Mutual encompassing and model equivalence

(1)

EUI WORKING PAPERS

(2)

EUROPEAN UNIVERSITY INSTITUTE 3 0001 0026 2712 5 © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(3)

ECONOMICS DEPARTMENT

EUROPEAN UNIVERSITY INSTITUTE, FLORENCE

EUI Working Paper ECO No. 97/17

Mutual Encompassing and Model Equivalence

m a o z u Lu and GRAYHAM E. MIZON ° $ d o \£ & © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(4)

No part o f this paper may be reproduced in any form without permission of the authors.

©Maozu Lu and Grayham E. Mizon Printed in Italy in May 1997 European University Institute

Badia Fiesolana I - 50016 San Domenico (FI)

(5)

Mutual encompassing and

model equivalence

Maozu Lu

Department of Economics, University of Southampton, UK Department of Economics, University of Southampton, UK and

European University Institute, Florence, Italy

Grayham E. Mizon

February 1997

Summary

This paper analyses the properties of mutual encompassing and its rela tionship to the K LIC equivalence between statistical models. It is shown that models are K LIC equivalent if and only if they are mutually encompassing and mutually Cox-encompassing. Further, within the exponential family encompassing implies Cox-encompassing and so mutual encompassing is ne cessary and sufficient for KLIC equivalence in this family. In addition, it is shown th at mutual encompassing is transitive for models in the exponential family.

Some key words: Cox-encompassing; Encompassing; Kullback information

criteria. © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(6)

(7)

1 In t r o d u c t io n

In comparing statistical models using the encompassing principle one pos sible outcome is that the models are mutually encompassing. This raises the question as to what the conditions are for two models to encompass each other, and also whether there is a sense in which they are equivalent. Since a congruent nesting model will encompass models nested within it (see Bon- temps & Mizon 1996), an example of mutual encompassing arises whenever a nesting model is both congruent and parsimoniously encompassed by a nested model. While the implications of mutual encompassing are straight forward in the case of nested models they are less so for nonnested models. Hendry (1995) points out that when two models mutually encompass each other they are isomorphic after suitable re-parameterizations and elimination of redundant parameters. Similarly, Gourieroux & Monfort (1995) argue that mutual encompassing suggests that both models under consideration can be reduced to a common or a core model, possibly after re-parameterization.

While it is possible that the mutual encompassing based on sample test statistics maybe the result of weak evidence, in this paper attention is con fined to the concept of mutual encompassing in the population. Heuristically, models th at mutually encompass each other can be expected to have equal ability in explaining and predicting the behaviour of rival models, and in this sense to be encompassing equivalent. The Kullback information criteria

(K L IC ) provides an alternative concept of equivalence in th at two models

are said to be K LIC equivalent when the KLIC for them is zero. It is estab lished th at models which are mutually encompassing and Cox-encompassing are K LIC equivalent. The difference between encompassing and Cox encom

passing is discussed. The latter becomes a special case of the former when

both models belong to the linear exponential family, but they contain dif ferent information otherwise. This reinforces the result in Kent (1986) that the Wald test statistic for the joint hypothesis of complete parametric en compassing and Cox encompassing has a degenerate null distribution when both models are nonnested and belong to the linear exponential family. Fur ther, the transitivity and other characteristics of the mutually encompassing models are discussed in the framework of the KLIC.

In the next section particular forms of encompassing are defined, and the relationship between them discussed. This is followed in section 3 by analysis of the relationship between mutual encompassing and equivalence in

(8)

the sense of the Kullback information criterion. Section 4 contains discussion of the properties of mutual encompassing generally, and in the particular case of models within the linear exponential family. Section 5 summarises and presents conclusions.

2 Encompassing and mutual encompassing

The encompassing analysis in this paper is based on the following multivari ate conditional models M \ and M2:

Mi : _{{ f ( y t | y t-}1, x,; (?) , 9 € © C f t1 j , (2-1) M2 : {3 (yt 1Vt-u £<; 7) , 7 € r C 7 ^ ]

where, 1 = (yt-1,1 Vt—2i * ' *)5 (%t) Xt— 1, Xt—2) * ‘ *) with x t strongly exogenous for the parameters of interest (see Engle et al 1983) thus sustaining conditional modelling. Both models are conditional on the same information set although they can be nonnested. The unknown data generating process (DGP) is assumed to have the form:

M0 : j/ t (y t | yt- i , x t;a ] , a € A c 1 l m} (2-2) when M0 is not necessarily contained in Mi or M2, or any linear combination of the two. The models in (2-1) and (2-2) are referred to as f (0), g (7) and

h (a) throughout the paper. The sample data are assumed to have been

generated from h (oo), where Qo € A. It is noted that the DGP need not be treated parametrically, but all th at would be required in such a case is that the density h (y t \ yt~i,Xi_', cvj no longer be indexed by a.

When the data are not generated by Mi or M2 the maximum likelihood estimators 9-r and 7t converge in probability respectively to the pseudo true values 0 (<*o) and 7 (qo) which are defined as:

6 (a 0) = 90 = arg max EXEQ0 log f (6),

#6©

7 (q0) = 70 = arg max ExEao log 3(7)

7€L

where “Eao” denotes the expectation taken under /i(a0), and “Ex” the ex pectation taken with respect to the marginal density for x t. In the following discussion Ex may be omitted when there is no confusion, but all expect ations are to be understood as being unconditional as a result of the joint

(9)

operations of EQ0 and Ex. Similarly, the pseudo true value of with respect to Mi (defined by Gourieroux & Monfort (1995) as a binding function) is defined as:

7i (0O) = arg max E ^ , , log3(7) -rer

where 60 £ 0, and consequently:

plim 7 r = 7x (0O) Ml

The concept of pseudo true value plays an important role in the following discussion of encompassing.

The essence of encompassing is the comparison of alternative models of the same phenomenon, and so an important decision concerns the basis for these comparisons. The following definitions of encompassing illustrate the major types of encompassing contrast.

Definition 1. Encompassing

Mi encompasses M2 with respect to 7 6 T if and only if:

7(Q0) = 7i(0o) (2-3)

denoted as Mi £ (7) M2 where M\ and M2 are as defined in (2-1).

Note th a t the basis for comparison in this definition of encompassing is the complete parameter vector of M2 and as such this hypothesis was defined to be complete parametric encompassing by Mizon (1984) and Mizon k. Richard (1986). An alternative basis for an encompassing contrast, which was used by Cox (1961, 1962) in the derivation of a generalized likelihood ratio test statistic for testing nonnested hypotheses, is the log likelihood ratio. The following definition employs this contrast.

Definition 2. Cox-Encompassing

Mi Cox-encompasses M2, denoted as M\ £c M2, if and only if:

f(00)

S{7i0?o)}. (2-4)

where “Eao” and “E^0” denote expectations under Mo and Mi respectively.

The encompassing contrast in (2-4) is the implicit null hypothesis of the test statistic originally developed by Cox (1961, 1962) as a generalized like lihood ratio test for nonnested hypotheses (see also Pesaran (1987) for a discussion of nested and nonnested hypotheses). Hendry & Richard (1982), Mizon (1984) and Mizon & Richard (1986) have shown th at the Cox non nested test statistic has an encompassing interpretation, and is equivalent to

(10)

variance encompassing in linear regression models. However, in more general contexts the encompassing contrasts in (2-3) and (2-4) contain independent information (see Kent 1986 and Theorem 1 below).

The encompassing relation between two models M\ and M2 will fall into one of the following three categories: (i) either Mi £ (7) M2 or M2 £ (8) Mi;

(ii) neither Mi £ (7) M2 nor M2 £ (6) Mi; and (iii) both M x £ (7) M2 and M2 £{8) Mi. Category (i) covers cases in which one model is inferentially dominant, making the other model inferentially redundant. Category (ii) arises when both models capture something of importance but not all relevant features, and so each model is inadequate in some direction. Heuristically, if Mi and M2 are equivalent then the encompassing relation between them cannot be in either category (i) or (ii). Were the models in some sense equivalent then they might be expected to be mutually encompassing, a concept th at is formalized in the following definition.

Definition 3. Mutual Encompassing

Mi and M2 are mutually encompassing, if and only if

Mi £ (7) M2 and M2 £ (8) Mi (2-5)

at 80 G B and 70 € F. Further, they are mutually Cox-encompassing, if and

only if

Mi £c M2 and M2 £c M\ (2-6)

at 80 € © and 70 € T. (Also see section 3.3 of Gourierioux & Monfort 1995). The above definitions of encompassing are applicable independently of whether the models are nested or nonnested. The next definition applies only to nested models, but is an important concept in the theory of encompassing.

Definition 4. Parsimonious encompassing

Mi parsimoniously encompasses M2, denoted as Mi £p M2, if and only

if

Mi £ (7) M2 and Mi C M2 where “C ” denotes that M x is nested in M2

-Having defined the major concepts of encompassing it is now possible to analyse the relationship between mutual encompassing and a particular form of model equivalence - K LIC equivalence.

3 Mutual Encompassing and K LIC Equivalence

The information criterion suggested by Kullback (1959) has often been used as a measure of distance between models, even though it is not strictly a

(11)

distance measure. The Kullback information criterion (KLIC) has also been used as the basis for the pseudo maximum likelihood method (see White 1982), and various model selection criteria have been developed based on different approximations to it (see Akaike 1973 and Schwartz 1978). It is also possible to use the K LIC to define a form < f model equivalence, and thus discuss the relationship between it and mutual encompassing.

The K LIC for M\ and M2 at 60 € 0 and 706 T is defined as: IC'J, 9> = E,.{loEg }

where denotes the expectation under M] at 60 € 0 . Thus defined

K .(f, g) > 0 with the equality holding if and only if f (6) = g{7) a.s. for

9 — 60 € 0 and 7 = 70 € T (see Kullback 1959). When /C (/, g) = 0 then Mi and M2 are defined to be KLIC equivalent for 60 € 0 and 70 € T.

The following theorem reveals the relationship between mutual encom passing and K LIC equivalence of M1 and M2.

THEOREM 1. Suppose models Mo, Mj and M2 are as defined in (2T) and (2-2), then Mi and M2 are KLIC equivalent at 70 € T and 90 € 0 , i.e.

IC{H0o), ff(7o)} = E*0 j l o g ^ l } = 0

if and only if M \ and M2 are mutually encompassing and mutually Cox-

encompassing at 70 € T and 60 € 0.

Proof: If Mi encompasses M2 in the sense of Cox, Mi £c M2, then from

Definition 2:

where q0 € A is the parameter of the data generating process Mo. If Mi also encompasses M2 with respect to 7 then

7o = 7i(0o)

where 70 and 7i(#o) are the pseudo true values of 77- under Mo and Mi, respectively. Hence, (3-1) becomes,

log m ) y h i ^ o ) } (3-1) © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(12)

On the other hand, if the reverse encompassing relations between M\ and M2 also hold, i.e. M2 £ (8) Mi and M2 £c M \, then the similar results follow:

8o = #2(70) (3-3)

and

S

t w

}

log

5(70)

f { 0 2(70)} (3-4)

where 80 and 82(70) are the pseudo true values of 8r under Mo and M2,

respectively.

Substituting (3-3) into (3-4) yields:

{log /(flj} E

70 {

1 os

/(?

o

)}

which when combined with (3-2) gives:

-<?0

_{( " ■ S I}

+ E,

{

, .9(7o) 1 _ n 108 7 ® ) / - ° (3-5) (3-6) Since —0Q

_{{ loes S}

} £0

E™{log

77 ^ } -

0

the equality holds in (3-6), if and only if:

-0Q 0

i.e. Mi and M2 are K LIC equivalent at 806 O and 70 € F. Hence, the necessity of the theorem is established.

On the other hand, if Mi and M2 are KLIC equivalent then:

0 — Eflo E#o (3-7) log f(00) > 0 and hence 9{7i(^o)}. 7o = 7l (^0) <=7 Mi £(7) M2. © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(13)

Similarly,

■

logM } ì “ E

log g( 7o) / { W } J see > 0 g(7o)]

**/(*) J**

so that 6>0 = g2(7o) <=> M2 £{0) Mi

To show that Cox-encompassing is also implied note that (3-7) implies th at /(^o) = g(7o) a.s. holds under any absolutely continuous measure, so that: log < sup m n g(7o)J log g(

7i

/(*o g(7o) < E,, logm ) g(7o) ^a(^o) = 0

where ffo is the zero-measure set on which /(go) 7= g(7o)- It then follows:

/ ( « o ) \ and Therefore,

{

7

g(7o) Ì . g(7o) 1 S /(^ o )I / (0o )j Mi £c M2 and M2 £c Mx This thus establishes sufficiency. ■

Note th at models M x and M2 are encompassing equivalent when they are mutually encompassing, and Cox-encompassing equivalent when they m utu ally Cox-encompass each other. However, in order to be K LIC equivalent M x and M2 must mutually encompass and mutually Cox-encompass each other. The following lemma weakens this requirement for models th at belong to the linear exponential family i.e. the density functions /(g ) and g{7) have the following form:

/ (x ; g) = exp {g • s (x) — k {g} + a (a:)} (3-8) where g is the canonical parameter and s = s (a:) the canonical statistic (Barndorff-Nielsen and Cox 1989).

(14)

Lemma 1. I f both M\ and, M2 belong to the linear exponential family,

then M\ £ (7) M2 implies Mi Ec M2.

PROOF: See Cox (1962), Gourieroux k. Monfort (1995) and Kent (1986). ■ The following corollary to Theorem 1 is an immediate consequence of

Lemma 1:

Corollary 1. I f in Theorem 1 both models Mi and M2 belong to the linear exponential family, then M i and M2 are K LIC equivalent at 60 and 7o, so that

if and only if they are mutually encompassing, i.e. M i £ (7) M2 and M2 E (0) Mi.

4 Properties of mutual encompassing

In this section some properties of mutually encompassing models are dis cussed. The first result shows th at mutual encompassing is transitive in the linear exponential family.

Theorem 2. Consider the following three multivariate conditional mod-els:

f (yt\yt-i,3±,o) ,

M i : 0 € 0 (4-1)

M2 : g (yt\yt- i , x t r / ) , 7

e r

M3 : k [yt\yt^ , 6 € A

when the unknown DGP belongs to:

M0: h (y t\y t-i,x t;a ) , a € A (4-2) I f both M i and M2 belong to the linear exponential family and are mutually

encompassing at do € 0 and 70 € T, and in addition M2 and M3 are mutually encompassing at 70 G T and 60 € A, then Mi and M3 are also mutually encompassing at 0o € 0 and 60 € A.

PROOF: Because both Mj and M2 belong to the linear exponential family, M i and M2 are mutually encompassing at 0O € Q and 70 € T if and only if:

M / ( U f f ( 7 o ) } = E*0 j l o g ^ } = 0 © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(15)

A similar relationship holds for M2 and M3 at 706 T and <50 € A:

IC{gho), k(60)} = E, 0 { l o g f | | y } = 0 It then follows that

i c{ f ( e0),*(*>)] E«0 <logf(0 o)l

k(60) J = Ee„ log U(0o) 1 9(70) 9(7o)Y k(6o)j. E*0 <logm ) } 9(70).• + E*0 - log g(7o) l k(S0)J log9(7o)]

k(60) J' + (Efl0 - E ro) jlog g(7o)1 *(*»)/ (E»0 - E

7o){ 10Sfc(-50) i n.

where the subscription “fl0” denotes the integral is taken on fi0 on which

f (90) ± 5(70)- flo is a zero-measure set under any absolutely continuous measure . Let l e d ™ sup k(60) Q < 00 Therefore, / C { / W , ^ o ) } = E#0| l o g { m j . 9i.lo) ' < ( U - E7o) log < sup log = Q-0 = 0 ff( 7o) k(60) n0 k{6o) {vg (Oq) + Vy ( ^0)}

where vg (f]0) and isy (fi0) are measures of set O0 under M x and M2, respect ively. This shows th at M1 and M3 are mutually encompassing at 60 and 7o- ■

Next, it is shown that if two mutually encompassing models belonging to the linear exponential family, are exponentially combined with a third model, then the resulting models remain mutually encompassing. The exponentially augmented model is discussed in Atkinson (1970) and Lu and Mizon (1996).

(16)

Theorem 3. Suppose that models My and M2 in (4-1) are augmented by M3, such that and where

m

; : {r (

7?) =

f ( 9 ) Tk ( 6 ) l~T Q l 3 T K /n l-T M*2 ■■ < fl*(0 = 9 (7Y k (6) 0 2 3 6 € 0 , 6 € A, r € [0, 1] > (4-3) 7 € r J € A, t 6 [0, 1] } (4-4) Q13 = E* { / ( ^ r 1 k {6)l~r } Q23 = E7 {3(7) ^ k (6)1- T}

If My, M2 and M3 all belong to the linear exponential family, and My and M2 are mutually encompassing at 90 € 0 and 70 € T, then, M* and M£ are

mutually encompassing at 9q € 0 , 70 € F and V<5 € A, Vr € [0, 1],

Proof: The augmentation of My and M2 by M3 in (4-3) and (4-4) ensures th at My and M | are also linear exponential. So, My and M2 are mutually encompassing, if and only if K, (My, M.2) = 0. The KLIC of My and M2 can be written as: where K. ( M[, M*) Ei;(qo) T Er|(ao)

(iog[{“ a ^ } / {

{ l o ® f < S } - C l o g C i , g(70 VH6)1-' Q23 - log 023)

}])

< 0 1 3 ~ 0 2 3 1

E* { / (^o) ^ 1 k (6) ^ } no - E* {3(7o)t_1 k («)1' T}no|

sup |fc (<$)1~T| {sup | / (0o)T_1| Ve (O0) + sup |g (7o)T_1| (fio)} 0

and O0 is the zero-measure set on which / {9 (ao)} / g {7 (do)}- Therefore,

IC(M{, M*)

= T E^(ao) {log - (log 013 - log 023) < T E^(a„) | l O g { H } | + log | 013/ 0231 < r s u p | l o g { ^ | | | i / , ( f t 0) = 0 © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(17)

where url (fi0) is the measurement of f20 under M*. Therefore, M [ and M2

are mutually encompassing. ■

Finally, let M \ and M2 be the completing models generated by Mi and M2, and M2 and M3, respectively (see Atkinson 1970, Lu & Mizon 1996). The following result shows that if M x and M2, and, M2 and M3 are mutually encompassing, then Mi and M2 are also mutually encompassing.

Theorem 3. Let Mi and M2 be the completing models defined as

M x : _{<f / ( 0 =} / W Tff( 7 )1_T Ql2 , 0 E 0 , 7 € r , r € [0,

l|)

► (4-5) and M2 : _{| s(v>) =} ff(7rA :(6)1- ' Q23 7 e r , <5 E A, " e [0, | (4-6) where Q n — E* { / W r_1y('y)1~T} Q23 = E-, {<7(7r (4-7)

If Mi and M2 are mutually encompassing at 0q E © and 70 6 T, and

M i and M3 are mutually encompassing at 70 € T and <50 E A, then M \ and M2 are mutually encompassing at 90 € 0 , 70 E P, <50 E A, r (a0) € [0, 1]

and v (ao) E [0, 1].

PROOF: First, we consider QJ2 and Q23. If Mi and M2 are mutually encom passing at f)0 E 0 and 70 E F and r (cc0) € [0, 1], we have,

Q12 — 1 + E#0

n0

where fl0 is zero-measure set on which / (f?0) / g (70). Since,

<

Therefore, Q]2 = 1, and similarly Q23 = 1. It then follows that:

< K. (Mi, M2) r ( « . ) E , | l o g S | „ i + ( l - O E , | l o g { g } | 0i © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(18)

where fix and fl2 are sets on which f (8o) ^ 9(70) and 5(70)) ^ fc(^o))> respectively. They are zero-measure sets under any absolutely continuous measure, hence,

K, (Mi,M 2)

< r ( a0)E, | l o g ® |fti + (1 - i/)E, |l o g { f ^ } } |^

< r ( a 0) sup | l o g ^ | + (1 - i/)su p |lo g fk2i| i/< (fi2) = 0

Therefore, Mi and M2 are mutually encompassing. ■

5 Conclusion

For models outside the linear exponential family tests for complete para metric encompassing and Cox-encompassing contain complementary inform ation, and this motivated Kent (1986) to propose a joint test statistic for these two hypotheses. In this paper it has been shown th at models are mu tually encompassing and mutually Cox-encompassing if and only if they are

K LIC equivalent. Hence the implicit null hypothesis (see Mizon & Richard

1986) of the joint application of Kent’s test statistic (firstly with M\ as the null hypothesis and then with Af2 as the null hypothesis) is K LIC equi valence. For models in the linear exponential family complete parametric encompassing implies Cox-encompassing and so mutual complete parametric encompassing is a necessary and sufficient condition for K LIC equivalence. This also means th at in the linear exponential family complete parametric encompassing is the implicit null hypothesis of the Kent test statistic. Fur ther, it has been shown that mutual encompassing is transitive for models in the linear exponential family, and that the exponential tilts of mutually encompassing models are also mutually encompassing.

Acknowledgements

We are grateful to the participants at the Econometrics Workshop in Southamp ton for their comments. We also wish to thank John Aldrich for his valuable comments on an earlier version of the paper. Financial support from the Research Council of the European University Institute and from the UK Economic and Social Research Council under grant R000233447 is grate fully acknowledged. © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(19)

Re f e r e n c e s

Ak a ik e, H. (1973). Information theory and an extension of the maximum likelihood principle. In Second International Symposium on Information

Theory, Ed. B.N. Petrov and F. Csaki, pp. 26"" 281. Budapest: Akademiai

Kiado.

At k in so n, A.C. (1970). A method for discriminating between models.

Journal of the Royal Statistical Society, Series B 32, 323-344.

Barndorff-Nielsen, O.E. & Cox, D.R. (1989). Asymptotic Techniques for Use in Statistics. London: Chapman and Hall.

BONTEMPS, C. & Mizon, G.E. (1996). Congruence and encompassing. Mi meograph June 1996. Forthcoming as Discussion Paper in Economics, European University Institute, Florence, Italy.

Cox,

D.R. (1961). Tests of separate families of hypotheses. In Proceedings

of the Fourth Berkeley Symposium on Mathematical Statistics and Prob ability, 1, Ed. J. Neyman, pp. 105-123. Berkeley: University of California

Press.

Cox,

D.R. (1962). Further results of tests of separate families of hypotheses.

Journal of the Royal Statistical Society, Series B 24, 406-424.

Engle, R.F., Hendry, D.F. & Richard, J.-F. (1983). Exogeneity. Econo- metrica 51, 277-304.

Gourieroux, C. & Monfort, A. (1995). Testing, encompassing and sim ulating dynamic econometric models. Econometric Theory 1 1, 195-228.

Hendry, D.F. & Richard, J.-F. (1982). On the formulation of empirical models in dynamic econometrics. Journal of Econometrics 20, 3-33.

Kent, J.T. (1986). The underlying structure of nonnested hypothesis tests. Biometrika 73, 333-43.

Kullback, S. (1959). Information Theory and Statistics, New York: Wiley.

Lu, M. & Mizon, G.E. (1996). The encompassing principle and specification tests. Econometric Theory 12, 845-858.

Mizon, G.E. (1984). The encompassing approach in econometrics. Chapter

6 in Econometrics and Quantitative Economics, Eds. D.F. Hendry and K.F. Wallis, pp. 135-172. Oxford: Basil Blackwell.

Mizon, G.E. & Richard, J-F. (1986). The encompassing principle and its Application to non-nested hypothesis tests. Econometrica 54, 657-678. Pesaran, M.H. (1987). Global and partial non-nested hypotheses and

asymptotic local power. Econometric Theory 3, 69-97.

Schw artz, G. (1987). Estimating the dimension of a model. Annals of Stat

istics 6, 461-464. © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(20)

Wh it e, H. (1982). Maximum likelihood estimation of misspecified Models. Econometrica 50, 1-25. © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(21)

EUI

WORKING

PAPERS

EUI Working Papers are published and distributed by the European University Institute, Florence

Copies can be obtained free o f charge - depending on the availability of stocks - from:

The Publications Officer European University Institute

Badia Fiesolana

1-50016 San Domenico di Fiesole (FI) Italy © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(22)

Publications of the European University Institute

To The Publications Officer European University Institute Badia Fiesolana

1-50016 San Domenico di Fiesole (FI) - Italy Telefax No: +39/55/4685 636

E-mail: publish@datacomm.iue.it

From N am e... Address...

□ Please send me a complete list of EUI Working Papers □ Please send me a complete list of EUI book publications □ Please send me the EUI brochure Academic Year 1997/98 Please send me the following EUI Working Paper(s): No, Author ... Title: ... No, Author ... Title: ... No, Author ... Title: ... No, Author ... Title: ... Date ... Signature © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(23)

Working Papers of the Department of Economics Published since 1994

E C O No. 96/1 Ana Rute CARDOSO

Earnings Inequality in Portugal: High and Rising?

E C O No. 96/2 Ana Rute CARDOSO

Workers or Employers: Who is Shaping Wage Inequality?

E C O No. 96/3

David F. HENDRY/Grayham E. MIZON The Influence of A.W.H. Phillips on Econometrics

E C O No. 96/4 Andrzej BANIAK

The Multimarket Labour-Managed Firm and the Effects of Devaluation

E C O No. 96/5 Luca ANDERLINI/Hamid SABOURIAN

The Evolution of Algorithmic Learning: A Global Stability Result

E C O No. 96/6 James DOW

Arbitrage, Hedging, and Financial Innovation

E C O No. 96/7 Marion KOHLER

Coalitions in International Monetary Policy Games

E C O No. 96/8

John MICKLEWRIGHT/ Gyula NAGY A Follow-Up Survey of Unemployment Insurance Exhausters in Hungary E C O No. 96/9

Alastair McAULEY/John

MICKLEWRIGHT/Aline COUDOUEL Transfers and Exchange Between Households in Central Asia E C O No. 96/10

Christian BELZIL/Xuelin ZHANG Young Children and the Search Costs of Unemployed Females

ECO No. 96/11 Christian BELZJ1,

Contiguous Duration Dependence and Nonstationarity in Job Search: Some Reduced-Form Estimates

ECO No. 96/12 Ramon MARIMON

Learning from Learning in Economics E C O No. 96/13

Luisa ZANFORLIN

Technological Diffusion, Learning and Economic Performance: An Empirical Investigation on an Extended Set of Countries

E C O No. 96/14

Humberto L6PEZ/Eva ORTEGA/An gel UBIDE

Explaining the Dynamics of Spanish Unemployment

E C O No. 96/15 Spyros VASSILAKIS

Accelerating New Product Development by Overcoming Complexity Constraints E C O No. 96/16 Andrew LEWIS On Technological Differences in Oligopolistic Industries E C O No. 96/17 Christian BELZIL

Employment Reallocation, Wages and the Allocation of Workers Between Expanding and Declining Firms E C O No. 96/18

Christian BELZIL/Xuelin ZHANG Unemployment, Search and the Gender Wage Gap: A Structural Model E C O No. 96/19

Christian BELZIL

The Dynamics of Female Tune Allocation upon a First Birth

E C O No. 96/20 Hans-Theo NORMANN

Endogenous Timing in a Duopoly Model

(24)

E C O No. 96/21

Ramon MARIMON/Fabrizio ZILIBOTl'I ‘Actual’ Versus ‘Virtual’ Employment in Europe: Is Spain Different?

E C O No. 96/22 Chiara MONFARDINI

Estimating Stochastic Volatility Models Through Indirect Inference

E C O No. 96/23 Luisa ZANFORLIN

Technological Diffusion, Learning and Growth: An Empirical Investigation of a Set of Developing Countries

E C O No. 96/24 Luisa ZANFORLIN

Technological Assimilation, Trade Patterns and Growth: An Empirical Investigation of a Set of Developing Countries

E C O No. 96/25

Giampiero M.GALLO/Massimiliano MARCELLING

In Plato’s Cave: Sharpening the Shadows of Monetary Announcements

E C O No. 96/26 Dimitrios SIDERIS

The Wage-Price Spiral in Greece: An Application of the LSE Methodology in Systems of Nonstationary Variables E C O No. 96/27

Andrei SAVKOV

The Optimal Sequence of Privatization in Transitional Economies

E C O No. 96/28

Jacob LUNDQUIST/Dorte VERNER Optimal Allocation of Foreign Debt Solved by a Multivariate GARCH Model Applied to Danish Data

E C O No. 96/29 Dorte VERNER

The Brazilian Growth Experience in the Light of Old and New Growth Theories E C O No. 96/30

Steffen HORNIG/Andrea LOFARO/ Louis PHLIPS

How Much to Collude Without Being Detected

E C O No. 96/31 Angel J. UBIDE

The International Transmission of Shocks in an Imperfectly Competitive

International Business Cycle Model E C O No. 96/32

Humberto LOPEZ/Angel J. UBIDE Demand, Supply, and Animal Spirits E C O No. 96/33

Andrea LOFARO

On the Efficiency of Bertrand and Cournot Competition with Incomplete Information

ECO No. 96/34

Anindya BANERJEE/David F. HENDRY/Grayham E. MIZON The Econometric Analysis of Economic Policy

E C O No. 96/35 Christian SCHLUTER

On the Non-Stationarity of German Income Mobility (and Some Observations on Poverty Dynamics)

E C O No. 96/36 Jian-Ming ZHOU

Proposals for Land Consolidation and Expansion in Japan

E C O No. 96/37

Susana GARCIA CERVERO

Skill Differentials in the Long and in the Short Run. A 4-Digit SIC Level U.S. Manufacturing Study

-#• -:?f

it'-E C O No. 97/1 Jonathan SIMON

The Expected Value of Lotto when not all Numbers are Equal

E C O No. 97/2 Bernhard WINKLER

Of Sticks and Carrots: Incentives and the Maastricht Road to EMU

ECO No. 97/3 James DOW/Rohit RAHI Informed Trading, Investment, and Welfare © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(25)

E C O No. 97/4 Sandrine LAB OR Y

Signalling Aspects of Managers’ Incentives

E C O No. 97/5

Humberto L6PEZ/Eva ORTEGA/Angel UBIDE

Dating and Forecasting the Spanish Business Cycle

E C O No. 97/6

Yadira GONZALEZ de LARA

Changes in Information and Optimal Debt Contracts: The Sea Loan

E C O No. 97/7 Sandrine LABORY

Organisational Dimensions of Innovation E C O No. 97/8

Sandrine LABORY

Firm Structure and Market Structure: A Case Study of the Car Industry E C O No. 97/9

Elena BARDASI/Chiara MONFARDINI The Choice of the Working Sector in Italy: A Trivariate Probit Analysis E C O No. 97/10

Bernhard WINKLER

Coordinating European Monetary Union E C O No. 97/11

Alessandra PELLONI/Robert WALDMANN

Stability Properties in a Growth Model E C O No. 97/12

Alessandra PELLONI/Robert WALDMANN

Can Waste Improve Welfare? E C O No. 97/13

Christian DUSTMANN/Arthur van SOEST

Public and Private Sector Wages of Male Workers in Germany

E C O No. 97/14 S0ren JOHANSEN

Mathematical and Statistical Modelling of Cointegration

ECO No. 97/15

Tom ENGSTED/S0ren JOHANSEN Granger’s Representation Theorem and Muldcointegration

ECO No. 97/16 S0ren JOHANSEN/ Ernst SCHAUMBURG Likelihood Analysis of Seasonal Cointegration

ECO No. 97/17

Maozu LU/Grayham E. MIZON Mutual Encompassing and Model Equivalence © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(26)

(27)

(28)