Economic Modelling. Tests for structural change, aggregation, and homogeneity. Esfandiar Maasoumi a,, Asad Zaman b, Mumtaz Ahmed b

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1 Economic Modelling 7 (00) Content lit available at ScienceDirect Economic Modelling journal homepage: Tet for tructural change, aggregation, and homogeneity Efandiar Maaoumi a,, Aad Zaman b, Mumtaz Ahmed b a Department of Economic, Emory Univerity, Atlanta, GA 303, United State b International Ilamic Univerity, Ilamabad, Pakitan article Keyword: Structural change Aggregation info abtract Structural change can be conidered by breaking up a ample into ubet and aking if thee can be aggregated or pooled. Strategie for contructing tet for aggregation and tructural change in thi etting have not received ufficient attention in the literature. Our methodology for teting generalize to multiple regime a dicuion of Pearan et al. (985) for the cae of two regime. Thi treatment permit a unified approach to a large number of teting problem dicued eparately in the literature, a pecial cae or a part of a tet of homogeneity. We alo provide a imple alternative to much more complex teting trategie currently being reearched and developed in teting for tructural change. 00 Elevier B.V. All right reerved.. Introduction Concern for contancy of model parameter i central to model fitting and prediction. Random coefficient model pioneered in the work of P.K Swamy and hi contemporarie i a frequentit repone to thi concern in which heterogeneity in lope coefficient i acknowledged and modelled explicitly a random draw from a ditribution. Teting for contancy of mean and variance parameter ideally precede a well a follow the deciion to model regime change. Coniderable challenge preent themelve in thi area, not the leat of which i repreented by the choice of teting trategie and the conequence of equencing in uch tet. Thi i partly due to the fact that, in the general cae, there i no UMP tet, and alternative approache mut be explored and compared. The claical regreion etting with many regime allow u to concentrate on the main iue. Technically precie extenion to more complex etting of dependent data and/or nonlinear-nonparametric model are an ongoing reearch agenda. In a claical regreion model y t =x t β+ t for t=,,,t, where ε t are i.i.d. N(0,σ ), the hypothei that the K+ parameter (β,σ ) remain table i crucial to the inferential validity and predictive performance of empirical model. One way to approach the problem i to plit the data into everal ubgroup. In thi etting, allowing for tructural change mean allowing each ubgroup of the data to have it own K+ parameter. For =,,,S define a typical ubgroup of the data a: y t = x t β + t for t = T +; ; T where t are i:i:d:n 0; σ : Correponding author. addre: EMAASOU@emory.edu (E. Maaoumi). Here T 0 =0,T S =T and T are potential breakpoint where the tructure of the model change. Each of the ubgroup of data will be called a regime. The quetion of central interet in thi paper i: can we pool, or aggregate, all the data in the different regime into one regime? Equivalently, are the regreion coefficient and the variance the ame in all the S regime? In a time erie context, thi i a quetion of tability of the regreion model. The ame problem can arie in different context. For example given firm level production function data, can we aume a common production function and pool all the data? Or, given cro ection data on a given relationhip in different region, can we aggregate the data, auming that the relationhip i the ame in all region? A previou review by Pearan et al. (985) ummarized the literature of the period for the two regime cae. The aumption of a ingle, fixed, and known breakpoint i highly retrictive, and much reearch ha been done to deal with unknown breakpoint and multiple breakpoint. Hanen (00) give a review in the context of dynamic model. The cae we conider here, which i that of a fixed number of known potential breakpoint, ha not received much attention. There are everal reaon why thi etup i of fundamental importance, and deerve a thorough invetigation. Firt, multiple fixed point regime change etting i le demanding of data without giving much up that i empirically relevant. The number of change point can be large, and they can be varied! Secondly, the tatitical theory i ubtantially impler, and many elegant finite ample reult are available, or well approximated. Undertanding thee reult provide the building block, and provide intuition, for tudying the much more complex cae with unknown multiple breakpoint, and more complex data/model etting. Thirdly, a we hall ee, thi problem can be broken down into five ubproblem, each of which ha been tudied eparately. Thu thi problem of pooling (or aggregation) provide a unified way of looking at a number of different /$ ee front matter 00 Elevier B.V. All right reerved. doi:0.06/j.econmod

2 E. Maaoumi et al. / Economic Modelling 7 (00) hypothei tet in the literature, and i uggetive of teting trategie and equential approache. Fig. below, which extend the one in Pearan et al. (985) to the multiple regime cae, i helpful. Let M tand for mean (the regreion coefficient, β ) and let V tand for the variance σ Let EM tand for the hypothei of equality of mean, and EV for the equality of variance. Let UM and UV tand for unretricted mean and variance. Different degree of departure from, and paage to a ingle regime may be gleaned a route from a model with (UM,UV) to one with (EM,EV). It i evident that Swamy' random coefficient model are included in thi cheme. The cae where regime change i repreented a change in the entire data generating law i conidered by method uch a the one decribed recently in Li et al. (009) where equality of nonparametrically etimated ditribution function i teted. Thoe approache require ditribution metric and are ditinct from the preent, more common etting. Fig. how that, broadly peaking, there are three route to mean-variance homogenou model... The VM trategy Firt tet equality of Variance and then tet equality of Mean. In the VM trategy, we tet to ee if the general model (UM,UV) can be reduced to (UM,EV); thi i the tet repreented by (A), the upper horizontal line in the rectangle diagram. The unretricted mean are nuiance parameter in both the null and the alternative. If thi tet doe not reject the null, we may aume that the null hypothei (UM, EV) hold. We can then proceed to the tet repreented by (B) (the right hand ide vertical line in the rectangle diagram) to ee if (UM, EV) can be further reduced to (EM,EV). Thi approach to equential teting ha been dicued and defended by Phillip and McCabe (983) in the cae of two regime for dynamic model. Thi approach i not a common a it may eem!.. The joint teting trategy (J) Thi i to make a joint tet, where the retriction on mean and variance (EM, EV) are imultaneouly teted againt the unretricted alternative (UM,UV). Thi approach i quite uncommon, and i cloer to the tet for whole ditribution change a exemplified in Li et al. (009). Our examination of thi cae i new in it extent and practical guidance..3. The MV trategy Firt tet for equality of Mean and then tet equality of Variance. In the MV trategy, we tet to ee if the general model (UM,UV) can be reduced to (EM, UV), a repreented by the vertical line (C) in the rectangle diagram. Auming that thi tet pae, we tet for EV conditional on EM to reduce further to (EM,EV); thi i repreented by (D) in the rectangle diagram. One may regard HAC procedure a UM, UV (C) EM UV EM, UV (A) EV UM (J) EM,EV (D) EV EM Fig.. Three route to tet for tructural change. UM, EV (B) EM EV EM, EV falling in thi category, but if HAC etimation i applied within each regime, it may be followed by a tet for equality of mean, and finally a White (980) type tet of heterokedaticity to the entire ample. Once et in our multiple regime context, however, HAC method within each regime may be een a perhap unneceary, and ometime not feaible for regime that are too hort. HAC method alo require unlikely large ubgroup ample ize for the accuracy of large ample theory that underpin uch method. The two equential tet involve two et of different tet, while the joint tet doe everything all at once. There are five tet involved, and they can be given a unified treatment via the likelihood ratio teting principle which clarifie their propertie and their role. Let θ tand for the (K+) S vector of all the parameter in all the regime with no contraint. Let θ UM, UV,θ UM, EV,θ EM, UV,θ EM, EV be the etimator of thee parameter under the contraint indicated by the upercript. The likelihood ratio tet tatitic can be written in term of thee etimate, which maximize the likelihood under different contraint. For our dicuion to follow it will be convenient to ditinguih between the hypothee being teted, and the tatitic which implement the tet. For the Joint route, the likelihood ratio teting principle can be implemented via the MZ tatitic, defined a: MZ = log l y; θ UM;UV + log l y; θ EM;EV : Thi tatitic implement the joint tet of all retriction required for aggregation imultaneouly. It i aymptotically Chi-quared with degree of freedom equal to the number of retriction under the null, or (K+) (S ). We can rewrite thi tet tatitic in two different way, one of which repreent the VM trategy, while the other repreent the MV trategy: MZ = ½ logðlðy; θ UM;UV Þ +logðlðy; θ UM;EV ÞŠ h i + log l y; θ UM;EV +log l y; θ EM;EV : By adding and ubtracting the likelihood under the contrained maxima θ UM, EV we repreent the MZ a the um of two tatitic, each of which i a likelihood ratio tet tatitic for the appropriate branch of the diagram above. The firt difference between θ UM, UV and θ UM, EV i the likelihood ratio tet for (A): equality of variance with uncontrained mean. The econd difference i the likelihood ratio tet for (B): equality of mean given that the variance are contrained to be equal. In equence, thee two hypothee combine to produce the null hypothei of aggregation or pooling. A econd decompoition of the MZ tet tatitic can be made by adding and ubtracting likelihood under the contrained EM, UV maxima θ i MZ = ½ logðlðy; θ UM;UV Þ +log l y; θ EM;UV h i + log l y; θ EM;UV +log l y; θ EM;EV : Again, the MZ i the um of two likelihood ratio tet. The quantity in the firt quare bracket i the likelihood ratio tatitic for teting (C): the hypothei of the equality of mean without any contraint on the variance. Thi i in fact a generalization of the Behren Fiher problem. The econd tatitic i the likelihood ratio tet for (D): equality of variance under the hypothei that the regreion coefficient are the ame in all regime. Thi etup i cloe to the one conidered by Goldfeld and Quandt (965) in teting for heterokedaticity. We will now tudy each of the three route to teting the hypothei of pooling or aggregation. We will call route J the joint tet of the null hypothei EV,EM. The high route i labeled VM, where we do a equential tet. Firt (A): EV UM that i, equality of variance with uncontrained mean, and econd (B): EM EV that i equality

3 384 E. Maaoumi et al. / Economic Modelling 7 (00) of mean under the maintained hypothei of equality of variance. The low route i labeled MV, where we do the equential teting in a different order. Firt (C) EM UV that i equality of mean with uncontrained variance, and econd (D) EV EM that i equality of variance under the maintained hypothei that the regreion coefficient are the ame in all regime.. Route J: the joint tet We firt conider the likelihood ratio tet of the joint null hypothei H 0 :EV,EM. A uual, computation of the likelihood ratio tatitic require the ML etimate under the null and the alternative. Thi i more or le trivial in the preent intance, becaue we face tandard regreion model under both the null and the alternative. If the null hypothei i true, then all the data can be aggregated into a ingle regreion model atifying tandard aumption, where the ML etimate are the claical OLS etimate. If the alternative hold, then each regime i an independent and eparate regreion model which atifie full ideal condition, and hence the ML etimate reduce to OLS etimate in each model eparately. Here we will aume that T i NK in each regime. We now et up the notation and definition required for the detailed calculation of the tet tatitic and their ditribution. Define the vector y 0 =(y,y,,y ), and imilarly β =(β, β,,β ), and the vector σ =(σ,,σ ). LetN =T T,be the number of obervation in regime.letn 0 =T,anddefine X 0 to be the T K matrix obtained by tacking the X, X,, X, and let β 0 and σ 0 be the common value of the coefficient β i and σ i under the null. The retricted model i then y 0 =X 0 β 0 +, where ~N(0,σ 0 I T ). For the regreion model defined above, the likelihood function for the obervation y i given by: lyjβ; ð σþ = N πσi i = exp σi jjy i X i β i jj : ðþ Before deriving the likelihood ratio tet, we derive the ML etimate under the null and the alternative. The uual OLS etimate are ˆβ i = X i X X i i y i, and ˆσ i = y i X i ˆβ i = ðn i KÞ, where, for i=0,,,. The uncontrained ML etimate coincide with OLS for the regreion coefficient ˆβ = ˆβ ; ˆβ ; ; ˆβ Þ, but are biaed for variance etimate: ˆσ ML; = v = N. Under the null hypothei, the etimate are the tandard OLS etimate for the aggregated model with pooled data: ˆβ 0 = ˆβ 0 ; ˆβ 0 ; ; ˆβ 0 Þ and ˆσ 0 = ˆσ 0 ; ˆσ 0 Þ, where ˆσ 0 = ky 0 X 0 ˆβ 0 k = ðt 0 KÞ: The ML etimate are the ame a OLS for the regreion coefficient, but biaed for the variance: ˆσ ML;0 = ky 0 X 0 β 0k = N 0. The likelihood ratio tet i the ratio of the maximized likelihood under the null and the alternative: Max β0 ;σ 0 πσ T0 = 0 exp σ 0 X 0 β LR = 0ky 0k : Max β;σ Π πσ Ni = i exp σ i ky i X i β ik We can immediately write down the likelihood ratio tet tatitic for the tet of the joint null hypothei of pooling all of the regime have identical regreion coefficient a well a variance. Subtituting ML etimate of β i and σ i into the expreion above lead to: LR = πˆσ N0 = ML;0 exp f T0 = g Ni = = πˆσ ML;i exp f Ni = g Π S ˆσ N0 = ML;0 : Ni = Π S ˆσ ML;i Becaue N 0 = N i, the π and the exponential in the numerator and denominator cancel. A preliminary analyi by imulation howed that ue of unbiaed variance etimate lead to omewhat improved performance of the LR tatitic. Below we give the ditribution of an unbiaed verion of the tet tatitic. Thi i the ame a log(lr) above, except for adjutment for degree of freedom. Define the um of quared reidual v i = ky i X i ˆβ i k for each regime i=,,,s eparately. Alo define v 0 = ky 0 X 0ˆβ 0 k a the um of quared reidual for the pooled data. Theorem MZ tet for aggregation. A tet aymptotically equivalent to the likelihood ratio for teting the null hypothei that H 0 :EV,EM that all regreion coefficient and variance are the ame in all the regime, i given by: MZ = ðn 0 K Þlog ˆσ S 0 = ðn KÞlog ˆσ : ðþ Under the null hypothei, thi ha an aymptotic Chi-quare denity with K(S ) degree of freedom. The exact finite ample null ditribution can be characterized a follow. Let Z i ~χ Ni K for i=,,,s and Z 0 ~χ K(S ) be independent Chi-quared variable. Then MZ defined below ha the ame ditribution a MZ in Eq. () under the null hypothei: MZ T = ðn 0 K S ÞlogZ 0 = ðn KÞlogZ : ð3þ Remark: Thi how that the ditribution of the joint tet depend only on the N and K and not upon the matrix of regreor. The form of the LR tatitic i quite cloe to that of a Bartlett tet tatitic, the exact ditribution of which ha been computed by everal author. None of thee appear to be directly applicable to LR or MZ above (ee Section 3.). However, the above characterization i ufficient to enable eay calculation of the critical value of the tet tatitic via imulation. Proof. Except for adjutment in contant to match degree of freedom, MZ i log(lr). Since thee adjutment do not affect the aymptotic denity, the reult i a tandard conequence of the aymptotic theory of the likelihood ratio tatitic ee for example, Zaman (996, Section 3.8). The degree of freedom i the number of retriction under the null, which i eaily counted to be K(S ). Becaue convergence can be low, it would be adviable to ue finite ample critical value from imulation baed on the exact ditribution of the σ under the null, a derived below. Firt, we need the following ueful characterization of the MLE of β 0. Lemma. If the above model i etimated under the null hypothei, the MLE of the parameter β 0 can be written a ˆβ0 = X i X i X ix i ˆβ i ð4þ where ˆβ i i the ML (OLS) etimator for the i-th regreion regime. The proof i traightforward. Note that the lemma diplay the etimate under the null a a matrix weighted average of the ˆβ i from each regreion model, where the weight are the invere of the covariance matrice. Let W i = X i X i and W = W i, then ˆβ 0 = W i W ˆβ i i. Next we compute the um of quared reidual under the null v 0 a follow: v 0 = ky i X iˆβ 0 k k = ky i X iˆβ i + X i ˆβ i ˆβ 0 = v i + = Z i + Z 0 ; Wi ˆβ i ˆβ 0 ˆβ i ˆβ 0 ay: ð5þ

4 E. Maaoumi et al. / Economic Modelling 7 (00) Note that v i =Z i are independent Chi-quared variable with T i K degree of freedom. Thee variable are alo independent, repectively, of the ˆβ i for i=0,,,. Thu, to prove the theorem it uffice to how that: Lemma. Suppoe, under the null hypothei, that β i =β 0 and σ i =σ 0 for all i. Then the quantity Z 0 /σ 0 ha a Chi-quare denity with K( ) degree of freedom. Proof. Let V i be a non-ingular K K matrix uch that W i =V i V i. Define ˆγ i = V i ˆβ i and note that ˆγ i en γ i ; σ0 I K, where γi =V i β i. Thi follow from the fact that V i (X i X i ) V i =V i (V i V i ) V i =I. We can rewrite the tatitic Z 0 a follow: Z 0 = kv i ˆβ i V i ˆβ 0 k = kˆγ i V i j =! V j V j k = V kˆγk : Let ˆγ be the KS vector obtained by tacking the ˆγ i and let ν be the KS K matrix obtained by tacking the V i. Then we have: ν Z 0 = k I ν ν ν ˆγ k k Π ν ˆγ k : It i eaily verified that under the null hypothei Π ν γ i zero, where γ = Eˆγ. Since Π ν i an idempotent matrix with trace K(S ) it follow that Z 0 /σ 0 i Chi-quared with K(S ) degree of freedom. Q.E.D. Several remark on the ignificance of thi reult are offered below: Remark. Anderon (003, Chapter 0) ha developed a joint tet for imultaneou equality of mean vector and covariance matrix in a collection of multivariate normal ditribution. The regreion etting dicued above, which offer ubtantial implification over thi mot general cae, appear not to have been tudied in the literature. Heavy analytical machinery available for variant of thi problem make it a worthwhile problem for deeper reearch. Remark. Similarity in form ugget that thi tatitic will hare the characteritic of the Bartlett' tet for heterokedaticity (dicued in detail in Section 3.); in particular, we expect that it will be very enitive to the aumption of normality. However, the iue ha not been explored in the literature. If the tet i enitive, it hould be amenable to treatment by method imilar to thoe ued for the Bartlett tet. The main idea i to aume that error in different regime have a common kurtoi. The principal effect of nonnormality come through kurtoi differing from the normal value of 3. The tet tatitic can be robutified by etimating the common non-normal kurtoi and adjuting the critical value appropriately. Remark 3. If there i no natural choice of breakpoint, how many regime hould be elected to provide ome ort of an optimal tet for tructural change? More regime are more enitive to maller departure from the null, but alo reult in reduced power becaue of maller ample and larger number of parameter. Thi i an open reearch problem. Note that the random coefficient model of Swamy repreent the limiting cae where each obervation i a eparate regime. See Zaman (00) for a detailed treatment of thi cae. 3. Route VM: the eay alternative A we hall ee, the VM route provide an eay alternative to the joint tet. Eay refer to the fact that the tatitic and ditribution involved are eay to calculate and have certain optimality propertie dicued below. Furthermore, the tet tatitic involved are readily available in exiting oftware package. A oppoed to thi, the MV ð6þ ð7þ route i difficult. The VM route conit of two tet carried out in equence. The firt i (A) EV UM, a tet for equality of variance with no retriction on the regreion coefficient. If thi tet reject the null, then we are done, becaue pooling cannot be done. If thi tet doe not reject the null of EV, then we carry out a tet for the equality of the regreion coefficient under the maintained hypothei of equality of variance acro the regime. Thi i the tet (B) EM EV. Some methodology and propertie of equential teting are dicued by Anderon (003, Section 9.6) under the heading of Step-Down Procedure. In any cae, given the independence of thee two tet, for example Phillip and McCabe (983) or Pearan et al. (985), the ize of the equential tet will be ( α )( α ) where α i are the ize of the contituent tet. 3.. (A): Teting for equality of variance with uncontrained mean A uual, we firt need to calculate the ML etimate under thee contraint. A before, the uncontrained OLS etimate are ˆβ i = X i X X i i y i, and ˆσ i = v i = ðn i KÞ, where v i = ky i X i ˆβ i k are the um of quared reidual for i=,,. Under the null hypothei (EV,UM) that all the variance are equal, with no contraint on the mean, it i eay to ee that an unbiaed etimate of the common variance i ˆσ S EVUM = ðn 0 SKÞ = ν i S = = ðn KÞ ðn 0 SK Þˆσ : ð8þ Conidering the OLS reidual from each regime a a ample, and the entire et of OLS reidual a a collection of S ample, the likelihood ratio tet in thi ituation i a tandard one way ANOVA tet for homogeneity of variance acro the ample, originally propoed by Neyman and Pearon. Bartlett (937) modified the tet by multiplying by a contant to provide a better approximation to the aymptotic ditribution. Thi form i popularly known a Bartlett' tet for equality of variance and i the mot commonly ued tet for thi purpoe. It i well known to be enitive to the aumption of normality, and numerou more robut alternative have been developed. A comprehenive tudy of thee alternative, and their robutne and power, i given in Conover et al. (98). In thi ituation, ue of the bia-adjuted LR*, which offer uperior performance, can be jutified via an invariance argument. Conider the um of quared reidual from each regime: v i = ky i X i ˆβ i k for i=,,. Thee are independent and have Chi-quare ditribution: ν i are σ i χ N i K. If we calculate a likelihood ratio tet for the equality of variance directly from the ditribution of the OLS reidual, the reulting tet i unbiaed. We can jutify a reduction to the conideration of the OLS reidual alone on the bai of an invariance argument. Tranlation of the regreion coefficient β do not affect the null and the alternative hypothei regarding the variance. Any tet tatitic which i invariant can depend only on the OLS reidual. Looking at the OLS reidual from each regime, the um of quared reidual ν i form ufficient tatitic for the variance σ i. Thi i preented below. Theorem. Baed on the ditribution of the um of quared reidual ν i (which i σ i χ N i K), the likelihood ratio tet for the null hypothei H 0 : σ =σ =.=σ veru the unretricted alternative reject H 0 for large value of the tatitic: LR = S = ν ð N K S = ν Þ= ðt SKÞ Proof. We omit the proof which i a traightforward calculation. Several remark on the ignificance of thee reult are offered below. ð9þ

5 386 E. Maaoumi et al. / Economic Modelling 7 (00) Note that the numerator of the tet tatitic i a weighted geometric mean (with weight proportional to the degree of freedom), while the denominator i proportional to the arithmetic mean of the um of quared reidual: ln LR + lnðt SK S N Þ = i K = T SK lnν ln T SK S =ν : ð0þ Remark. Conider the tatitic λ=c( loglr ), where C i a contant which define the Bartlett correction factor. The idea of the Bartlett correction i to make the ditribution of the tet tatitic λ cloer to it aymptotic Chi-quare ditribution, o that the ize or level of the tet i cloer to it aymptotic level in finite ample. The concept of Bartlett correction i really a pre-computer age idea, where it wa impoible to compute critical value on the fly. Tet could only be ued if table of critical value exited. Computer age technique provide two alternative which are uperior to Bartlett correction. In the firt place, if normality of error can afely be aumed, then exact finite ample critical value can be computed. Several author have derived different form of the exact denity of the tet tatitic LR* derived above. Tang and Gupta (986), Nagarenker (984), Glaer (976), Chao and Glaer (978), and Dyer and Keating (980) give method for finding the exact ditribution of Bartlett' tet tatitic. The econd cae i when we are not ure about normality. It ha been hown that the ditribution of the tet tatitic i enitive to the kurtoi of the common ditribution under the null. A robut boottrapping technique i baed on etimating the common kurtoi of the error, and i detailed by Boo and Brownie (989). Under normality, exact critical value dominate Bartlett correction. When normality fail, boottrapping dominate, ince the Bartlett correction aume, and i very enitive to the normality. Remark. The normality aumption amount to auming that the kurtoi i 3 for all error in all regime. Boo and Brownie (989) how how to ue the boottrap under the aumption that all of the S ubample have common kurtoi different from 3. Many other technique for robutifying the Bartlett tet, dicued in Conover et al. (98) are baed on etimating thi common kurtoi. One might conider making the tet even more robut by teting for equality of variance of the error while allowing for differing kurtoe in the different regime. Moe (963) how that thi i not poible. Without any commonality retriction acro ubample, tet for equality of variance mut have ome pathological behavior. Remark 3. In the cae that there are only two regime, it can be etablihed that thi tet i UMP invariant and equivalent to the Goldfeld Quandt tet for the cae =. In thi cae we have EV = x T = = ð+xþ N 0 = : Thi i monotone increaing in x v /v which i the Goldfeld Quandt tet tatitic. Furthermore, ince the tranformation β i β i +γ i for i=, leave the null hypothei invariant, the data can be reduced to the maximal invariant (v,v ). And, ince the hypothei of equality of variance i invariant under cale change, thi reduce the data further to the maximal invariant v /v. Since thi ha a noncentral F denity which ha monotone likelihood ratio, there i a unique mot powerful tet baed on the maximal invariant. For detail of thi argument, ee Zaman (996, Section 0.5). Remark 4. In general for N thi tet i not UMP among invariant tet and no UMP tet exit. However, the tet doe have the important property of unbiaedne. That i, the probability of rejecting the null i larger than the ize of the tet for each parameter in the alternative. To prove thi, reduce the problem by invariance to the tatitic v,, v. Taking log turn σ i into a cale parameter for the problem. With ome computation, we can alo how that the unbiaed tet propoed by Pitman for equality of cale parameter i equivalent to the tet above. For a proof and ome propertie of the tet, the reader i referred to Pitman (938), Dutta and Zaman (989), or Zaman (996, Section 8.6). Briefly, no UMP tet exit in thi cae, but the tet tatitic propoed i a natural tatitic which maximize average power over the pace of alternative, where the averaging i done with repect to Lebegue meaure on vector pace of tranlation parameter ln σ i. Remark 5 Lehmann' aymptotic UMP invariant tet. Define V i = ln ˆσ i = lnν i = ðn i KÞ), a i =/(N i K), and their weighted average a W = j =V i = a i = : a i j = Lehmann (986) propoed the following tet of equality of variance which i approximately UMP invariant in large ample: L = a i ðv i WÞ : Becaue we have focued on the likelihood ratio, we did not tudy thi tet in thi paper. 3.. (B): Equality of mean under the aumption of homokedaticity Tet for equality of regreion coefficient under homokedaticity were long known to tatitician under the name of analyi of covariance ; ee for example Scheffe (959). Chow (960) introduced the ubject to econometrician with the innovation of allowing for inufficient obervation (N K) in one of the two regime being teted, and thi tet became known a Chow' tet in general. In the cae where =, o there are only two regreion, Chow' tet for aggregation ( equality of regreion ) i the following: CH T = ˆβ K ˆσ ˆβ X X + X X : ˆβ ˆβ EVUM Thi i an optimal (UMP invariant) tet for the null hypothei H 0 when the alternative hypothei i H *:β β ; σ =σ. In other word, the equality of the variance acro the regime i a maintained hypothei. The generalization to the preent cae of everal regime ha alo been familiar to tatitician ince Kullback and Roenblatt (957), a follow: CH = KS ð S Þˆσ EVUM ˆβ i ˆβ 0 X ix i ˆβ i ˆβ 0 : ðþ Thi i the likelihood ratio tet tatitic for EM EV cae. Like the tet tatitic in the two regime cae, thi ha an F ditribution under the null with N 0 K and K(S ) degree of freedom. Dufour (98) ha given the extenion to the mot general cae, where all except one of the regime may have inufficient obervation to permit etimation by OLS. Going even further to the cae of inufficient obervation in all regime lead naturally to the random coefficient model of Swamy. In thee cae, Eq. (9) above doe not work ince the OLS etimate are not well defined. In our preent paper, we aume ufficient obervation in each regime o thi cae i not relevant to u. Anderon (003, Section 8.8) dicue teting the equality of everal vector mean under the aumption of identical covariance matrice for the multivariate normal ditribution. Again, thee reult are not directly applicable.

6 E. Maaoumi et al. / Economic Modelling 7 (00) Together, a equential route to pooling which ue ome robutified verion of the Bartlett' tet for heterokedaticity, followed by the generalization of Chow' tet for multiple regime, i relatively eay to implement computationally and alo ha nice invariance and optimality propertie. Surpriingly, the third route, dicued below, i rather different. 4. The route MV: the difficult alternative The other equential procedure involve teting equality of mean (EM) with uncontrained variance (UV) firt. If thi doe not reject the null, then we can tet for equality of variance (EV) under the aumption of EM. We now dicu the detail of thi econd equential route to teting for pooling. Depite imilarity in appearance to VM, thi route involve ubtantial complication. 4.. (C): Equality of regreion coefficient under heterokedaticity There i an extenive literature on Chow-type tet of equality of mean where heterokedaticity i a maintained hypothei. See Thurby (99) for a comparative tudy of many of uch tet and earlier reference. In econometric, thi literature began with the realization that Chow' original tet perform poorly when variance are unequal; ee Toyoda (974), Schmidt and Sickle (977) and Kochat and Weerahandi (99). Thi i actually a generalization of the Behren Fiher problem. It ha been proven that good claical olution to thi problem do not exit. That i, tet do not exit which atify all three requirement of (i) invariance treat all obervation the ame, (ii) imilarity that i, ditribution do not depend on nuiance parameter (which are the differing variance acro regime), and (iii) fixed level alpha of type I error probability. Interetingly, one can get tet atifying any two out of the three requirement.. Dropping invariance. Thi involve dropping ome obervation, or treating ome of the regreor aymmetrically in an arbitrary fahion. Tet of thi type were firt adapted for ue in regreion model by Jayatia (977) and other. By doing thi, one can achieve imilarity the ditribution of the tet become independent of the value of the variance. However, the lack of invariance i unappealing; we make arbitrary choice which affect the outcome. Alo, imulation tudie in Thurby (99) report poor propertie for thi cla of tet, which i not dicued further here.. Dropping imilarity. If we allow the ditribution of tet to depend on the nuiance parameter, then good approximate tet can be found. Welh and Apin firt developed uch olution in the context of ANOVA, and thee olution can eaily be generalized to the regreion context. Specific detail are available from Thurby (99) and Zaman (996). 3. Dropping fixed ize. Uing a randomized rejection region with an exact but randomized p-value lead to a new type of olution introduced by Kochat and Weerahandi. Thurby (99) report good reult from thi type of tet. Since our focu in thi paper i on likelihood ratio, we develop thi tet tatitic below. To the bet of our knowledge, thi tet ha not been ued in the literature, perhap due to the following difficultie. Firt, the likelihood ratio tet tatitic cannot be computed explicitly. Second, the ditribution of the LR depend on the nuiance parameter. Thi mean that exact critical value cannot be computed, and even approximate one cannot be tabulated. Furthermore it require an empirical invetigation to explore how table the ize i in different data context. Regularity condition hold, o the aymptotic ditribution i the tandard one for the likelihood ratio; thu, thi problem hould not be eriou in large ample. An Iterative Algorithm for Computing ML etimate under EM UV: If the variance σ i were known, the optimal etimator for β i under the contraint that thee are equal acro regime would be given by a preciion weighted average of the eparate OLS etimator of each regime: β =! X i X i σ i X i X i ˆβ i!: ðþ σ i Thi etimate alo maximize the likelihood conditional on the variance. Given the ML β, one can etimate σ i by σ i = T i ky i X i βk : ð3þ Maximum likelihood etimate for β i and σ i can be obtained by iterating between thee two equation. Start with ay ˆβ and calculate σ i from Eq. (3). Given etimate for σ i plug thee into Eq. () to get a new etimate for β i. Iterate back and forth until convergence. The likelihood ratio tatitic can be decribed in term of thee etimate a follow. Theorem 3 EM UV. The likelihood ratio tet for equality of mean acro regime, with unretricted variance, reject the null of equality for high value of the tatitic below: LREM = S = σn i S =ˆσ N i : Proof. Thi fall out of the likelihood ratio immediately after ubtituting the maximizing value under the null and the alternative. Remark. It i likely that the unbiaed verion of thi tatitic will perform better. Thi involve replacing N by N K in the denominator. In the numerator, it i unclear how to debia. A total of K parameter are being etimated acro all the regime, o one could allocate the hare K/S a the lo of degree of freedom in each regime, and replace N by N (K / S). Cloer calculation are poible, but it i not clear if they would be worthwhile, ince the ditribution in quetion are no longer Chi-quared and there i dependence in the variance etimate acro the regime. Remark. The likelihood ratio tatitic diplayed above i very different from the tandard tet tatitic for thi ituation, large number of which have been dicued in Thurby (99). Since the null hypothei ay that all the regreion coefficient are the ame, tandard tet concentrate on the difference ˆβ i ˆβ 0 between eparate etimate for each regime and the aggregated etimate. Intead, the LR tet look at the increaed variance caued by replacing the variance minimizing value ˆβ i by the null hypothei etimate ˆβ 0. Thi i an indirect meaure of the lo caued by impoing the retriction. Remark 3. Unlike the Bartlett type tatitic in the VM route, the ratio of variance tend to be robut in analogou ituation, o we may conjecture that thi tet will be le enitive to normality. Remark 4. Since the tet i not imilar, computing critical value i not traightforward. The null denity depend on the nuiance parameter, the differing variance, and i alo intractable. A imple trategy i to etimate common regreion coefficient and different variance for each regime. Treat thee etimated parameter a true value, and ue thi (boottrapping) trategy to generate imulated value of the tet tatitic under the null. Critical value calculated from thi method provided ufficiently table ize under the null to be uable for the imulation reported in Section 5. Note that thi mean that critical value cannot be tabulated but mut be computed

7 388 E. Maaoumi et al. / Economic Modelling 7 (00) Table Power of MZ tet (diagonal route via J) Boldface entrie how greatet power for the MZ tet. directly for each data et on the bai of the etimate for unequal variance under the null. Experience from imilar problem ugget that critical value will be table for large ample ize and when the variance are not too far from equal. 4.. (D): Teting for equality of variance, auming equality of regreion coefficient acro regime If the tet in (C) fail to reject the equality of regreion coefficient, we may proceed to tet for equality of variance, which will allow u to aggregate the data. Thi i a tet for EV conditional on the maintained hypothei EM. Theorem 4 EV EM. The LR tatitic to tet H 0 :β =.=β and σ =.=σ veru the alternative H :β =.=β and σ i σ j for ome i and j, i given below: LREV = σ N i i : ð4þ N0 = ˆσ 0 Proof. The proof i a traightforward computation and hence i omitted. Several remark are given below: Remark. Thi i the uual etup for the Goldfeld Quandt (GQ) tet, where the firt tep i to ort the obervation in order of increaing variance. The ample i plit into two halve and variance etimate are compared acro the two halve. There are two ource of inefficiency in uing the GQ tet for heterokedaticity in uch ituation. The ability to ort obervation mean that we know of ome variable/ which can be ued to order variance. Directly teting for aociation between uch variable and the quared error (proxy for variance) will uually be more efficient than thi indirect method. Secondly, plitting the data into two (or more) regime and looking at the variance ratio ignore the equality of the regreion coefficient acro the regime. Not impoing thi contraint i extremely inefficient ince it amount to throwing away a good deal of the data when etimating the beta in each regime eparately. Numerical comparion in Tomak (994) how ubtantial gain achieved over GQ in the cae that the equality of the regreion coefficient i valid and impoed. Remark. The ditribution of the LREV tatitic i unavailable, and likely to be difficult to derive. Thi i becaue the variance etimate are no longer independent acro regime the ue of a common beta, which itelf depend on the variance etimate, ubtantially complicate the ditribution theory. Luckily, imulation take uch matter in it tride, and i eaily done to obtain critical value for thi tet. Greene (007) ha utilized thi tet in the context of an example dicued in greater detail below. He ha ued aymptotic critical value for the LR tet. A we hall ee, thee differ quite a bit from the finite ample critical value calculated uing the method dicued above. 5. Comparion of the three route For illutrative purpoe, we computed the power of the three different method of teting for aggregation within the context of a imple empirical example. Data from Greene (007, Appendix F, page 949) Table F6.: Cot Data for U.S. Airline, provide 90 obervation on 6 firm for 5 year, We take obervation on cot a a linear function of fuel price for three different firm, and ak if thee cot function are the ame for all ix firm. With C a cot, Q a output in paenger-mile, PF a the price of fuel, and LF a the load factor, the regreion equation ued by Greene (007, Chapter ) i a follow: lnðcþ = β + β lnðqþ + β 3 lnðpfþ + β 4 LF + : For detailed definition of the variable and jutification of thi regreion equation, the reader i referred to Greene. Greene introduce dummy variable to allow for the poibility that the contant varie acro the firm, aume that the three coefficient of the regreor are the ame acro the firm and tet for the equality of variance acro firm. To achieve comparability with Greene, and avoid firm pecific dummie, we adjuted the cot data by the firm Table Power of VM tet (top route via A,B) Boldface entrie how greatet power for the VM tet.

8 E. Maaoumi et al. / Economic Modelling 7 (00) Table 3 Power of MV tet (bottom route via C,D) Boldface entrie how greatet power for the MV tet. pecific contant o a to achieve equality of the contant under the unretricted mean hypothei. Thi i a rough and ready way to remove the contant from conideration while teting for equality of cot function acro firm. The reult of the three tet to ae whether the cot function are the ame for all 6 firm are a follow: The J route: We have LMZ=LEV+LCH. The computed tatitic are 0.4= The p-value for LMZ i 0.000, o we trongly reject the null of aggregation (EM,EV). The VM route: Thi i baed on firt teting equality of variance via LEV=0.07. Thi ha a p-value of 6.% which mean that we cannot reject the null of equality of variance (with unretricted mean under both null and alternative). Therefore we proceed to the tet for equality of mean, auming homokedaticity. Thi i baed on the tet tatitic LCH=0.349 which ha a p-value of under the null of equality of mean. Thu we trongly reject equality of mean. The MV route: The tet tatitic LREM i.67 which ha a p-value of Thu the hypothei of equality of mean (with unretricted variance) i trongly rejected. There i no need to proceed to the econd tet of equality of variance. Nonethele, if we do proceed, we can compute LREV= 0.8. Thi ha a high p- value (97%) which fail to reject the null. To compare with Greene' analyi, we note that Greene (007, Example.6, page 36) carrie out a tet of (D): Equality of Variance under the aumption of equality of mean. In our methodology, we never get to thi econd portion of the MV route. Our analyi indicate that thi tet i not jutified becaue the hypothei EM i trongly rejected at tage (C). If we ignore thi rejection of EM by that data and proceed to do the LREV tet anyway, then we get p-value of 97% by our finite ample method. The p-value obtained by Greene baed on the aymptotic Chi-quare denity for the likelihood ratio i , which i a very trong rejection. Thi mean there i a conflict between our reult and Greene on the outcome of the tet for (D). To explore thi further, we look at Greene' variance etimate for the ix regime. Under the EMUV cenario, thee etimate are (0.005, , 0.009, , 0.003, and ). Uing the HET meaure of ditance from equality of all 6 variance defined in Eq. (5) below, we calculate HET=0., which i a very mall departure from null of equality of variance. Simulation from Greene' data how that with identical variance in all regime, larger departure of etimated variance from the null of equality will occur with 97% probability, o that the data doe not provide any evidence againt equality of variance. Greene' rejection come from the ue of aymptotic Chiquare denity of the likelihood ratio, which i not applicable at the mall ample ize of 5 obervation per firm. Ue of aymptotic critical value can and often doe lead to wrong reult in mall ample; for another example from publihed literature, ee Zaman (996, Section 6.). 5.. Power of the three tet. To compute power, we treat OLS etimate under the null a the true null parameter. We then vary the tandard error to make them different for each regime. With weight w i =N i /N 0, a natural meaure of the degree of heterokedaticity i: H = log w i σ i w i logσ i : ð5þ For the preent data et, the weight are equal ince there are 5 point of data in each regime. Similarly, a natural meaure of the degree of departure from the null of equality of regreion coefficient acro the regime i the noncentrality parameter defined a: D = ðβ i β 0 Þ X ix i ð βi β 0 Þ In thi ection we firt preent three table of ize/power, one for each teting trategy/route. The power in boldface are thoe entrie for which the tet i mot powerful among the three tet tabulated here. Except for ome Monte Carlo imulation error, thee how the expected pattern. The power of all three tet increae quite rapidly to 00% a the degree of departure from the null increae. The VM tet prioritize variance, and doe well at low and intermediate dicrepancie from equality of variance (meaured by H). The MV tet prioritize mean and doe well at low and intermediate value of Table 4 % Difference, MZ minu VM (middle route minu top route) Table 5 % Difference, MV minu MZ (bottom route minu middle route)

9 390 E. Maaoumi et al. / Economic Modelling 7 (00) D which meaure dicrepancie from equality of the regreion coefficient. The Joint tet doe bet near the diagonal, where both dicrepancie are about equal. At firt glance, there i not much to chooe between the three trategie. A more detailed comparion i offered in the next ection (Table 3). 5.. Pairwie comparion of the tet The table below lit the percentage point difference between the power of the MZ tet and the equential VM tet. An entry of 9 mean that MZ ha 9% greater power than the VM tet, while an entry of 0 indicated 0% greater power for the VM tet over MZ. Tet EV firt prioritize the variance, which i reflected in the power comparion in the table. Since VM tet variance firt, it pick up maller departure from EV more effectively than the MZ, which i the joint tet. Thi account for it uperior power for low value of D and intermediate value of H. At higher value of D, thi ame equencing act againt the EV, and the MZ offer uperior performance. Otherwie there i not much to chooe between the two method of teting. Of coure, the VM tet i more convenient in that it i eaier to implement on conventional econometric oftware package (Table 4). Thi table of difference in power confirm the ame picture which emerge from our initial analyi: VM prioritize variance inequality over the joint tet J. However power difference are not major and decline for large value of difference which make both tet capable of picking them up (Table 5). Comparing MV to J, the prioritization of mean in MV relative to the joint tet how clearly in thi table. However the gain of MV over J are relatively mall for larger value of D, while the J tet ha ubtantially uperior power for larger value of H. The reveral of prioritie between the VM and MV tet i clearly diplayed in the table below which compare their power. A H increae, the relative performance of VM increae, while a D increae the relative performance of MV improve. For ufficiently large value, both tet are equivalent with 00% power. Above the diagonal, where dicrepancie in variance are larger relative to dicrepancie in mean, VM perform much better becaue it prioritize variance. Below the diagonal, MV perform better at picking up dicrepancie from the null hypothei of equality of mean (Table 6). 6. Concluion In thi paper, we have outlined trategie for teting for aggregation, homogeneity, and tructural change. Earlier extenive literature on tructural change i reviewed in Kramer (989) and, with Bayeian orientation, in Broemeling and Turumi (986). Our approach build on thi claical tradition: we pecify the number and location of potential point of tructural change a priori, and work with a imple i.i.d. error tructure. More recently, a Hanen (00) decribe, attention ha hifted to model with complex error procee with unknown timing Table 6 % Difference, VM minu MV (top route minu bottom route) and number of breakpoint. The complexity of thee model make an analytic analyi of the probabilitie and conequence of type I and II error very difficult. In addition, one mut urely pay a cot in preciion for the added uncertainty of unknown breakpoint. Varying the breakpoint in our propoed approach i very imilar to what need to be done in data dependent earche. Our tet can be ued in averaging and maximum earch trategie that guide the unknown breakpoint approache; for example ee Andrew (993), Chu and White (99), and Ploberger et al. (989). There are large number of open quetion regarding thi approach which are amenable to analytic analyi; many of thee have been raied in the coure of the dicuion above. Acknowledgment In earlier verion of thi paper, circa 990, we received programming aitance from Arif Zaman, and reearch aitance from Craig Walker and David Oppedahl. Thee are gratefully acknowledged. Reference Anderon, T.W., 003. An Introduction to Multivariate Statitical Analyi. Wiley, New York. Andrew, D.W.K., 993. Tet for parameter intability and tructural change with unknown change point. Econometrica 6 (4), Jul. Bartlett, M.S., 937. Propertie of Sufficiency of Statitical Tet. Proceeding of the Royal Statitical Society (Serie A) 60, Boo, D.D., Brownie, C., 989. Boottrap method for teting homogeneity of variance. Technometric 3 (), Broemeling, Lyle D., Turumi, Hiroki, 986. Econometric and Structural Change. Marcel-Dekker, New York. Chao, M., Glaer, R.E., 978. The exact ditribution of Bartlett' tet tatitic for homogeneity of variance with unequal ample ize. Journal of the American Statitical Aociation 73, Chow, G.C., 960. 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