RECENT DEVELOPMENTS IN QUANTITATIVE COMPARATIVE METHODOLOGY:


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1 Federco Podestà RECENT DEVELOPMENTS IN QUANTITATIVE COMPARATIVE METHODOLOGY: THE CASE OF POOLED TIME SERIES CROSSSECTION ANALYSIS DSS PAPERS SOC 302
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3 INDICE 1. Advantages and Dsadvantages of Pooled Analyss... Pag The Estmaton Issue: GLS vs. OLS Tme and Space Effects Poolng Dlemma and Casual Heterogenety Pooled TSCS analyss n STATA software Bblography... 41
4 The frst draft of ths paper has been wrtten at McDonough School of Busness (Georgetown Unversty) n November For ths reason, I would lke to thank Professor Denns Qunn for the opportunty he allowed me.
5 Students of the poltcal economy have tended to nvestgate relatonshp between nsttutons and economc varables by comparng observatons across space or observatons over tme. Untl recently, the space and the tme domans have rarely been combnated n the comparatve research. However, new quanttatve methods stress senstvty to tme as well as space. Pooled tme seres crosssecton analyss (TSCS) s probably the most mportant way to examne smultaneously these dmensons. In ths paper, I wll try to descrbe the state of the art of ths approach dscussng frst the characterstcs of TSCS data and advantages and dsadvantages of ths statstcal technque (Secton 1). Hence, I wll dscuss man ssues that relate to the estmaton method (secton 2). After that, I wll address the most mportant problems that relate to the model specfcaton by concentratng frst on effects of the tme and the space (Secton 3), and then on the poolng dlemma and causal heterogenety ssue (Secton 4). Fnally, I wll present mplementatons and commands n STATA software to analyze TSCS data (Secton 5). Recent Developments n Quanttatve Comparatve Methodology 5
6 1. Advantages and Dsadvantages of Pooled Analyss Pooled analyss combnes tme seres for several crosssectons 1. Pooled data are characterzed by havng repeated observatons (most frequently years) on fxed unts (most frequently states and natons). Ths means that pooled arrays of data are one that combnes crosssectonal data on N spatal unts and T tme perods to produce a data set of N T observatons. Here, the typcal range of unts of analyzed would be about 20 (f we examne developed countres), wth each unt observed over a relatvely long tme perod, lke years. However, when the crosssecton unts are more numerous than temporal unts (N>T), the pool s often conceptualzed as a crosssectonal domnant. conversely, when the temporal unts are more numerous than spatal unts (T>N), the pool s called temporal domnant (Stmson 1985). Gven ths preamble, we can wrte the generc pooled lnear regresson model estmable by Ordnary Least Squares (OLS) procedure k 1 + β k k =2 y t = β xkt + et (1) Where = 1,2,.; N; refers to a crosssectonal unt; t = 1,2,.; T; refers to a tme perod and k = 1,2,.; K; refers to a specfc explanatory varable. Thus, y t and x t refer respectvely to dependent and ndependent varables for 1 Another name of pooled TSCS analyss s panel analyss, but t can be confused wth panel research n survey studes. 6 Recent Developments n Quanttatve Comparatve Methodology
7 unt and tme t; and e t s a random error and β 1 and β k refer, respectvely, to the ntercept and the slope parameters. 2 Moreover we can denote the NT NT varancecovarance matrx of the errors wth typcal element Ε e e t ) by Ω. ( js Estmatng ths knd of model and some of ts varants (see below), solves many problems of tradtonal methods of the comparatve research (.e. tme seres analyss and crosssectonal analyss). Several reasons support ths. The frst reason concerns the small N problem suffered by both tme seres and crosssectonal analyss. The lmted number of spatal unts and the lmted number of avalable data over tme led data sets of these two technques to volate basc assumpton of standard statstcal analyss. Most specfcally, the small sample of conventonal comparsons shows an mbalance between too many explanatory varables and too few cases. Consequently, wthn the contest of the small sample the total number of the potental explanatory varables exceeds the degree of freedom requred to model the relatonshp between the dependent and ndependent varables. In contrast, thanks to pooled TSCS desgns, we can greatly relax ths restrcton. Ths s because, wthn the pooled TSCS research, the cases are countryyear (NT observatons) startng from the country n year t, then country n year t+1 through country z n the last year of the perod under nvestgaton Ths allow us to test the mpact of a large number of predctors of the level and change n the dependent varable wthn the framework of a multvarate analyss (Schmdt 1997, 156). Second, pooled models have ganed popularty because they permt to nqury nto varables that elude study n smple crosssectonal or tme 2 I assume the dependent varable, y, s contnuous. In the case of bnary Recent Developments n Quanttatve Comparatve Methodology 7
8 seres. Ths s because ther varablty s neglgble, or not exstent, across ether tme or space. In practce, many characterstcs of natonal systems (or nsttutons) tend to be temporally nvarant. Therefore, regresson analyss of pooled data combnng space and tme may rely upon hgher varablty of data n respect to a smple tme seres or crosssecton desgn research (Hcks 1994, ). A thrd reason to support pooled TSCS analyss concerns the possblty to capture not only the varaton of what emerges through tme or space, but the varaton of these two dmensons smultaneously. Ths s because, nstead of testng a crosssecton model for all countres at one pont n tme or testng a tme seres model for one country usng tme seres data, a pooled model s tested for all countres through tme (Pennngs, Keman e Klennjenhus 1999, 172). Gven these advantages, n the last decade pooled analyss has became central n quanttatve studes of comparatve poltcal economy. Several authors have utlzed pooled models to answer to classcal questons of ths dscplne. An accumulatng body of research has used ths statstcal technque to test the man hypothess concernng the poltcal and nsttutonal determnants of macroeconomc polces and performances (Alvarez, Garrett, Lange 1991; Hcks 1991; Swank 1992). Most specfcally, regardng the study of publc polcy, we can cte emprcal works on poltcal and socoeconomc causes of the welfare state development (Pampel and Wllamson 1989; Huber Ragn and Stephen 1993; Schmdt 1997). Regardng research on both economc polces and performances, researchers have tred to verfy and characterze a dependent varable, see Beck et al. (1998). 8 Recent Developments n Quanttatve Comparatve Methodology
9 macroeconomc partsan strategy. In partcular, they have shown that, once n offce, dfferent partes attempt to manage the economc cycle usng the standard fscal and monetary nstruments. However, these same studes have dscovered that the ablty of partes to pursue ther most preferred macroeconomc strateges depends on nsttutonal structures of the domestc labor market (Comptson 1997; Oatley 1998), and ncreasngly nternatonalzed markets (Garrett 1998; Garrett and Mtchell 1999). Fnally, several authors have utlzed TSCS analyss to examne the mpact of poltcal and economc varables on the fnancal openness of domestc markets (Alesna et al. 1994; Qunn and Inclan 1997). Therefore, pooled TSCS analyss s an nalenable nstrument for the development of the comparatve poltcal economy. However, the popularty of ths statstcal technque does not depend only on ts applcaton n substantve research, but also recent papers dscussng methodologcal ssues that t mples (Stmson 1985; Hcks 1994; Beck and Katz 1995; 1996). In partcular, ths latter lterature s more numerous now because pooled TSCS desgns often volate the standard OLS assumptons about the error process. 3 In fact, the OLS regresson estmates, used by socal scentsts commonly to lnk potental causes and effects, are lkely to be based, neffcent and/or nconsstent when they are appled to pooled data. 4 Ths s because the errors for regresson 3 For OLS to be optmal t s necessary that all the errors have the same varance (homoschedastcty) and that all of the errors are ndependent of each other. 4 An unbased estmator s one that has a samplng dstrbuton wth a mean equal to the parameter to be estmated. An effcent estmator s one that has the smallest dsperson, (.e., one that one whose samplng dstrbutonhas the Recent Developments n Quanttatve Comparatve Methodology 9
10 equatons estmated from pooled data usng OLS procedure and pooled data tend to generate fve complcatons (Hcks 1994, ). Frst, errors tend to be no ndependent from a perod to the next. In other terms, they mght be serally correlated, such that errors n country at tme t are correlated wth errors n country at tme t+1. Ths s because observatons and trats that characterze them tend to be nterdependent across tme. For example, temporally successve values of many natonal trats (.e., populaton sze) tend not to be ndependent over tme. Second, the errors tend to be correlated across natons. They mght be contemporaneously correlated, such that errors n country at tme t are correlated wth errors n country j at tme t. As Hcks (1994, 174) notes, we could not expect errors n the statstcal model for Sweden to lack some resemblance to those for the Norway or errors for Canada and the Unted Stated to be altogether ndependent. Instead, we would expect dsturbances for such natons to be crosssectonally correlated. In ths way, errors n Scandnavan economes may be lnked together but reman ndependent wth errors of North Amercan countres. Thrd, errors tend to be heteroschesdastc, such that they may have dfferng varances across ranges or sub sets of natons. In other words, natons wth hgher values on varables tend to have less restrcted and, hence, hgher varances on them. For example, the Unted Stated tends to have more volatle as well as hgher unemployment rates than the Swtzerland. Ths means that the smallest varance). Fnally, an estmator s sad to be consstent f ts samplng dstrbuton tends to become concentrated on the true value of the parameter as sample sze ncreases to nfnte (Kmenta 1986,123). 10 Recent Developments n Quanttatve Comparatve Methodology
11 varance n employment rates wll tend to be greater for bgger natons wth large heterogeneous labor forces than for small, homogeneous natons (Hcks 1994, 172). Moreover, errors of a TSCS analyss may show heteroschesdastcty because the scale of the dependent varable, such as the level of government spendng, may dffer between countres (Beck and Katz 1995, 636). Fourth, errors may contan both temporal and crosssectonal components reflectng crosssectonal effects and temporal effects. Errors tend to conceal unt and perod effects. In other words, even f we start wth data that were homoschedastc and not autocorrelated, we rsk producng a regresson wth observed heteroschestastc and autocorrelated errors. Ths s because heteroschedastcy and autocorrelaton we observe s a functon also of model msspecfcaton. The msspecfcaton, that s pecular of pooled data, s the assumpton of homogenety of level of dependent varable across unts and tme perods. In partcular, f we assume that unts and tme perods are homogeneous n the level (as OLS estmaton requres) and they are not, then least squares estmators wll be a compromse, unlkely to be a good predctor of the tme perods and the crosssectonal unts, and the apparent level of heteroschedastcty and autocorrelaton wll be substantally nflated (Stmson 1985, 919). Ffth, errors mght be nonrandom across spatal and/or temporal unts because parameters are heterogeneous across subsets of unts. In other words, snce processes lnkng dependent and ndependent varables tend to vary across subsets of natons or/and perod, errors tend to reflect some causal heterogenety across space, tme, or both (Hcks 1994, 172). Therefore, ths Recent Developments n Quanttatve Comparatve Methodology 11
12 complcaton, lke the prevous one, could be nterpreted as a functon of msspecfcaton. If we estmate constantcoeffcents models, we cannot capture the causal heterogenety across tme and space. In the last decade, several models have been developed to deal wth these complcatons and dfferent solutons have been jonted because problems usually do not appear alone. However, for reasons of the clearness, I wll present the dfferent models tryng to separate varous solutons utlzed to deal wth sngle problems. In the next secton, I wll dscuss ParksKmenta method and Beck and Katz s (1995; 1996) proposal. They represent two dfferent approaches to tackle the complcatons of seral correlaton, contemporaneous correlaton and heteroschedastcty (respectvely problem 1, 2 and 3). After that, I wll address specfcaton problems by dstngushng between the ssue of tme and space effects (problem 4) and causal heterogenety (problem 5). 12 Recent Developments n Quanttatve Comparatve Methodology
13 2. The Estmaton Issue: GLS vs. OLS ParksKmenta method has been the most utlzed approach for TSCS analyss n comparatve poltcal economy untl the mdnnetes (see for example, Pampel and Wllamson 1989; Alvarez, Garrett, Lange 1991; Hcks 1991; Swank 1992; Huber, Ragn, and Stephen 1993). Nevertheless, from those years, when the two papers of Beck and Katz (1995; 1996) suggested an alternatve approach to the ParksKmenta method, ths latter proposal has probably became the one most utlzed by socologsts and poltcal scentsts (see for example, Qunn and Inclan 1997; Oatley 1998; Garrett 1998; Garrett and Mtchell 1999). Accordng to an hstorcal reason, let me start by dscussng the Parks Kmenta method. Ths method frst elaborated by Parks (1967) and then dscussed by Kmenta (1971; 1986) (here referred to as ParksKmenta method) uses an applcaton of the generalzed least squares (GLS) estmaton. The regresson equaton for ths method may be wrtten n the same form of the equaton 1: k 1 + β k k =2 y t = β xkt + et (2) Thus, t s an equaton where a sngle ntercept and slope coeffcent are constant across unts and tme ponts. However, accordng to ParksKmenta method, ths equaton must be estmated by GLS because ths estmaton procedure s based on less restrctve assumptons concernng the behavor of Recent Developments n Quanttatve Comparatve Methodology 13
14 regresson dsturbance and, thus, concernng the varancecovarance matrx, Ω, than the classcal regresson model (Kmenta 1986, 607). Therefore, the GLS estmaton has a specal nterest n connecton wth tme seres and crosssecton observatons. Regardng the problem of estmatng parameters β of the generalzed lnear regresson model, we can wrte the followng expresson: ' 1 1 ' 1 ( x Ω x) x Ω y (2.1) Ths estmaton s based on the assumpton that the varancecovarance matrx of the errors, Ω, s known. However, snce n many cases the varancecovarance matrx s unknown, we cannot use GLS but feasble generalzed least squares (FGLS). It s feasble because t uses an estmate of varancecovarance matrx, avodng the GLS assumpton that Ω s known. Consequently, we need to fnd a consstent estmate of Ω, say, Ω ), to substtute Ω ) for Ω n the formula to get a coeffcent estmator β (Kmenta 1986, 615). Thus we denote the FGLS estmates of β by β ). Let me now consder the problem of error complcatons. The Parks Kmenta method combnes the assumptons concernng seral correlaton, contemporaneous correlaton and panel heteroschedastcty of errors. The partcular characterzaton of these assumptons are (Kmenta 1986, 622): 2 t Ε ( e ) = σ (2.2) Ε ( e ) = σ (2.3) t e jt j 14 Recent Developments n Quanttatve Comparatve Methodology
15 e t = ρ e 1 + ν (2.4) t t In the other words, ths approach deals wth errors complcatons by specfyng respectvely a model for heteroschedastcty (equaton 2.2), a model for contemporaneous correlaton (equaton 2.3), and a model for seral correlaton so called AR(1) (I.e., frstorder autoregressve model), where ρ s a coeffcents of frstorder autoregressveness. In ths model we allow the value of the parameter another(equaton 2.4). ρ to vary from one crosssecton unt to 2 We now need to fnd consstent estmators of ρ and σ (.e., elements of the varancecovarance matrx of the errors). Accordng to ths am, Parks Kmenta method conssts of two sequental FGLS transformatons. Frst, t elmnates seral correlaton of the errors then t elmnates contemporaneous correlaton of the errors. 5 Ths s done by ntally estmatng equaton 2 by OLS. The resduals from ths estmaton are used to estmate the untspecfc seral correcton of the errors, whch are then used to transform the model nto one wth serally ndependent errors. Resduals from ths estmaton are then used to estmate the contemporaneous correlaton of the errors, and the data s once agan transformed to allow for the OLS estmaton wth now errors wthout any complcatons. 5 As Beck and Katz (1995, 637) note, accordng to ParksKmenta method the correcton for the contemporaneous correlaton of the errors automatcally corrects for any panel heteroschedastcty. Consequently we need only consder correctons for contemporaneous correlaton and seral correlaton of errors. Recent Developments n Quanttatve Comparatve Methodology 15
16 2 Havng obtaned consstent estmators of ρ and σ, we have completed the task of dervng consstent estmators of elements of the Ω. Hence, by substtutng Ω ) for Ω, we can obtan desred estmates of coeffcents and of ther standard errors (Kmenta 1986, 620). From ths pont Beck and Katz revew the ParksKmenta method. They (1995, 694) clam that, whle GLS are optmal propretes for TSCS data, the really appled FGLS does not do the same. Ths s because, although FGLS uses an estmate of the error process, the FGLS formula for standard errors assumes that the varancecovarance matrx of the errors s known, not estmated. Ths s a problem for TSCS models because the error process has a large number of parameters. Ths oversght causes estmates of standard errors of the estmated coeffcents to understate ther true varablty. In partcular, Beck and Katz show that the overconfdence n the standard errors makes the ParksKmenta method unusable unless where there are more tme ponts than there are crosssecton unts. In other words, t he problem of the ParksKmenta method s most evdent for the types of TSCS data typcally analyzed by poltcal scentsts and socologsts. Here, Beck and Katz propose to use a less complex method. Ths because t s well known that even though OLS estmates of TSCS model parameters may not be optmal, they often perform well n practcal research stuatons. If the errors meet one of more of the TSCS error assumptons, the OLS estmates of β wll be consstent but neffcent. Moreover, t s well known that the OLS estmates of the standard errors may be hghly naccurate n such stuatons. Consequently, Beck and Katz propose to retan OLS parameter estmators but replace OLS standard errors wth panelcorrected standard errors (PCSEs) that 16 Recent Developments n Quanttatve Comparatve Methodology
17 take nto account the contemporaneous correlaton of the errors and perforce heteroschedastcty. 6 However, any seral correlaton of the errors must be elmnated before PCSEs are calculated. Seral correlaton may be modeled by ncludng a lagged dependent varable n the set of ndependent varables or corrected by estmatng a model for autoregressveness as proposed by Parks Kmenta method. Beck et al. (1993, 946) revew ths latter soluton. Ths s because t s hart to see why the parameters of the equaton 2 should be constant across crosssecton unts, whle the nusance seral correlaton parameters should vary from unt to unt. In a recent paper, Beck and Katz (1996) argue that seeng the seral correlaton as a nusance that obscures the true relatonshp and transformng the data to remove seral correlaton, many approaches to pooled TSCS data analyss can be msleadng. In other words, they argue that overtme persstence n the data consttutes substantve nformaton that should be ncorporated n the model. Therefore, they argue that t s best model dynamcs va a lagged dependent varable rather than va seral correlaton errors. Incorporatng a lagged value of the dependent varable on the rght hand sde of the equaton yelds an explct estmate of the extent of stckness or persstence n the dependent varable. Ths allows us to stay closer to the orgnal data than transformed data would. Consequently, we can develop the equaton 1 n the followng form: 6 Beck and Katz (1995, 634) argue that snce s not possble to provde analytcal formula for the degree of overconfdence ntroduced by the Parks Kmenta method, they provde evdences from Monte Carlo experments usng smulated data. At the same tme, by usng Monte Carlo analyss they show that OLS wth PCSEs allow for accurate estmaton of varablty n the presence n the presence of the TSCS errors structures Recent Developments n Quanttatve Comparatve Methodology 17
18 k 1 + β 2 y t 1+ β k k = 3 y t = β xkt + et (3) Where y t 1 stands for the frst lag of the dependent varable and β 2 stands for ts slope coeffcent. Once the dynamcs are accounted for, TSCS analysts can estmate model parameters by OLS and ther standard errors by PCSEs n order to take nto account contemporaneous correlaton of the errors and heteroschedastcty. The correct formula for the samplng varablty of OLS estmates s gven by the roots of the dagonal term of the followng expresson (Beck and Katz 1995, 638): ' 1 ' ' 1 Cov( β v ) = ( x x) ( x Ωx)( x x) (3.1) The mddle term of ths equaton contans the correcton for the panel data. Under the condtons that the resduals are contemporaneously correlated and heteroschedastc, the matrx of covarance of the errors Ω s an NT NT block dagonal matrx wth an N N matrx of the contemporaneous covarance, Θ along the dagonal. Thus, to estmate equaton 3.1 we need a consstent estmate of Θ. Snce the OLS estmates of the equaton 1 are consstent, we can use OLS resduals from that estmaton to provde a consstent estmate of Θ. The dea of usng OLS wth PCSEs s fne and smple and has been known to have numerous applcatons n comparatve poltcal economy. However, as Maddala (1997, 3) argues, Beck and Katz s prescrptons are not, strctly 18 Recent Developments n Quanttatve Comparatve Methodology
19 speakng, correct. They suggest OLS estmaton wth panel corrected covarance matrx estmaton, as suggested n ther earler paper (Beck and Katz 1995) for a model wth no lagged dependent varables. Wth lagged dependent varables, t s well known that OLS estmators are nconsstent n the presence of seral correlaton n errors. Thus the problem s not merely gettng the correct standard errors but also to get consstent estmates of the parameters. In other words, the soluton offered by Beck and Katz can be categorzed n the what not to do f there are lagged dependent varables. Accordng to Maddala ther crtcsm of the ParksKmenta method s vald but not ther suggested soluton. Therefore, although the BeckKatz approach has been heavly appled n comparatve research, the estmaton debate could not be conclusve yet. However, snce BeckKatz argument addresses the problem of standard error nflaton and, hence, to avod callng somethng sgnfcant when t mght not be, let me now dscuss the nference ssue n comparatve research. Tests of statstcal sgnfcance are generally used n regresson analyss to evaluate the relablty of estmaton results. These tests calculate the probablty that a random sample n whch the regresson coeffcents are as estmated could be drawn from a parent populaton n whch the regresson coeffcent was zero. However, for TSCS data sets,used n comparatve research, the countres and years under nvestgaton are not a representatve sample of a larger populaton of countres and years. They are the populaton. Ths means that regresson estmate of the populaton regresson s a coeffcent of tself. Therefore, once we have carry out the regresson therefore we know whether the populaton parameter s zero (or not) wthout the need for recourse Recent Developments n Quanttatve Comparatve Methodology 19
20 to probablty theory (Comptson 1997, 7412). Nevertheless, as Western and Jackman (1994, 4123) suggest, we can adopt a Bayesan approach of statstcal nference rather than the conventonal statstcal nference to address ths problem. In fact, the comparatve researchers dscomfort wth frequentstc nference s well founded because s not applcable to a no stochastc settng. It s smply rrelevant for ths problem to thnk of observatons as drawn from a random process when further realzatons are mpossble n practce, and lack meanng even as abstract propostons. In contrast, the Bayesan model of the statstcal nference s a vald soluton to ths problem of the comparatve research. Ths s because the probablty s conceved subjectvely as characterzng a researcher s uncertanty about the parameters of statstcal model rather than a fact characterzng an object n the external world. Consequently, for the Bayesan approach t s not relevant that data are not generated by a repeatable mechansm such as con flp. 20 Recent Developments n Quanttatve Comparatve Methodology
21 3. Tme and Space Effects As we argued above, error complcatons can be also caused by model msspecfcatons. If we assume that the level of the dependent varable s homogeneous across tme perods and unts, we rsk that error contans both temporal and crosssecton components reflectng, respectvely, tme effects and crosssecton effects. In partcular, f dfferent tme perods and crosssecton are consstently hgher or lower on the dependent varable, the common ntercept β 1 estmated n OLS regresson wll be a average of all tme perod and unts that may not be representatve for any one of the sngle groups of observatons. To deal wth ths problem, we can use ether the covarance model or the error component model. Both these models use a varyng ntercept term n order to capture the dfferences n behavor over tme and space (Judge et al. 1985, 519). Consequently, for both models, we can wrte the followng equaton: k t = 1 + µ + λt ) + k =2 y ( β β x + e (4) k kt t Wth ntercept β + 1 t = β 1 + µ λt. Where 1 β s the mean ntercept, µ represents the unt effects and λ t represents tme effects. However, f we are nterested n stable dfference across crosssecton unts only, we use the µ term and drop the λ t from the equaton. Alternatvely, f we are nterested n change over tme only, we use λ t and drop the µ term from the equaton. Recent Developments n Quanttatve Comparatve Methodology 21
22 If the term µ and λ t are fxed, the equaton 4 s a covarance model (or a dummy varable model). Conversely, when they are random, t s an error component model. In other words, n the case of covarance model, the specfc characterstc of a crosssecton unts and of a tme perod are parameters; but, usng error component model the specfc characterstc of a crosssecton unts and of a tme perod are normally dstrbuted random varables. Thus, n the statstcal lterature, the error component model s kwon as a random effect model, and the covarance model s referred to as a fxed effect model. The reasonng underlyng the covarance model s that n specfyng the regresson model we have faled to nclude relevant explanatory varables that do not change over tme and/or others that do not change across crosssecton unts, and hence the ncluson of dummy varables s a coverup of our gnorance (Kmenta 1986, 633). Conversely, the reasonng underlyng the error component model s that the relevant explanatory varables that we have omtted random varables and, thus, µ and/or λ t are drawn from a normal dstrbuton. Regardng the covarance model as one wth a varyng ntercept appears reasonable because we address the unt and/or the perod effects through the ad hoc addton of dummy varables for crosssecton unts and/or tme perods. On the other hand, regardng error component model as one wth a varyng ntercept could appear arbtrary (Judge et al 1985, 522). In fact, t could also be vewed as one where all coeffcents are constant and the regresson dsturbances are composed by three ndependent models (one component assocated wth the tme, one component assocated wth the space and the thrd assocated wth both dmensons) (Kmenta 1986, 633). Consequently, for the 22 Recent Developments n Quanttatve Comparatve Methodology
23 error component model we can reparameterze the equaton 4 n the followng form: k 1 + β k k =2 y t = β xkt + e t t (4.1) Where et = µ + λt + ω t and µ are random over crosssecton, λ t are random over tme and ω t are random over space and tme. The three components, µ, λ t, and ω t are normally dstrbuted and each has propertes lke those assumed for OLS regresson. Each component also has a constant error varance such that the varance for the summary error e t s constant, or homoschedastc. Thus, ths model cannot deal wth the heteroschedatc error complcaton (Hcks 1994, 177). Moreover, each component s also free of autocorrelaton. However, despte to ths apparent neglect of problem of autoregressveness, a coeffcent of correlaton between of a gven crosssecton unt at two dfferent pont of tme (between e t and e s ) s mpled by the formulaton of ths model At the same tme, the error component model address the contemporaneous correlaton of the error by ncludng a coeffcent of correlaton between the dsturbance of two dfferent crosssectonal unts at a gven pont of tme (between e t and e s ) (Kmenta 1986, ). Therefore, usng ths knd of unrestrctve assumpton concernng the dsturbance, the most approprate estmaton procedure for the error component model s the GLS (and most specfcally FGLS) method recommended by ParksKmenta approach. In contrast, the alternatve method prescrbng the Recent Developments n Quanttatve Comparatve Methodology 23
24 OLS wth PCSEs s not especally approprate for the error component model. In fact, gven that the error component model s heavly used n crosssectonal domnant data set, Beck and Katz (1995, 645) do not consder ths model. Ther proposal s lmted to temporally domnant models. Alternatvely, for the covarance model, we can use ether BeckKatz approach or ParksKmenta method. In ths case the dsturbance e t s supposed to satsfy the assumpton of the classcal lnear regresson model. Moreover, as Beck and Katz (1995, 645) note, ths model presents no specal problems, especally when t s utlzed n a temporally domnant model and t s allowed ntercepts to vary by unt only. Ths s because the number of untspecfc dummy varables requred s not large and, thus, the fxed effects do not use an absurd number of degree of freedom. However, we could allow e t to be autoregressve and heteroschedastc, and then use the GLS (FGLS) estmaton procedure for a covarance model (Kmenta 1986, 630). In other words, the ParksKmenta method can be made to address the unt and perod effects through the ad hoc addton of dummy varables for crosssecton unts, tme perods, or both (Hcks 1994, 175). Fnally, let me brefly dscuss the problem of the choce between these models. Snce assumng unt and/or tme perod effects to be fxed or random s not obvous, the estmaton procedure could not be chosen accordngly (Judge et al.1985, 527). For example, consder a research where the dependent varable, n addtonal to explanatory varables, s affected by a varable whch vares across unts yet remans constant over tme (as usually happens n poltcal economy research). Here, the nference concernng coeffcents of relevant explanatory varables could be uncondtonal wth the respect to other 24 Recent Developments n Quanttatve Comparatve Methodology
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