Encompassing in stationary linear dynamic models

JOURNALOF

Econometrics Journal

ELSEVIER

of Econometrics

Encompassing Bernadette

63 (1994) 245-270

in stationary

Govaert9,

linear dynamic

models

David F. Hendry*vb, Jean-Franqois

Richard”

“Solvay & Cie, Bruxelles, Belgium bNuJield College, Oxford OXI INF, UK ‘University of Pittsburg. Pittsburg. PA, USA (Received

October

1993)

Abstract A model M 1 encompasses

a rival model M z if MI can explain M 2’s results. A Wald checks if a statistic of interest to M, coincides with an estimator of its predicted value under M 1. We propose techniques for evaluating WETS in stationary, linear, dynamic, single equations with weakly exogenous regressors, extending results for strong exogeneity. Dynamics can constrain MI’s predictions of M,‘s findings, so encompassing tests can differ from existing tests as examples illustrate. Their asymptotic power functions are compared with the outcomes in a small Monte Carlo. The results support the use of parsimonious encompassing tests. Encompassing

Test

Kq

words;

IEL

classijcation:

(WET)

Encompassing; Cl;

Dynamics;

Weak

exogeneity;

Parsimony;

Monte

Carlo

c4

1. Introduction

One model M 1 is said to encompass a rival model M 2 of the same variable y if the former can account for the results obtained by the latter. This notion is a natural component of a progressive research strategy and has been formalized in the econometric literature: see Hendry and Richard (1990) for an overview and bibliographic perspective.

*Corresponding

author.

This research was financed in part by grants BOO220012 and ROW231184 from the U.K. Economic and Social Research Council. Helpful comments from Jean-Pierre Florens, Grayham Mizon, Mark Steel, and two anonymous referees are gratefully acknowledged.

0304-4076/94/$07.00 0 SSDI 030440769301567

1994 Elsevier Science B.V. All rights reserved 6

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et al. i Journal

of Econometrics

63 (1994)

245-

270

We will first define encompassing at the level of estimated models, emphasizing dynamic equations, relate it to a more heuristic concept and to nesting, discuss testing for encompassing, then extend the definitions to conditional settings. Our discussion draws on Florens, Hendry, and Richard (1991) to which the reader is referred for a formal analysis. Let M, and M, denote two competing parametric dynamic models with no exogenous variables. Their (sequential) density functions for a common vector y, are fbt 1 Y,_ 1, a) for a E d and g(yt 1 Y,_,,6) for 6 E 53. The matrix Y,_ 1 = (JJ- 1, . , y,), thereby grouping observations prior to period t. Observations start at t = 1, and initial conditions are assumed to be known.’ The models are called estimated in that they are provided with estimators i& and 8, for any finite sample YT. Let ti 1 = (M i, hT) and tin, = (M2, &). De$inition I. ti i exactly encompasses tiz if and only if there exists a sequence of functions {& } such that & = &(G,), M i-almost surely. When attention is restricted to a subset 6i of 6, Definition 1 applies subject to the qualification ‘with respect to 8,,‘. In general, we would not expect encompassing to hold exactly for finite sample sizes, even if Mi were the data generation process (DGP). A weaker requirement is that of asymptotic encompassing, which we refer to as encompassing when no ambiguity arises: De$nition 2. m 1 asymptotically encompasses l?l z (denoted by Q 18 m 2) if and only if there exists a function i, such that & = A(&) + O,(T- ‘/‘), M i-almost surely.2 The concept of asymptotic encompassing captures the heuristic notion that the DGP ought to encompass all competing models. If M1 were the DGP, asymptotic encompassing would generally hold under appropriate technical conditions, with A(a) being given by the plim of & under M l. In particular, if & were a pseudo-maximum likelihood (ML) estimator, A(a) would coincide with the pseudo-true value associated with the Kullback-Leibler information criterion (KLIC): see, e.g., Sawa (1977) Kent (1986), or Gourieroux and Monfort (1991).

‘This assumption is largely for notational convenience: extended to incorporate sampling distributions for Y,. ’ The order of convergence appropriate for the present

the specification

can be adapted to the class of models under paper where we consider stationary processes.

of M, and

consideration.

M, can be

I:,:?

is

B. Govaerts et al. / Journal 01’Econometrics 63 (1994) 245- 270

241

If we define nesting such that it implies the existence of a function 6 = d(a), and use an estimation procedure which is equivariant over such a function, then nesting implies encompassing. However, nesting is often defined by the weaker requirement that the KLIC of M2 relative to M 1 be zero (see, e.g., Pesaran, 1987, or Gourieroux and Monfort, 1991). Then nesting also implies asymptotic encompassing for estimators that are asymptotically equivalent to (pseudo-)ML. Naturally, encompassing does not imply nesting as illustrated by the DGP encompassing all rival models, whether formally nested or not within it. The distinction between the two concepts of nesting is irrelevant to the analysis below and so M 1 c M 2 reads as ‘M 1 is nested in M Z’ without qualification. From the perspective of econometric modelling, an important concept is parsimonious encompassing which determines whether a ‘simple’ model is capable of capturing the salient features of a more ‘general’ model within which it is nested. 3. ti 1 parsimoniously M1 c M2 and (ii) Ml&M,.

Dejinition

encompasses

M, (Q 18, ti 2) if and only if (i)

The concept of parametric encompassing in Mizon and Richard (1986) is closely related to Definitions 1 and 2 (asymptotically), but restricts & to be a pseudo-true value. That restriction creates conceptual problems (such as loss of transitivity) but delivers an operational concept that has important unifying features for testing nested and nonnested hypotheses. Tests of whether or not M 1 encompasses MZ are generically based on measures of divergence between & and A,(&.) for some suitable choice of {2,) (see Florens et al., 1991). Typically one aims at selecting a sequence {A,} which minimizes the divergence of M 2 relative to M 1, but this often results in intractable functional optimization problems, so that approximate solutions must be considered. Given the previous discussion, possible candidates are (finite sample or asymptotic) pseudo-true values. Wald encompassing test (WET) statistics, introduced by Mizon and Richard (1986) and central to the objectives of the present paper, rely upon pseudo-true values to examine whether or not the encompassing difference fi[& - A(&,)] is significant on M, (see Section 2 below). Consider the case where the competing models include a vector of weakly exogenous variables. As discussed in Engle, Hendry, and Richard (1983) weak exogeneity implies that there can be no efficiency gain in designing estimators which depend on the specification of the exogenous variables’ process. Hence no such estimators will be considered in the present paper. Nevertheless, the sampling distribution of 3, on Ml is bound to depend on the characteristics of the exogenous process since, in particular, M 2 is misspecified from the viewpoint of M 1. Hence generalizations of Definitions l-3 for conditional models require explicit consideration of the exogenous process and an issue of robustness arises.

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Let M, denote a sequential model for the exogenous variables rt, with density function h(v, ) S,_ [, 7) where S,_ , regroups past observations on s; = (y, : v;) and r E Y is a nuisance parameter. Let 0: = (pi, ti,) for i = 1, 2. DeJinition 4. 0,

encompasses

Q 2 given Q, if and only if Q ; d M 5 relative to &.

It is unreasonable in most applications to require that M, be more than just an auxiliary model, which could be severely misspecified for the DGP, so we extend the definitions to a class of completing models MC (if only for consideration of robustness): Dejnition

5. ti 1 encompasses

such that

ti ‘1d 0 5 relative

Q, given 2, to &.

if and only if there exists m, in 2,

The analysis of the choice of regressor problem in Mizon and Richard (1986) and Florens, Hendry, and Richard (1988) can be reinterpreted within the context of these definitions. However, since they only considered cases where y, does not Granger-cause Y,(see Granger, 1969), so that Y,is strongly exogenous in the sense of Engle et al. (1983), the analysis simplifies considerably, and for practical purposes can be treated as a ‘fixed regressor’ analysis. The question naturally arises as to whether their results remain valid once Granger causality feedbacks from y, to Y, are allowed. To simplify notation for the rest of the analysis, we do not explicitly distinguish between models and their estimated counterparts, using Mi as a shorthand for both. Following Hendry and Richard (1982), the limit of mi under the DGP, equivalent to the reduction of Mi from the DGP, is called an empirical model (Hendry, 1994), and it is the characteristics of the empirical model which determine the outcome of any modelling exercise. The two objectives of our paper can now be formulated. First, we propose general techniques for the evaluation of WET statistics for single equations estimated from stationary, linear, dynamic systems where the regressors contain lagged dependent and weakly exogenous variables. We derive conditions under which results obtained for strong exogeneity extend to weak exogeneity. Because of feedbacks from the endogenous to the conditioning variables, the rival models do not fully characterize the joint data density, which needs to be completed by an auxiliary system linking the nonmodelled variables. The formulation of the completing model is important for the validity and power of the encompassing tests and Section 3 proposes a general approach which ensures at least a consistent test procedure. The other objective is to explore the fact that dynamics often constrain the predictions which one model can make of another’s findings, and encompassing tests that exploit such information can differ from existing tests. Consider the

B. Govaerts et al. ! Journal of Econometrics

rival dynamic

63 11994) 245- 270

249

models:

M,:

Yr = BY,-

M2:

Y, = YYt-2 +

1 +

Elrt

&2t,

where )/I 1< 1 (to ensure stationarity), and each model assumes its error to be independent normal, mean zero, with variance of, denoted IN (0, cf ). Then, Ml predicts 7 to be /12, but also predicts the presence of residual autocorrelation in M, since it views M, as: MT:

y, = /12yr-2 + e,, + B&it-1.

These factors determine M i’s prediction of the estimate of ?: in M2. The WET statistic is equivalent to the usual F-test for deleting y,_ 2 from the linear nesting model: M,:

yt = by,-,

+ cy,-,

+ v,.

Both tests are approximations in finite samples due to the dynamics. If we switch the roles of the competing models, then M2 predicts /I to be zero, and predicts the presence of an autoregressive error in M 1. The F-test has the same form, but now tests if b = 0, whereas the form of the WET statistic is noticeably different from the previous case due to the regressor in M, being at a longer lag than that in M 1. Such considerations extend to more interesting dynamic models which include regressors that are weakly exogenous for the parameters of interest under M,. The structure of the paper is as follows. Section 2 reviews results on encompassing tests in previously studied models, and Section 3 extends those findings to stationary, linear dynamic models. The examples in Section 4 illustrate implementation issues, including test power considerations. Section 5 concludes, and comments on extensions to integrated processes. The paper draws on Govaerts (1987) where additional results, proofs and examples are found (a copy is available from the first author upon request). 2. Encompassing

test statistics

This section summarizes the definitions and results needed for our later analysis. Proofs and additional details can be found in the literature noted in the introduction. We focus on parametric encompassing tests in the sense of Mizon and Richard (1986). 2.1.

WET statistics

Ignoring exogenous

for the present any complications variables, let 6 denote a consistent

arising from the treatment of estimator of a under Mi and

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&a statistic M, is

of interest

63 (1994) 24% 270

of Ml. The pseudo-true

in the context

value & of &under

4. = plim J. M,

(2.1)

Let i$ denote the encompassing difference relative to 6, namely between 6 and an estimate of its pseudo-true value &:

the difference

&=&&. The limiting

(2.2) distribution

of ,,/?i&

on Mi is

where 3 means M, Also, V,[fid”,]

‘is asymptotically

distributed

is the asymptotic

Using the estimated given by

variance

variance

V,[,,/?d”,],

where p is the rank of Vi[fiz4] Moore-Penrose inverse.

on M, as’. matrix

of @z,++.

a WET statistic

and

the

superscript

with respect to &is

’

denotes

the

2.2. The choice qf regressors problem The choice of regressors

problem el-

usually

M,:

y=Xj?+el,

N(O,a’I,),

M,:

y = Zy + cZ, c2 - N(0,r21T),

takes the form: a=(/l,02),

(2.4)

6 = (Y,T2 ),

(2.5)

where X and Z are full column rank matrices of dimensions T x k and T x 1 respectively. We assume for convenience that X and Z have no common regressors; the general case is easily treated at the cost of more cumbersome algebra. In the rest of the paper, the overscript _ denotes (pseudo-)ML estimators, which coincide with ordinary least squares (OLS), except that the sums of squares in variance estimators are divided by T.

B. Gouaeris rt ul. / Journal qf Econometrics 63 11994) 245- 270

Another MZ: M,:

251

possible rival model for M 1 is M,, the linear nesting model of M 1 and y = Xb + Zc + e,

e - N (0, v2 IT),

d = (6, c, v2).

(2.6)

We consider four classes of WET statistics, reflecting which parameters in M 2 or M, are of interest to the builder of M, . WET statistics against M2 (nonnested encompassing) will be denoted by ‘1. and against M, (parsimonious encompassing) by II:. (a) Complete

WET statistic

(b) Simplification

(CWET)

WET statistic

(SWET)

(c) Hausman

WET statistic

(HWET)

(d) Variance

WET statistic

(VWET)

Their precise expressions are found, e.g., in Florens et al. (1988). The statistics qH and r; have been dropped because the Hausman WET statistic is not applicable in M2 (since M 1 and M2 have no common regressors), and v; is not defined because the encompassing difference for the variance of the nesting model M, is O(T-‘). A further important test in this context is the conventional F-test of the null hypothesis c = 0 in M,. Two versions of it can be given: Fc, the conventional small sample F-statistic, and qF, an asymptotically equivalent version: --2

YIF= CJ Y

,

ex-w PX-T’Z

where Qx = I - X(X/X)Fc =

T-k-l Tl

d

Qxy + j12(r), M,

(2.7)

‘X’, so that

~~(1 - Tm’yF)m’.

(2.8)

The WET statistics defined above and the F-statistic qr are closely related as shown in Proposition 2.1 below. In the rest of the paper, equations of the forms:

respectively denote (a) finite sample equality, (b) large sample equality [q, = ~~ + O,(Tml) on M r], and (c) weak asymptotic equivalence [q. = ylr; + o,(l) on M,], where O,(T-“) and op(T-‘) are defined as in Mann and Wald (1943) and White (1984). Using this notation, the main results available in the literature are:

B. Goaarrts rt al. / Journal of Econometrics

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Proposition have:

2.1.

Under the assumption

63

(1994J

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270

that X and Z are strongly exogenous,

jj - Ye = (Z’Z)_‘Z’Q,y, -2 z

-

Ilc

7;

=

M’,

-

f@

+

(2.9) Xa”)‘Z(ZfZ)-

‘Z’Qxy,

et al. (1988) for proofs.

(2.10) (2.11)

Ils=vF=fl;=YIP.

See, e.g., Florens

we

In other words we find:

(9 The SWET statistic of M , against

M 2 is equal to the F-statistic, and is asymptotically equivalent to the CWET statistic. (ii) The SWET statistic of M 1 against M, is equal to both the F-statistic and the CWET statistic. (iii) Ml&TM2 if and only if Mlb,M,. encompassing entails variance encompassing in static linear (iv) Simplification models; or equivalently, the implicit null hypothesis of qs is included in the implicit null hypothesis of vV (see Florens et al., 1988, for similar results for VH

and

rl;).

A major objective of the present paper is to investigate under what conditions Proposition 2.1 remains valid for weak exogeneity. For reasons of space, we restrict attention to the modified F-statistic qf, together with the four WET statistics ns, II:, qc, and vE.

3. Stationary

linear dynamic models

3.1, The models Let y, denote the value of the dependent variable, Y, the set of all current variables believed weakly exogenous for the parameters of interest by either investigator, and p the maximum lag length. Define s; = (y,, u;), s’(,_ i) = (St- i, . . . ,s,_,) and f; = (s;, & i)). Initial conditions are assumed to be known. The rival models M1 and M2 and the nesting model M, are formulated as

MI:

~tIrt,+u-

IN(B’x,,~2),

(3.1)

M2:

~rlrt,

IN(y’z,,z2),

(3.2)

M,:

q-l)-

ytI~t,+l)

_ IN(b’x, + c’z,,v2),

(3.3)

B. Gocaerrs et al. / Journal of Econometrics 63 f 1994j 245- 270

253

where x, and z, are selections of k and I variables in (Y,,+ 1J). To derive the WET statistics, an auxiliary process is needed for the nonmodelled variables. The completing model is defined as M,:

yt I +

1, - IN(r’w,,E),

(3.4)

where wt c s,,_ 1I and r is a matrix of unrestricted parameters. This model may, but need not have, an economic interpretation as it is purely instrumental in the method used to derive the WET statistics. From (3.1) and (3.4), the completed model M’, = (M 1, M,) is a vector autoregressive (VAR) process in s,. Let a denote the complete set of parameters of M ‘1: a = (/?, 02, c Z) and suppose that they ensure the stationarity of M’, The notation matches that of Section 2, in that a denotes the parameters of the augmented M, model needed to derive the predicted values of the statistics in M2 when there is feedback from the dependent variable onto the conditioning variables of either model. Two reformulations can be given for M;; first, the Markovian representation (i.e., the companion form): M’,: where n(a) distribution M;:

N-1

and O(a) are known off;: fr-

(3.5)

_ lN(n(a)ftm~,Q(a)),

N(O,Y(a)),

functions

of a; and secondly,

the marginal

(3.6)

where Y(a) = Q(a) + h’(a) Y(a)n(a) is the marginal varianceecovariance matrix of the variables evaluated under M;. The ML estimator of Y(a) is obtained by replacing a by 6 and is denoted below by p = Y(G). The unconditional variance
of WET

statistics

We now derive the WET statistics relative to 8= (p, S2) for (complete) encompassing of MZ. The derivation is conceptually straightforward and technicalities are omitted unless they are essential for the development of the argument (see Govaerts, 1987, for detailed derivations of the relevant expressions). Encompassing methodology in conditional dynamic models requires that statistics are explicitly derived under the joint M ‘1 = (M Ir M,) and not just under MI. Because the completing model M, is instrumental in the analysis, it

B. Goruerr.~ et ul. 111 Journal of’ Econowwrric.s 63 (1994) 245- 270

254

influences the values of the WET statistics and hence the outcomes of the tests. Consequently, a careful choice of M, is required. This is discussed in Section 3.3. The method is based on the following result in Hannan (1970). Let A denote the unconstrained estimator of the second-order moment matrix offt: (3.7) then the asymptotic

fivec(A -

distribution

Y(a)) i

of vet A, on M;, is given by

N@,@(a)),

(3.8)

where @(a) is an expression in Y(a) which follows from Hannan’s formulae. Govaerts (1988) proposes an algorithm for evaluating @(a) which is based on a Jordan canonical form representation of the system (3.5). Since 8 is a known function of A, the encompassing differences & = J-

6, = (I - &,72 - ri)

are as follows. First:

so that jj - y4 = (Z’Z))‘Z’y

-

!P/,-,’ Q&p

= (z’z)~lz’QXy+((z’z)-‘Z’X-

@&r P&

(3.9)

Next:

so that r,’ 2 02, showing

that variance

dominance

remains

necessary.

Further:

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63 (1994) 245- 270

Compared to (2.9) and (2.10) these expressions show that the difference between the dynamic and static case is principally based on the values of the differences

D7 = (Z'z)m‘Z’X - pi; Of2= ;X'QzXor, collecting

(3.11)

!&,,

(3.12)

!&,.,,

these together

in a sufficient

(but not necessary)

formulation, (3.13)

This is the difference between an unrestricted estimator of the joint regressor second-moment matrix, and an estimator thereof restricted by the implications from M 1 for the dynamic behaviour of the regressors. For testing parsimonious encompassing of M1 against M,, Z is replaced by the total set of regressors (X, z) in the previous formulae. In that case, all the formulae can be simplified, since projecting X on 2 = (X, z) ensures that

(Z z)- ‘2x

= Pi; !Pz, = (I: O),

;Xf QzX =

!?xx.z= 0.

(3.14)

Thus, in nested models with valid completing assumptions, encompassing differences are identical in static and dynamic cases, but that does not imply the equality of the corresponding WET statistics as shown below. The formula to derive V,[fi(z -- S,)] = V,[J’?~~] follows from the fact that J8, as a function of G and 8, is a known, continuous, and differentiable function of A in (3.7): (3.15)

& = h(A). Hence, the asymptotic

Jr& :

distribution

of L?~on M; is given by

N(0,Wa) @(aW(a)’ ),

(3.16)

where H(a) = plim,(ah(A)/avecA’). Analytical evaluation of H(a) is impractical for most models (even simple ones), but its numerical evaluation is general and based on elementary matrix operations. Nevertheless, under restricted conditions (detailed almost known,

below) the analytical expression of V,[J?z8] is known, and a direct comparison with the static case can be made.

3.3. Choice qf the completing

or

model

The choice of different completing models in the comparison of M 1 to M 2 can lead to different expressions for !i’, D,, D+, D, and hence can affect the

encompassing statistics, occasionally inducing different conclusions for the corresponding encompassing tests. Consequently, the issue of robustness of the encompassing analysis against alternative choices of M, arises. The orders in probability of the differences D,, Or*,D are at the root of the discussion. If D = 0,the WET will be equivalent to the F-test; when D # 0,different cases must be studied. Six cases are retained, corresponding to the possible values of the encompassing differences, and the following concepts are defined: (i)

M; is exactly

moment

(ii)

M; is exactly projection

efficient (EME) against efficient (EPE) against

(iii) M; is strongly

moment

efficient (SME) against

(iv) M; is strongly

projection

efficient (SPE) against

(v) M’, is weakly moment

efficient (WME)

(vi) M’, is weakly projection

against

efficient (WEP) against

M2 iff D = 0. M, iff D7 = 0 and Or2= 0. M, iff D z 0.

.K

M, iff D, z 0 and D+ =z0.

.I/

M, iff D'z 0.

.x

M, iff D;.2 0 and D,z = 0.

.K, These concepts Proposition

3.1.

are related

to each other as summarized

.@,

in:

EME + EPE 1 1 SME + SPE 1 1 WME 4 WPE

There are counterexamples concepts. If none of the properties WET statistics is seriously

DT #O .K

or

showing

that no other relation

exists between

these

in (i))(vi) holds, the robustness of the corresponding compromised. In fact, if either of

D+ #O, .*(

then there exists at least one (M,, M,*) E J@‘, such that plim, 2; # 0 or plim, if2 # 0 under (M , , M,* ). As a consequence, when M, is misspecified, M ‘1can fail to encompass M2 despite the fact that M 1 is the true model: Example 4.7 illustrates this phenomenon. Fortunately, WPE can always be achieved as stated in: Proposition 3.2. For any pair of’rical models M, and M2 of the,fi)rm (3.1) und (3.2), there exists at least one completing model M, such that M; is WME and WPE against M2.

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63 (1994) 24% 270

A sufficient condition to ensure WME (and so WPE) is that plim D = 0, so M’, must generate the correct number and type of second moments of the joint set of regressors, which requires that M; include a sufficiently rich dynamic specification. Parsimonious encompassing automatically ensures minimal robustness for WET statistics: Proposition completing

3.3. model

IfM, c M2,

then

M’, is EPE

against

More important are the properties of the WET possible values of DY, Dr2, and D summarized in: Proposition

3.4.

M=I tf

‘Ic

.?,

statistics

of

for the different

M’1 t,

M2 F

SWET

CWET EME

Mz for uny choice

M,.

qs

=

r/F

CWET

SWET =

?/;

M,

=

@

SPE

In every case, the WET statistics are analytically known and are easily computed. The first row (EME) directly generalizes (2.11) to dynamic models; SME ensures strong asymptotic equivalence under JL’ across all the tests; SPE only delivers weak asymptotic equivalence under M; ; but SPE and WME together deliver weak asymptotic equivalence under JH. The important question is whether or not, given a pair of models M 1 and M 2, there exists a completing model M, that ensures SPE or a stronger property. Unfortunately, such a model does not always exist, and as stated in Proposition 3.2, the best property which can always be achieved is WME. Following Govaerts (1987) an algorithm to build a completing model for a given pair of models Ml, M,, based on a systematic comparison of the two lists of regressors in M , and Ml, is described in the appendix. The outcome is a list of regressors defining a minimal completing model, denoted here by M,“, with a number of features: (i) If M 1 and M2 are such that there exists a completing model M, under which (M,, M,) is EME (or SME), then (M,, M,” ) is EME (or SME). (ii) The model (M r, M,“) is always WME and, hence, WPE.

(iii)

M,” is minimal in that its list of regressors is included in the list associated with any other model M, that satisfies EME, SME, or WME. (iv) If there exists a model M, for which SPE is achieved, it will be nested in M,” but the latter will not necessarily be SPE. (v) The minimal completing model is invariant to the choice of rival model M ,* nesting M i and nested in M,. It follows that, unless the proprietor of Ml has particular reasons for selecting a specific completing model M,, selecting M: as the completing model achieves robustness within the class of VAR completing models. Other choices for M, need not achieve even this minimal property. Finally, based on the appendix algorithm, a sufficient condition for SME is available which does not need any additional calculations to check if M’, is SME: Proposition 3.5. Jf(i) the number qf’parameters of (MI, M,) is equal to the number of data second-order moments used in the ecaluation qf the estimators sf’ these parameters (neglecting terms of’O,( 1IT)), and (ii) all the cross-products included in (l,JT)X’X, (I/T)X’Z, and (l/T)Z’Z appear in the estimators qf the parameters of M’,, then (M,, M,) is SME against MI. Intuitively, the stated conditions ensure that D is at most 0,(1/T) members of JZ%‘,which in turn was shown above to entail SME.

for all

3.4. PottIer comparisons Although Proposition 3.4 relates the various tests qc, qs, qF, ~15,and q: under the null, it does not follow that they have the same power properties when Ml is false. In particular, care is required in formulating the maintained hypothesis within which power is studied. There are three distinct ways in which M 1 might be misspecified relative to the assumed DGP. First, the alternative (denoted M,) may differ from M 1 in the direction of M2 but retain elements of M 1; second, M, could be Mz itself; finally, none of the elements in either M 1 or Mz may be present in M,. In each case, to correctly evaluate the power, M 1 must remain congruent with the data evidence despite being misspecified, and in the context of dynamic models this aspect is not easy to achieve. When M, corresponds to M, (e.g., the union of M, and Mz less any redundant elements), then the F-test is bound to have the highest power in large samples since its error will be an innovation with minimum variance in the class, whereas M 1 will fail to have an innovation error if M2 is dynamic. However, tests for innovation errors seem more appropriate for such a scenario than the encompassing tests discussed here. Example 4.2 illustrates this, and points up the advantages of parsimonious encompassing tests and general-to-simple modelling strategies.

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63 (1994) 245- 270

When M, does not correspond to M,, power rankings are less clear-cut, in small samples may depend on degrees of freedom. Example 4.3 illustrates case.

and this

4. Examples The aim of this section is to illustrate the concepts and properties defined above using some simple examples. The choice of the completing model is discussed, explicit forms of the WET differences and statistics are given, most of the possible cases of moment and projection efficiency are illustrated, and the minimal completing model is given for each example. The F-test against M, is asymptotically valid (i.e., it has the correct size asymptotically and is consistent against fixed alternatives), and is the parsimonious encompassing statistic. Example

4.1: The stutic case

Static linear models

were formulated

in Section

2 as:

MI:

y1 = p’xl + clt

with the claim that

.sr, - IN(0, a:),

Mz:

y, = y’zr + ~2,

with the claim that

cl1 - IN(O,oi).

The minimal completing model derived from the algorithm minimally the unrestricted white-noise process: MY’:

wt = U,

where

u, - IN(0, Z)

when

is simply

and

w; = (xi, z;).

EME, EPE, etc. are verified in this case since, for example, Yww = plim, T- ’ W’ W = Z and @,,+, = Z = T- ’ W’ W, so that D = 0. The results given in Proposition 3.4 apply and coincide with those given in Proposition 2.1. However, as discussed in Section 1, invalidly restricting .Z can lead to rejection of MI even when it is the DGP. Mr can be written in an equivalent projection form for the conditional distribution of X, given zt, jointly with the marginal process for zI: M;*:

x, = lczt + uTt

where

zt

with

=

U2t

uTt - IN(O,Zxx.z), ~21

-

IN(O,Zzz),

where n = ZXZ Z;,’ and urt I u2,. The first part of M,“* is the completing model used by Hendry and Richard (1982) to introduce the encompassing principle in

260

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of

et al. / Journal

Econometrics

63

i 1994)

the static case. They show that defining the projection sufficient to ensure EPE to M’, We have: = L,-,’ zzx = 1~’ and

V,-,’ ul,,

and

245-

270

of x, on zt in this way is

k’ = (Z’z)-‘Z’X, k,,.,

= f X’ Qz X.

(4.1)

so that only strong exogeneity needs to be postulated for the marginal process of zl. This claim remains true even if in fact zr is Granger-caused by y, so complete robustness is achieved. Example

4.2: Az~torr~gres.riue models

Reconsider

the simple dynamic

model from Section

MI:

J’, = /?y,_ 1 + clr

where

M,:

yr = ~_y_~ + ~~~

with

Y,,

~~~- IN(0, a’); E2f

IN(O,?),

-

M , is itself the completing

such that I/3 < 1. Then, = pltm$-2J’_2

model, and we have

= /IcJ’/(~ - fi2),

where (e.g.) y’_ 1 = ( y2

(4.2)

. yT_ 1), and

Y,, = plimfp’_2y_2

= [fl’/(l

- f12)] = Yx,,

so that p&l ez, = ii, and hence the restricted by M, alone. Thus: (Z’Z))‘Z’X=

1.

(y’~2y~2)~ly’~2p_, = (4”LrJ’-1)-1y’-,y

where the terms of 0,(1/T)

fX’QzX = !&,.,

0,(1/T).

are determined

= 3 + 0,(1/T) + 0,(1/T),

are due to lagging.

+

second moments

(4.3)

(4.5)

Similarly: (4.6)

and hence MI is SPE against M,. One can show that MI is also SME on replacing /I and CJ’ by their estimators in the above formulae. This result guarantees that, in this simple dynamic case, the WET statistics are equivalent to the F-statistic as stated in Proposition 3.4. If we switch the two competing models so the rival models become: Mf:

Jr =

M 5:

y, = :‘~‘t-, + c2r,

BY,-2

+

Elrr

B. Gocarrts 6’1al. : Journal ~f’Gonometric.s 63 11994) 245- 270

261

then the forms of the statistics change noticeably and Mf is no longer SME but only WME. This example illustrates the differences that can occur between qF, qs, and q; when including lagged dependent variables in the list of regressors. The WET statistics can be evaluated (unconditionally on the regressors) either directly or by taking advantage of the general technique discussed in Section 3.2 above. The completing model is now M 1, and the complete encompassing differences relative to the parameters of M 5 and M,, (defined in Section 1) are 1: -&

& =

(y’y)_‘y’y_,,

(

(4.7)

)

whereQ,-, =

IT - y-i(JJ-,J'-i)‘y’mi. The variance-covariance matrices of d”, and d; on MI are both singular rank 1 and so:

2 x2(1X

rlc = ~~'(Y'~,~~Y~~)~(Y'Y)~'

v: = ~-2(~‘Q,~2~~,)2(~‘Q,~,~)-2~‘~ qF

=

(z-

“(JJ’Q,_,Y-I)‘(Y’Q,_,JT’

; x’(l),

:

x’(1).

of

(4.9)

(4.10)

(4.11)

In this example, the three statistics are ordered as qc d ~~ < q$. They are asymptotically equivalent on any model for which Tm 1y’y- , + p 0 as T-t ;c (i.e., in particular on M, or on local alternatives to M, which do not introduce first-order An interesting difference between the two orderings of the hypotheses here is that the implicit projection of the F-test switches from the realizable dynamic model y, = 4y,_, + 11,to the forward projection of Y,-~ on yml, although M,” remains the same. Govaerts (1987) examined the power of the three statistics in (4.9)-(4.11) by Monte Carlo simulations under local alternatives to MI of the form: y, = T- l!* ,IlY,-1 =

UyIml

+

+

\'2y,-2

v2yrm2 + [,

+

;,

where

[, _ IN(O,crt).

(4.12)

After correction for size, qc turns out to have lower finite sample power than either qF or ~6. The asymptotic power functions for qc and qF under the local

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B. Gooaerrs et al. / Journal of Econometrics

63 (1994) 24% 270

alternative in (4.12) can be derived as follows. First, the plims of the data second moments under (4.12) are Z,,, = (1 - l’&;/[(l &_,

= e&.,/(1

+ l’z)j(l

- \J2)2 - HZf],

- I’*),

_2 = [1’2(1 - v2) + e”]Z$J(l = YY Estimation of Ml 2 = plim 5’:

leads

- VJ.

to population

parameter

values

p,, = plim

p and

OP

BP

=

‘q;’

z;l’_

0; = a;(1

2

=

[1’2(1 - r2) + 0’]/(1

+ [(l - v2)@/(l

- vz),

(4.13)

+ \J2){(l - r2)2 - 02}]).

(4.14)

On local alternatives like (4.12) the q. tests become noncentral (see Mizon and Hendry, 1980) $(VF)

= VI/(1 - 11:) = TH2/(1 - r;,,

$(qc)

= a;${(1

- \,2)2 - Q’)/cJ,‘(l

- vZ)(l - vZ)2

= p2(nF)[1

- 2Q2/(1 - v;)(l

- v2)] + O(T-‘)

x2( 1, p2) where

(4.15)

< p2(rlF).

(4.16)

Two factors contribute to the power loss arising from the smaller noncentrality parameter: on (4.12) gi 3 ot, which yap avoids; and b, # 8, so the residuals in M 1 are not white noise, whereas (if} is, so M, ceases to be congruent and is not a good basis for testing. Even though 0 is O(T- 1’2), the asymptotic power loss is quite large as Fig. 1 shows for the parameter values 0 = 0.1 and v2 = 0.8 over T = 70, . ,300. The small sample power of the F-test is also shown, based on recursive Monte Carlo using PC-NAIVE (see Hendry, Neale, and Ericsson, 1991) to illustrate the applicability of the asymptotic formulae. Example

4.3: Strong moment

Let the two rival models

@iciency

(SME)

be

Ml:

y, = /Is, + ~11,

Mz:

y, = :i’yt-i + e2!,

with the usual claims that Eir - IN(O,o,). It seems a completing model, the realizable projection model: M;:

xr = 4.bI

+ k

where

natural

to consider

as

u, - IN(0,72).

The appendix algorithm for the minimal completing model also yields Mr. Verifying that M’, is SME in this case is tedious if one wants to explicitly calculate the order of the difference D in Section 3.3(i)-(vi) above, but

263

B. Gocaerts et al. ! Journal of Economerrics 63 (1994) 245- 270

.8 ///A(I-z .72-

.64P(F)*
:’

__.. ,,: ,,~. _,-' __: .....d '. __.*' ,... ..'. .' / ,,., ....J *";'P(F)" ,,,,,...+ ,,I..,... .” _,I_,....;' ". __.'_

,*'. /,..'

.48-

/.-../

/-.

,;’

.4

,’

24_

” ,,,,,,,,.,..,..../‘../‘. .,...J

F .. ;: ;’ ...’,_’ ,/” _:

;/

.32

,‘ ...' ._,.I ,',,,,... ‘.

,;.: .. P
..

II./ ,..' _.I ,,,'

.16 4"' T+ .08 170

120

270

220

Fig. 1. Asymptotic and finite sample power functions of the F-test and the encompassing and P(E)* are asymptotic, P(F)-is based on recursive Monte Carlo: Example 4.2

test: P(F)*

fortunately the sufficient condition for SME in Proposition 3.5 is applicable. The number of parameters in M’, is four: p, c2, 4, r2. The data second-order moments used in their estimation are (l/T)x’x, (l/T)x’y_ 1, (l/T)y’y, and (l/T)x’y, and hence condition (i) is verified. Second, in present notation, the cross-products appearing in (l/T)X’X, (l/T)X’Z, and (l/T)Z’Z are (l/T)x’x, (l/ T)x’y _ 1, and (l/T)y’y which ensures that (ii) is satisfied. M; is then SPE and the equivalences between WET and F-statistics given in Proposition 3.4 hold. If we switch the rival models Mi:

y, = fiy,_r

M;:

yt = yx, + c2,

the completing model local alternative: M,:

y, = J1yr_,

+ elf

remains

where

eIt - IN(0, a:),

where

&21- IN(O,o:),

the projection

+ i2zr + i,

where

model

M,” above.

Consider

the

;, - IN(O,af),

when 2,

=

px, + vt

with

I’, - IN(O,a<).

(4.17)

264

B. Goourrts

In this example, Mf are fl = (i,

direct

et ul.

i

Journal

of Econometrics

substitution

+ izp&)

and

of (4.17) into

Further,

= (I&:p’a,z)/(A:o;

the asymptotic

plimZ+

24%

M, shows that /I and I-:,~in

the asymptotic

(4.18) noncentrality

encompassing

p’(qc)

variance

difference

of ~,‘(I~~~~ + of)/crz,

= (n;p’o:)l(J.:o,Z

of the

(4.19)

+ C?,. is

= plim(T - yB) = i.,pcr,2/0.~,

with a limiting

270

zIf = (i, + i,,v, + i.,pu,),

whereas e, in M, is (i, + A2rt), so that F-test is $(rlF)

63 i 1994)

(4.20) so that (4.2 1)

+ +,,

and so the two tests are asymptotically equivalent under a local alternative that keeps M 1 congruent. The use of other completing models may generate WET statistics that satisfy neither strong projection nor moment efficiency conditions, and even worse choices can lose weak efficiency. For example, consider: M,:

Under

x, = u,.

M;,

YX,, = YXY_, = 0, and the encompassing

difference

on 7 becomes

7-- ;‘? = (y’_,y_,)-‘y’_,y, from which vs can be derived. Since M ‘1is neither WME nor WPE, there exists at least one MT = (M,, M:) E A1 such that D, Mf:0. If the DGP is (M,, M:), where M,*:

x, = +x,-r

+ u,,

then qs -+ x under the DGP when T increases, and hence the WET will, in most cases, reject even though the model MI is correct. Such an outcome demonstrates the dangers of using inappropriate completing models and encourages the design of completing models with the aim of ensuring a minimum of robustness for the resulting WET statistics. Example

4.4:

Exact

projection

eficiency

Let:

Ml:

Y, = Bx, + &I,,

M2:

Y, =

YIY,-I

+

~2yt-2

+

~2t.

(EPE)

B. Gooaert.7 et al. / Journal

The minimal

completing

of Econometrics

63 (1994)

245-

270

265

of x, on y,_ 1 and

model is given by the projection

Yr-2:

MT:

xt =

41yr-1

+

$zy,-2

+

~1.

Formula (4.1) in Example 1 can be applied to check that EPE is satisfied here. This does not imply the equality of the WET statistics with the F-statistic because V,[@d”] case. Nevertheless, because WME.

differs from the corresponding matrix from Proposition 3.4 the encompassing

the choice of the minima1

Example

4.5:

Strong projection

This example by one period:

Mz:

completing

obtained statistics

mode1 ensures

that

in the static are ,F to qF MC, 1s also

tlfji’cienq~ (SPE)

differs from the last in lagging the regressors

of the two models

yt = >‘lJ’lm2 + Y2J’t-3 + C2t.

The projection M,:

mode1 remains

s, = 4,y-1

and is included MZ’:

+

42~1-2

in the minima1

xr=

4,

Y,-I

+

~z.Y-2

the same: +

tl,,

completing +

mode1 given by

&~t-3

+

dv-,

+

u,.

Then (MI, M,) is SPE for the same reasons as before. The difference between EPE and SPE is due to lagging; for example, (l/T)x’_ ,y_, in (Z’Z)- ’ Z’X is estimated by (l/ T)x’y_ 1 in !?;i Fz,. It can also be shown that WME is not achieved for M,, which implies that qs and vF are not equivalent under any mode1 of A%?~:for example, consider (M 1, M ,“) E AI. Despite that difficulty, M, probably remains a good choice for the completing model since My ensures no more than WME. Example

4.4:

Weak moment e#iciency

Consider: MI:

Y, =

B+l

+

<>

M,:

yt =

YY,-1

+

~ztr

with ME”: X( = 4x,-r

+ U,.

(WME)

For these Ml and M, models, no completing model exists such that SME or SPE is satisfied. MT ensures WME. The resulting simplification WET statistic vs is completely different from the F-statistic. For example, the encompassing difference on y has the form

;7-

?1’0i =

WY-‘WQ~_,Y-I

+y'Qx_,

+_W-,d,>

(4.22)

which tests if COV( y,, y, _ 1, 1x, _ I) + ,!ICOV( y,, x, 1x, _ I ) = 0. The F-statistic tests if COV(y,, yt_ 1Ix,_ 1) is zero. The different possible WET statistics are given in Govaerts (1987) jointly with an analysis of their asymptotic power against local alternatives and Monte Carlo simulations to compare their small sample properties. For example, when M 1 relates y, to y, _ 2 and M 2 relates y, to ytm3, the encompassing statistic can be more powerful than the F-test.

5. Conclusion The general analysis in Section 3 and the examples in Section 4 reveal that encompassing in linear stationary dynamic processes raises new issues. The need to take account of feedbacks from lagged dependent variables forces the explicit introduction of a completing model, and the choice of its formulation is important if the encompassing tests are to be robust to how the completing model is specified. Six levels of efficiency of the completing model were distinguished, and illustrated by examples that highlighted which features induced which consequences. The resulting analysis reproduces that previously established for strong exogeneity when the models are static. However, in dynamic systems, a poor choice of the completing model can lead to rejection of the correct hypothesis as shown in the Example 4.3. Stationarity is an important assumption in the approach adopted here because of the central role played by (3.7) and (3.8). In integrated systems with cointegrated relationships, Hendry and Mizon (1993) develop asymptotically valid encompassing tests for linear equations or subsystems against each other, or the VAR when the latter is the DGP. Similar generalizations should hold for the class of equations of interest above. In particular, parsimonious encompassing is the final check on a reduction sequence, by which stage, mapping to I(0) variables will usually have occurred. However, weak exogeneity only holds in cointegrated systems if the error corrections in the equation of interest do not enter other equations of the system, so that it enforces a necessary condition for the present analysis to apply. Further, when weak exogeneity is violated due to the presence of common cointegrating vectors, the limiting distributions are no longer linear mixtures of normals and inference can be distorted (see, e.g., Phillips and Loretan, 1991; Hendry, 1994). Nevertheless, both the present analysis and cointegration theory emphasize the primary role

B. Goracvts et al. / Journal qf Economrtric.s 63 (1994)

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261

of the system in sustaining inference even when interest is in individual equations. Overall, our analysis favours the use of parsimonious encompassing tests or F-tests as these are more robust to the specification of the completing model, are not restricted to paired comparisons between models, and complement Further, they are invariant to general-to-specific modelling strategies. extensions in the specification of the rival model up to the union of all the nonredundant regressors in both models. Finally, when the alternative hypothesis makes the model under test noncongruent, there is a potentially serious power loss in encompassing tests (including nonnested tests) relative to parsimonious encompassing tests.

Appendix: Minimal completing

models

Sample moments are denoted by either A (matrices) appropriately subscripted. In particular, let

or 4 (scalars and vectors)

(A.11 The ML estimator of v(a) is given by 4 = t,~(&)where & = (&6-,p, 2) regroups the OLS estimators of the parameters in M, and M, as given by (3.1) and (3.4) respectively. Hence EME requires that A = w(k), while SME (WME) requires that A z w(G) on JV(J%!~). The following relationships hold: Axl. = Ax&

AWR = A,+&

A,,=BZ+~Axx~,

ARR=z+f.‘AWWf-.

(A.9

Let LB denote a list of all the distinct elements in (A,,, AX,,, A,,, AWR). Conditionally on Axx and Aww, (A.2) defines a l-l mapping between LB and i& subject to the usual restrictions for moment matrices (symmetry and positivity). However, in order for A itself to be a function of 6, it has to be the case that Axx and Aww, which are included in A, can be retrieved from LB. For ease of discussion, we restrict attention to the case where all the restrictions between A, A ww, and LB originate from the exclusion of regressors in Mi and M,.3 Let LA denote a list of all the distinct elements in A and Aww; then L, c L,

64.3)

3 Accounting for more general linear restrictions among the variables in M; raises no conceptual problems but necessitates additional algebraic manipulations of the relevant moment matrices.

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et al. / Journal yf’Econometric.s

63 (1994) 245- 270

is - for all practical purposes ~ necessary and sufficient for A to be a function of k. A formal proof of that assertion requires explicitly stating a number of technical conditions and goes beyond the objectives of the current paper. At a more heuristic level, if condition (A.3) holds, then (A.2) can be solved (recursively) for A: at stepj of the recursion, A xx and Aww are set at the values obtained on step (j - 1) and (A.2) is solved for LB; Axx and A,, are then updated and the procedure is repeated until convergence (which follows from a fixed point theorem). If, on the other hand, A is a function of G, then (A.3) holds. If, indeed, an element of L, were not included in LB, then we could assign an arbitrary value to that element and proceed as just described. That element would never be revised, contradicting the assertion that A is a function of 6 alone. The algorithm we propose aims to select a (minimal) set of regressors for w, in such a way that condition (A.3) holds. Two additional issues deserve attention before we can describe the algorithm: (1) The only difference between EME and SME lies in the treatment of initial and terminal observations in the sample. For example, under EME moments such as Tm ’ x’y_, and Tm ’ x’- ,_v- z are treated as different entities, while under SME they are conflated with each other. Our proposed algorithm trivially accommodates that distinction. (2) w, consists of la+~rll variables only, while Y,regroups all current exogenous variables. It follows that ARW cannot include cross-moments between x’s and leading y’s. If any such moments are included in L,, then EME (SME) cannot be obtained. Let X, consist of all regressors that are excluded from M i. The following asymptotic equivalence holds on M 1: A,,

z A,, ./o

A ij

a,,

.(A.4)

Both Axx and a,, are already covered by an analysis of condition (A.3). Hence, for the purpose of achieving WME, we can replace components of A,,, in LA by the corresponding elements of Axx and proceed. The proposed algorithm follows from the above discussion. We initially set w, = 0 and accordingly define LA and L,. We then examine whether condition (A.3) holds. Each time an element of L, is found to be missing in L,, w, is ‘augmented according to one of the following two (mutually exclusive) schemes: Type-A auymentation: Direct augmentation is included in the augmented LLI.

of w, such that the missing element

Tjjpr-B augmentation: The missing element belongs to arL; it is replaced by the corresponding elements in Axx, and w, is then augmented in such a way that the latter are included in the augmented Lg.

B. Govaerts

et al. / Journal

of Econometrics

63 (1994)

245-

270

269

The algorithm is finite and generates a ‘minimal’ completing model under which WME holds. Further, if it necessitates only type-A augmentations, then EME (SME) obtains. All the minimal completing models My, to which we refer in Section 4, have been obtained by application of this algorithm.

References Cox, David R., 1961, Tests of separate families of hypotheses, in: Proceedings of the fourth Berkeley symposium on mathematical statistics and probability, Vol. I (University of California Press, Berkeley, CA) 105- 123. Cox, David R., 1962, Further results on tests of separate families of hypotheses, Journal of the Royal Statistical Society B 24, 406424. Engle, Robert, F., David F. Hendry, and Jean-Fran+ Richard, 1983, Exogeneity, Econometrica 51, 277-304. Florens, Jean-Pierre, David F. Hendry, and Jean-Francois Richard, 1988, Parsimonious encompassing: An application to non-nested hypotheses and Hausman specification tests, ISDS discussion paper (Duke University, Durham, NC). Florens, Jean-Pierre, David F. Hendry, and Jean-Francois Richard, 1991, Encompassing and specificity, Cahier 91c (GREMAQ, University of Toulouse, Toulouse). Gourieroux, Christian, Alain Monfort, and Alain Trognon, 1984, Pseudo-maximum likelihood methods: Theory, Econometrica 52, 68 l-700. Gourieroux, Christian and Alain Monfort, 1991, Testing non-nested hypotheses, Mimeo. (INSEE, Paris). Gourieroux, Christian and Alain Monfort, 1992, Testing, encompassing and simulating dynamic econometric models, Mimeo. (INSEE, Paris). Govaerts, Bernadette, 1987, Application of the encompassing principle to linear dynamic models, Ph.D. dissertation (Universite Catholique de Louvain, Louvain). Govaerts, Bernadette, 1988, A note on a method to compute the asymptotic distribution of the sample second order moments of dynamic linear normal variables, Communications in Statistics 18. 4799488. Granger, Clive W.J., 1969, Investigating causal relations by econometric models and cross-spectral methods, Econometrica 37, 4244438. Hannan, Edward G., 1970, Multiple time series (Wiley, New York. NY). Hendry. David F., 1994, Dynamic econometrics (Oxford University Press, Oxford) forthcoming. Hendry. David F. and Grayham E. Mizon, 1993, Evaluating dynamic econometric models by encompassing the VAR, in: Peter C.B. Phillips, ed.. Models, methods and applications of econometrics (Basil Blackwell, Oxford) 2722300. Hendry, David F. and Jean-Francois Richard. 1982, On the formulation of empirical models in dynamic econometrics, Journal of Econometrics 20, 3-33. Hendry, David F. and Jean-Francois Richard, 1990, Recent developments in the theory of encompassing, in: Bernard Cornet and Henry Tulkens, eds., Contributions to operations research and econometrics: The XXth anniversary of CORE (M.I.T. Press. Cambridge, MA) 3933440. Hendry, David F., Adrian J. Neale, and Neil R. Ericsson, 1991, PC-NAIVE: An interactive program for Monte Carlo experimentation in econometrics (Oxford Institute of Economics and Statistics, Oxford). Kent, John T., 1986, The underlying nature of nonnested hypothesis tests, Biometrika 73, 333-343. Mann, H.B. and Abraham Wald, 1943, On stochastic limit and order relationships, Annals of Mathematical Statistics 14, 217-226.

270

B. Gornerts et cd. i Journal qf Ecowonwtrics

63 (1994) 245- 270

Mizon, Grayham E., 1984, The encompassing approach in econometrics, in: D.F. Hendry and K.F. Wallis, eds., Econometrics and quantitative economics (Basil Blackwell, Oxford) 1355172. Mizon, Grayham E. and David F. Hendry, 1980, An empirical application and Monte Carlo analysis of tests of dynamic specification, Review of Economic Studies 47, 21-46. Mizon, Grayham E. and Jean-Francois Richard, 1986, The encompassing principle and its application to non-nested hypotheses, Econometrica 54, 6577678. Pesaran, Hashem M., 1987, Global and partial non-nested hypotheses and asymptotic local power, Econometric Theory 3, 69990. Phillips, Peter C.B. and Mica Loretan, 1991. Estimating long-run economic equilibria, Review of Economic Studies 53, 4733495. Sawa, Takeshi, 1978, Information criteria for discriminating among alternative regression models, Econometrica 46, 1273 1292. White, Halbert, ed., 1983, Non-nested models. Special Issue of the Journal of Econometrics 21. White, Halbert, 1984, Asymptotic theory for econometricians (Academic Press, New York, NY).