model selection strategies in a spatial context

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Model selection strategies in a spatial context

Article · January 2005

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Documento de Trabajo 2005-06

Facultad de Ciencias Económicas y Empresariales

Universidad de Zaragoza

Model selection strategies in a spatial context.

Jesús Mur (*)

Ana Angulo (**)

University of Zaragoza.

Department of Economic Analysis

Gran Vía, 2-4 (50005)

Zaragoza, (SPAIN)

Phone: +34-976-761815

(*) [email protected]

(**) [email protected]

Abstract: This paper follows on from the discussion of Florax, Folmer and Rey (2003) on the advantages and disadvantages of various specification strategies for econometric models in a spatial setting. Habitual practise has popularised a technique based on the well-known Lagrange Multipliers, which seems to give good results although its basis is entirely ad hoc. In this paper we also contemplate other alternatives which, from a strictly theoretical point of view, seem to be more elaborated. We focus attention on the problem of deciding which model should be specified once the initial one, generally static, presents symptoms of misspecification. There are two alternatives habitually contemplated, the Spatial Lag Model and the Spatial Error Model, which leads us to a classical decision problem. In the final part of the paper we present the results of a Monte Carlo exercise which has enabled us to clear up some doubts, although others still persist.

JEL Classification: C21

Keywords: Model selection; Spatial Econometrics; Cross-sectional dependence

Acknowledgements: This work has been carried out with the financial support of project SEC 2002-02350 of the Ministerio de Ciencia y Tecnología del Reino de España.

DTECONZ 2005-06: J.Mur and A.Angulo

1

1- Introduction

Generally speaking, it could be said that there are hundreds of different ways of

dealing with the problem of specifying an econometric model. Obviously, it is unlikely

that all of them would finish up with the same equation. Thus, the question of the

method of selecting, and discarding, models arises as one of the most important for us

when doing applied econometrics.

In a recent paper, Florax, Folmer and Rey (2003, FFR from now on), review the

strategies of specifying models in the area of Spatial Econometrics. According to them,

at present, a classical forward stepwise approach dominates, structured in three steps:

(i)- A simple model is specified, usually static, under ideal conditions.

(ii)-A series of tests (of spatial dependence) are applied to the estimated

equation.

(iii)- If the null is rejected in any of the tests, some adjustments will be applied:

reformulating the equation, filtering the variables, incorporating elements of spatial

dynamics, etc.

This method could be called Specific to General Modelling and is very popular

among econometricians. Implicitly, the reliability of the procedure depends on the

number and quality of the misspecification tests carried out during the process, which

may explain the wide range of such tests habitually reported. As FFR indicate, we could

adopt exactly the opposite approach in a ‘Hendry-like specification strategy’ structured

in the following steps:

(i)- Estimate the most ample model consistent with theory.

(ii)-Test a series of simplifying assumptions until no more restrictions are

accepted.

(iii)- Test the overall robustness of the final specification using theoretical as

well as statistical considerations. In case of doubts, return to (i).

The paper of FFR focuses on comparing both approaches. They find that the first

approach tends to perform better than the second. This is an interesting result, but the

literature on econometric model selection is not limited to just these two broad

approaches. There are other techniques that have proven useful in different situations.


2

For example, the role of the Bayesian methodology cannot be ignored because it

occupies a central position in the discussion. In applied econometrics, an approach

based on the Kullback-Leiber information measure dominates. The AIC or the SBIC

statistics are well-known criteria related to this measure and they are produced

routinely.

In this paper we will focus our attention on the proposals of Vuong (1989) and

Clarke (2003). The first introduces another way of dealing with the Kullback-Leiber

information measure, whereas the second produces a very simple criterion of model

selection, rooted in the maximum likelihood estimation. The problem posed in both

cases involves only two different Data Generating Processes (DGP). These models

could be nested, overlapped or non-nested in the case of Vuong but must be non-nested

in the analysis of Clarke. The restriction of having only two DGPs may appear too tight

for both methods to be useful, although this situation is very common in applied spatial

econometrics. This gives the Vuong and the Clarke test a role.

In the second section we summarise a series of results, well established in

mainstream econometrics, on the problem of how to compare models, paying special

attention to the above-mentioned proposals of Vuong and Clarke. The third section

describes the search strategies that dominate in a spatial context, as discussed in FFR.

Next, in the fourth section of the paper, we will solve a brief simulation that may help

us to compare the performance of the different approaches discussed previously. Section

5 presents some conclusions and prospects for future research.

2- Discriminating among econometric models. Some (well-known) keys.

The question of how to discriminate between rival models has always had an

important place in the research agenda of Econometrics. The decisions adopted in this

respect will severely condition both the method of the research itself and the results

derived from it. Nevertheless, practice continues to be very heterogeneous and, on many

occasions, not very well-reasoned, which means that it is necessary to delve even deeper

into the discussion.

In spite of the surprising claim of Hausman (1992, p. 32) when he states that ‘the

Economic method is deductive’, during recent decades there has been a strengthening of


3

the consensus on the positivist tradition, in which the preferentialist approach

predominates over the normative. As Popper (1979, p. 13) says: ‘The theoretician, I will

assume, is essentially interested in truth, and especially in finding true theories. But

when he has fully digested that we can never justify empirically -that is, by tests

statements- the claim that a scientific theory is true, and that we are therefore at best

always faced with the question of preferring, tentatively, some guesses to others, then he

may consider from the point of view of a seeker for true theories, the question: What

principles of preference should we adopt? Are some theories ‘better’ than others?’.

This reasoning requires ‘falsifying’, systematically confronting the theories with the

data and among themselves, which has contributed to consolidating a series of basic

principles, such as the Information Criterion (Akaike, 1973) or that of Encompassment

(Mizon, 1984). Furthermore, the consistency of the Bayesian approaches has

generalised the use of concepts like Loss Function, Prior Distribution or Decision

Criteria. We do not intend to follow this question any further as it clearly goes beyond

the modest objectives of this paper (for a review of the state of the question, see, for

example, Morgan, 1990, Ripley, 1996, or Burnham and Anderson, 2002). We only wish

to highlight certain questions relevant to our case.

In general terms, two broad strategies to resolve the problem of model

discrimination can be identified. Dastoor and McAleer (1989) refer to them as Model

Selection strategy, with which we look for the best model among various alternatives,

and Hypotheses Testing strategy, where we test whether one model in particular is

admissible. The difference between them, as is demonstrated in Aznar (1989), depends

on the loss function assumed by the researcher. In normal circumstances (unless there

are very well-defined preferences a priori in favour of one specification) the first

strategy seems preferable, although in habitual practice the second type dominates. The

approach that FFR call ‘‘classical’ specification search’ in spatial econometrics belongs

to the latter category: it involves a succession of nested models using a stepwise

forward method.

The problems arise when we try to use the same approach with a collection of

non-nested models because, as Chow (1983) indicates, the adequate tests are not the

same and the interpretation of the results also differs. In this case, four different

techniques can be identified to deal with the problem of discrimination between non-

nested models (Clarke, 2004). The best-known are the Bayesian approaches and the


4

traditional model selection criteria. The others are the test of Cox (1961), and its

derivations in the J and JA tests of Davidson and MacKinnon (1981) and Fisher and

McAleer (1979), respectively, together with model selection tests, particularly those of

Vuong (1989) and the Distribution-Free test of Clarke (2003).

The test of Cox is complex, especially when the equations are non-linear, and

confusing in many cases because, as a result of the logic of the test, it allows both

models to be accepted or rejected simultaneously. However, its use has become popular

through the J and JA tests. The Bayesian reasoning is very attractive, though it is also

not exempt from criticism. The posterior odds combine the prior odds with the Bayes

factor, which measures the change in the priors due to the sampling information. The

first problem is to define the prior odds (Lesage, 2004, proposes very interesting

solutions). However, as Clarke (2004, p.3) indicates: ‘(...) the Bayes factor does not

provide a measure of support for one model over another, but rather it measures “the

change in the odds in favor of the hypothesis when going from prior to the posterior”1’.

The most extended technique to compare models in a context of non-nested

models uses one, or various, of the wide range of model selection criteria available in

the literature. Possibly the most popular is the AIC of Akaike (1973), or its Bayesian

version, the SBIC (Schwarz, 1978), included systematically in almost all the software

applications. The main advantage of this line lies in its simplicity and the clarity of its

results. The selection criterion used will always choose the best model according to the

internal philosophy of the criterion. Nevertheless, in many situations the differences

between the models are hardly appreciable and will not be reflected in the working of

the selection criterion.

The novelty that the model selection tests bring with respect to the criteria is that

they explicitly contemplate a situation of indifference between the alternatives to choose

from. If the available evidence is not sufficiently clear in favour of one of the models, it

is important to transmit this information to the user so that he can decide accordingly.

The main difference lies in the firmer exploitation of the statistical elements associated

with the problem of the decision to be carried out in the second option. The final results

are not, necessarily, more complex. Otherwise, the essential reasoning underlying the

approaches of the criteria and of the tests of selection are basically the same.

1 In the original, quoted from Lavine and Schervish (1999).


5

The test of Voung (1989) can be presented as a reinterpretation of the test of Cox

(1961). In the latter we have two families of conditioned density functions:

{ }pY|X( );f fθ = θ θ ∈ Θ ⊂ ℜ and { }q

Y|X( );g gγ = γ γ ∈ Γ ⊂ ℜ , and we want to test one

against the other. The null hypothesis corresponds to one of the families while the other

is the alternative ( 0 :H f θ vs. A : gH γ ). The Cox statistic is a centred and typified

version of the traditional Likelihood Ratio:

( )

( )( )n Z f nn nn n

0as

n Z f nn nn n

( ; ) ( ; )LR E E LR(f ) ~ N 0,1CV ( ; ) ( ; )LR E E LR

−γ γθ θ=⎡ ⎤−γ γθ θ⎣ ⎦

% %% %

% %% % (2.1)

Where nθ% and nγ% are the respective ML estimations of θ and γ and LRn is the

Likelihood Ratio ( n f gn nn n( ; ) ( ) ( )LR L L= −γ γθ θ% %% % ). Next, the test should be repeated

taking the other density function g γ in the null hypothesis. The problem with this

reasoning is that the model of the alternative hypothesis only has power to accept or

reject the model of the null hypothesis, which is a potential source of conflict.

Vuong (1989) uses the structure of inference of the test of Cox, but redefines the

content of the null and alternative hypotheses. Now, the former is associated with a

situation of indifference between the models (taking into account the sample evidence),

while the alternative is bilateral and identifies the model most favoured by the data. In

analytical terms:

t t00

t t

t t0f

t tA

t t0g

t t

f ( | ; )Y X: lg 0H E g( | ; )Y X

f ( | ; )Y X: lg 0H E g( | ; )Y X:H

f ( | ; )Y X: lg 0H E g( | ; )Y X

⎫⎡ ⎤θ= ⎪⎢ ⎥γ⎣ ⎦ ⎪

⎪⎧ ⎡ ⎤θ ⎪> ⎬⎪ ⎢ ⎥γ⎪ ⎣ ⎦ ⎪⎨ ⎪⎡ ⎤θ⎪ ⎪<⎢ ⎥⎪ γ ⎪⎣ ⎦⎩ ⎭

(2.2)

This change of perspective allows us to obtain additional results of convergence

in probability and in distribution with respect to the LRn statistic of (2.1). The most

important are summarised in the following expression:


6

( )( )

( )( )

( )( )

( )( )

Dt tn n n 20

t t

22t t2 0 0

t t

t t t t0

t t t t

f | ;( ; ) Y XLRn lg N(0; )En g | ;Y X

f | ; f | ; f | ;Y X Y X Y Xlg lg lgV E Eg | ; g | ; g | ;Y X Y X Y X

⎫⎧ ⎫⎡ ⎤θγ⎪ ⎪θ − → ⎪ω⎨ ⎬⎢ ⎥γ ⎪⎪ ⎪⎣ ⎦⎩ ⎭ ⎪⎬⎡ ⎤ ⎛ ⎞⎡ ⎤ ⎛ ⎞ ⎡ ⎤θ θ θ ⎪⎢ ⎥= = − ⎜ ⎟ω ⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎪⎜ ⎟ ⎜ ⎟γ γ γ⎢ ⎥⎣ ⎦ ⎝ ⎠ ⎣ ⎦⎝ ⎠ ⎪⎣ ⎦ ⎭

% %

(2.3)

Finally, Theorem 5.1 of Vuong (1989, p. 318) establishes that: ‘.... if Fθ and Gγ

are strictly non-nested, then:

(i) under [ ] )1;0(~/~;~: 2/10 NLRnH

D

nnnn →ωγθ−

(ii) under [ ] ∞+→ωγθ−as

nnnnf LRnH ~/~;~: 2/1

(iii) under [ ] ∞−→ωγθ−as

nnnng LRnH ~/~;~: 2/1 (...)’

Given that the two models are different, the acceptance of the null should be

interpreted as meaning that the available evidence does not permit us to discriminate

between both alternatives. With slight variations, the test can also be used in the case

where the models are nested or overlapped.

The Distribution-Free Test of Clarke is even more intuitive because it ‘(…)

applies a modified paired sign test to the differences in the individual log-likelihoods

from two nonnested models. Whereas the Vuong test determines whether or not the

average log-likelihood ratio is statistically different from zero, the proposed test

determines whether or not the median log-likelihood ratio is statistically different from

zero. If the models are equally close to the true specification, half the individual log-

likelihood ratios should be greater than zero and half should be less than zero’ (Clarke,

2004, p.6). In other words, the reasoning reproduces the discussion of Vuong, just

substituting the average with the median. In this way:


7

t t t t0 0

t t t t

t tf

t tA

t tg

t t

f ( ; ) f ( ; )Y X Y X: Median lg 0 : Pr lg 0 0.5H Hg( ; ) g( ; )Y Z Y X

f ( ; )Y X: Median lg 0H g( ; )Y X:H

f ( ; )Y X: Median lg 0H g( ; )Y X

⎡ ⎤ ⎡ ⎤θ θ= ⇒ > =⎢ ⎥ ⎢ ⎥γ γ⎣ ⎦ ⎣ ⎦

⎧ ⎡ ⎤θ>⎪ ⎢ ⎥γ⎪ ⎣ ⎦

⎨⎡ ⎤θ⎪ <⎢ ⎥⎪ γ⎣ ⎦⎩

(2.4)

Its resolution is particularly simple. It is only necessary (i) to estimate the model

corresponding to the family { }pY|X( );f fθ = θ θ ∈ Θ ⊂ ℜ retaining the individual log-

likelihoods { }t t t nlg f ( ; ) ; t 1, 2, , nf Y X= =θ% K ; (ii) to estimate the model corresponding

to the family { }qY|X( );g gγ = γ γ ∈ Γ ⊂ ℜ maintaining the individual log-likelihoods

{ }t t nt lg g( ; ); t 1, 2, , ng Y X= =γ% K ; (iii) To obtain the differences between the log-

likelihoods { }n,,2,1t;gfd ttt K=−= ; (iv) Calling B the number of positive differences

dt, its distribution under the null hypothesis of (2.4) is a Binomial(n;0.5).

The conditions under which both tests are applicable differ slightly. In the case

of the Clarke test, it is only necessary that there is independence among the log-

likelihoods ratios and that each of them comes from a continuous population (‘not

necessarily the same’ as Clarke, 2004, p.6, indicates). In the case of the Vuong test, it is

necessary to observe the iid (independent and identically distributed log-likelihoods)

clause, as well as the habitual conditions of regularity that guarantee the existence of the

ML estimators2. In no case is either the normality or the lineality of the equation

necessary.

3- Discriminating among spatial econometric models. Some (arguable) keys.

From our point of view, the adjective spatial does not stamp a special character

on purely econometric work. Evidently, there are certain peculiarities derived from the

spatial dimension that must be taken into account when interpreting the results.

Nevertheless, the research programme consolidated in mainstream Econometrics should

2 Rivers and Vuong (2002) discuss the use of the Vuong test for GMM and non maximum likelihood estimators and models with weakly dependent heterogeneous data. However, the manipulation of the test becomes more complex.


8

prevail. That is to say, the content of the previous section is directly applicable to the

spatial case, with some minor nuances. This is the spirit which seems to underlie the

above-mentioned work of FFR, when they revise the specification searches which

dominate and/or, may be useful in spatial econometrics.

The peculiarities come from the type of data on which the models with spatial

structure are built. As has been said many times before, space is an abstract entity that is

malleable at the will of the administrator. The problem is that there are numerous

administrators spread throughout space, with programmes and objectives that are hardly

compatible, which means that ‘the generalizations made at one level do not necessarily

hold at another level and conclusions we derived at one scale may result invalidated at

another’. (Hagget, 1965, p. 138). The sensibility of statistical results to the problem

known as MAUP (Modifiable Areal Unit Problem) has made the modelling of the error

terms of an equation a routine procedure. Furthermore, in space, there are externalities,

spillover effects and multiple cross-flows whose inclusion is inevitable in any

econometric specification with the objective of being useful in applied analysis.

In short, the problem of cross-sectional dependencies occupies a central position

in the process of the elaboration of an econometric model for spatial data. FFR

recognise this importance which is essential to appreciate the differences that exist

between the four strategies contemplated in their work. The first three are based on a

static initial model in which the omission of elements of spatial dependency is tested

for, either in the equation errors or in the main equation. The three share the philosophy

of the approach that Charemza and Deadman (1997) call Specific to General Modelling.

In the so-called classical approach the traditional LM-ERR and LM-LAG (see

Appendix A) tests are used. The problem is that, being designed to test for specification

errors only in one direction (in the errors or in the equation, respectively), they also

react to errors in the other, which, very often, leads to a situation of uncertainty. The

solution, as FFR (2003, p. 561) recognise, is ad hoc: ‘If both tests are significant,

estimate the specification pointed to by the more significant of the two tests. For

example, if LM-LAG>LM-ERR then estimate the spatial lag model using MLLAG. If

LM-LAG<LM-ERR then estimate the spatial error model using MLERROR’. Obviously,

if both are not significant or if one of them is but the other is not, the model for the

second stage is perfectly identified.


9

The second strategy replicates the previous one, but uses the LM-EL and LM-LE

tests instead of the traditional LM-ERR and LM-LAG. The advantage of these tests is

that, though they are simple to obtain because they are based on the LS estimation of the

model, Anselin et al. (1996) demonstrate that they are robust to local specification

errors. However, this sensibility does not disappear completely, generating surprises

that are difficult to explain, as will be seen later. FFR propose a third strategy,

denominated hybrid, combining the previous two: the initial analysis of the static

equation is carried out with the traditional tests (LM-ERR and LM-LAG) and, if both

are significant, the model that shows the highest value of the robust statistic (LM-EL

and LM-LE) is chosen. In Appendix B we demonstrate that this strategy leads

necessarily to selecting the same model as the first.

The fourth strategy of specification contemplated by FFR develops a line of

analysis very close to the methodological suggestions of Hendry. In this case, too, the

tests of spatial dependence play an essential role because the ample model that,

supposedly, nests the other specifications is ‘the spatial regressive/autoregressive

model’ proposed by Burridge (1981) to resolve the common factor test (LR-COM):

y Wy X WX= ρ + β + η + ε (3.1)

The common factor hypothesis implies that: H0: ρβ=-η, which results in the

spatial error model (SEM):

y X uu Wu

= β + ⎫⎬= ρ + ε⎭ (3.2)

The alternative hypothesis (HA: ρβ ≠ -η) refers to the spatial lag model (SLM)

with (possible) externalities in the exogenous variables and white noise structure in the

error term. The resolution of the test is simple. Given that (3.1) nests (3.2), it is enough

to estimate both specifications (using ML methods) and compare the log-likelihoods

through a Likelihood Ratio:

[ ] 20 A

asLR COM 2 l( ) l( ) ~ (k)H H− = − χ (3.3)

Where k is the number of parameters (or restrictions in this case) included in vector β.

After this test, the following step is, either in (3.1) or in (3.2), to examine the relevance

of the spatial effects still present in the equation.


10

The aspect we wish to highlight is that, apparently, the key to constructing a

good econometric model in this context lies in the correct choice of the mechanisms of

cross-sectional dependence that should be introduced3. In this respect, we believe that

the so-called model selection tests can help us to take the correct decisions. It should be

borne in mind that it is relatively normal to have a lot of evidence against the initial

specification, generally static, without any clear direction predominating on the nature

of the effects omitted. For example, if the two traditional Lagrange Multipliers (LM-

ERR and LM-LAG) are significant, it is not clear how to proceed; neither is the

previous situation unusual when using the Robust Multipliers (LM-LE and LM-EL). A

possible solution consists, as we have seen, of taking into account the values of these

tests and using them as selection criteria.

However, there are other alternatives. In particular, model selection tests adjust

relatively well to the conditions of the problem: only two rival models (SEM or SLM)

are contemplated; they are non-nested and there are no very defined preferences in

favour of either of the two alternatives.

The requirement of independence among the individual log-likelihoods for the

two non-nested models can more problematic. To circumvent the implication of this

clause, it is sufficient to filter the variables of the model, using the eigenvectors of W.

As this weighting matrix is binary and symmetric, it will admit the basic spectral

decomposition:

[ ]⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

λλλ=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

λ

λλ

=Λ

⇔Λ=

qqq

qqq

qqqqqq

Q

diag

QQW

R

RRRR

R

R

R

R

K

KKKKK

KK

KKKKK

KK

21

21

22212

12111

2121

),...,,(

00

0000

' (3.4)

where {λr, r=1, 2,..., R} are the eigenvalues of W and the eigenvectors are in the

columns of matrix Q. Using this matrix in the SLM, we obtain that:

3 There are other aspects to which attention must also be paid such as those associated with the instability of the relationships, the specification of the contiguity matrices or the univariate characteristics of the series. However, the problem of the cross-sectional dependencies seems to worry more than the others.


11

⎪⎭

⎪⎬⎫

σ

+β+Λρ=⇒

⎭⎬⎫

σ

+β+ρ=

)I,0(N~uuXyy

)I,0(N~uuXWyy

21

*

****

21

(3.5)

where y*=Q’y, X*=Q’X and u*=Q’u are the filtered series. The final model of (3.5)

does not have relationships of cross-sectional dependency because matrix Λ is diagonal.

The log-likelihood function of that model is:

( )[ ] ( )∑ λρ−+∑σ

β−λρ−−σ−π−=ϕ ==

R1r r

R1r 2

1

2211

* 1ln2

xy1ln

2R2ln

2R)|y(l

*'r

*rr

(3.6)

where [ ]σρβ=ϕ 211 ;; and [ ]'x;....;x;xx *

kr2*

r2*r1

*r = . In the case of the SEM, the filtering

process results in:

⎪⎭

⎪⎬

⎫

σεε∆=+β=

⇒⎪⎭

⎪⎬

⎫

σεε+ρ=

+β=−

)I,0(N~u

uXy

)I,0(N~Wuu

uXy

22

*

*1*

***

22

(3.7)

where ε=ε '* Q . The filter leads to a heteroskedastic error term ( ),0(~ 222

* ∆σ −Nu ,

where ∆=Ι−ρΛ), which is independent in a cross-sectional setting. Once again, the log-

likelihood function of model (3.7) is:

( )[ ] ( )∑ λθ−+∑

σ

λθ−β−−σ−π−=ϕ ==

R1r r

R1r 2

2

2222

* 1ln2

1)xy(ln

2R2ln

2R)|y(l r

*'r

*r (3.8)

with [ ]σρβ=ϕ 222 ;; . In both cases, (3.6) and (3.8), the individual log-likelihoods are

independent because the filter has removed the cross-sectional dependencies.

The likelihood ratio which is the basis for the test of Vuong can be expressed as:

( )2

* * 1 rRR r 11 2 1 2 2

2 r

R 1ˆ ˆ ˆ ˆ; l( | ) l( | ) ln lny yLR 2 1=− ρσ λ= − = − +ϕ ϕ ϕ ϕ ∑− θσ λ

%%%%

(3.9)

The test maintains its general expression:

( ) ( )

DR 1 2 *2ˆ ˆ;LR N 0;Rϕ ϕ

→ ϖ (3.10)


12

Where:

( )( )

( )( )

( )( )

2 2

R2R r 1

22

R2R r 1

* *r r1 1r rR

r 1* *r r2 2r r

*r 1r R 1 2

*r 2r

ˆ ˆ| |y yL L1 1lg lgR Rˆ ˆ| |y yL L

ˆ|yL ˆ ˆ( , )1 LRlgR Rˆ|yL

=

=

∑ =

⎛ ⎞ ⎛ ⎞ϕ ϕ⎜ ⎟ ⎜ ⎟= − ⇒∑ω ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ϕ ϕ⎝ ⎠ ⎝ ⎠

⎛ ⎞ϕ ϕ ϕ⎛ ⎞⎜ ⎟⇒ = −∑ω ⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟ϕ⎝ ⎠

%

%

(3.11)

Furthermore, the obtaining of the Clarke test is immediate by just considering

the following results:

( )( )

( )( )

2

2* 1

r r1 21

2

2* 2

r r2 22

* * *r rr 1 2 1 2

* *'1 y xrr rln 2 lnˆ( | ) ln 1yl 2 2 2

* *'( ) 1y xr rrln 2 lnˆ( | ) ln 1yl 2 2 2

ˆ ˆ ˆ ˆ( | , ) ( | ) ( | )y y yd l l

⎫⎡ ⎤−ρ − βλ ⎪⎢ ⎥π σ ⎣ ⎦= − − − + − ρϕ λ ⎪⎪σ⎬

⎡ ⎤ ⎪− β −θλ⎢ ⎥π σ ⎪⎣ ⎦= − − − + − θϕ λ ⎪σ ⎭⇒ = −ϕ ϕ ϕ ϕ

%%% %

%

% %% %

%

(3.12)

In the following section, we examine the behaviour of these two selection tests

against the Lagrange Multipliers. In spite of the popularity of the latter, we understand

that the situation is confusing when both Multipliers, be they traditional or robust, are

significant. The solution of following the direction indicated by the most significant

statistic is, as we said before, merely ad hoc and sustained only by the good results

obtained in various simulation exercises (Anselin and Florax, 1995). Nevertheless, we

should remember that there are other alternatives, more elaborate from a theoretical

point of view.

The Cox test has attracted little attention in the literature on spatial

econometrics, perhaps because of its analytical complexity. However, we believe that

Cox's approach to the problem is relevant and deserves an additional research effort. On

the other hand, the Bayesian analyses have been well-received in this field and very

interesting results have been produced during recent years including the topic of

discrimination between non-nested models (for example, in Hepple, 1995a and b). The

other two lines, model selection criteria and tests, end up mixing together in the strict

problem we are dealing with here. For example, if we take the AIC criterion:


13

[ ]AIC 2log Maximized likelihood 2k= − + (3.13)

where k is the number of parameters of the model. The AIC criterion will consist of

choosing the model that minimises (3.13). In our case, k will be the same for both the

rival models (the SLM and the SEM), so that the choice will depend exclusively on the

maximised likelihood, a circumstance that takes us back to Vuong's test. The SBIC

criterion is consistent:

[ ] k log(R)SBIC log Maximized likelihood2

= − + (3.14)

Nevertheless, the conclusion is still the same: it takes us back to Vuong's test. As

a consequence, neither are we going to follow this line in the following section, in spite

of its indubitable interest in a more general context of discrimination.

4- Discriminating between the Spatial lag model and the Spatial error model. Some

Monte Carlo evidence.

In this section we are going to present the results of a Monte Carlo exercise

designed to analyse the behaviour of two model selection strategies, that based on the

selection tests of Vuong and of Clarke and that habitually used in the field of spatial

econometrics which we call Robust strategy following the terminology introduced in

FFR (2003).

4.1- Premises of the simulation and additional information

In the exercise, whose results we present below, we have taken a Simple Linear

Model as a point of reference:

r rry ux= α + β + (4.1)

It is immediate, taking the base of the static model of (4.1), to obtain an SLM or

an SEM. In matrix terms:

( ) ( )2

2

y x uy Wy x u

SLM : SEM : u Wuu~iid 0; I

~iid 0; I

⎧ = β += ρ + β +⎧ ⎪⎪ = ρ + ε⎨ ⎨

σ⎪⎩ ⎪ε σ⎩

(4.2)


14

The most relevant characteristics of the exercise are the following:

(i)- We have used two pairs of values for α and β in (4.1). The first, (α=10;

β=0.5), guarantees an average determination coefficient, in the absence of spatial

effects, of 0.2 while the second, (α=10; β=2.0), raises it to 0.8.

(ii)- The observations of the variable x and, where necessary, of the random terms

ε and u have been generated using univariate normal distributions with mean

zero and unit variance. That is, σ2 is equal to one in (4.2).

(iii)- We have used three different sample sizes: 25, 100 and 225.

(iv)- The contiguity matrix that intervenes in (4.2) has always been specified as of

binary type, using rook movements in a regular lattice system of (5x5), (10x10)

or (15x15).

(v)- The range of values admissible for parameter � depends on the contiguity

matrix used in the exercise. For the matrix of the (5x5) system, the interval is (-

0.274; 0.274), of the (10x10) it is (-0.248; 0.248) and of the (15x15) it is (-0.229;

0.229). In each case, 40 values of the parameter, distributed regularly over the

whole interval, have been simulated.

(vi)- Each combination has been repeated 1000 times.

The simulation exercise has generated an abundant quantity of information

which we are going to structure in three cases of interest. In the first two, one of the two

models that has intervened in the simulation, the SEM or the SLM, is the true one (that

is, the data have been generated with one of the two models, the true one, and in the

decision problem this same model is contemplated together with the other one which is,

obviously, false).

The difference between the first and the second case lies in the approach used to

resolve the decision problem. In the first, we speak of the Testing Approach, given that

we apply the selection tests of Vuong and of Clarke as they have been presented in the

previous section: the null hypothesis means that both models are not distinguishable,

according to the sample available, and the alternative is bilateral. The Lagrange

Multipliers are used in the same way. In this case, if both tests coincide (both are

significant or both are not significant), we understand that we are not capable of

adopting a decision so the problem will remain unresolved. On the other hand, if the


15

LM-EL test is significant but the LM-LE is not, an SEM model is identified. When the

relation is produced in the opposite direction, the identification is of an SLM.

The second case we call the Criterion Approach and we use the same

instruments but disconnected from their probabilistic structure. That is, if the Vuong

statistic (obtained as the typified difference between the log-likelihood of the SLM

model and that corresponding to the SEM model) is negative, we identify the SEM

model and, if it is positive, the SLM. In the case of Clarke, we will choose that model to

which the highest number of differences between the individual log-likelihoods

corresponds. We also replicate the traditional strategy in terms of the Robust

Multipliers, as FFR suggest: if LM-EL>LM-LE we choose the SEM model and, if the

inequality is in the opposite sense, the SLM.

Lastly, the third case that we contemplate in the exercise is characterised by the

true model not belonging to the catalogue of decision alternatives. The data have been

generated with a mixed model like the following:

( )

1

22

y Wy x uu Wu

~iid 0; I

⎫= + β +ρ⎪

= + ερ ⎬⎪ε σ ⎭

(4.3)

However, the decision options that are contemplated are limited to the SEM or

to the SLM of (4.2), both false in this setting.

Figures (4.1) to (4.4) summarise the results obtained in the first case (Testing

Approach). In Figures (4.1) and (4.2) the data have been generated using an SEM model

with a high signal-to-noise ratio (that is, a high R2) in Figure (4.1) and a low one in the

second. Figures (4.3) and (4.4) reproduce the results obtained when the data have been

generated using an SLM. The data we present in each case indicate the number of times,

in percentages, in which the corresponding decision method chooses one of the two

models. Moreover, VU(SLM) means Vuong test as the method and SLM as selected

model, CL indicates Clarke test and LM is reserved for the Robust Multipliers. We do

not present, in order to make the reading of the figures easier, the percentages

corresponding to the case of indetermination. Lastly, in the four figures we have also

included the LR-COM test of common factors. The series denominated LRCF

corresponds to the number of rejections of the hypothesis of common factors or, put

another way, the percentage of decisions taken by this test in favour of the SLM model.


16

Figures (4.5) to (4.8) have the same structure but are associated with the Criteria

Approach. The change of focus means that there are no longer situations of

indetermination (the sum of the series VU(SEM) and VU(SLM), for example, is 1,

while in the previous case the result of the sum was less than or equal to 1). In this

collection of figures we have not included the LR test of common factors.

Lastly, in Figures 4.9 to 4.17 we summarise the results corresponding to the

third case, in which the data have been generated with the mixed model of (4.3) and

high Signal-to-Noise Ratio. The figures we include are bi-dimensional. Horizontally we

reproduce the range of ρ1 (the coefficient that accompanies the spatial lag of y in the

main equation of 4.3) and vertically the range of ρ2 (the dependence coefficient of the

error term in 4.3). Each figure reflects the number of times that each test selects the

model identified in the heading. In Figures (4.9) to (4.14) the so-called Testing

Approach is developed, in which situations of indifference are relevant, while in the last

three the Criteria Approach is used (consequently, there is no place for uncertainty).

4.2- Main results of interest

It is not very encouraging to begin this conclusions section by saying that ‘there is still a

lot of work to do’ though, in our case, the comment seems inevitable. Some aspects of the

difficulties involved in the discrimination between models in a spatial context seem clearer now,

but there remain many questions which we have been unable to answer. Among the former we

want to mention the following:

* There is an evident sample size effect that conditions the behaviour of all the

methods examined. Their reliability improves substantially when the size of the sample

increases from 25 to 100 observations. The leap from 100 to 225 observations also

introduces improvements, although less pronounced.

* The intensity of the autocorrelation, residual or substantive, is an even more

determining factor than the sample size. The behaviour of the different method is

deficient when the symptoms of autocorrelation are weak, but improves as we approach

the extremes of the range of values admissible for the parameter.

* Relevant changes in the results are not appreciable with respect to the weight

of the systematic and random part of the equation. That is, the Signal-to-Noise Ratio

factor is secondary.


17

* The methods of discrimination seem biased towards models that contain a

strong autocorrelation structure. That is, when SEM models are simulated, the zones of

no determination, or of erroneous selections are often very broad, but the margins are

sharp when the data has been obtained from SLM models. Generally speaking, we can

say that there is a certain predisposition to select SLM models more frequently than is

necessary.

* The Robust Multipliers are not reliable when used in a Testing Approach. At

the extremes of the parameter interval considered, situations of indefinition, in which

both tests are significant, tend to dominate. This is why the percentage of correct

decisions corresponding to the Robust Multipliers in Figures 4.1 and 4.2 falls at the

extremes of the interval. If the data come from an SLM, the anomalies occur in a zone

of intermediate values in the positive branch of the interval of parameters. The

dimensions of the ‘bubbles’ produced in this case increase with the sample size. The

consequences are dramatic when using a sample of 225 observations. Moreover, the

reliability of the Multipliers deteriorates at the extremes of the parameter interval.

* The Clarke test suffers from a similar effect when the data come from an SLM,

although the size of the bubble decreases with the sample size.

* The Vuong test seems to work (relatively) better then the Clarke test in the

Testing Approach, which contradicts the results obtained by Clarke himself (Clarke

2004; see also Clarke and Signorino, 2004). In any case, the best option in this setting is

the LR-COM.

* The bubbles disappear when these techniques are used in a Criteria Approach.

The evolution of the series in Figures 4.5 to 4.8 is more consistent, although proximity

to the extremes of the interval has pernicious effects on the Robust Multipliers. In this

case, the falls tend to decrease with the sample size as well as with the Signal-to-Noise

Ratio: the criterion based on the multipliers works better for equations with a high

explicative power.

* The Clarke test, except in certain cases associated with the SEM model, seems

generally worse than the other two alternatives. It could be considered the third option

in order of preference.

* The differences between the results of the Vuong test and those of the Robust

Multipliers are small, although the former is more consistent in the range of parameter


18

considered and, generally, outperforms the latter. The Vuong test should be taken as the

best option in this case.

* The bias of all the discrimination methods considered in favour of the SLM

alternative is evident in Figures 4.9 to 4.17. The bias is accentuated in the case of the

LR-COM test, as is evident in Figure 4.14.

* Vuong's test and Clarke's test tend towards indefinition (correct decision in this

case) when the sample size is reduced. As it increases, the zones of no determination are

compressed towards the diagonals of the parametric space. Simultaneously, there is an

increase in the number of times that the SEM model and, very especially, the SLM are

selected. This last option is the dominant one with a sample size of 225.

* The direction of the movements is less clear in the case of the Robust

Multipliers. The preference for the SLM model also dominates, but the increase of the

sample size does not compress the zones of indefinition. Both tests tend to be non-

significant, as we expected, around point (0,0), while both tend to be significant in the

interior regions of the four quadrants.

* The preference for SLM structures is much clearer when the analysis is

resolved with a Criteria Approach. In the three cases represented in Figures 4.15 to

4.17, the decisions taken in favour of this model clearly dominate, relegating the SEM

option to a thin band concentrated around the value zero for the parameter that

accompanies the spatial lag in the main equation of the model.

5- Conclusions

With this paper we want to vindicate the importance of a stage that has occupied

a secondary position in the process of elaboration of econometric models with a spatial

context. During recent years, many results on estimation and test methods in this setting

have been published, though references to how to discriminate between rival models

have been scarce.

In our paper we contemplate a very particular problem: how to discriminate

between an SLM and an SEM model, once there are clear signs of misspecification in

the initial static equation. Day to day practice has ended up consolidating some methods

that, seemingly, work reasonably well. In the exercise that we have resolved, the simple


19

comparison of the values of the Robust Multipliers, as a guide to choosing the most

appropriate model, has obtained good results. It is a simple proposal, cheap (in terms of

calculations) and relatively reliable. However, it is not the only available option. In the

paper we have examined other alternatives that adapt well to the problem proposed.

We have concentrated, in particular, on the model selection tests and the results

are not disappointing. When we resolve the analysis in a Testing Approach, the

behaviour of the tests of Vuong and of Clarke is more consistent than that of the Robust

Multipliers. Nevertheless, the preferable option in this case is the traditional Likelihood

Ratio of Common Factors. In a Criteria Approach, the test of Vuong is perfectly

competitive against the Robust Multipliers, accepting the cost, evidently, of a greater

complexity of calculation.

The simulation exercise, and the previous discussion, have allowed us to resolve

some questions though there are still many doubts that require additional work. The

nature of the ‘bubbles’ that arise in the power functions of the Robust Multipliers is one

of them. After discarding the existence of coding errors, we have not found a

satisfactory explanation either for their cause or for the fact that their dimension

increases with the sample size. There are also many other topics that we have not

explicitly contemplated: heterogeneity, outliers, non normality, etc. The use of

estimators other than the ML, more and more frequent in this setting, introduces new

unknowns. Another question that still seems interesting to us is the apparently low

reliability of what FFR (2003) call the ‘Hendry-like specification strategy’. It seems

necessary to take up all of these aspects again at some time in the future.


20

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22

Appendix A: Misspecification tests used in the analysis.

The tests described here always refer to a static model, such as: y = X� + u.

This model has been estimated by LS, where 2σ and β correspond to the LS

estimations and u to the residual series. These tests are the following (see Florax and de

Graaff, 2004, for the details):

LM-ERR Test: 2

2 1

ˆ û 'Wu 1LM ERRTˆ

⎛ ⎞− = ⎜ ⎟

σ⎝ ⎠; [ ]1 tr W 'W WWT = + (A.1)

LM-EL Test:

21

2 2-

21

1-

ˆ ˆ û 'Wu u 'WyTˆRˆ ˆJ

LM ELT

T ˆRJ

ρ β

ρ β

⎛ ⎞⎜ ⎟−⎜ ⎟σ σ⎝ ⎠− =

−

(A.2)

LM-LAG Test: 2

2ˆ1 u 'WyLM LAG

ˆR ˆJρ−β

⎛ ⎞− = ⎜ ⎟

σ⎝ ⎠ (A.3)

LM-LE Test:

2

2 2

1-

ˆ ˆ û 'Wy u 'Wuˆ ˆLM LE

ˆR TJρ β

⎛ ⎞−⎜ ⎟

σ σ⎝ ⎠− =−

(A.4)

Moreover, 21

ˆ ˆˆR (β'X'WMWXβ) / ˆTJρ−β = + σ and M=[I-X(X’X)-1X’]. The four

Lagrange Multipliers have an asymptotic )(12χ distribution.


23

Appendix B: The Hybrid and the Classical strategy of FFR

(2003)

These strategies differ in the way that they solve situations of uncertainty, where

the LM-ERR and the LM-LAG are statistically significant. In the so-called ‘classical

strategy’, we compare the value of both statistics in order to specify the model

associated with the more significant: the SEM if LM-ERR>LM-LAG and the SLM if

LM-ERR<LM-LAG. The hybrid strategy replicates the previous one but uses the

Robust Multipliers, that is the LM-EL and the LM-LE instead of the LM-ERR and the

LM-LAG. In this Appendix we show that both inequalities are, at last, the same. That is:

LM-ERR > LM-LAG ⇔ LM-EL > LM-LE. The proof is simple although a bit tedious.

Using the expressions of Appendix A:

LM-EL>LM-LE ⇒

22

12 2 2 2-

2 1-11

-

ˆ ˆ û 'Wu u 'WyT ˆ ˆ û 'Wy u 'WuˆRˆ ˆJ ˆ ˆ

ˆR TT JT ˆRJ

ρ β

ρ β

ρ β

⎛ ⎞ ⎛ ⎞⎜ ⎟− −⎜ ⎟⎜ ⎟σ σ σ σ⎝ ⎠ ⎝ ⎠>−

−

(B.1)

Simplifying common terms, (B.1) reads as:

⇒ ( )2

- 21

1 -

ˆRJ Tˆ ˆ ˆ ˆ ˆ û 'Wu u 'Wy u 'Wy u 'WuˆRT J

ρ β

ρ β

⎛ ⎞⎜ ⎟− > −⎜ ⎟⎝ ⎠

(B.2)

After solving for the squares and grouping terms, we obtain:

⇒ ( ) ( )- 2 2

1

ˆRJ ˆ ˆ û 'Wu u 'WyT

ρ β > (B.3)

That is:

2 2

2 21 -

ˆ ˆ ˆ1 u 'Wu 1 u 'WyLM EL LM LEˆRT ˆ ˆJ

LM EL LM LE LM ERR LM LAG

ρ β

⎛ ⎞ ⎛ ⎞− > − ⇒ >⎜ ⎟ ⎜ ⎟

σ σ⎝ ⎠ ⎝ ⎠

− > − ⇔ − > −(B.4)


24

Figure 4.1: Testing Approach. DGP: SEM. High Signal-to-Noise Ratio. Figure 4.1A:. R=25.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-0.27 -0.24 -0.21 -0.18 -0.15 -0.12 -0.09 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27

VU(SLM) VU(SEM) CL(SLM) CL(SEM) LRCF LM(SLM) LM(SEM) Figure 4.1B:. R=100.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-0.25 -0.22 -0.19 -0.16 -0.13 -0.10 -0.07 -0.04 -0.01 0.02 0.05 0.08 0.11 0.14 0.17 0.20 0.23

VU(SLM) VU(SEM) CL(SLM) CL(SEM) LRCF LM(SLM) LM(SEM) Figure 4.1C:. R=225.

0.0

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-0.23 -0.20 -0.17 -0.14 -0.11 -0.08 -0.05 -0.02 0.01 0.04 0.07 0.10 0.13 0.16 0.19 0.22

VU(SLM) VU(SEM) CL(SLM) CL(SEM) LRCF LM(SLM) LM(SEM)


25

Figure 4.2: Testing Approach. DGP: SEM. Low Signal-to-Noise Ratio. Figure 4.2A:. R=25.

0.0

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-0.27 -0.24 -0.21 -0.18 -0.15 -0.12 -0.09 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27


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-0.25 -0.22 -0.19 -0.16 -0.13 -0.10 -0.07 -0.04 -0.01 0.02 0.05 0.08 0.11 0.14 0.17 0.20 0.23


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-0.23 -0.20 -0.17 -0.14 -0.11 -0.08 -0.05 -0.02 0.01 0.04 0.07 0.10 0.13 0.16 0.19 0.22



26

Figure 4.3: Testing Approach. DGP: SLM. High Signal-to-Noise Ratio. Figure 4.3A:. R=25.

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-0.27 -0.24 -0.21 -0.18 -0.15 -0.12 -0.09 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27


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-0.25 -0.22 -0.19 -0.16 -0.13 -0.10 -0.07 -0.04 -0.01 0.02 0.05 0.08 0.11 0.14 0.17 0.20 0.23


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27

Figure 4.4: Testing Approach. DGP: SLM. Low Signal-to-Noise Ratio. Figure 4.4A:. R=25.

0.0

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-0.27 -0.24 -0.21 -0.18 -0.15 -0.12 -0.09 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27


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-0.25 -0.22 -0.19 -0.16 -0.13 -0.10 -0.07 -0.04 -0.01 0.02 0.05 0.08 0.11 0.14 0.17 0.20 0.23


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-0.23 -0.20 -0.17 -0.14 -0.11 -0.08 -0.05 -0.02 0.01 0.04 0.07 0.10 0.13 0.16 0.19 0.22VU(SLM) VU(SEM) CL(SLM) CL(SEM) LRCF LM(SLM) LM(SEM)


28

Figure 4.5: Criteria Approach. DGP: SEM. High Signal-to-Noise Ratio. Figure 4.5A:. R=25.

0.0

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-0.27 -0.24 -0.21 -0.18 -0.15 -0.12 -0.09 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27

VU(SLM) VU(SEM) CL(SLM) CL(SEM) LM(SLM) LM(SEM) Figure 4.5B:. R=100.

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-0.25 -0.22 -0.19 -0.16 -0.13 -0.10 -0.07 -0.04 -0.01 0.02 0.05 0.08 0.11 0.14 0.17 0.20 0.23

VU(SLM) VU(SEM) CL(SLM) CL(SEM) LM(SLM) LM(SEM) Figure 4.5C:. R=225.

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-0.23 -0.20 -0.17 -0.14 -0.11 -0.08 -0.05 -0.02 0.01 0.04 0.07 0.10 0.13 0.16 0.19 0.22

VU(SLM) VU(SEM) CL(SLM) CL(SEM) LM(SLM) LM(SEM)


29

Figure 4.6: Criteria Approach. DGP: SEM. Low Signal-to-Noise Ratio. Figure 4.6A:. R=25.

0.0

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-0.27 -0.24 -0.21 -0.18 -0.15 -0.12 -0.09 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27


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-0.25 -0.22 -0.19 -0.16 -0.13 -0.10 -0.07 -0.04 -0.01 0.02 0.05 0.08 0.11 0.14 0.17 0.20 0.23


Figure 4.6C:. R=225.

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30

Figure 4.7: Criteria Approach. DGP: SLM. High Signal-to-Noise Ratio. Figure 4.7A:. R=25.

0.0

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-0.27 -0.24 -0.21 -0.18 -0.15 -0.12 -0.09 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27


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31

Figure 4.8: Criteria Approach. DGP: SLM. Low Signal-to-Noise Ratio. Figure 4.8A:. R=25.

0.0

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-0.27 -0.24 -0.21 -0.18 -0.15 -0.12 -0.09 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27


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-0.25 -0.22 -0.19 -0.16 -0.13 -0.10 -0.07 -0.04 -0.01 0.02 0.05 0.08 0.11 0.14 0.17 0.20 0.23


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32

Figure 4.9: Testing Approach. DGP: Mixed. High Signal-to-Noise Ratio.

Vuong and Clarke Test. Identification: SEM.

Vuong test Figure 4.9A: R=25. Clarke test

Vuong test Figure 4.9B: R=100. Clarke test

Vuong test Figure 4.9C: R=225. Clarke test


33


Vuong and Clarke Test. Identification: SLM..





34


Vuong and Clarke Test. Identification: Indiference.





35


Robust LM. Identification: Positive.

Robust LM: SEM Figure 4.12A: R=25. Robust LM: SLM

Robust LM: SEM Figure 4.12B: R=100. Robust LM: SLM

Robust LM: SEM Figure 4.12C: R=225. Robust LM: SLM


36


Robust LM. Identification: Not Conclusive.

Robust LM: NONE Figure 4.13A: R=25. Robust LM: BOTH

Robust LM: NONE Figure 4.13B: R=100. Robust LM: BOTH

Robust LM: NONE Figure 4.13C: R=225. Robust LM: BOTH


37


LR-COMMOM Factors. Identification: SEM.

Figure 4.14A: R=25.

Figure 4.14B: R=100.

Figure 4.14C: R=225.


38

Figure 4.15: Criteria Approach. DGP: Mixed. High Signal-to-Noise Ratio.

Vuong test.

Identification: SLM Figure 4.15A: R=25. Identification: SEM

Identification: SLM Figure 4.15B: R=100. Identification: SEM

Identification: SLM Figure 4.15C: R=225. Identification: SEM


39


Clarke test.





40


Robust LM.





41

Documentos de Trabajo

Facultad de Ciencias Económicas y Empresariales. Universidad de Zaragoza.

2002-01: “Evolution of Spanish Urban Structure During the Twentieth Century”. Luis Lanaspa, Fernando Pueyo y Fernando Sanz. Department of Economic Analysis, University of Zaragoza.

2002-02: “Una Nueva Perspectiva en la Medición del Capital Humano”. Gregorio Giménez y Blanca Simón. Departamento de Estructura, Historia Económica y Economía Pública, Universidad de Zaragoza.

2002-03: “A Practical Evaluation of Employee Productivity Using a Professional Data Base”. Raquel Ortega. Department of Business, University of Zaragoza.

2002-04: “La Información Financiera de las Entidades No Lucrativas: Una Perspectiva Internacional”. Isabel Brusca y Caridad Martí. Departamento de Contabilidad y Finanzas, Universidad de Zaragoza.

2003-01: “Las Opciones Reales y su Influencia en la Valoración de Empresas”. Manuel Espitia y Gema Pastor. Departamento de Economía y Dirección de Empresas, Universidad de Zaragoza.

2003-02: “The Valuation of Earnings Components by the Capital Markets. An International Comparison”. Susana Callao, Beatriz Cuellar, José Ignacio Jarne and José Antonio Laínez. Department of Accounting and Finance, University of Zaragoza.

2003-03: “Selection of the Informative Base in ARMA-GARCH Models”. Laura Muñoz, Pilar Olave and Manuel Salvador. Department of Statistics Methods, University of Zaragoza.

2003-04: “Structural Change and Productive Blocks in the Spanish Economy: An Imput-Output Analysis for 1980-1994”. Julio Sánchez Chóliz and Rosa Duarte. Department of Economic Analysis, University of Zaragoza.

2003-05: “Automatic Monitoring and Intervention in Linear Gaussian State-Space Models: A Bayesian Approach”. Manuel Salvador and Pilar Gargallo. Department of Statistics Methods, University of Zaragoza.

2003-06: “An Application of the Data Envelopment Analysis Methodology in the Performance Assessment of the Zaragoza University Departments”. Emilio Martín. Department of Accounting and Finance, University of Zaragoza.

2003-07: “Harmonisation at the European Union: a difficult but needed task”. Ana Yetano Sánchez. Department of Accounting and Finance, University of Zaragoza.


42

2003-08: “The investment activity of spanish firms with tangible and intangible assets”. Manuel Espitia and Gema Pastor. Department of Business, University of Zaragoza.

2004-01: “Persistencia en la performance de los fondos de inversión españoles de renta variable nacional (1994-2002)”. Luis Ferruz y María S. Vargas. Departamento de Contabilidad y Finanzas, Universidad de Zaragoza.

2004-02: “Calidad institucional y factores político-culturales: un panorama inter.-nacional por niveles de renta”. José Aixalá, Gema Fabro y Blanca Simón. Departamento de Estructura, Historia Económica y Economía Pública, Universidad de Zaragoza.

2004-03: “La utilización de las nuevas tecnologías en la contratación pública”. José Mª Gimeno Feliú. Departamento de Derecho Público, Universidad de Zaragoza.

2004-04: “Valoración económica y financiera de los trasvases previstos en el Plan Hidrológico Nacional español”. Pedro Arrojo Agudo. Departamento de Análisis Económico, Universidad de Zaragoza. Laura Sánchez Gallardo. Fundación Nueva Cultura del Agua.

2004-05: “Impacto de las tecnologías de la información en la productividad de las empresas españolas”. Carmen Galve Gorriz y Ana Gargallo Castel. Departamento de Economía y Dirección de Empresas. Universidad de Zaragoza.

2004-06: “National and International Income Dispersión and Aggregate Expenditures”. Carmen Fillat. Department of Applied Economics and Economic History, University of Zaragoza. Joseph Francois. Tinbergen Institute Rotterdam and Center for Economic Policy Resarch-CEPR.

2004-07: “Targeted Advertising with Vertically Differentiated Products”. Lola Esteban and José M. Hernández. Department of Economic Analysis. University of Zaragoza.

2004-08: “Returns to education and to experience within the EU: are there differences between wage earners and the self-employed?”. Inmaculada García Mainar. Department of Economic Analysis. University of Zaragoza. Víctor M. Montuenga Gómez. Department of Business. University of La Rioja

2005-01: “E-government and the transformation of public administrations in EU countries: Beyond NPM or just a second wave of reforms?”. Lourdes Torres, Vicente Pina and Sonia Royo. Department of Accounting and Finance.University of Zaragoza

2005-02: “Externalidades tecnológicas internacionales y productividad de la manufactura: un análisis sectorial”. Carmen López Pueyo, Jaime Sanau y Sara Barcenilla. Departamento de Economía Aplicada. Universidad de Zaragoza.

2005-03: “Detecting Determinism Using Recurrence Quantification Analysis: Three Test Procedures”. María Teresa Aparicio, Eduardo Fernández Pozo and Dulce Saura. Department of Economic Analysis. University of Zaragoza.

2005-04: “Evaluating Organizational Design Through Efficiency Values: An Application To The Spanish First Division Soccer Teams”. Manuel Espitia Escuer and Lucía Isabel García Cebrián. Department of Business. University of Zaragoza.


43

2005-05: “From Locational Fundamentals to Increasing Returns: The Spatial Concentration of Population in Spain, 1787-2000”. María Isabel Ayuda. Department of Economic Analysis. University of Zaragoza. Fernando Collantes and Vicente Pinilla. Department of Applied Economics and Economic History. University of Zaragoza.

2005-06: “Model selection strategies in a spatial context”. Jesús Mur and Ana Angulo. Department of Economic Analysis. University of Zaragoza.

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model selection strategies in a spatial context

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