North-Holland)
problems, such as those considered by Powell. the resultant semiparametric estimator is root-n consistent and possesses a limiting Normal distribution. properties shared with the MLE in the absence of misspecification. In the context of this paper. the assumptions of such semiparametric techniques. implicit or otherwise, are the maintained hypothesis for the specification tests considered below. For the Tobit model with normally distributed errors, Newey (1987) proposed a Hausman (1978) specification test for this distributional assumption based on the difference of the Tobit MLE and Powell’s (lY86) symmetrically censored least-squares estimator (SCLSE). Clearly. such an approach has wide applicability. In this papcl-. a similar approach is taken: however, a score or Lagrange multiplier statistic is proposed which only requires the parametric or null MLE in contradistinction to the Hausman statistic which requires both null and alternative (semiparametric) estimators. Section 2 outlines the basic components which undcrly the suggested statistic and the limiting distribution theory for both parametric and semiparametric estimation methods. The Hausman score test statistic is presented along with some variants in section 3. Section 4 consists of the results of some Monte Carlo experiments on the Hausman and Hausman score statistics for the Tobit model comparing the Tobit MLE and the SCLSE, whereas section 5 examines the Hausman score statistic using as alternative estimator the censored least absolute deviations estimator (CLADE) due to Powell (19X4). Section 6 contains some concluding remarks.
2. Parametric Given
and semiparametric
a sample
estimation
of independently
and identically distributed observations joint density function of the random variable y and random k-vector s. a usual objective is to perform inference on some aspects of the conditional density function of y given .I-. for example, conditional moments or quantiles of y given x. The parametric estimation procedure most commonly adopted is to fully specify a conditional density function for the random variable y given .x. f’tylx; HI, known up to and differentiable in the p-vector of parameters 0. and to assume that the random vector .‘r is weakly exogenous for 0. Generally, some partial reparameterisation of the parameter vector 0, ch = (b(H), captures the parametric components of those conditional moments or quantiles of interest. Under the correctness of these distributional assumptions. an appropriate estimator for the true value H,, of 0 is the maximum likelihood estimator (MLE) 6,!, which satisfies the first-order conditions or likelihood equations: ( y,. x,), i = 1.. . , II, from an (unknown)
S. Peters and R.J. Smith, Distributional
specification
tests
177
where ~~(0) = d Inf(yiIxi; O)/#, the score vector for observation i, i = 1, . . . , II. Under suitable regularity conditions, see Huber (1967) and the append&, and satisfaction of the assumption of correct distributional specification, 8, is a root-n consistent estimato_r for f3”. In particular, $n is a weakly On= e,,, and obeys a Central Limit consistent estimator for e,, plim.,, Theorem:
n”2(6n - e,) : N,(O,j,-'),
(2)
where jO=
limn-’ n~m
~E,,,[Si(e”)Si(eo)‘] i=l
and E,[ .I denotes the expectation of [ .] taken with respect to the assumed conditional density function f(y IX;0). The limiting distributional result (2) is obtained via a stochastic first-order Taylor series expansion of S, of (1) about e,,:
n1/2s,=/on1/2(fin - eo)
+op( i),
(3)
where
For limited dependent variable models where, typically, the random variable y may be regarded as a censored observation on some underlying latent variable y*, failure of the conditional distributional assumption f(ylx; 0) may cause inconsistency in the MLE O,,, that is plim, j_ 8, # t9”. An alternative approach is to partially specify parametrically a particular feature of interest of the (unknown) conditional density of y or that of the underlying latent variable y*, such as the conditional mean or median of y*, and to derive an appropriate criterion function based on the censoring mechanism linking y to y*. As above, let this semiparametric specification be parameterised by the q-vector of parameters 4 with true value &. If the fully parametric specification f(y lx; 0) were correct, then 4 = 4(O), with & = 4(0,>, where the q-vector of functions $4.) is continuously differentiable in 0 and the (p,q) derivative matrix @ = a@/ae is full column rank at least in a neighbourhood of 0”. It is also assumed that the random vector x is weakly exogenous for 4, as in the classical case. The semiparametric estimator (SPE) 4, satisfies: ,. 4, = argminn-’ 4
t i=l
R( yi, xi; 4),
where the right-hand side denotes a suitable criterion function; for example, see Powell (1984,1986). In most cases, the above criterion function will be only left differentiable and the first-order condition may only be stated asymptotically; that is:
where g,(b) = dR( y,. x,; c#I)/~c$. i = I. , II. Under appropriate conditions, it $,, is weakly consistent for ch,, and satisfies the asymptotic first-order condition given above, it may be shown that
11
’ 2 c s,( d,,) I-
+
I?’ 2(E[g(d,)]),!,~,;,,,=(),‘(1).
(4)
I
where E[.] denotes expectation of [ ‘1 taken with respect to the true ,joint density of y and x and ~(4) = dR(~t, s: d)/dd; see Huber (1967, theorem 3, p. 230: 1981, theorem 3.1, p. 13.1) and Powell (1984, lemma A..?. p. 317). Assuming that E[g(b)l possesses a nonsingular derivative matrix at cb,,. this allows a Central Limit Theorem to be stated for ch,, given the weak exogeneity of .Y for ch: viz.:
where
and
where E,,,,[.l denotes expectation taken with respect to the true conditional density of y given x. In a similar manner to the use of eq. (3), the limiting distribution result (5) is obtained from (4) as
I”$&, = -Qz~2(f& where
- cb,,)+
O/J 1).
(7)
S. Peters and R.J. Smith, Distributional specification tests
Examples of particular sections 4 and 5. 3. The Hausman
criterion
score test statistic
179
functions and models are presented
for distributional
in
specification
Denote the hypothesis of correct distributional specification f(y(x; 0) by H, with corresponding alternative hypothesis Hr. The Hausman _(1978) statistics discussed below consider the null hypothesis H,*: plim. _+m4, = 4,, (and associated alternative hypothesis H;“: plim,, em 4, Z &) concerning the parameters of interest 4, where 4, = +(e,). These test statistics have a well-defined limiting distribution under H, and, hence, are pure si&‘kznce tests for H “; see Cox and Hinkley (1974). In general, plim. _m 6, = +(0, ), where 0* satisfies lim,,, E+Js,,l = 0, s,, = n-‘C~=,si(e,) and E+,I[*l denotes expectation with respect to the true conditional density of y given X; see White (1982). Although H, (H;F) is sufficient for H,* (H,), Hz (H,) may not imply H, (H:); see Holly (1982). Therefore, under particular distributional misspecifications, the degree of inconsistency of 4, for & may be small or null. See Arabmazar and Schmidt (1982) for evidence concerning the Tobit model. Thus, these Hausman statistics are generally inconsistent tests of H, but, as rejection of H,* implies rejection of H,, may still provide useful specification tests for H,. As in section 2, assume that there exists an SPE 4, with the limiting distributional properties described in section 2 underA Hr. Then a Hausman test statistic based on the difference between 6, and 6, provides a test of H, against H, [see Hausman (1978) Hausman and Taylor (1981) Holly (1982) and Ruud (1984)]: HT, = n(&, - &)‘f,-(6,
- 6n),
(8)
where +a Ais a consistent estimator for the asymptotic variance I’,, of n’j2($,, - 4,) under H,, and [ .I- denotes a g-inverse for [ .I. Under appropriate conditions [see Andrews (1987)] and H,, HT, has a limiting x2 distribution with r degrees of freedom with r = rk(l/,).’ Under H,, 6, is asymptotically efficient which allows V. to be simply expressed as the difference of the asymptotic variance matrices of $J,, and 6,, viz.:
‘The following
conditions
are sufficient
for a statistic
X,‘A,X,
to have a limiting
,y*[rk(A)I
P[rk(A.) = rk(A)] = 1; (iv) distribution: (i) X,, 5 N(O, A); (ii) A 5 A; (iii) lim P[“A = A’ ] = 1. Co:zzons (i), (ii), (iv), and (v) are lim n_” P[X,, ER(A,,)I = 1; (v) lim,,,, relatively easy to verify for statistic (8); condizon C% must be assumed.
180
S. Peters and R.J. Smith, Distributional
specificatiorl
test\
It is necessary to choose the estimator q, of V,, to be positive semidefinite to avoid negative values of HT,, in finite samples. Choosing a common estimator for the unknown parameters in V,, is often sufficient. Newey (1987) has applied HT,, in the context of a Tobit model using the difference of the Tobit MLE and the symmetrically censored least-squares estimator of Powell (1986), which we consider further in section 4. The form of test statistic (8) suffers from the same computational disadvantage as the likelihood ratio statistic in that it requires the two estimators d;,, quite burdensome, and J,,. the second of which may be computationally especially for limited dependent variable models; see Paarsch (1984) for the censored least-absolute deviations estimator of Powell (1984). In the classical context of parametric restrictions, Ruud (1984) presented score test versions of HT,, based on the alternative hypothesis MLE. In contradistinction to Ruud (19841, the test statistic to be presented below uses the parametric MLE &,,, which in a semiparametric framework may offer considerable computational advantages over HT,,; in the familiar classical setting of Ruud (1984). the statistic would reduce to the usual score or Lagrange multiplier statistic for parametric restrictions. See White (1987) for a similar approach when testing dynamic specification. Consider the asymptotic first-order conditions. eqs. (4) and (7). Similar arguments to those of Huber (1967, 1981) allow the following proposition to be stated: Proposition. Under assumptions ( N-l). ( N-2 1, ( N-3). pendiv and if n’/‘(6,, - @,,I = O,,( 1). then:
and ( N-4 ) of’ the up-
li &(&,)
lz”2g,i=nm”2
I I = II
Proqf.
If2 &I,) i-l
+rr’/?(E[g(~)]),,~_,l,,+o,,(l).
See the appendix.
Expanding
the second
term in eq. (10) about
$,, gives
~~“‘(E[~(cb)])d,~~~,, =n”‘(E[g(d)]
)_
+(dE[g,(~)]/a~‘),-~~,,n’/‘(~,,-~,,) --n--l/”
kg,(&) I-l
+z,, n”Q,,
+o,,(l) -&)
+ o,>,(1).
( 10)
S. Peters and R.J. Smith, Distributional specification tests
181
which after substitution into eq. (10) leaves nl/2-
g, =
?T ‘i2( 6, - 4,) + OJ 1). #on
(11)
A further substitution into eq. (8) for n 1/2($ - 4,) from (111, using the weak consistency of 6,, for B0 and dropping asymitotically negligible terms, allows the statement of the following theorem for the efficient Hausman score statistic EHST,: Theorem. appendix :
(N-2), (N-3), (N-4), and (N-5) of the
Under assumptions (N-I),
has a limiting x2 distribution with r degrees of freedom under H,, where ‘In indicates evaluation at 6, and r = rk(Z,, - ECj@hA;‘@,,E,j).2
The second equality of (12) is obtained by noting the invariance to the choice of g-inverse; see Rao and Mitra (1971, lemma 2.2.4(n), (iii), p. 21). A number of alternative variants of the Hausman score statistic are available using different consistent estimators for V, of eq. (9). Firstly, define:
s,= (s,(e,),..., s&in))‘,
G,,= (gl(~n),...,g,<~,>>‘.
Secondly, using a similar result to the familiar information
matrix equality
‘Let h()?lx; QO,$O,iO) denote the true conditional density function of y given x after reparameterising 0 to (4. +) such that f( y Ix; 0) = h(y Ix; 4, I/J,0). Under local alternatives where 0 < [‘i < m, the Hausman statistic (8) and the efficient Hausman score 4, = i/n’/‘, statistic (12) will have a limiting x2 distribution with r degrees of freedom and noncentrality parameter
where
~~~(~,,,cL,,,o)-aInh(y,lx,;b,,cL,,O)/a~, and AP compare
= [ Am0
Holly (1982).
- A+W%&%‘,m”]
- ‘;
i=l,...,n,
etc.,
S. Peters and R.J. Smtth, Distrihutior~ul specificutim
IX2
evaluated
under
test.!
H ,,:
E,I,[K,(~(B,,))s,(H,~)‘] = (~E,[g,(~(Hll))l/~H'),~,,, =
-(“E&,(d)]/~~‘)
,,,_,,,,, @‘;I.
(13)
where EJ .] indicates expectation taken with respect to the H,, conditional density f(~\x; 0); compare Newey ( 1987. eq. (3.8), p. 131). Thus. from eq. (6):
and a consistent estimator for 5,, under distributional specification H ,, is given by ~
/I
Substitution alternative
where
the
null
hypothesis
of correct
s,,/y;,’ q,, ( c&p,, ’ G,,) ’
’ G,;
of this estimator for .?,, in EHSI;,of eq. (12) produces form of the Hausman score statistic:
L is an n-vector
each of whose elements
the
are unity and
_ _
L, = (‘0. q,q,/“,,- ’ )‘. The statistic regression:
(14) may be computed
as UK’ (uncentred)
from the artificial
noting that L$,‘,~ = 0. Making use of the fact that, under H,,. S,;S,,/fl is a consistent /“;,, a further alternative form of EHST,,is given by HST,;” = where
ld,,(
c,;M,
?/
estimator
&L.
- X(X'X) 'X'. This statistic M,y= I,,
may also be computed
for
( 15)
as ru?’
S. Peters and R.J. Smith, Distributional specification tests
(uncentred)
183
from the regression:
Finally, if we introduce the auxiliary (p - q)-vector of parameters 4 such that (4, $,> represents a bijective (one-to-one onto) transformation of 8, then similarly to eq. (13):
where E ,+,,I[.I denotes expectation taken with respect density f( y IX; c$, $1 after reparameterisation, and
to the H,, conditional
as
(aE,,l,,,,[s,(~)l/a~‘),=,,,,,=,,l = 0. Thus, another asymptotically under H, is given by
equivalent
form of the Hausman
score statistic
using the invariance of score-type statistics to bijective parameter transformations. This outer product of gradients form of the Hausman score statistic, as is usual, may be computed as the nR2 (uncentred) from the regression:
It should be emphasized that although all four forms of EHST, (121, (141, with r degrees of (151, and (16) possess the same limiting x * distribution freedom under H,, the finite-sample characteristics of these statistics may be very different. In the next two sections, the finite-sample performance of the efficient form of the Hausman score statistic (12) together with that of the Hausman statistic (8) itself is examined.
S. t’cters and R.J. Smith. Distributional
IXJ
specrfication
4. Monte Carlo experiments for the Tobit model censored least-squares estimator Consider
the familiar
Tobit
y, = max( 0, s;j3,, +
mode1 defined
u,}.
i=
I,
using
tern
the symmetrically
as follows: I1.
( 17)
where (Y,. x,) are assumed to constitute a random sample and the U, to be independently and identically distributed independently of x,, i = 1. . tt. Alternatively, the generating mechanism, eq. (17). may be cast in terms of a latent variable, viz.: !‘,* =
x;p,, + II, (
j’,
1( ?‘,*> 0) ).,*.
=
/=
I.
. Il.
where l(. ) denotes the indicator function. The null hypothesis H,, of correct distributional to the assumption that 14, -
IN(0.
CT,;,),
i= l.....,l.
which allows the use of the Tobit the likelihood equations:
+wt,+,
( 18)
specification
corresponds
( 19)
MLE for t/3,,, v~,:,). (a,,, 6,,!,,). which solves
:P,, i “;i;, I I =(I .y (?
,,,,
,
where c, = 1 if J, > 0 and 0 otherwise, 6, and 6, are the standard Normal density and cumulative distribution functions. respectively, evaluated at x:&/4,,,. As has been tion (19) may Schmidt (1982). function of U,
noted elsewhere, failure of the above distributional assumprender the Tobit MLE inconsistent: see Arabmazar and An alternative estimator which only assumes that the density is symmetric about zero is the symmetrically censored least-
S. Peters and R.J. Smith, Distributional specification tests
185
squares estimator (SCLSE) due to Powell (1986). In the context of section 3, the assumption of symmetric errors constitutes the alternative hypothesis H, and x,‘& is the conditional median of y* given xi, i = 1,. . . , n; that is:
Eb,,[1(Y; >xlPo)]= t,
i=l
>..*> II.
(20)
Returning to the notation of section 2, the criterion to be minimised is given by
n-1 CR(YjYxi;4) i=l
+n-’
5 1(~,>2x;p)((+y~)~-
(max{O,x;P})‘),
(21)
i=l
where, as above, lc.1 denotes the indicator function; the SCLSE B,, minimises (21). Thus, from section 2, the parameter vector 4 equals p and the (k + 1, k) derivative matrix @ = (I,, 0)‘. Differentiating (21) gives
S,(P) = l(x:P > O)(min{y,,2x:D)-x,P)xi,
i=l
,...,
n,
(22)
as in Powell (1986, eq. (2.9), p. 1439) and eq. (4). Correspondingly,
(23)
s,, =
lim n-’ i 77-a
l( x/PO > O)x,x,‘E+,,[rAI
(24)
2
r=l
where r ,”
=
mid Y,, 2x;Po)-x;&
= l(0 2x;&)
- l( y; < o))x;p,,, i=l
9 . .
see Newey (1987, p. 129) and Powell (1986, theorem 1, p. 1443).
. 3
n;
Under H,, or ey. (191, the expressions somewhat:
(23) and
(24) may be simplified
( 75 )
+2( I - ‘1’,)(
x:/3,,)‘)*
(76)
whcrc 4, and @, denote the standard Normal density and cumulative distribution functions, respectively, evaluated at .x~‘/~,,/v,,,,. Both the matrices of (25) and (26) may be consistently estimated under H,, by their sample counterparts evaluated at the Tobit MLE (p,,. C,,!,,). which provides the component matrices g,, and s,, of H7;, and EHST,, given in eqs. (8) and (12), respectively. In addition:
(I;,,,= ( I,
,O)‘.
(27)
(‘8) where the subscripts refer to the appropriate subblocks of /:,,. In this cast. the variance matrix V,, is full rank; that is. I’ = k. The experiments reported in this and the subsequent section set Jo = 500 and use 8000 replications. The number of regressor variables k = 2 and includes a constant term; that is, s,’ = (1. x,,). The design considered for x,, consists of the expected order statistics from the uniform distribution normalised to have mean zero and unit variance. The use of expected order
S. Petem and R.J. Smith, Distributional
specification
tests
187
Table 1 Point statistics
LQ X:
0.58
Median 1.39
I II III IV
0.57 0.54 0.91 1.11
1.39 1.33 2.26 2.49
I II III IV
0.58 0.70 0.68 1.92
I II III IV
0.58 0.99 0.64 3.13
for HT,, and EHST,,,
UQ
Mean 2.00
2.77
(A) Full Hausman
(B) Efficient 1.38 1.69 1.67 4.19 (C) Efficient
“I: II: III: IV: LQ: SK:
1.39 2.29 1.52 6.83
2.03 2.02 3.11 4.18
Hausman 2.81 3.40 3.41 7.78
2.06 2.12 2.93 9.07
score test EHST, 2.01 2.45 2.40 5.74
Hausman
Normal errors, Laplace errors, Mixture of Normal errors, t2 errors. Lower quartile, UQ: Upper’quartile, Skewness, KU: Kurtosis.
SD 2.00
SK 2.00
KU 6.00
2.02 2.13 1.66 26.39
6.02 6.49 3.74 1276.82
2.02 2.03 1.81 3.34
6.17 6.25 4.59 29.69
2.04 1.98 1.91 2.90
7.03 5.78 5.69 15.29
test HT, (MLE and SCLSE)
2.81 2.76 4.43 4.75
2.83 4.58 3.03 13.16
n = 500, 8000 replications.a
(SCLSE) 2.02 2.46 2.35 5.75
score test EHST,, (CLADE) 2.01 3.30 2.18 9.85
SD: Standard
2.00 3.27 2.14 10.28
deviation,
statistics gives a representative sample for random sampling from a particular distribution of the exogenous variable. The parameter vector is defined such that p’ = (0.0,l.O). The error distributions considered were drawn from I: Normal, II: Laplace, III: mixture of Normals, IV: r2. Distribution III constitutes a nonsymmetric alternative error density function formed as a mixture of two Normal densities, viz.: f(u;)
=0.4N(-1.0,O.l)
+0.6N(0.66,0.5),
i=l
,...,
IZ.
(29)
Note that distributions I, II, and III have moments of all orders, whereas IV possesses no integer moments apart from the first. Distributions I, II, and III were appropriately located and scaled to have mean zero and unit variance, whereas IV had the same interquartile range as that of a standard Normal variate which, together with the distribution for the exogenous variable should give, approximately, censoring of fifty percent and an R* of a similar order in the latent model (18). Panels A of tables 1 and 2 present point statistics and distribution function results, respectively, for the Hausman test statistic HT,, eq. (8), formed from
S. Peten and R.J. Smith, Disfrihufiorul
lxx
Table
Distribution
functions
0.7500
0.5000
0.2500
I II 111 IV
0.7455 0.7363 0.8355 0.X65.5
1 II 111 IV
0.73YY 0.7Y42 0.7847 0.0243
I II 111 IV
0.7522 0.X436 0.7710
0.5WY 0.6630 0533x
0.256’) 0.4245 0.2841
O.YShX
0.89
0.77x1
(A) Full Hausman 0.50 10 0.4885 0.64Xh OhY4 I
(B) Efficient
CC)
“1: II: III: IV:
Hausman
Efficient
Ilausman
13
Normal errors. Laplace error,. Mixture of Normal I. error,.
0.1050 0.1041 0.2355 0.2607
score teat EHS7;,
0.254Y 0.3 193 0.3225 0.64YY
0.4YUX 0.5627 0.5603 0.x1
0. IO00
II = 500. X000 replication\.
O.O5(lO
test 117;, (MLE and SCLSE):
0.2545 0.2493 0.428 I 0.4SSY
IWS
2
for H’f;, and EHSI;,.
(I
specificu!iorf
O.OYXX 0. IS.14 0.1.53x 0.4615
score test EffST,,
0.051’) 0.056‘l 0.1180 0.1803 (SCLSE):
0.0100
1).0050
P(HT,, t ki(ru)) 0.0101 0.0133 0.0440 0.0x5x P{EHST,
Il.Oll55 ll.oo8x 0.07hl I).fkli5
11.0015 0.001 x 0.00x0
0.0400
2 \:t~)l
0.050h 0.0x53
O.OOXY 0.0730
1i.0014 Il.0135
0.0x14 O..1S?J
O.OlY3 I~.lXYO
I).OlOi II.IJ?i
((‘LADE):
11.0010
0.00I0 O.OOiS 0.0071
0.07hI
P{EIIST,, ‘_ yj(cu))
0.00x0 0.24x0
O.OSl3 (I. I fl’5
o.I)oxo 0.0h00
11.0045 11.03Xh
l).OOli II.0 Ii
0.1’31 0.0374
0.062 I (I.S4XJ
0.01’0 l1.3XSH
l).lNK23 ll.330’~
O.OOl I).233
I I
rrror,
the difference of the Tobit MLE and SCLSE, together with components (75). (26), (27), and (28) appropriately evaluated at the Tobit MLE (/?,,,G(,!,,): this statistic was originally suggested by Newey (1987). It is evident that the finite-sample distribution of HT,, under H,, (I). eq. (19), for sample size II = 500, is in close accord with that predicted by asymptotic theory. Concerning the behaviour of HT,, under the Laplacc (double cxponcntial) distribution (II) for the errors of the latent model (181, that is. a symmetric distribution under which the SCLSE is root-r? consistent, table 2 indicates that HT,, has power differing only slightly from empirical size. Thus. in this case, HT,, provides a very weak test of the distributional assumption ot normality. The performance of HT,, under the mixture of Normals (III). eq. (29), indicates substantially more power than for the Laplace error distribution (II) but still of relatively small magnitude. A marginal increase in powel is evident under the f, error distribution (IV); note that SCLSE is root-n consistent under t, errors. Panels B of tables I and 2. respectively, present analogous results for EHST,, of (12). This statistic uses eq. (20) evaluated at the Tobit MLE. Similarly, the results (I) show that asymptotic theory is a reasonable guide to the H,, performance of EHST,,, although the tail behaviour of this statistic is
S. Peters and R.J. Smith, Distributional specification tests
189
Table 3 Point statistics
for slope parameter
estimates,
p, = 1.0, n = 500, 8000 replications.”
MLE
LQ Median UQ Mean SD RMSE
SCLSE
I
11
III
IV
I
II
III
IV
0.96 1.00 1.04 1.oo 0.06 0.063
1 .oo 1.04 1.09 1.05 0.07 0.084
0.94 0.98 1.02 0.98 0.06 0.066
1.22 1.36 1.57 1.51 0.70 0.867
0.90 1.00 1.12 1.03 0.18 0.183
0.92 1.00 1.09 1.02 0.13 0.135
0.90 1.05 1.25 1.11 0.31 0.33 1
0.90 1.00 1.14 1.04 0.22 0.219
“I: Normal errors, II: Laplace errors. III: Mixture of Normal errors. IV: t2 errors. LQ: Lower quartile, UQ: Upper quartile. RMSE: Root mean squared error.
SD: Standard
deviation,
slightly worse than that for HT,. A slight improvement in power over the performance of HT, is evident for the Laplace error distribution (II). However, for the mixture of Normal errors distribution (III) this ranking is reversed between HT, and EHST,,. Under the t, error distribution (IV), EHST,, is moderately powerful and clearly dominates the performance of HT,? To rationalise these results, table 3 presents point statistics from the distributions of the Tobit MLE and SCLSE for the error distributions I, II, III, and IV considered above. It is immediately apparent that it is only in the case of error distribution IV (t,> that the Tobit MLE is substantially biased. In fact, the Tobit MLE dominates SCLSE in root mean squared error (RMSE) terms for the other distributions I, II, and, in particular, III. Examining the results for distribution II (Laplace) provides evidence for the poor performance of both HT, and EHST, as the Tobit MLE is only mildly mean and median biased, whereas SCLSE is approximately median unbiased and slightly mean biased.4 Turning to distribution III (mixture of Normals), it ‘Another design for the regressor variable was examined where x,,, i = 1,. , n, were the expected order statistics from the standard Normal distribution also normalised to have zero mean and unit variance. In this case, under H,,, both HT,, and EHST, perform slightly worse than under the uniform order statistic design. Additional experiments for EHST, indicated great and marginal increases in power against the Laplace alternative for ten percent censoring and R*. respectively, whereas, against the mixture of normals alternative, power respectively declined and marginally increased. Details are available from the authors on request. The symmetry of these exogenous variable designs implies that the results of this and the following section are also valid for p = (0.0, - 1.0). We are grateful to an anonymous referee for this point. ‘Similar results are reported for the Tobit MLE by Paarsch (1984), for sample sizes of n = 50. 100,200, where the relative bias of the Tobit MLE was relatively small. These conclusions are also evident from the simulations of Powell (1986. table II, p. 1451).
is the substantial mean bias of SCLSE which accounts for the improvement in power of HT,,, whereas that of the Tobit MLE is small accounting for the poor performance of EHS7;,. Distribution IV (t,) induces a large bias (both median and mean) in the Tohit MLE and thus both HT,,and EHS7;, show increases in power. Whereas the improvement in the power of EHS'I;,is dramatic. it is somewhat disappointing that although SCLSE is approximately median unbiased and only moderately mean biased, the power of HT,,is still rather low. On the basis of these results. the Hausman score statistic EHST,, is to be recommended over the Hausman statistic HT,, itself. For those casts in which the Tobit MLE dominates SCLSE in RMSE (distributions I, II, and III) EHS~, displays low power whereas for distribution IV (t,) it has moderately good power propcrtics.
5. Monte Carlo experiments for the ‘i’ohit model censored least-absolute deviations estimator
using
EHST,,
and the
As in section 4, the model is the Tobit model. cqs. ( 17) or (IX): H,, corresponds to the distributional assumption ( IO). Another alternative cstimator for ,3,, which imposes weaker distributional assumptions than those of SCLSE is the censored least-absolute deviations estimator (CLADE) due to Powell ( 1084) which minimises the criterion function:
As for SCLSE, CLADE maintains the conditional median assumption (70) but drops the symmetric error assumption of SCLSE: as above denote these CLADE assumptions H ,. Under H,, CLADE is root+ consistent; see Powell (19841. There is a lack of suitable computer routines to locate (‘LADE: see Paarsch (1984). Thus. only the test statistic form EHS7;, is considered in the experiments. The matrices required for EHS7;, arc given as follows:
- = lim II EC, II >I
\’ -0
’ k 1( _v;/q, > 0) x,_\-,‘.f( 0). I I
( 30)
= lim II ’ c I( _\-;/I$,> o)x,.r;/3. I, I7 i I
g,(O) = I(x;p,,>
o,{C -
I(
?‘,
(311
i= I.....)?.
(32)
S. Peters and R.J. Smith, Distributional
specijication
tests
191
where f<.> denotes the (unknown) density of u,, i = 1,. . . , n. Again, the variance matrix V, has full rank; that is, r = k. Consistent estimators for (30) and (31) under H,, may be obtained by using their sample counterparts, substitution of 1/(27r)“2q,o for f(O) and the Tobit MLE (p,, G,‘n> for the unknown parameters (PO, a,g>. As in section 4, 4=p and @=(Z,,O). The experiments follow the same format as section 4 in that the same sample size, n = 500, number of replications, 8000, design and distributions (Normal, Laplace, mixture of Normals, and t2) are used. Panels C of tables 1 and 2 report point statistics and distribution function results, respectively, for EHST,, eq. (12), formed from (27), (28), (301, (311, and (32) evaluated at the Tobit MLE. Again, the H, (1) finite-sample distribution of EHST, agrees closely with that predicted by asymptotic theory. Results for error distribution II (Laplace) indicate that EHST, has some (but small) power; however, the CLADE-based EHST, is superior to the SCLSE-based HT, and EHST, for Laplace alternatives, but not overwhelmingly so. The performance of EHST, for the nonsymmetric (mixture of Normals) alternative distribution III of eq. (29) shows that the power of EHST, differs very little from size and is dominated by both the SCLSE based HT, and EHST,,. For distribution IV (fz), the CLADE EHST, has good power superior to that of both the SCLSE-based HT, and EHST,.’ Turning to table 3, it is clear that the power of the CLADE-based EHST, increases with the bias and RMSE of the Tobit MLE, that is according to the distributional ranking I, III, II, and IV. In particular, the low power of this statistic for distributions II and III (Laplace and mixture of Normals) is explained by the small bias (both mean and median) of the Tobit MLE, whereas its high power against distribution IV (t,> is associated with large biases of the Tobit MLE. Thus, in this sense, the CLADE-based EHST,, is a reasonable distributional specification test statistic for the Tobit model.
6. Concluding
remarks
This paper has been concerned with providing convenient specification tests for parametric or likelihood based models against distributional alternatives where the model is only partially specified parametrically. The procedure is a variant of the Hausman test statistic but the proposed test statistics have the advantage that they need only the maximum likelihood estimator computed for the parametric model. The statistics have a similar basis to those proposed by White (1987) for testing dynamic specification. In contradistinction to the statistics of Ruud (1984) for the classical case of parametric restrictions, our statistics to not require an estimator under the ‘Additional
experiments
as in footnote
3 gave similar
results
for the CLADE-based
EHST,,.
alternative hypothesis; in such circumstances, our statistics would reduce to familiar score or Lagrange multiplier test statistics for parametric restrictions To explore the efhcacy of the proposed statistics and to compare them with the corresponding Hausman statistic, some Monte Carlo experiments were conducted for the Tobit model using various error distributions. For moderate sample sizes in cross-section econometrics, the null distributions of both the Hausman and proposed efficient Hausman score statistics accorded well with that predicted by asymptotic theory. Against the alternative distributions considered, the Hausman test itself performed poorly, whereas the Hausman score test statistics based on the asymptotic first-order conditions for the symmetrically censored least-squares estimator and the censored leastabsolute deviations estimator had some power in detecting those circumstances in which the Tobit estimator performed badly. In particular, the censored Icast-absolute deviations estimator based Hausman score statistic behaved especially well in this regard and can therefore be recommended as a useful distributional specification test statistic for the Tobit model on the basis of these results although more extensive experiments arc required for different exogenous regressor designs and other alternative distributions to ascertain whether these results arc of wider generality. Various regression-based forms of the Hausman score test statistic were also obtained. It still remains to examine their finite-sample properties.
Appendix The following (1967). Let
assumptions
arc
identical
to those
where E[ .] denotes expectation taken with respect ~1and x and ~(4) = 3Rty, X; ~$)/iJd. (N-l)
For each tied G!I,g(4) is measurable Doob: see Huber ( 1967. p. 222).
(N-2) There
is some d,, such that A($,,) = 0.
employed
by Huhcr
to the true joint density
and is separable
of
in the sense of
S. Peters and R.J. Smith, Distributional specification tests
193
(N-3) There are strictly positive numbers a, b, c, and d, such that (i) llA($)ll 2 all+ - &II for II4 - &II 54, (ii) E[p(+, d)] I bd for II+ - 4Jl+ d I d,, d 2 0, (iii) E[p(+, d121I cd for II4 - 4+jll+ d I d,, d 2 0. (N-4) h(b) possesses a nonsingular derivative in a neighbourhood
of 4 = 4”.
(N-5) E[ llg(&)112] is finite. Define
Z,(T,4)
=
II
1 + n1’211A(T)II
Then, as in Huber (1967, lemma 3, p. 227): Lemma.
Under
assumptions (N-1 ), (N-2 1, and (N-3 ):
Using this lemma, the proof of the proposition is straightforward: Proof of proposition. O*(l):
From the lemma, (N-4) together with n’/2(&n - Bo) =
Given that n’12(6,, - Bo) = O,(l) and (N-4):
References Andrews, D.W.K., 1987, Asymptotic results for generalized Wald tests, Econometric Theory 3, 348-358. Arabmazar, A. and P. Schmidt, 1982, An investigation of the robustness of the Tobit estimator to non-normality, Econometrica 50, 1055-1063. Cox, D.R. and D.V. Hinkley, 1974, Theoretical statistics (Chapman Hall, London).
Hausman. J.A., IY7X, Specitication Hau5man. J.A. and W.E. Taylor. 230-245.
teats in rconometrics. 19X1. A generalized
Econometrica 46. 1251-1271. specification test. Economics Letter\
S.
HOI&. /\.. IYX7. A remark on Hausman‘r specltication teat. Econometrlca 50. 7JYC75Y. Fluber, P.J.. IYh7. The behaviour of maximum likelihood estimates under nonstandard condtions. Proceedings of the Fifth Berkeley Symposium 1. 22I-733. Huber. P.J., 19X1. Robust statistics (Wile>. NCA~ York. NY). Newry. W.K.. 19x7. Specification teats for the distributional assumptions III the Tohit model. Journal of Econometrics 34. 125- 115. Paarach. H.J.. 19X4. A Monte Carlo compariwn 01 e\tunator\ tar censored regression modrls. Journal of Econometrics 24, 107~211. Powell. J.L.. 10x4. Least ahsolute deviations estimation tar the ccnwrrtl rrrgreasion model. Journal of Econometric\ 2.5. 303-325. Powell. J.L.. 10X6. Symmetrically trimmed Icast quare\ r\tlmuticm tot- TohIt modela. Econometrica 53. I435- 1460. Rae, (‘.R. and S.K. Mitra. 1971. Grneralizcd inwrw of matricc\ and it\ application\ (Wllq. No\ York. NY). Ruud. PA.. IYX4. Te$t\ 01 spccilication in cconometrlc\. Econometric Rwieus 3. 1 I I~-21. White. H.. IYX?. Maximum likelihood estimation of miaspecified models. Econometrica 50. I -3 White. H.. 19X7. Specification testing in dynamic model\. in: Truman F. Bewley. rd.. Advances in econometrics, Fifth world congres\ (<‘amhritl_ce Ilnivcrsit\ Press. New York. NY) 1-5s
of Econometrics
47 (1991) 175-194.
North-Holland
Distributional specification tests against semiparametric alternatives* Simon Peters Unir~ersityof Bristol, Bristol BS8 IW,
Richard
U.K.
J. Smith
Unkersity of Cambridge, Cambridge CB3 900,
U.K.
Score or Lagrange multiplier versions of a Hausman test for distributional specification are presented for limited dependent variable models. These tests are based on the first derivatives of semiparametric criterion functions associated with robust estimation of such models and only require the computation of the maximum likelihood estimator. Various regression forms of the statistic are also presented. Monte Carlo results for the Tobit model indicate that such tests may be efficacious in detecting situations in which the maximum likelihood estimator is seriously biased due to incorrect distributional specification.
1. Introduction
This paper considers distributional specification tests for parametric models estimated by the method of maximum likelihood. As is now widely appreciated for limited dependent variable models, failure of the distributional assumption is usually fatal for the consistency property of the maximum likelihood estimator (MLE). Recently, therefore, there has been an upsurge of interest in robust estimation techniques for such models. In particular, such methods typically specify only part of the generating process for the data, leaving the remainder to be dealt with in a nonparametric way. Such methods are termed semiparumetric; for example, see inter alia Powell (1984,1986) for two approaches to the semiparametric estimation of a Tobit model. For some *We are grateful for the helpful comments of the Editors, two anonymous referees, Halbert White, and other participants at the ESRC Econometric Study Group Conference, 1989, University of Warwick, where an earlier version of this paper was presented. Research for this paper was supported partially by the ESRC under grant B 0023 2150.
0304-4076/91/$03.500
l991-Elsevier
Science
Publishers
B.V. (North-Holland)
problems, such as those considered by Powell. the resultant semiparametric estimator is root-n consistent and possesses a limiting Normal distribution. properties shared with the MLE in the absence of misspecification. In the context of this paper. the assumptions of such semiparametric techniques. implicit or otherwise, are the maintained hypothesis for the specification tests considered below. For the Tobit model with normally distributed errors, Newey (1987) proposed a Hausman (1978) specification test for this distributional assumption based on the difference of the Tobit MLE and Powell’s (lY86) symmetrically censored least-squares estimator (SCLSE). Clearly. such an approach has wide applicability. In this papcl-. a similar approach is taken: however, a score or Lagrange multiplier statistic is proposed which only requires the parametric or null MLE in contradistinction to the Hausman statistic which requires both null and alternative (semiparametric) estimators. Section 2 outlines the basic components which undcrly the suggested statistic and the limiting distribution theory for both parametric and semiparametric estimation methods. The Hausman score test statistic is presented along with some variants in section 3. Section 4 consists of the results of some Monte Carlo experiments on the Hausman and Hausman score statistics for the Tobit model comparing the Tobit MLE and the SCLSE, whereas section 5 examines the Hausman score statistic using as alternative estimator the censored least absolute deviations estimator (CLADE) due to Powell (19X4). Section 6 contains some concluding remarks.
2. Parametric Given
and semiparametric
a sample
estimation
of independently
and identically distributed observations joint density function of the random variable y and random k-vector s. a usual objective is to perform inference on some aspects of the conditional density function of y given .I-. for example, conditional moments or quantiles of y given x. The parametric estimation procedure most commonly adopted is to fully specify a conditional density function for the random variable y given .x. f’tylx; HI, known up to and differentiable in the p-vector of parameters 0. and to assume that the random vector .‘r is weakly exogenous for 0. Generally, some partial reparameterisation of the parameter vector 0, ch = (b(H), captures the parametric components of those conditional moments or quantiles of interest. Under the correctness of these distributional assumptions. an appropriate estimator for the true value H,, of 0 is the maximum likelihood estimator (MLE) 6,!, which satisfies the first-order conditions or likelihood equations: ( y,. x,), i = 1.. . , II, from an (unknown)
S. Peters and R.J. Smith, Distributional
specification
tests
177
where ~~(0) = d Inf(yiIxi; O)/#, the score vector for observation i, i = 1, . . . , II. Under suitable regularity conditions, see Huber (1967) and the append&, and satisfaction of the assumption of correct distributional specification, 8, is a root-n consistent estimato_r for f3”. In particular, $n is a weakly On= e,,, and obeys a Central Limit consistent estimator for e,, plim.,, Theorem:
n”2(6n - e,) : N,(O,j,-'),
(2)
where jO=
limn-’ n~m
~E,,,[Si(e”)Si(eo)‘] i=l
and E,[ .I denotes the expectation of [ .] taken with respect to the assumed conditional density function f(y IX;0). The limiting distributional result (2) is obtained via a stochastic first-order Taylor series expansion of S, of (1) about e,,:
n1/2s,=/on1/2(fin - eo)
+op( i),
(3)
where
For limited dependent variable models where, typically, the random variable y may be regarded as a censored observation on some underlying latent variable y*, failure of the conditional distributional assumption f(ylx; 0) may cause inconsistency in the MLE O,,, that is plim, j_ 8, # t9”. An alternative approach is to partially specify parametrically a particular feature of interest of the (unknown) conditional density of y or that of the underlying latent variable y*, such as the conditional mean or median of y*, and to derive an appropriate criterion function based on the censoring mechanism linking y to y*. As above, let this semiparametric specification be parameterised by the q-vector of parameters 4 with true value &. If the fully parametric specification f(y lx; 0) were correct, then 4 = 4(O), with & = 4(0,>, where the q-vector of functions $4.) is continuously differentiable in 0 and the (p,q) derivative matrix @ = a@/ae is full column rank at least in a neighbourhood of 0”. It is also assumed that the random vector x is weakly exogenous for 4, as in the classical case. The semiparametric estimator (SPE) 4, satisfies: ,. 4, = argminn-’ 4
t i=l
R( yi, xi; 4),
where the right-hand side denotes a suitable criterion function; for example, see Powell (1984,1986). In most cases, the above criterion function will be only left differentiable and the first-order condition may only be stated asymptotically; that is:
where g,(b) = dR( y,. x,; c#I)/~c$. i = I. , II. Under appropriate conditions, it $,, is weakly consistent for ch,, and satisfies the asymptotic first-order condition given above, it may be shown that
11
’ 2 c s,( d,,) I-
+
I?’ 2(E[g(d,)]),!,~,;,,,=(),‘(1).
(4)
I
where E[.] denotes expectation of [ ‘1 taken with respect to the true ,joint density of y and x and ~(4) = dR(~t, s: d)/dd; see Huber (1967, theorem 3, p. 230: 1981, theorem 3.1, p. 13.1) and Powell (1984, lemma A..?. p. 317). Assuming that E[g(b)l possesses a nonsingular derivative matrix at cb,,. this allows a Central Limit Theorem to be stated for ch,, given the weak exogeneity of .Y for ch: viz.:
where
and
where E,,,,[.l denotes expectation taken with respect to the true conditional density of y given x. In a similar manner to the use of eq. (3), the limiting distribution result (5) is obtained from (4) as
I”$&, = -Qz~2(f& where
- cb,,)+
O/J 1).
(7)
S. Peters and R.J. Smith, Distributional specification tests
Examples of particular sections 4 and 5. 3. The Hausman
criterion
score test statistic
179
functions and models are presented
for distributional
in
specification
Denote the hypothesis of correct distributional specification f(y(x; 0) by H, with corresponding alternative hypothesis Hr. The Hausman _(1978) statistics discussed below consider the null hypothesis H,*: plim. _+m4, = 4,, (and associated alternative hypothesis H;“: plim,, em 4, Z &) concerning the parameters of interest 4, where 4, = +(e,). These test statistics have a well-defined limiting distribution under H, and, hence, are pure si&‘kznce tests for H “; see Cox and Hinkley (1974). In general, plim. _m 6, = +(0, ), where 0* satisfies lim,,, E+Js,,l = 0, s,, = n-‘C~=,si(e,) and E+,I[*l denotes expectation with respect to the true conditional density of y given X; see White (1982). Although H, (H;F) is sufficient for H,* (H,), Hz (H,) may not imply H, (H:); see Holly (1982). Therefore, under particular distributional misspecifications, the degree of inconsistency of 4, for & may be small or null. See Arabmazar and Schmidt (1982) for evidence concerning the Tobit model. Thus, these Hausman statistics are generally inconsistent tests of H, but, as rejection of H,* implies rejection of H,, may still provide useful specification tests for H,. As in section 2, assume that there exists an SPE 4, with the limiting distributional properties described in section 2 underA Hr. Then a Hausman test statistic based on the difference between 6, and 6, provides a test of H, against H, [see Hausman (1978) Hausman and Taylor (1981) Holly (1982) and Ruud (1984)]: HT, = n(&, - &)‘f,-(6,
- 6n),
(8)
where +a Ais a consistent estimator for the asymptotic variance I’,, of n’j2($,, - 4,) under H,, and [ .I- denotes a g-inverse for [ .I. Under appropriate conditions [see Andrews (1987)] and H,, HT, has a limiting x2 distribution with r degrees of freedom with r = rk(l/,).’ Under H,, 6, is asymptotically efficient which allows V. to be simply expressed as the difference of the asymptotic variance matrices of $J,, and 6,, viz.:
‘The following
conditions
are sufficient
for a statistic
X,‘A,X,
to have a limiting
,y*[rk(A)I
P[rk(A.) = rk(A)] = 1; (iv) distribution: (i) X,, 5 N(O, A); (ii) A 5 A; (iii) lim P[“A = A’ ] = 1. Co:zzons (i), (ii), (iv), and (v) are lim n_” P[X,, ER(A,,)I = 1; (v) lim,,,, relatively easy to verify for statistic (8); condizon C% must be assumed.
180
S. Peters and R.J. Smith, Distributional
specificatiorl
test\
It is necessary to choose the estimator q, of V,, to be positive semidefinite to avoid negative values of HT,, in finite samples. Choosing a common estimator for the unknown parameters in V,, is often sufficient. Newey (1987) has applied HT,, in the context of a Tobit model using the difference of the Tobit MLE and the symmetrically censored least-squares estimator of Powell (1986), which we consider further in section 4. The form of test statistic (8) suffers from the same computational disadvantage as the likelihood ratio statistic in that it requires the two estimators d;,, quite burdensome, and J,,. the second of which may be computationally especially for limited dependent variable models; see Paarsch (1984) for the censored least-absolute deviations estimator of Powell (1984). In the classical context of parametric restrictions, Ruud (1984) presented score test versions of HT,, based on the alternative hypothesis MLE. In contradistinction to Ruud (19841, the test statistic to be presented below uses the parametric MLE &,,, which in a semiparametric framework may offer considerable computational advantages over HT,,; in the familiar classical setting of Ruud (1984). the statistic would reduce to the usual score or Lagrange multiplier statistic for parametric restrictions. See White (1987) for a similar approach when testing dynamic specification. Consider the asymptotic first-order conditions. eqs. (4) and (7). Similar arguments to those of Huber (1967, 1981) allow the following proposition to be stated: Proposition. Under assumptions ( N-l). ( N-2 1, ( N-3). pendiv and if n’/‘(6,, - @,,I = O,,( 1). then:
and ( N-4 ) of’ the up-
li &(&,)
lz”2g,i=nm”2
I I = II
Proqf.
If2 &I,) i-l
+rr’/?(E[g(~)]),,~_,l,,+o,,(l).
See the appendix.
Expanding
the second
term in eq. (10) about
$,, gives
~~“‘(E[~(cb)])d,~~~,, =n”‘(E[g(d)]
)_
+(dE[g,(~)]/a~‘),-~~,,n’/‘(~,,-~,,) --n--l/”
kg,(&) I-l
+z,, n”Q,,
+o,,(l) -&)
+ o,>,(1).
( 10)
S. Peters and R.J. Smith, Distributional specification tests
181
which after substitution into eq. (10) leaves nl/2-
g, =
?T ‘i2( 6, - 4,) + OJ 1). #on
(11)
A further substitution into eq. (8) for n 1/2($ - 4,) from (111, using the weak consistency of 6,, for B0 and dropping asymitotically negligible terms, allows the statement of the following theorem for the efficient Hausman score statistic EHST,: Theorem. appendix :
(N-2), (N-3), (N-4), and (N-5) of the
Under assumptions (N-I),
has a limiting x2 distribution with r degrees of freedom under H,, where ‘In indicates evaluation at 6, and r = rk(Z,, - ECj@hA;‘@,,E,j).2
The second equality of (12) is obtained by noting the invariance to the choice of g-inverse; see Rao and Mitra (1971, lemma 2.2.4(n), (iii), p. 21). A number of alternative variants of the Hausman score statistic are available using different consistent estimators for V, of eq. (9). Firstly, define:
s,= (s,(e,),..., s&in))‘,
G,,= (gl(~n),...,g,<~,>>‘.
Secondly, using a similar result to the familiar information
matrix equality
‘Let h()?lx; QO,$O,iO) denote the true conditional density function of y given x after reparameterising 0 to (4. +) such that f( y Ix; 0) = h(y Ix; 4, I/J,0). Under local alternatives where 0 < [‘i < m, the Hausman statistic (8) and the efficient Hausman score 4, = i/n’/‘, statistic (12) will have a limiting x2 distribution with r degrees of freedom and noncentrality parameter
where
~~~(~,,,cL,,,o)-aInh(y,lx,;b,,cL,,O)/a~, and AP compare
= [ Am0
Holly (1982).
- A+W%&%‘,m”]
- ‘;
i=l,...,n,
etc.,
S. Peters and R.J. Smtth, Distrihutior~ul specificutim
IX2
evaluated
under
test.!
H ,,:
E,I,[K,(~(B,,))s,(H,~)‘] = (~E,[g,(~(Hll))l/~H'),~,,, =
-(“E&,(d)]/~~‘)
,,,_,,,,, @‘;I.
(13)
where EJ .] indicates expectation taken with respect to the H,, conditional density f(~\x; 0); compare Newey ( 1987. eq. (3.8), p. 131). Thus. from eq. (6):
and a consistent estimator for 5,, under distributional specification H ,, is given by ~
/I
Substitution alternative
where
the
null
hypothesis
of correct
s,,/y;,’ q,, ( c&p,, ’ G,,) ’
’ G,;
of this estimator for .?,, in EHSI;,of eq. (12) produces form of the Hausman score statistic:
L is an n-vector
each of whose elements
the
are unity and
_ _
L, = (‘0. q,q,/“,,- ’ )‘. The statistic regression:
(14) may be computed
as UK’ (uncentred)
from the artificial
noting that L$,‘,~ = 0. Making use of the fact that, under H,,. S,;S,,/fl is a consistent /“;,, a further alternative form of EHST,,is given by HST,;” = where
ld,,(
c,;M,
?/
estimator
&L.
- X(X'X) 'X'. This statistic M,y= I,,
may also be computed
for
( 15)
as ru?’
S. Peters and R.J. Smith, Distributional specification tests
(uncentred)
183
from the regression:
Finally, if we introduce the auxiliary (p - q)-vector of parameters 4 such that (4, $,> represents a bijective (one-to-one onto) transformation of 8, then similarly to eq. (13):
where E ,+,,I[.I denotes expectation taken with respect density f( y IX; c$, $1 after reparameterisation, and
to the H,, conditional
as
(aE,,l,,,,[s,(~)l/a~‘),=,,,,,=,,l = 0. Thus, another asymptotically under H, is given by
equivalent
form of the Hausman
score statistic
using the invariance of score-type statistics to bijective parameter transformations. This outer product of gradients form of the Hausman score statistic, as is usual, may be computed as the nR2 (uncentred) from the regression:
It should be emphasized that although all four forms of EHST, (121, (141, with r degrees of (151, and (16) possess the same limiting x * distribution freedom under H,, the finite-sample characteristics of these statistics may be very different. In the next two sections, the finite-sample performance of the efficient form of the Hausman score statistic (12) together with that of the Hausman statistic (8) itself is examined.
S. t’cters and R.J. Smith. Distributional
IXJ
specrfication
4. Monte Carlo experiments for the Tobit model censored least-squares estimator Consider
the familiar
Tobit
y, = max( 0, s;j3,, +
mode1 defined
u,}.
i=
I,
using
tern
the symmetrically
as follows: I1.
( 17)
where (Y,. x,) are assumed to constitute a random sample and the U, to be independently and identically distributed independently of x,, i = 1. . tt. Alternatively, the generating mechanism, eq. (17). may be cast in terms of a latent variable, viz.: !‘,* =
x;p,, + II, (
j’,
1( ?‘,*> 0) ).,*.
=
/=
I.
. Il.
where l(. ) denotes the indicator function. The null hypothesis H,, of correct distributional to the assumption that 14, -
IN(0.
CT,;,),
i= l.....,l.
which allows the use of the Tobit the likelihood equations:
+wt,+,
( 18)
specification
corresponds
( 19)
MLE for t/3,,, v~,:,). (a,,, 6,,!,,). which solves
:P,, i “;i;, I I =(I .y (?
,,,,
,
where c, = 1 if J, > 0 and 0 otherwise, 6, and 6, are the standard Normal density and cumulative distribution functions. respectively, evaluated at x:&/4,,,. As has been tion (19) may Schmidt (1982). function of U,
noted elsewhere, failure of the above distributional assumprender the Tobit MLE inconsistent: see Arabmazar and An alternative estimator which only assumes that the density is symmetric about zero is the symmetrically censored least-
S. Peters and R.J. Smith, Distributional specification tests
185
squares estimator (SCLSE) due to Powell (1986). In the context of section 3, the assumption of symmetric errors constitutes the alternative hypothesis H, and x,‘& is the conditional median of y* given xi, i = 1,. . . , n; that is:
Eb,,[1(Y; >xlPo)]= t,
i=l
>..*> II.
(20)
Returning to the notation of section 2, the criterion to be minimised is given by
n-1 CR(YjYxi;4) i=l
+n-’
5 1(~,>2x;p)((+y~)~-
(max{O,x;P})‘),
(21)
i=l
where, as above, lc.1 denotes the indicator function; the SCLSE B,, minimises (21). Thus, from section 2, the parameter vector 4 equals p and the (k + 1, k) derivative matrix @ = (I,, 0)‘. Differentiating (21) gives
S,(P) = l(x:P > O)(min{y,,2x:D)-x,P)xi,
i=l
,...,
n,
(22)
as in Powell (1986, eq. (2.9), p. 1439) and eq. (4). Correspondingly,
(23)
s,, =
lim n-’ i 77-a
l( x/PO > O)x,x,‘E+,,[rAI
(24)
2
r=l
where r ,”
=
mid Y,, 2x;Po)-x;&
= l(0
- l( y; < o))x;p,,, i=l
9 . .
see Newey (1987, p. 129) and Powell (1986, theorem 1, p. 1443).
. 3
n;
Under H,, or ey. (191, the expressions somewhat:
(23) and
(24) may be simplified
( 75 )
+2( I - ‘1’,)(
x:/3,,)‘)*
(76)
whcrc 4, and @, denote the standard Normal density and cumulative distribution functions, respectively, evaluated at .x~‘/~,,/v,,,,. Both the matrices of (25) and (26) may be consistently estimated under H,, by their sample counterparts evaluated at the Tobit MLE (p,,. C,,!,,). which provides the component matrices g,, and s,, of H7;, and EHST,, given in eqs. (8) and (12), respectively. In addition:
(I;,,,= ( I,
,O)‘.
(27)
(‘8) where the subscripts refer to the appropriate subblocks of /:,,. In this cast. the variance matrix V,, is full rank; that is. I’ = k. The experiments reported in this and the subsequent section set Jo = 500 and use 8000 replications. The number of regressor variables k = 2 and includes a constant term; that is, s,’ = (1. x,,). The design considered for x,, consists of the expected order statistics from the uniform distribution normalised to have mean zero and unit variance. The use of expected order
S. Petem and R.J. Smith, Distributional
specification
tests
187
Table 1 Point statistics
LQ X:
0.58
Median 1.39
I II III IV
0.57 0.54 0.91 1.11
1.39 1.33 2.26 2.49
I II III IV
0.58 0.70 0.68 1.92
I II III IV
0.58 0.99 0.64 3.13
for HT,, and EHST,,,
UQ
Mean 2.00
2.77
(A) Full Hausman
(B) Efficient 1.38 1.69 1.67 4.19 (C) Efficient
“I: II: III: IV: LQ: SK:
1.39 2.29 1.52 6.83
2.03 2.02 3.11 4.18
Hausman 2.81 3.40 3.41 7.78
2.06 2.12 2.93 9.07
score test EHST, 2.01 2.45 2.40 5.74
Hausman
Normal errors, Laplace errors, Mixture of Normal errors, t2 errors. Lower quartile, UQ: Upper’quartile, Skewness, KU: Kurtosis.
SD 2.00
SK 2.00
KU 6.00
2.02 2.13 1.66 26.39
6.02 6.49 3.74 1276.82
2.02 2.03 1.81 3.34
6.17 6.25 4.59 29.69
2.04 1.98 1.91 2.90
7.03 5.78 5.69 15.29
test HT, (MLE and SCLSE)
2.81 2.76 4.43 4.75
2.83 4.58 3.03 13.16
n = 500, 8000 replications.a
(SCLSE) 2.02 2.46 2.35 5.75
score test EHST,, (CLADE) 2.01 3.30 2.18 9.85
SD: Standard
2.00 3.27 2.14 10.28
deviation,
statistics gives a representative sample for random sampling from a particular distribution of the exogenous variable. The parameter vector is defined such that p’ = (0.0,l.O). The error distributions considered were drawn from I: Normal, II: Laplace, III: mixture of Normals, IV: r2. Distribution III constitutes a nonsymmetric alternative error density function formed as a mixture of two Normal densities, viz.: f(u;)
=0.4N(-1.0,O.l)
+0.6N(0.66,0.5),
i=l
,...,
IZ.
(29)
Note that distributions I, II, and III have moments of all orders, whereas IV possesses no integer moments apart from the first. Distributions I, II, and III were appropriately located and scaled to have mean zero and unit variance, whereas IV had the same interquartile range as that of a standard Normal variate which, together with the distribution for the exogenous variable should give, approximately, censoring of fifty percent and an R* of a similar order in the latent model (18). Panels A of tables 1 and 2 present point statistics and distribution function results, respectively, for the Hausman test statistic HT,, eq. (8), formed from
S. Peten and R.J. Smith, Disfrihufiorul
lxx
Table
Distribution
functions
0.7500
0.5000
0.2500
I II 111 IV
0.7455 0.7363 0.8355 0.X65.5
1 II 111 IV
0.73YY 0.7Y42 0.7847 0.0243
I II 111 IV
0.7522 0.X436 0.7710
0.5WY 0.6630 0533x
0.256’) 0.4245 0.2841
O.YShX
0.89
0.77x1
(A) Full Hausman 0.50 10 0.4885 0.64Xh OhY4 I
(B) Efficient
CC)
“1: II: III: IV:
Hausman
Efficient
Ilausman
13
Normal errors. Laplace error,. Mixture of Normal I. error,.
0.1050 0.1041 0.2355 0.2607
score teat EHS7;,
0.254Y 0.3 193 0.3225 0.64YY
0.4YUX 0.5627 0.5603 0.x1
0. IO00
II = 500. X000 replication\.
O.O5(lO
test 117;, (MLE and SCLSE):
0.2545 0.2493 0.428 I 0.4SSY
IWS
2
for H’f;, and EHSI;,.
(I
specificu!iorf
O.OYXX 0. IS.14 0.1.53x 0.4615
score test EffST,,
0.051’) 0.056‘l 0.1180 0.1803 (SCLSE):
0.0100
1).0050
P(HT,, t ki(ru)) 0.0101 0.0133 0.0440 0.0x5x P{EHST,
Il.Oll55 ll.oo8x 0.07hl I).fkli5
11.0015 0.001 x 0.00x0
0.0400
2 \:t~)l
0.050h 0.0x53
O.OOXY 0.0730
1i.0014 Il.0135
0.0x14 O..1S?J
O.OlY3 I~.lXYO
I).OlOi II.IJ?i
((‘LADE):
11.0010
0.00I0 O.OOiS 0.0071
0.07hI
P{EIIST,, ‘_ yj(cu))
0.00x0 0.24x0
O.OSl3 (I. I fl’5
o.I)oxo 0.0h00
11.0045 11.03Xh
l).OOli II.0 Ii
0.1’31 0.0374
0.062 I (I.S4XJ
0.01’0 l1.3XSH
l).lNK23 ll.330’~
O.OOl I).233
I I
rrror,
the difference of the Tobit MLE and SCLSE, together with components (75). (26), (27), and (28) appropriately evaluated at the Tobit MLE (/?,,,G(,!,,): this statistic was originally suggested by Newey (1987). It is evident that the finite-sample distribution of HT,, under H,, (I). eq. (19), for sample size II = 500, is in close accord with that predicted by asymptotic theory. Concerning the behaviour of HT,, under the Laplacc (double cxponcntial) distribution (II) for the errors of the latent model (181, that is. a symmetric distribution under which the SCLSE is root-r? consistent, table 2 indicates that HT,, has power differing only slightly from empirical size. Thus. in this case, HT,, provides a very weak test of the distributional assumption ot normality. The performance of HT,, under the mixture of Normals (III). eq. (29), indicates substantially more power than for the Laplace error distribution (II) but still of relatively small magnitude. A marginal increase in powel is evident under the f, error distribution (IV); note that SCLSE is root-n consistent under t, errors. Panels B of tables I and 2. respectively, present analogous results for EHST,, of (12). This statistic uses eq. (20) evaluated at the Tobit MLE. Similarly, the results (I) show that asymptotic theory is a reasonable guide to the H,, performance of EHST,,, although the tail behaviour of this statistic is
S. Peters and R.J. Smith, Distributional specification tests
189
Table 3 Point statistics
for slope parameter
estimates,
p, = 1.0, n = 500, 8000 replications.”
MLE
LQ Median UQ Mean SD RMSE
SCLSE
I
11
III
IV
I
II
III
IV
0.96 1.00 1.04 1.oo 0.06 0.063
1 .oo 1.04 1.09 1.05 0.07 0.084
0.94 0.98 1.02 0.98 0.06 0.066
1.22 1.36 1.57 1.51 0.70 0.867
0.90 1.00 1.12 1.03 0.18 0.183
0.92 1.00 1.09 1.02 0.13 0.135
0.90 1.05 1.25 1.11 0.31 0.33 1
0.90 1.00 1.14 1.04 0.22 0.219
“I: Normal errors, II: Laplace errors. III: Mixture of Normal errors. IV: t2 errors. LQ: Lower quartile, UQ: Upper quartile. RMSE: Root mean squared error.
SD: Standard
deviation,
slightly worse than that for HT,. A slight improvement in power over the performance of HT, is evident for the Laplace error distribution (II). However, for the mixture of Normal errors distribution (III) this ranking is reversed between HT, and EHST,,. Under the t, error distribution (IV), EHST,, is moderately powerful and clearly dominates the performance of HT,? To rationalise these results, table 3 presents point statistics from the distributions of the Tobit MLE and SCLSE for the error distributions I, II, III, and IV considered above. It is immediately apparent that it is only in the case of error distribution IV (t,> that the Tobit MLE is substantially biased. In fact, the Tobit MLE dominates SCLSE in root mean squared error (RMSE) terms for the other distributions I, II, and, in particular, III. Examining the results for distribution II (Laplace) provides evidence for the poor performance of both HT, and EHST, as the Tobit MLE is only mildly mean and median biased, whereas SCLSE is approximately median unbiased and slightly mean biased.4 Turning to distribution III (mixture of Normals), it ‘Another design for the regressor variable was examined where x,,, i = 1,. , n, were the expected order statistics from the standard Normal distribution also normalised to have zero mean and unit variance. In this case, under H,,, both HT,, and EHST, perform slightly worse than under the uniform order statistic design. Additional experiments for EHST, indicated great and marginal increases in power against the Laplace alternative for ten percent censoring and R*. respectively, whereas, against the mixture of normals alternative, power respectively declined and marginally increased. Details are available from the authors on request. The symmetry of these exogenous variable designs implies that the results of this and the following section are also valid for p = (0.0, - 1.0). We are grateful to an anonymous referee for this point. ‘Similar results are reported for the Tobit MLE by Paarsch (1984), for sample sizes of n = 50. 100,200, where the relative bias of the Tobit MLE was relatively small. These conclusions are also evident from the simulations of Powell (1986. table II, p. 1451).
is the substantial mean bias of SCLSE which accounts for the improvement in power of HT,,, whereas that of the Tobit MLE is small accounting for the poor performance of EHS7;,. Distribution IV (t,) induces a large bias (both median and mean) in the Tohit MLE and thus both HT,,and EHS7;, show increases in power. Whereas the improvement in the power of EHS'I;,is dramatic. it is somewhat disappointing that although SCLSE is approximately median unbiased and only moderately mean biased, the power of HT,,is still rather low. On the basis of these results. the Hausman score statistic EHST,, is to be recommended over the Hausman statistic HT,, itself. For those casts in which the Tobit MLE dominates SCLSE in RMSE (distributions I, II, and III) EHS~, displays low power whereas for distribution IV (t,) it has moderately good power propcrtics.
5. Monte Carlo experiments for the ‘i’ohit model censored least-absolute deviations estimator
using
EHST,,
and the
As in section 4, the model is the Tobit model. cqs. ( 17) or (IX): H,, corresponds to the distributional assumption ( IO). Another alternative cstimator for ,3,, which imposes weaker distributional assumptions than those of SCLSE is the censored least-absolute deviations estimator (CLADE) due to Powell ( 1084) which minimises the criterion function:
As for SCLSE, CLADE maintains the conditional median assumption (70) but drops the symmetric error assumption of SCLSE: as above denote these CLADE assumptions H ,. Under H,, CLADE is root+ consistent; see Powell (19841. There is a lack of suitable computer routines to locate (‘LADE: see Paarsch (1984). Thus. only the test statistic form EHS7;, is considered in the experiments. The matrices required for EHS7;, arc given as follows:
- = lim II EC, II >I
\’ -0
’ k 1( _v;/q, > 0) x,_\-,‘.f( 0). I I
( 30)
= lim II ’ c I( _\-;/I$,> o)x,.r;/3. I, I7 i I
g,(O) = I(x;p,,>
o,{C -
I(
?‘,
(311
i= I.....)?.
(32)
S. Peters and R.J. Smith, Distributional
specijication
tests
191
where f<.> denotes the (unknown) density of u,, i = 1,. . . , n. Again, the variance matrix V, has full rank; that is, r = k. Consistent estimators for (30) and (31) under H,, may be obtained by using their sample counterparts, substitution of 1/(27r)“2q,o for f(O) and the Tobit MLE (p,, G,‘n> for the unknown parameters (PO, a,g>. As in section 4, 4=p and @=(Z,,O). The experiments follow the same format as section 4 in that the same sample size, n = 500, number of replications, 8000, design and distributions (Normal, Laplace, mixture of Normals, and t2) are used. Panels C of tables 1 and 2 report point statistics and distribution function results, respectively, for EHST,, eq. (12), formed from (27), (28), (301, (311, and (32) evaluated at the Tobit MLE. Again, the H, (1) finite-sample distribution of EHST, agrees closely with that predicted by asymptotic theory. Results for error distribution II (Laplace) indicate that EHST, has some (but small) power; however, the CLADE-based EHST, is superior to the SCLSE-based HT, and EHST, for Laplace alternatives, but not overwhelmingly so. The performance of EHST, for the nonsymmetric (mixture of Normals) alternative distribution III of eq. (29) shows that the power of EHST, differs very little from size and is dominated by both the SCLSE based HT, and EHST,,. For distribution IV (fz), the CLADE EHST, has good power superior to that of both the SCLSE-based HT, and EHST,.’ Turning to table 3, it is clear that the power of the CLADE-based EHST, increases with the bias and RMSE of the Tobit MLE, that is according to the distributional ranking I, III, II, and IV. In particular, the low power of this statistic for distributions II and III (Laplace and mixture of Normals) is explained by the small bias (both mean and median) of the Tobit MLE, whereas its high power against distribution IV (t,> is associated with large biases of the Tobit MLE. Thus, in this sense, the CLADE-based EHST,, is a reasonable distributional specification test statistic for the Tobit model.
6. Concluding
remarks
This paper has been concerned with providing convenient specification tests for parametric or likelihood based models against distributional alternatives where the model is only partially specified parametrically. The procedure is a variant of the Hausman test statistic but the proposed test statistics have the advantage that they need only the maximum likelihood estimator computed for the parametric model. The statistics have a similar basis to those proposed by White (1987) for testing dynamic specification. In contradistinction to the statistics of Ruud (1984) for the classical case of parametric restrictions, our statistics to not require an estimator under the ‘Additional
experiments
as in footnote
3 gave similar
results
for the CLADE-based
EHST,,.
alternative hypothesis; in such circumstances, our statistics would reduce to familiar score or Lagrange multiplier test statistics for parametric restrictions To explore the efhcacy of the proposed statistics and to compare them with the corresponding Hausman statistic, some Monte Carlo experiments were conducted for the Tobit model using various error distributions. For moderate sample sizes in cross-section econometrics, the null distributions of both the Hausman and proposed efficient Hausman score statistics accorded well with that predicted by asymptotic theory. Against the alternative distributions considered, the Hausman test itself performed poorly, whereas the Hausman score test statistics based on the asymptotic first-order conditions for the symmetrically censored least-squares estimator and the censored leastabsolute deviations estimator had some power in detecting those circumstances in which the Tobit estimator performed badly. In particular, the censored Icast-absolute deviations estimator based Hausman score statistic behaved especially well in this regard and can therefore be recommended as a useful distributional specification test statistic for the Tobit model on the basis of these results although more extensive experiments arc required for different exogenous regressor designs and other alternative distributions to ascertain whether these results arc of wider generality. Various regression-based forms of the Hausman score test statistic were also obtained. It still remains to examine their finite-sample properties.
Appendix The following (1967). Let
assumptions
arc
identical
to those
where E[ .] denotes expectation taken with respect ~1and x and ~(4) = 3Rty, X; ~$)/iJd. (N-l)
For each tied G!I,g(4) is measurable Doob: see Huber ( 1967. p. 222).
(N-2) There
is some d,, such that A($,,) = 0.
employed
by Huhcr
to the true joint density
and is separable
of
in the sense of
S. Peters and R.J. Smith, Distributional specification tests
193
(N-3) There are strictly positive numbers a, b, c, and d, such that (i) llA($)ll 2 all+ - &II for II4 - &II 54, (ii) E[p(+, d)] I bd for II+ - 4Jl+ d I d,, d 2 0, (iii) E[p(+, d121I cd for II4 - 4+jll+ d I d,, d 2 0. (N-4) h(b) possesses a nonsingular derivative in a neighbourhood
of 4 = 4”.
(N-5) E[ llg(&)112] is finite. Define
Z,(T,4)
=
II
1 + n1’211A(T)II
Then, as in Huber (1967, lemma 3, p. 227): Lemma.
Under
assumptions (N-1 ), (N-2 1, and (N-3 ):
Using this lemma, the proof of the proposition is straightforward: Proof of proposition. O*(l):
From the lemma, (N-4) together with n’/2(&n - Bo) =
Given that n’12(6,, - Bo) = O,(l) and (N-4):
References Andrews, D.W.K., 1987, Asymptotic results for generalized Wald tests, Econometric Theory 3, 348-358. Arabmazar, A. and P. Schmidt, 1982, An investigation of the robustness of the Tobit estimator to non-normality, Econometrica 50, 1055-1063. Cox, D.R. and D.V. Hinkley, 1974, Theoretical statistics (Chapman Hall, London).
Hausman. J.A., IY7X, Specitication Hau5man. J.A. and W.E. Taylor. 230-245.
teats in rconometrics. 19X1. A generalized
Econometrica 46. 1251-1271. specification test. Economics Letter\
S.
HOI&. /\.. IYX7. A remark on Hausman‘r specltication teat. Econometrlca 50. 7JYC75Y. Fluber, P.J.. IYh7. The behaviour of maximum likelihood estimates under nonstandard condtions. Proceedings of the Fifth Berkeley Symposium 1. 22I-733. Huber. P.J., 19X1. Robust statistics (Wile>. NCA~ York. NY). Newry. W.K.. 19x7. Specification teats for the distributional assumptions III the Tohit model. Journal of Econometrics 34. 125- 115. Paarach. H.J.. 19X4. A Monte Carlo compariwn 01 e\tunator\ tar censored regression modrls. Journal of Econometrics 24, 107~211. Powell. J.L.. 10x4. Least ahsolute deviations estimation tar the ccnwrrtl rrrgreasion model. Journal of Econometric\ 2.5. 303-325. Powell. J.L.. 10X6. Symmetrically trimmed Icast quare\ r\tlmuticm tot- TohIt modela. Econometrica 53. I435- 1460. Rae, (‘.R. and S.K. Mitra. 1971. Grneralizcd inwrw of matricc\ and it\ application\ (Wllq. No\ York. NY). Ruud. PA.. IYX4. Te$t\ 01 spccilication in cconometrlc\. Econometric Rwieus 3. 1 I I~-21. White. H.. IYX?. Maximum likelihood estimation of miaspecified models. Econometrica 50. I -3 White. H.. 19X7. Specification testing in dynamic model\. in: Truman F. Bewley. rd.. Advances in econometrics, Fifth world congres\ (<‘amhritl_ce Ilnivcrsit\ Press. New York. NY) 1-5s