RECENT DEVELOPMENTS IN QUANTITATIVE COMPARATIVE METHODOLOGY:

Transcription

1 Federco Podestà RECENT DEVELOPMENTS IN QUANTITATIVE COMPARATIVE METHODOLOGY: THE CASE OF POOLED TIME SERIES CROSS-SECTION ANALYSIS DSS PAPERS SOC 3-02

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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 cross-secton 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 cross-sectons 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 cross-sectonal 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 cross-secton unts are more numerous than temporal unts (N>T), the pool s often conceptualzed as a cross-sectonal 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 cross-sectonal 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 varance-covarance 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 cross-sectonal analyss). Several reasons support ths. The frst reason concerns the small N problem suffered by both tme seres and cross-sectonal 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 country-year (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 cross-sectonal 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 cross-secton 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 cross-secton 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 soco-economc 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 macro-economc 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 cross-sectonally 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 1986,12-3). 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 cross-sectonal components reflectng cross-sectonal 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 auto-correlated, we rsk producng a regresson wth observed heteroschestastc and auto-correlated errors. Ths s because heteroschedastcy and auto-correlaton 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 cross-sectonal unts, and the apparent level of heteroschedastcty and auto-correlaton 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 constant-coeffcents 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 Parks-Kmenta 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 Parks-Kmenta method has been the most utlzed approach for TSCS analyss n comparatve poltcal economy untl the md-nnetes (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 Parks-Kmenta 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 Parks-Kmenta 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 Parks-Kmenta 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 varance-covarance 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 varance-covarance 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., frst-order autoregressve model), where ρ s a coeffcents of frst-order autoregressveness. In ths model we allow the value of the parameter another(equaton 2.4). ρ to vary from one cross-secton unt to 2 We now need to fnd consstent estmators of ρ and σ (.e., elements of the varance-covarance 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 unt-specfc 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 Parks-Kmenta 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 Parks-Kmenta 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 varance-covarance 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 Parks-Kmenta method unusable unless where there are more tme ponts than there are cross-secton unts. In other words, t he problem of the Parks-Kmenta 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 panel-corrected 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 cross-secton 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 over-tme 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 Parks-Kmenta method s vald but not ther suggested soluton. Therefore, although the Beck-Katz approach has been heavly appled n comparatve research, the estmaton debate could not be conclusve yet. However, snce Beck-Katz 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, 741-2). Nevertheless, as Western and Jackman (1994, 412-3) 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 cross-secton components reflectng, respectvely, tme effects and cross-secton 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 cross-secton 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 cross-secton unts and of a tme perod are parameters; but, usng error component model the specfc characterstc of a cross-secton 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 cross-secton unts, and hence the ncluson of dummy varables s a cover-up 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 cross-secton 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 cross-secton, 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 cross-secton 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 cross-sectonal 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 Parks-Kmenta 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 cross-sectonal 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 Beck-Katz approach or Parks-Kmenta 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 unt-specfc 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 Parks-Kmenta method can be made to address the unt and perod effects through the ad hoc addton of dummy varables for cross-secton 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

25 varables, or t could be condtonal on the other varables. The advantage of usng the error component model s that we save a number of degree of freedom and, then, obtan more effcent estmates of the regresson parameters. The dsadvantage of usng the error component model s that f the crosssecton characterstc s correlated wth ncluded explanatory varables, the estmated regresson coeffcents are based and nconsstent. The advantage of the covarance model s that t protects us aganst a specfcaton error caused by such a correlaton, but ts dsadvantage s a loss of effcency due to the ncreased number of parameters to be estmated. Therefore, the crucal consderaton s the possblty of a correlaton between cross-sectonal and/or tme perod characterstcs and ncluded explanatory varables (Kmenta 1986, 634). If there s doubt about the correlaton between the cross-sectonal characterstc and the ncluded explanatory varables, we may carry out a test of the null hypothess that not such correlaton exsts aganst the alternatve hypothess that there s a correlaton. For ths purpose we can use the Hausman s test. Under the null hypothess that Ε( x µ ) = 0 the GLS estmator of β of the t random effect model should not be very dfferent from the least squares estmator of β of the fxed effects model. Provded no other classcal assumpton s volated, a statstcally sgnfcant dfference between these two estmators ndcates that Ε x µ ) s dfferent from zero (Kmenta 1986, 635). ( t Ths test s formally a test of equalty of the coeffcents estmated by the fxed and the random effect estmator. If the coeffcents dffer sgnfcantly, ether Recent Developments n Quanttatve Comparatve Methodology 25

26 the model s msspecfed or the assumpton that the random effect µ are correlated wth the regressor x t s ncorrect. Nevertheless, snce for several poltcal economy models the fxed effects cause serous econometrc problems, the ssue created becomes fxed effects vs. no fxed effects, and not fxed effects vs. random effects. Ths s so because these cases have many ndependent varables that change slowly over tme, and then the fxed effects are hghly collnear wth some of them. Consequently, n many poltcal economy researches, the analysts tend to not control for country fxed effects (Beck 2000, 5). Nevertheless, Garrett and Mtchell (1999, 19) consder ths a mstake. Ths s because f a regressor vares only lttle over tme, but greatly across countres, and f the ncluson of country dummy has a substantal effect on the drecton, magntude, or statstcal sgnfcance of the varable, the approprate response s not to exclude the country dummes. Rather, the analyst should conclude that the relevant varable s part the underlyng hstorcal fabrc of a country that affects the dependent varable and that s not captured by any of the tme and country-varyng regressors. When these fxed effects are taken nto account, the apparent effect of year-to-year fluctuatons n the varable could well be very dfferent than when country dummes are not ncluded. 26 Recent Developments n Quanttatve Comparatve Methodology

27 4. Poolng Dlemma and Causal Heterogenety These models do not address the problem 5. They do not consder that the error tends to be nonrandom across spatal and/or temporal unts because parameters (lke the underlyng processes that they reflect) are heterogeneous across subsets of unts. In fact, for the fxed effect model, the random effect model and the smpler pooled model, the slope coeffcents are assumed to be equal over tme and space. The homogenety of slope coeffcents s often an unreasonable assumpton (Maddala et al. 1997, 90). The soluton frequently suggested s to apply a prelmnary test of sgnfcance to test the equalty of the coeffcents and decde not to pool f ths hypothess s rejected and to pool f ths hypothess s not rejected. Consequently, the queston becomes: to pool or not to pool? The queston s whether to estmate the models separately for dfferent cross-secton unts or for dfferent tme seres, or to estmate the model by poolng the entre data set and, thus, by estmatng a model wth coeffcents that are constant across unts and tme perods (Maddala 1991, 302). Kttel s proposal concerns these two extreme cases of complete homogenety and complete heterogenety. He (1999, ) argues that the constant coeffcent model n TSCS analyss (wthout any ncluson of fxed or random effects) represents the combned average partal effect for both tme and space. It does not yeld nformaton about the relatve contrbuton of two dmensons to ts value. In other words, wthout addtonal analyss the queston of the whether cross-country dfferences or cross-specfc developments account for the varaton goes unanswered. Consequently, to nspect the Recent Developments n Quanttatve Comparatve Methodology 27

28 development of the relaton over tme, we estmate repeated cross-secton regresson analyss. In partcular, by estmatng yearly cross-sectonal models, we can evaluate whether the relatonshp between the dependent varable and ndependent varables changes over the perod nvestgated or whether t remans constant as the constant coeffcent model prescrbes. A second method of valdatng pooled coeffcents compares the tme seres of the countres analyzed. Snce n the comparatve poltcal economy research several explanatory varables tend to vary across countres, but are constant or change wthn many countres, we cannot assess the relatonshp of the tme seres dmensons. However, ths ndcates as a further restrcton to the senstve use of the constant coeffcent approach to the poolng. Beng based on mostly constant data n the tme seres dmenson, the pooled coeffcents of these varables rely almost completely on the cross-secton dmenson. Therefore, accordng to Kttel (1999, 232-3), these problems do not mean that poolng s not worth the effort. They smply pont to the provso that the data set should be analyzed wth care and that reportng poolng coeffcents n the poolng constant coeffcents model wthout further evaluaton of the relatve contrbutons of the space and tme dmensons to the coeffcents can lead to unwarranted conclusons. However, between these two extreme cases of the complete homogenety of the constant coeffcent model and of the complete heterogenety of the separate estmaton of cross-secton or tme seres coeffcents, there are more approprate ntermedate solutons. The problem wth these two estmaton methods of ether poolng the data or obtanng separate estmates for each cross-secton or tme seres s that both are based on extreme assumptons. The 28 Recent Developments n Quanttatve Comparatve Methodology

29 parameters are assumed to be all the same or all dfferent n the each crosssecton and/or tme seres. The truth probably les somewhere n between.. The parameters are not exactly the same, but there s some smlarty between them (Maddala et al. 1997, 91). One way of allowng for the smlarty s to assume that the parameters all vary over tme or/and unts. However, n ths paper I wll concentrate only on the cross-secton heterogenety. Ths s because, although they are not sgnfcantly appled n the comparatve research yet, they represent crtcal methodologcal ssues regardng the pooled analyss. From a substantve perspectve, a key dea of comparatve research s that causal process vares across countres. The fundamental problem n the comparatve research contextual explanaton where the dfferences n the causal processes wthn countres are related to characterstcs that vares across countres. Ths contextual dea s expectably relevant to the comparatve poltcal economy. In ths area cross-natonal varaton n economc relatonshps orgnated wth endurng nsttutonal dfferences (Western 1998, 1255). Therefore, wth causal heterogenety dependng on cross-natonal varaton n the nsttutons, model wth slope coeffcents that vary over cross-sectonal unts provde a closer ft between nsttutonal theory and model specfcaton. Consequently, we assume that response of the dependent varable y t to an explanatory varable x kt s dfferent for dfferent unts, but for a gven cross-secton t s constant over tme. The equaton for ths knd of models may be wrtten as; k y t = β k xkt + et (5) k =1 Recent Developments n Quanttatve Comparatve Methodology 29

30 k = ( β k = 1 + α k ) x + e kt t In contrast to prevous equatons I no longer treat the constant term dfferently from the other explanatory varables. β= β... β ) can be vewed as ( 1 k the common-mean coeffcent vector and α = α... α ) as the ndvdual ( 1 k devaton from the common mean. When β k are treated as fxed and dfferent constants, equaton 5 can be vewed as the seemngly unrelated regresson model. Conversely, when β k are treated as random parameters, equaton 5 s equvalent to the random coeffcent model (Judge et al. 1985, 538-9; Hsao 1986, 130-1). The seemngly unrelated regresson model treats each cross-secton and the tme seres wthn that cross-secton as a separate equaton that s unrelated to any other cross-secton (and tme seres wthn the cross-secton) n the pooled data set. Most specfcally, ths model s nterpretable as a seres of a naton specfc regresson analyss that utlzes contemporaneous cross-equaton error correlatons among the error of a system of equaton to mprove the effcency of the equaton s estmates (Sayrs 1989, 39; Hcks 1994, 181). The random coeffcent model s due to Swamy (1970). Ths model assumes that each β are drawn from a common normal dstrbuton In other words, β = β + α are treated as random, wth a common mean. The model set up s: ^ β ~ ( β; I ) (5.1) 30 Recent Developments n Quanttatve Comparatve Methodology

31 Ε( α α ) = Ε[( β β )( β β )] = 0 f j (5.2) j j Ε( x α ) = Ε[ x ( β β )] = 0 (5.3) t t t e jt j Ε ( e ) = σ f I=j (5.4) Ε( e ) = 0 f j (5.5) e t jt Ths set up assumes that the β are drawn from a common normal dstrbuton (wth mean β and varance covarance I ^ ) where each of the drawns are ndependent from each other as well as of the x t s. Ths set up allows for the components of β to be correlated and also allows panel heteroschedastcty. Moreover, the set up can nclude dynamcs modeled wth a lagged dependent varable, but wthout seral correlaton of error (Hsao 1985, 131; Beck and Katz 1996b, 3). In ths model, we wll determne the mean β from whch we can estmate cross-sectonal ndvdual β. Wthout gong nto detals, to estmate the mean β and ts standard errors,we can use FGLS. Conversely, the estmates of β are a weghted combnaton of the OLS estmates of β and the common estmate of β. The weghts of two these estmates are a functon of the estmated varablty of the β s. But, whch of these models s the more approprate for the comparatve poltcal economy? Can students of ths dscplne use the random coeffcents model? Hsao (1986, 136) concludes that the queston of whether β should be assumed fxed and dfferent or random and dfferent depends on whether we are makng nferences condtonal on the ndvdual characterstcs or makng Recent Developments n Quanttatve Comparatve Methodology 31

32 uncondtonal nferences on the populaton characterstcs. In the former cases, fxed-coeffcents model should be used. In the latter cases, the random coeffcents model should be used. Such a concluson ndcates that the random coeffcents model s approprate for panel studes n survey research rather than for comparatve TSCS analyss. Ths s because n the panel data the observed people are of no nterest per se, wth all of nferences of nterest beng to the underlyng populaton that was sampled. TSCS data show the opposte stuaton. Here, all nferences of nterest are condtonal on the observed unts (Beck 2000, 3) However, one way to avod ths problem s to use Bayesan approach. As Beck and Katz (1996, 5) suggest, the advantage of Bayesan lnear herarchcal model s that the randomness resdes n the parameters, and not the unts. Hence, the dstncton between fxed and sampled unts s no longer relevant. Ths approach yelds dentcal results to the random coeffcents model. Here, we have a pror on the varablty of the β s, whch looks lke equaton 5.1. Ths assumes that β are exchangeable, snce a pror we cannot dstngush between the unts other than through the covarates. The Bayesan approach lets the data choose the pror usng the estmated structure of I ^ as the pror of parameter varablty. However, from the Bayesan perspectve lettng the data choose the pror appears a bt odd. Bayesans have pror belefs about the dversty of β. Thus, these prors can be combned wth the observed data (va lkelhood functon) to produce a new, posteror, set of belefs about the β. ^ The pror whch s represented by I, s based on the analyst s belef about the 32 Recent Developments n Quanttatve Comparatve Methodology

33 world rather than a parameter to be estmated. The pror I ^ than can be combned wth the OLS estmates of β. However, the Bayesan herarchcal model can be made more useful for comparatve research by allowng the β to be a functon of the other unt varables, whch allow modelng dfferental effects as a functon of dfferent nsttutons. In fact, as Western (1998, 1241) notes, n the Beck and Katz s (1996b) model nsttutonal effect and the ssue of contextual explanaton are omtted. Consequently, one can allow the β to be functon of other unt varable, z, whch allows for modelng effects as a functon of dfferng nsttutons. Most specfcally, to allow the chance for the contextual varaton, we can rewrte the model for a sngle country as follows: y β x + e (6) t = 1 + β 2 y t 1+ β 3 kt kt Hence, varaton n the tme seres coeffcents s wrtten as a functon of tme nvarant nsttutonal condton, β = + (6.1) 1 φ11 + φ12 z δ 1 β = + (6.2) 2 φ 21 + φ 22 z δ 2 β = + (6.2) 3 φ31 + φ32 z δ 3 The subscrpts on the φ coeffcents ndcate that the nsttutonal effects are constant across countres. Ths herarchcal model can be wrtten as a sngle Recent Developments n Quanttatve Comparatve Methodology 33

34 equaton wth nteracton terms by substtutng equaton 6.1, 6.2, and 6.3 n equaton 6 (Western 1998, 1237): y = ) x + e t ( φ11 + φ12 z + δ 1) + ( φ 21 + φ 22 z + δ 2 ) yt 1 + ( φ31 + φ32 z + δ 3 t t (6.1) = φ 11 + φ12z + φ21yt 1 + φ31xt + φ22z yt 1 + φ32z xt + xt + e t ( δ1 + δ 2 yt 1 + δ 3 ) Ths model s dentcal to the usual sngle-equaton regresson wth nteracton term except that the error term has a more complcated structure. Here, t ncludes two sources of uncertanty, e and δ, and wth random coeffcents on the lnear term only. Snce the tme seres coeffcents have a stochastc component, δ, we can consder these stochastc components as drawn from a sngle populaton dstrbuton shared by all the countres under study. Therefore, the random coeffcents model and, n partcular, Bayesan herarchcal model ndcate that one of the most mportant ssues of the TSCS methodologcal research s to carefully the model specfcaton. These models can represent a soluton to the trade-off between the nsttutonal approach of the poltcal economy and model specfcaton. 34 Recent Developments n Quanttatve Comparatve Methodology

35 5. Pooled TSCS analyss n STATA software Fnally, ths secton presents some mplementatons and commands n STATA software to analyze TSCS data. Several econometrc packages for pooled models are now wdely avalable (SAS and SHAZAM). However, I wll consder STATA only, assumng that the reader s famlar wth the bascs of ths statstcal software. The xt seres of STATA commands provde tools for analyzng crosssectonal tme seres data sets. Cross-sectonal tme seres (longtudnal) data sets are of form x t, where x t s a vector of observatons for unt and tme t. The partcular commands as such xtreg, xtgls and xtpcse allow us to estmate the majorty of pooled models dscussed n ths paper. Snce TSCS data sets are characterzed by both unt and tme t dmensons, correspondng STATA optons are usually requred to estmate pooled models usng these commands. The opton ( ) sets the name of the varable correspondng to the unt. The opton t( ) sets the name of the varable correspondng to the tme ndex t (STATA 1999, 317). Gven that n the comparatve analyss these varables are often represented by country and year, n the next examples I wll use these varable names. Moreover, for expostory purpose, let me suppose that we have data of the followng form: y (dependent varable), and x1, x2, x3 (respectvely, 1 st predctor, 2 nd predctor and 3 rd predctor). The smplest model estmable va OLS procedure (related to equaton 1) can be obtaned by usng the STATA command regress by typng:. regress y x1 x2 x3 Recent Developments n Quanttatve Comparatve Methodology 35

36 Hover, as noted, TSCS desgns often volate the standard OLS assumptons, frst we need to consder STATA mplementatons concernng Parks-Kmenta method and Beck-Katz approach. Regardng the Parks-Kmenta approach, the xtgls command estmates models usng FGLS procedure. Ths command allows estmaton n presence of AR(1) autocorrelaton wthn unts, cross-sectonal correlaton and/or heteroschedastcty across unts. In other words, the model related to the equaton 2 can be estmated by xtgls STATA command and the assumptons concernng the panel heteroschedastcty, contemporaneous correlaton and seral correlaton can be obtaned by specfyng partcular optons (STATA 1999, ). The heteroschedastc models obtaned by specfyng by:. xtgls y x1 x2 x3, (country) panels(heteroschedastc) However, we may wsh to assume that the error terms are correlated n addtonal to havng dfferent varances. Hence, we must specfy:. xtgls y x1 x2 x3, (country) t(year) panels(correlated) Fnally, xtgls allows dfferent optons so that you may assume seral correlaton wthn unts. If we assume a seral correlaton where the correlaton parameter s common for all unts, we must specfy the followng command:. xtgls y x1 x2 x3, (country) t(year) corr(ar1) 36 Recent Developments n Quanttatve Comparatve Methodology

37 Conversely, f we assume that each group has errors that follow a dfferent autoregressve process, we must use:. xtgls y x1 x2 x3, (country) t(year) corr(psar1) Obvously, these optons can be combned accordng to our assumpton about the error term. Regardng Beck-Katz proposal, xtpcse command produces panel corrected standard error (PCSE) estmates for TSCS lnear models. Ths command produces OLS estmates of the parameters when no autocorrelaton s specfed. 7 In computng the standard errors and the varance-covarance estmates, the dsturbances are, by default, assumed to be heteroschedastc and contemporaneously correlated across unts. However, n order to compute PCSEs, STATA must be able to dentfy the unts to each whch observaton belongs and also be able to match the tme perods across the unts. Thus, we tell STATA how to do ths matchng by specfyng the tme and the unt varables usng the followng command:. tsset country year, yearly Hence, to estmate a model related to the equaton 3, we must to type: 7 xtpcse command s mplemented n the updated of STATA 6.0. Conversely, STATA 6.0. Wthout any update allows producng OLS parameters wth Recent Developments n Quanttatve Comparatve Methodology 37

38 . xtpcse y ly x1 x2 x3 Where ly stands for a lagged dependent varable of one year. It can be obtaned by typng:. generate ly = y[_n-1] Let me now consder fxed and random effect models. For both these models we can refer to the equaton 4 when ncludes the unt effect term and drops the tme effect term. In fact, STATA consders the case n whch fxed and random effect model nclude the unt effects only. The command to estmate ths models s xtreg (STATA 1999, ). In partcular, to obtan an OLS fxed effect model, we must to type:. xtreg y x1 x2 x3, fe Conversely, to obtan a GLS random effect model, we must type:. xtreg y x1 x2 x3, re After xtreg, re estmaton we can obtan the Hausman test by typng:. xthaus PCSES by specfyng xtgls wth the ols or pcse opton (STATA 1999, 364). 38 Recent Developments n Quanttatve Comparatve Methodology

39 However, the dummy varables are not ncluded n the outputs correspondng to these STATA commands. But, snce fxed effect regresson s supposed to produce the same coeffcent estmates as standard error as ordnary regresson when dummy varables are ncluded for each unts, to obtan an output ncludng dummy varables we must type:. x: regress y x1 x2 x3.country Where x command allows us to create dummy varables. Consequently, by usng ths command wth ether xtgls or xtpcse respectvely Parks- Kmenta method or Beck-Katz approach can be made to address unt effects. Let me now consder the models wth slope coeffcents that vary across unts, represented by equaton 5. The STATA command to estmate a seemngly unrelated regresson model s sureg. However, n order to estmate a smultaneous equaton model usng sureg, we should frst reshape our data (STATA 1999, 415). In other words, we need to convert our data from long to wde form by typng:. reshape y x1 x2 x3, (year) j(country) Hence, we can estmate a seemngly unrelated regresson model by typng:. sureg (y1 x11 x21 x31) (y2 x12 x22 x32) (y3 x13 x23 x33) Recent Developments n Quanttatve Comparatve Methodology 39

40 Where the numbers 1, 2, 3 assocated to the varables y x1 x2 x3 represent the dfferent countres of an example wth three cases only. Conversely, to estmate a random coeffcent model, we do not need to reshape our cross-sectonal tme seres data set. To obtan such a model t s enough typng the followng command:.xtrchh y x1 x2 x3, (country) t(year) STATA uses FGLS procedure to estmate random coeffcents models, as suggested by Swamy (1971), assumng that all coeffcents are drawn from a common multvarate normal dstrbuton. 8 The requrement of the random coeffcents model that that all varables) wth random coeffcents) vary wthn unts, may cause dffculty for comparatve poltcs applcatons. where t s frequently the case that some mportant varables are tme nvarant natonal characterstcs (Beck and Katz 1996b, 9). Nevertheless, STATA does not allow assumng that the coeffcents of some varables are fxed. 9 Fnally, regardng herarchcal models, software s now avalable (Western 1998, 1244). An extensve revew of fve packages for herarchcal modelng s reported by Kreft et al. ( 1994). In addtonal to specalzed software, routne can also be found n the general statstcal software SAS and S-PLUS (see Pnhero and Douglas 2000). 8 Such a procedure can cause some problems to estmate the varance matrx of the error. For full detals see Beck (2000, 19-20). 9 Conversely, LIMPDEP 7.0. allows the nvestgator to specfy whether varables have fxed or random coeffcents (see Greene 1995). 40 Recent Developments n Quanttatve Comparatve Methodology

41 Bblography: Alesna A., Grll V. and Mles-Ferrett G. M., 1994, The Poltcal Economy of Captal Controls, n L. Lederman and A. Razn, (edted by), Captal Moblty, Cambrdge, Cambrdge Unversty Press. Alvarez M., Garrett G., and Lange P., 1991, Government Partnershp, Labour Organzaton and Macroeconomc Performance: , n Amercan Poltcal Journal Revew, 85, pp Beck N., 2000, Issues n the Analyss of Tme-Seres Cross-Secton Data n the Year 2000, Unversty of Calforna, Manuscrpt. Beck N. and. Katz J.N, 1995, What To Do (and Not To Do) wth Tme-Seres Cross-Secton Data, n Amercan Poltcal Journal Revew, 89, pp Beck, N. and Katz J.N, 1996, Nusance vs. Substance: Specfyng and Estmatng Tme-Seres-Cross-Secton Models, Poltcal Analyss, 6, Beck, N. and Katz J.N., 1996b, Lumpers and Spltters Unted: The Random Coeffcents Model, Presented at the annual meetng of the Methodology Poltcal Group, Ann Arbor, Mchgan. Beck, N. and Katz J.N., 1998, Takng Tme Serously: Tme-Seres-Cross-Secton Analyss wth a Bnary Dependent Varable, n Amercan Journal of Poltcal Scences, 42(4), pp Beck, N, Katz J.N., Alvarez M., Garrett G., and Lange P., 1993, Government Partsanshp, Labor Organzaton, and Macroeconomc Performance: A Corrgendum, Amercan Poltcal Journal Revew, 67(4). pp Recent Developments n Quanttatve Comparatve Methodology 41

42 Compston H., 1997, Unon Power, Polcy-Makng and Unemployment n Western Europe: , n Comparatve Poltcal Studes, 30(6), pp Garrett, G., 1998, Partsan Poltcs n the Global Economy, Cambrdge, Cambrdge Unversty Press. Garrett, G. and Mtchell D., 1999, Globalzaton and the Welfare State, Yale Unversty, Manuscrpt. Hcks A., 1991, Unon, Socal Democracy, Welfare and Growth, Research n Poltcal Socology, 5, pp Hcks A., 1994, Introducton to Poolng, n T. Janosk and A. Hcks (edted by), The Comparatve Poltcal Economy of the Welfare State, Cambrdge Unversty Press Hsao C., 1986, Analyss of Panel Data, Cambrdge Unversty Press Huber E., Ragn C., and Stephens J.D., 1993, Socal Democracy, Chrstan Democracy, Consttutonal Structure, and the Welfare State, Amercan Journal of Socology, 99(3) pp Judge G.G., Grffths W.E., Hll R.C., Lutkepohl H. and Lee T-C., 1985, The Theory and Practce of Econometrcs, New York, Wley, 2nd Ed Kttel B., 1999, Sense and Senstvty n Pooled Analyss n Poltcal Data, n European Journal of Poltcal Research, 35(2), pp Kmenta, J., 1986, 42 Recent Developments n Quanttatve Comparatve Methodology

43 Elements of Econometrcs. New York: Macmllan; London: Coller Macmllan, 2nd Ed. Maddala, G.S., 1994, To Pool or Not to Pool: That s the Queston, n Maddala G.S.,(edted by), Econometrc Method and Applcatons, Edward Elgart Maddala, G.S., 1997, Recent Developments n Dynamcs Econometrc Modelng: A Personal Vewpont, Oho State Unversty, Manuscrpt. Maddala, G.S., H. L, R.P. Trost and F. Joutz (1997), 'Estmaton of Short-Run and Long-Run Elastctes of Energy Demand From Panel Data Usng Shrnkage Estmators, Journal of Busness Economcs and Statstcs, 15, pp Oatley T., 1993, How Constranng s Captal Moblty? The Partsan Hypothess n an Open Economy, n Amercan Journal of Poltcal Scences, 43(4) pp Pampel F.C. and Wllamson G.B., 1989, Age, Class, Poltcs and the Welfare State, Cambrdge, Cambrdge Unversty Press Parks R.W., 1966, Effcent Estmaton of a System of Regresson Equaton When Dsturbance are Both Serally and Contemporaneously Correlated, n Journal of the Amercan Statstcal Assocaton, 62, Pennngs P., Keman H. and Klennjenhus: J., 1999, Dong Research n Poltcal Scence: An Introducton to Comparatve Methods and Statstcs, Sage, Qunn, D. P. and Inclan, C., 1997, The Orgns of Fnancal Openness, n Amercan Journal of Poltcal Scences, 41(3), pp Recent Developments n Quanttatve Comparatve Methodology 43

44 Sayrs, L.W., 1989, Pooled Tme Seres Analyss, Sage Publcatons Schmdt M.G., 1997, Determnants of Socal Expendture n Lberal Democraces, n Acta Poltca, 32(2) pp Stmson, J.A., 1985, Regresson n Space and Tme: A Statstcal Essay, n Amercan Journal of Poltcal Scences, 29(4) pp Swamy P.A.P.V. 1970, Effcent Inference n a Random Coeffcents Regresson Model, Econometrca, 32, pp Swank D.H., 1992, Structural Power and Captal Investment n the Captalsts Democraces, n Amercan Poltcal Journal Revew, 86, pp Western, B., 1998, Causal Heterogenety n Comparatve Research: A Bayesan Herarchcal Modelng Approach, n Amercan Journal of Poltcal Scences, 42(4), pp Western B. and Jackman S., 1994, Bayesan Inference for Comparatve Research, Amercan Poltcal Journal Revew, 88(2), pp Recent Developments n Quanttatve Comparatve Methodology