Modeling Dynamics in Time-Series Cross-Section Political ...users.cla.umn.edu/~uggen/beck_arps_11.pdfPL14CH16-Beck ARI 14 April 2011 17:11 Modeling Dynamics in Time-Series–Cross-Section

PL14CH16-Beck ARI 14 April 2011 17:11

Modeling Dynamics inTime-Series–Cross-SectionPolitical Economy DataNathaniel Beck1 and Jonathan N. Katz2

1Department of Politics, New York University, New York, New York 10003;email: [email protected] of the Humanities and Social Sciences, California Institute of Technology,Pasadena, California 91125; email: [email protected]

Annu. Rev. Polit. Sci. 2011. 14:331–52

First published online as a Review in Advance onMarch 17, 2011

The Annual Review of Political Science is online atpolisci.annualreviews.org

This article’s doi:10.1146/annurev-polisci-071510-103222

Copyright c© 2011 by Annual Reviews.All rights reserved

1094-2939/11/0615-0331$20.00

Keywords

lagged dependent variables, correlated errors, error correction model,non-stationarity, model specification

Abstract

This article deals with a variety of dynamic issues in the analysis oftime-series–cross-section (TSCS) data. Although the issues raised aregeneral, we focus on applications to comparative political economy,which frequently uses TSCS data. We begin with a discussion of spec-ification and lay out the theoretical differences implied by the varioustypes of dynamic models that can be estimated. It is shown that thereis nothing pernicious in using a lagged dependent variable and that alldynamic models either implicitly or explicitly have such a variable; thedifferences between the models relate to assumptions about the speedsof adjustment of measured and unmeasured variables. When adjust-ment is quick, it is hard to differentiate between the various models;with slower speeds of adjustment, the various models make sufficientlydifferent predictions that they can be tested against each other. As thespeed of adjustment gets slower and slower, specification (and estima-tion) gets more and more tricky. We then turn to a discussion of esti-mation. It is noted that models with both a lagged dependent variableand serially correlated errors can easily be estimated; it is only ordi-nary least squares that is inconsistent in this situation. There is a briefdiscussion of lagged dependent variables combined with fixed effectsand issues related to non-stationarity. We then show how our favoredmethod of modeling dynamics combines nicely with methods for deal-ing with other TSCS issues, such as parameter heterogeneity and spatialdependence. We conclude with two examples.

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iid: independentidentically distributed

1. INTRODUCTION

Time-series–cross-section (TSCS) data areperhaps the most commonly used data in com-parative politics (broadly defined) and particu-larly in comparative political economy.1 Thereare various issues related to TSCS data; herewe focus on some important ones related tothe dynamic (time-series) properties of the data.Obviously many of these issues are similar tothose for a single time series, but the contextof comparative political economy and the rel-atively short lengths of the TSCS time peri-ods make for some interesting special issues.We assume that the reader is familiar withthe basic statistical issues of time-series data.Because various specification issues are coveredfor political scientists elsewhere (Beck 1985,1991; Keele & Kelly 2006; De Boef & Keele2008), we go fairly quickly over the basic issues,spending more time on issues relating to thedynamic modeling and interpretation of thosemodels in political economy.

1.1. Notation and Nomenclature

Our interest is in modeling the dynamics ofTSCS models. By dynamics we mean any pro-cess where some variable has an impact that isdistributed over time. Let yi,t be an observa-tion for unit i at time t where i = 1, . . . N andt = 1, . . . T . We assume that y is measured asa continuous variable, or at least can be takenas continuous. In what follows, we typically donot care if we have one or more than one inde-pendent variable or variables, so let xi,t be eitheran observation on a single independent variableor a vector of such variables; if the latter, it is

1Comparative politics refers to any comparison of politicalunits, so it encompasses almost all of international relations,which has countries or country pairs as the unit of analysis,and any study that compares units (regions, states, or coun-ties) within a single country. Our language comes from com-parative political economy, but everything applies to all othertypes of TSCS studies (both within and beyond political sci-ence) so long as the data are observed over a long enoughperiod. Adolph et al. (2005) report that by the early 2000s∼5% of all political science articles in JSTOR journals usedTSCS data.

assumed that the dynamics apply similarly to allthe components of that vector.

Where we need to differentiate dynamics,we use a second variable (or vector of variables),zi,t. Because the constant term is typically ir-relevant for what we do, we omit it from ournotation (or include it in the vector of otherindependent variables). We assume that theseindependent variables are exogenous, which isclearly both a strong and critical assumption.We postpone discussing specifications for yi,t

until the next section.TSCS data are most commonly used in com-

parative political economy, and so we refer totime periods as years and units as countries.Given the yearly frequency, we focus on modelswhere explicit lags are only of one or two years;we would not expect to see the higher-orderdynamic processes common in standard time-series analysis of monthly or quarterly data.Although we focus on low-order processes, itis trivial to extend the interpretation, tests, andestimation to higher-order dynamics.

T must be large enough so that averagingover time makes sense. In our prior work (Beck& Katz 1995, 1996), we did not examine sit-uations where T < 15. Political economy dataoften span a sample period of 30 or more yearsand so there are no problems. This article doesnot discuss standard survey panels that typicallyhave five or fewer waves, and there is no reasonto believe that the methods discussed here ap-ply to such survey panels. We make no assump-tions about N, although in comparative politicaleconomy it is seldom less than 20 (advanced in-dustrial nations); it can be quite large (>100,000for some studies in international relations usingdyadic data).

We distinguish two types of error terms:νi,t refers to an independent identically dis-tributed (iid) error process, whereas εi,t refers toa generic error process that may or may not beiid. Unless specifically stated, we restrict non-iid processes to simple first-order autoregres-sions. This simplifies notation with no loss toour argument; it is simple to extend our ar-gument to more complicated error processes.Because coefficients are only interpretable in

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the context of a specific model, we superscriptcoefficients to indicate the specification they re-fer to whenever confusion might arise. We usethe terms specification and model interchange-ably, and refer to both as equations when refer-ring to a specific equation in the text.

When relevant, we use L as the lag operator,so that

Lyi,t = yi,t−1 if t > 1 1.

and Lyi,1 is missing. The first difference opera-tor is then � = 1 − L.

1.2. Stationarity

We initially, and for most of the article, as-sume that the data are drawn from a station-ary process. A univariate process is stationaryif its various moments and cross-moments donot vary with the time period. In particular,the initial sections assume that the data aredrawn from a “covariance stationary” process,that is,

E(yi,t) = μ 2a.

Var(yi,t) = σ 2 2b.

E(yi,t yi,t−k) = σk 2c.

(and similarly for any other random variables).Stationary processes are mean reverting, and

the best long-run forecast for a stationary pro-cess is that mean. Thus, we can think of themean as the “equilibrium” of a stationary pro-cess. Alternatively, we can think of the statisticalproperties of the data as not varying simply asa function of time (so, for example, there areno trends in the data). We briefly discuss non-stationary data in Section 4.

1.3. Missing Data

To simplify notation, we assume that thedataset is rectangular, that is, each country isobserved for the same time period (which iscalled the “sample period” even though it is nota sample of anything). It is easy to extend ev-erything to a world in which some countries arenot observed for a few years at either the begin-

ning or the end of the period under study, andthe only cost of so doing would be an additionalsubscript in the notation.

Missing data in the interior of the periodunder study is not benign. At a minimum, theirabsence causes all the standard problems asso-ciated with missing data (Little & Rubin 1987).The default solution, list-wise deletion, is wellknown to be an inferior solution to the missing-data problem for cross-sectional data. But theproblem is more severe for TSCS data becausethe specification invariably includes temporallags of the data; even if the model has onlyfirst-order lags, each observation with missingdata leads to the deletion of two data points.Thus, even more than with cross-sectional data,multiple imputation techniques are de rigueurfor dynamic models that have more than atrivial amount of “missingness.” Obviously theamount of missingness will vary as we movefrom studies of advanced industrial societies tostudies of poorer nations, and so the attentionpaid to missingness can vary.

Although it is easy to say that analystsshould use Little & Rubin’s multiple imputa-tions, the standard methods for cross-sectionalimputations (hot decking or assuming thatboth missing and non-missing observations areessentially multivariate normal) are not appro-priate for TSCS data. This is because we knowa lot about TSCS data. Thus, for example, wewould expect that missing economic variablesare likely to be highly related to observed valuesin nearby countries with similar economies, orthat observations on trending time series canbe imputed with interpolated values. Honaker& King’s (2010) Amelia II allows users thiskind of flexibility. But there is no mechanisticsolution to the missing-data problem inpolitical economy. To give but one example,missingness is often related to civil wars; ifwe simply use some complicated averagingmethod to impute missing economic dataduring a civil war, our imputations are likelyto be overly optimistic. Analysts using TSCSdatasets with significant missing data can onlybe warned that they must take extreme care.

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LDV: laggeddependent variable

FDL: finitedistributed lag model

AR1: first-orderautoregressive errorprocess

SC: first-order seriallycorrelated error model

1.4. Roadmap

The next section, on the interpretation of al-ternative dynamic specification, is the heart ofthe article. There we deal only with stationarydata. The third section briefly examines com-bining dynamics with cross-sectional issues, inparticular accounting for heterogeneity acrossunits. The fourth section extends the argumentto slowly moving and non-stationary data. Twoexamples of dynamic modeling with politicaleconomy TSCS data are discussed in the fifthsection, and we offer some general conclusionsin the final section.

2. DYNAMIC MODELS:STATIONARY DATA

There are various specifications for any time-series model; for reviews considering applica-tions to political science, see Beck (1991) andDe Boef & Keele (2008). All time-series speci-fications have identical counterparts in TSCSmodels. These specifications appear in anystandard text, so we discuss general specifica-tion issues without either citation or claim oforiginality.

In our prior work (Beck & Katz 1996) weargued that a lagged dependent variable (LDV)specification is often adequate; because that hassparked some discussion (Achen 2000, Keele &Kelly 2006), we spend some time on this issue.After discussing a variety of specifications, wediscuss issues of interpretation and estimation.2

2.1. Dynamic Specifications

The generic static (non-dynamic) specificationis

yi,t = β s xi,t + νi,t . 3.

This specification is static because any changesin x or the errors are felt instantaneously and

2We often refer to Achen’s (2000) critiques of the use ofLDVs. Although it is odd to spend time critiquing a decade-old unpublished paper, this paper has been influential (morethan 300 Google Scholar citations as of this writing). Weonly deal with the portions of Achen’s paper relevant to issuesraised here.

their effect dissipates instantaneously; there areno delayed effects. (It may be that xi,t is mea-sured with a lag, so the effect could be felt witha lag, but the model is still inflexible in thatthe effect is completely and only felt at the onespecified year.)

There are several ways to add dynamics tothe static specification. The simplest is the fi-nite distributed lag (FDL) model, which as-sumes that the impact of x sets in over two (ora few) periods but then dissipates completely.This specification has

yi,t = β f 1xi,t + β f 2xi,t−1 + νi,t 4.

with the obvious generalization for higher-ordered lags. Equation 3 is nested insideEquation 4 (Equation 3 is a special case ofEquation 4), so testing between the two is sim-ple in principle (although the correlation of xand its lags makes for a number of practicalissues).

Another commonly used dynamic specifica-tion is to assume that the errors follow a first-order autoregressive (AR1) process rather thanthe iid process of Equation 3. If we assume thatthe errors follow an AR1 process, we have a se-rially correlated (SC) error model:

yi,t = βSC xi,t + νi,t + ρεi,t−1 5a.

= βSC xi,t + νi,t

1 − ρL5b.

= βSC xi,t + ρyi,t−1 − βSCρxi,t−1 + νi,t . 5c.

The formulation in Equation 5c makes the dy-namics implied by the model clearer and alsomakes it easier to compare various models.

Another alternative is the LDV model (withiid errors):

yi,t = βLDV xi,t + φyi,t−1 + νi,t 6a.

= βLDV xi,t

1 − φL+ νi,t

1 − φL. 6b.

As Equation 6b makes clear, the LDV modelsimply assumes that the effect of x decays geo-metrically (and for a vector of independent vari-ables, all decay geometrically at the same rate).Note also that the compound error term is aninfinite geometric sum (with the same decay pa-rameter as for x); this error term is equivalent to

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a first-order moving average (MA1) error pro-cess, again with its decay parameter constrainedto equal φ, the rate at which the effect of x on ydecays.

Both the SC and LDV specifications are spe-cial cases of the autoregressive distributed lag(ADL) model,

yi,t = β ADLxi,t + θ yi,t−1 + γ xi,t−1 + νi,t, 7.

where Equation 5c imposes the constraint thatγ = −β ADLθ and Equation 6a assumes thatγ = 0. The nesting of both the LDV and SCspecifications within the ADL specification al-lows for testing between the various models.

For interpretative purposes, it can be help-ful to rewrite the ADL model in error correc-tion (EC) form (Davidson et al. 1978). To dothis, subtract yi,t−1 from both sides of the ADLmodel to get a first difference in y on the left-hand side, and add and subtract β ADLxi,t−1 fromthe right-hand side to get a first difference of xin the specification. This leads to

�yi,t = βEC�xi,t −λ(yi,t−1 −κxi,t−1)+νi,t, 8.

which allows for the nice interpretation thatshort-run changes in y are a function of bothshort-run changes in x and how much x andy were out of equilibrium last year, where theequilibrium y and x are given by yi,t = κxi,t

and the speed of equilibration (per year) is λ.The coefficients of the EC model can be easilyderived from the corresponding ADL model:βEC = β ADL, λ = θ − 1 and κ = γ+β ADL

θ−1 . Forcomparison with other models the ADL modelworks better, but for direct substantive inter-pretation of the coefficients the EC model iseasier to work with (since one can directly readoff the short-run impact of a change in x as wellas various long-run impacts). Because the twoare identical, either one can be estimated. Wereturn to the EC model when we deal with non-stationary data in Section 4.

2.2. Interpretation

To see how the various specifications differ, weturn to unit and impulse response functions.Since x itself is stochastic, assume the process

MA1: first-ordermoving averageprocess

ADL: autoregressivedistributed lag model

EC: error correctionmodel

IRF: impulseresponse function

URF: unit responsefunction

has run long enough for y to be at its equilib-rium value (stationarity implies the existence ofsuch an equilibrium). We can then think of aone-time shock in x (or ν) of one unit, with asubsequent return to equilibrium (zero for theerror) the next year; if we then plot y againstthis, we get an impulse response function (IRF).Alternatively, we can shock x by one unit andlet it stay at the new value; the plot of y againstx is a unit response function (URF).

The static specification assumes that all vari-ables have an instantaneous and only an instan-taneous impact. Thus, the IRF for either x orν is a spike, associated with an instantaneouschange in y, and if x or ν then returns to its pre-vious value in the next period, y immediatelyalso returns to its previous value. The URF issimply a step function, with the height of thesingle step being β s .

The FDL model generalizes this. The URFhas two steps, of height β f 1 and β f 1 + β f 2, andthe interval between the steps is one year. Thus,unlike in the simple static model, if x changes, ittakes two years for the full effect of the changeto be felt, but the effect is fully felt in those twoyears. For example, it may take one year for aparty to have an impact on unemployment, butit may be the case that after that year the newparty in office has done all it can and will doin terms of changing unemployment. Similarly,an institutional change may not have all of itsimpact immediately, but the full impact mayoccur within the space of a year.

We could add more lags to Equation 4,allowing for more complicated dynamics.But time series within a country are oftentemporally correlated, so multicolinearitymakes it difficult to get good estimates of thecoefficients of the FDL specification with manylags of x. Given the annual frequency of much ofthe data seen in comparative political economy,the problem of having to add too many lags toEquation 4 (say more than one additional lag)may not, in practice, be a problem. It may beunlikely that interesting institutional changeshave only an immediate impact, but the FDLmodel might be appropriate. It surely should beborne in mind in thinking about appropriate


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specifications, and, as we shall see, it combinesnicely with some other specifications.

The SC model has a different IRF for x andthe error. The IRF for x is a spike, identicalto that of the static model; the IRF for the er-ror has a declining geometric form with rate ofdecay ρ. It seems odd that all the omitted vari-ables have a declining geometric IRF but the xwe are modeling has only an instantaneous im-pact. Maybe that is correct, but this is not thefirst specification that would occur to us. TheSC specification can be generalized by addingmore lags of x, but we would still have very dif-ferent dynamics for x and the unobserved vari-ables in the “error” term. One should clearlyhave a reason to believe that dynamics are ofthis form before using the SC specification.

The LDV model has an IRF for both x andthe error that has a declining geometric form;the initial response is βLDV (or 1 for the error);this declines to zero geometrically at a rate φ.Although the effect never completely dissipates,it becomes tiny fairly quickly unless φ is almostone. The URF starts with an effect βLDV im-mediately, increasing to βLDV

1−φ. If φ is close to

one, the long-run impact of x can be 10 or moretimes the immediate impact.

Although the ADL specification appears tobe much more flexible, it actually has an IRFsimilar to that of the LDV specification, otherthan in the first year (and is identical for a shockto the error process). Initially, y changes byβ ADL units; in the next period the change isβ ADLθ + γ , which then dies out geometricallyat a rate θ . Thus, the ADL specification is onlya bit more general than the LDV specification.It does allow for the maximal impact of x to oc-cur a year later, rather than instantaneously (or,more generally, the effect of x after one periodis not constrained to be the immediate impactwith one year’s decay). This may be importantin some applications. A comparison of the vari-ous IRFs and URFs is in Figure 1, which clearlyshows that the difference between the specifi-cations has simply to do with the timing of theadjustment to y after a change in x.

Before we get to slightly more complicatedmodels, this analysis tells us several things. The

various models differ in the assumptions theyimpose on the dynamics that govern how x andthe errors impact y. None of the dynamic speci-fications can be more or less right a priori. Later,we discuss some estimation (see Section 2.5) andtesting (see Section 2.6) issues, but for now wecan say that various theories would suggest var-ious specifications. The most important issueis whether we think a change in some variableis felt only immediately or whether its impact isdistributed over time; in the latter case, do wethink that a simple structure, such as a declin-ing geometric form, is adequate? How wouldwe expect an institutional change to affect somey of interest in terms of the timing of that ef-fect? If only immediately or completely in oneor two years, the SC or FDL model seems right;if we expect some initial effect that increases tosome limit over time, the LDV or ADL modelwould be used. But there is nothing atheoreticalabout the use of a lagged dependent variable,and there is nothing that should lead anyoneto think the use of a lagged dependent variablecauses incorrect harm. It may cause “correct”harm, in that it may keep us from incorrectlyconcluding that x has a big effect when it doesnot, but that cannot be a bad thing. As has beenwell known, and as Hibbs (1974) showed threedecades ago for political science, the correctmodeling and estimation of time-series modelsoften undoes seemingly obvious findings.

A related way to say the same thing is thateach of the models (for stationary data) impliessome long-run equilibrium and a speed withwhich that equilibrium is reached after someshock to the system. It is easy to solve for equi-libria (if they exist) by noting that in equilibriumboth x and y are stable. Let yE and xE refer toequilibrium x and y. Then for the ADL modelit must be true that

yE = β ADLxE + θ yE + γ xE , 9.

yielding yE = (β ADL+γ xE1−θ

(|θ | < 1 by station-arity). This is easier to see in the EC form,where yi,t−1 = κxi,t−1 in equilibrium and λ is therate (per year) at which y returns to this equi-librium. All the models for stationary data im-ply both a long-run equilibrium and a speed of

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Year

Res

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e

0.0

0.5

1.0

1.5

2.0

2 4 6 8 10

ADLAR1FDLLDV

Year

Res

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0.5

1.0

1.5

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2 4 6 8 10

ADLAR1FDLLDV

B

A

Figure 1Comparison of impulse and unit response functions for four specifications: autoregressive distributed lag(ADL, yi,t = 0.2xi,t + 0.5yi,t−1 + 0.8xi,t−1), finite distributed lag (FDL, yi,t = 1.5xi,t + 0.5xi,t−1), laggeddependent variable (LDV, yi,t = 1.0xi,t + 0.5yi,t−1), and autoregressive errors (Static, yi,t = 2xi,t ).

equilibration, with the different parameter con-straints determining these long-run features.Each of these models implies different short-and long-run reactions of y to x, and standardeconometric methods (see Section 2.6) can beused to discriminate between them.

2.3. Higher-Order Processesand Other ComplicationsWe can generalize any of these models to allowfor non-iid errors and higher-order dynamics.However, because our applications typically useannual data, it is often the case that first-order


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error processes suffice, and it would be unusualto have more than second-order processes. Aswe shall see, it is easy to test for higher-ordererror processes, so there is no reason to sim-ply assume that errors are iid or only followa first-order process. For notational simplicity,we restrict ourselves to second-order processes,but the generalization is obvious.

Consider the LDV model with AR1 errors,in which

yi,t = βLDV xi,t + φyi,t−1 + νi,t

1 − ωL. 10.

After multiplying through by (1 − ωL), we geta model with two lags of y, x and lagged x andsome constraints on the parameters; if we gen-eralize the ADL model similarly, we get a modelwith two lags of both y and x and more con-straints. The interpretation of this model is sim-ilar to the model with iid errors.

We have already seen that the LDV modelwith iid errors is equivalent to a model wherethe effect of all the independent variables andthe error decline at the same geometric rate. Butif we assume that the “errors,” that is, omitted orunmeasured variables, follow an MA1 processwith the same decay rate, φ, as for the measuredvariables (which may or may not be a good as-sumption), then we have

yi,t = βLDV xi,t + φyi,t−1 + (1 − φL)νi,t, 11a.

which simplifies to

yi,t = βLDV xi,t

1 − φL+ νi,t . 11b.

That is, we have a model that combines a geo-metrically declining impact of x on y with iid er-rors. It is surely more likely that the “errors” arecorrelated than that they are independent. Ofcourse, the most likely case is that the errors areneither iid nor MA1 with the same dynamics asx, so we should entertain a more general spec-ification, where the effects of both measuredand unmeasured variables have a declining ge-ometric impact with different rates of decline.The simplest such specification is Equation 10.We return to this more general specification inSection 2.5.

2.4. More ComplicatedDynamics—MultipleIndependent Variables

We typically have more than one independentvariable. How much dynamic generality can orshould be allowed for? One easy generalizationis to allow for two independent (or sets of in-dependent) variables, x and z. Allowing also fora separate speed of adjustment for the errorsyields

yi,t = βxi,t

1 − φx L+ γ

zi,t

1 − φzL+ νi,t

1 − φν L. 12.

Obviously each new variable now requires usto estimate two additional parameters. Also, onmultiplying out the lag structures, we see thatwith three separate speeds of adjustment wehave a third-order lag polynomial multiplyingy, which means that we will have the first threelags of y on the right-hand side of the specifica-tion (and two lags of both x and z) and a second-order moving average error process. Althoughthere are many constraints on the parametersof this model, the need for three lags of y costsus three years’ worth of observations (assumingthe original dataset contained as many obser-vations as were available). With k independentvariables, we would lose k+1 years of data; fora typical problem, where T is perhaps 30 andk is perhaps 5, this is nontrivial. Thus, we areunlikely to ever be able to (or want to) estimatea model where each variable has its own speedof adjustment.

But we might get some leverage by allowingfor two kinds of independent variables: thosewhere adjustment (speed of return to equi-librium) is relatively fast (x) and those wherethe system returns to equilibrium much moreslowly. To simplify, assume the error processshows the same slower adjustment speed as z;we can obviously build more complex models,but they bring nothing additional to thisdiscussion. We then would have

yi,t = βx xi,t + βzzi,t

1 − φL+ νi,t

1 − φL13a.

= βx xi,t−φβx xi,t−1+βzzi,t+φyi,t−1+νi,t . 13b.

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Thus, at the cost of one extra parameter, we canallow some variables to have only an immediateor very quick effect, while others have a slowereffect, with that effect setting in geometrically.With enough years we could estimate morecomplex models, allowing for multiple dynamicprocesses, but such an opportunity is unlikelyto present itself in studies of comparativepolitical economy. We could also generalizethe model by allowing for the lags of x and z toenter without constraint. It is possible to testfor whether these various complications aresupported by the data, or whether they simplyask too much of the data. As always, it is easyenough to test and then make a decision.

2.5. Estimation Issues

As is well known, a specification with no laggeddependent variable but serially correlated er-rors is easy to estimate using any of several vari-ants of feasible generalized least squares, withthe Cochrane-Orcutt iterated procedure beingthe most well known. It is also easy to estimatesuch a model via maximum likelihood, breakingup the full likelihood into a product of condi-tional likelihoods.

The LDV model with iid errors is optimallyestimated by ordinary least squares regression(OLS). However, it is also well known thatOLS yields inconsistent estimates of theLDV model if the error process is seriallycorrelated. Perhaps less well known is thatCochrane-Orcutt or maximum likelihood pro-vides consistent estimates of the LDV modelwith serially correlated errors by accountingfor that serial correlation (Hamilton 1994,p. 226). Thus, it is easy to correctly estimatethe LDV model while allowing for seriallycorrelated errors if analysts wish to do so. Butwe hope that analysts will not wish to do so.

It is often the case that the inclusion of alagged dependent variable eliminates almost allserial correlation of the errors. To see this, startwith the SC equation:

yi,t = βSC xi,t + εi,t 14a.

εi,t = νi,t + ρεi,t−1. 14b.

Remember that the error term is simply all theomitted variables, that is, everything that de-termines y that is not explained by x. If we ad-join yi,t−1 to the specification, the error in thatnew specification is εi,t − φyi,t−1, where εi,t isthe original error in Equation 14a, not somegeneric error term. Since the εi,t are serially cor-related because they contain a common omittedvariable, and yi,t−1 contains the omitted vari-ables at time t − 1, including yi,t−1 will almostcertainly lower the degree of serial correlation,and often will eliminate it. But there is no reasonto simply hope for this; we can estimate (usingOLS) the LDV model assuming iid errors, andthen test the null hypothesis that the errors areindependent using a Lagrange multiplier test.(This only requires that OLS be consistent un-der the null of iid errors, which it is.) The testis trivial to implement, by simply regressing theresiduals from the OLS regression on the lag(s)of this residual and all the independent vari-ables including the lagged dependent variablewith NTR2 from this regression, being asymp-totically distributed χ2 with degrees of freedomequal to the number of lags tested. If, as oftenhappens, we do not reject the null that the re-maining errors are iid, we can continue withthe OLS estimates. If we do reject that null, weshould estimate a more complicated model.

Obviously, failing to reject the null of noserial correlation of the errors is not the samething as knowing there is no serial correlationof the errors. Is this incorrect logic in inter-preting a failure to reject the null likely tocause problems? There are two reasons to besanguine here. First, the large amount of datain typical TSCS studies gives the Lagrangemultiplier test good power. In our first example(Section 5.1), with ∼300 total observations, theLagrange multiplier test detected a serial cor-relation of the errors of ∼0.10. It is also the casethat ignoring a small amount of serial correla-tion (that is, estimating the LDV model withOLS as if there were no serial correlation) leadsto only small amounts of bias. As Achen (2000,p. 13) elegantly shows, the estimation bias inincorrectly using OLS to estimate the LDVmodel with serially correlated errors is directly


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ADLLDV2specification: theautoregressivedistributed lagspecification with asecond lag of thedependent variable

proportional to that serial correlation. Appliedresearchers make many assumptions to simplifyanalysis, assumptions that are never exactly cor-rect. Ignoring a small serial correlation of theerrors is surely one of the more benign mistakes.

As we shall see in Section 3, a number offruitful avenues of investigation are open if theerrors are either uncorrelated or sufficientlyuncorrelated that we can ignore their smallcorrelation. But what if a researcher is not sosanguine? As we have seen in Section 2.5, onecan easily estimate, using methods other thanOLS, the ADL model with serially correlatederrors. But a more fruitful approach, as shownin Section 2.3, is to include second-order lagsof the variables in the ADL specification; this“ADL2” specification can be appropriatelyestimated by OLS, once again allowing theresearcher to more easily examine other inter-esting features of the data. Of course the sameLagrange multiplier testing procedure shouldfirst be used to test for remaining serial corre-lation, but with annual data we can be hopefulthat we will not need highly complicated lagstructures in the preferred specification.

Obviously more parsimonious specificationsare easier to interpret (and convey to thereader), and so more complicated specificationswith higher-order lags come at a cost. Thus,we might want to consider models intermedi-ate between the ADL and ADL2 models. Oneobvious choice is to simply append a second lagof the dependent variable to the ADL speci-fication; this is analogous to moving from thestatic to the LDV specifications, as discussedabove. This simpler specification, ADLLDV2,should be tested to see if the errors are iid. TheADLLDV2 specification may be a good com-promise between parsimony and fidelity to im-portant features of the data; in our first examplethis is our preferred model. In other cases evensimpler models may provide a better tradeoffbetween the various goals.

2.6. Discriminating Between Models

We can use the fact that the ADL modelnests the LDV and SC models to test which

specification better fits the data. The LDVmodel assumes γ = 0 (in Equation 7), whereasthe SC model assumes γ = −θβ ADL. Thus,we can estimate the full ADL model and testwhether γ = 0 or γ = −θβ ADL.3 If both sim-plifications are rejected, we can retain the morecomplicated ADL model.4 Even in the absenceof a precise test, the ADL estimates will oftenindicate which simplification is not too costlyto impose.

For fast dynamics (where θ is close to zero),it will be hard to distinguish between the LDVand SC specifications—or, alternatively, it doesnot make much difference which specificationwe use. To see this, note that if the SC modelis correct, but we estimate the LDV model, weare incorrectly omitting the lagged x variablealthough it should be in the specification, butwith a constrained coefficient θβ. As θ goes tozero, the bias from failing to include this termgoes to zero. Similarly, if we incorrectly esti-mate the SC model when the LDV model iscorrect, we have incorrectly included in thespecification the lagged x variable, with co-efficient −θβ. Again, as θ goes to zero, thisgoes to zero. Thus, we might find ourselvesnot rejecting either the LDV or SC specifi-cations in favor of the more general specifica-tion, but for small θ it matters little. As θ growslarger the two models diverge, and so we havea better chance of discriminating between thespecifications.

This view is different from the conventionalwisdom on omitted-variable bias. It is normallythought to be worse to incorrectly excludethan to incorrectly include a variable. This

3The first test is an ordinary t-test. The second is easiest viaa linear approximation to the nonlinear constraint using aTaylor series (Greene 2008, pp. 96–98); this test is imple-mented in some common statistical packages such as Stata.4Starting with the ADL model and then testing whether sim-plifications are consistent with the data is part of the idea ofgeneral-to-simple testing (also called the encompassing ap-proach) espoused by Hendry and his colleagues (Hendry &Mizon 1978, Mizon 1984). This approach could start witha more complicated model with higher-order specifications,but given annual data, the ADL model with no more than twolags is often the most complicated specification that need beconsidered.

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difference is because both models constrain thecoefficient of the lagged x, and so the SC model“forces” the lagged x to be in the specification.But if we start with the ADL model and thentest for whether simplifications are consistentwith the data, we will not be misled. This test-ing of simplifications is easy to extend to morecomplicated models, such as Equation 13b.

3. COMBINING DYNAMIC ANDCROSS-SECTIONAL ISSUES

Modeling dynamics with TSCS data is onlyhalf the job; clearly analysts also need to modelthe cross-sectional properties of the data. Wehave discussed various cross-sectional issues forTSCS data elsewhere (Beck & Katz 1995, Beck2001). Here we discuss some issues that re-late to the interaction of modeling dynam-ics and cross-sectional issues. For reasons ofspace, we omit discussion of dynamics withdiscrete dependent variables (see Beck et al.1998). Dynamics are no less important in mod-els with discrete dependent variables, but therecommended modeling is different for thatsituation.

3.1. Independent Errors SimplifyCross-Sectional Modeling

We have advocated modeling dynamics by in-cluding appropriate current and lagged valuesof the x’s and lagged values of the dependentvariable so that the resulting errors appear tobe serially independent, enabling easy inter-pretation and estimation. This approach makesit much simpler to model cross-sectional sit-uations. Most standard programs that allowfor modeling complicated cross-sectional situ-ations do not allow for temporally correlatederrors. Although this is a practical rather thana theoretical issue, some estimation methodsare sufficiently complex that one really wantsto use a “canned” program (see Sidebar “WhyNot Just Correct the Standard Errors?”).

In particular, realistic political economymodels often should allow for spatial ef-fects, that is, they should recognize that vari-

WHY NOT JUST CORRECT THESTANDARD ERRORS?

Most standard programs that allow for modeling complicatedcross-sectional situations do not allow for temporally correlatederrors. Some researchers try to solve this problem by using sim-ple models and then correcting the standard errors using somevariant of Huber’s (1967) “robust” standard errors. This is thereasoning behind our recommendation to use PCSEs (panel cor-rected standard errors) to deal with some difficult cross-sectionalcomplications of the error process. There are similar autocor-relation consistent standard errors (Newey & West 1987). Wedo not recommend these because failing to account for seriallycorrelated errors often leads to substantial inefficiencies in esti-mation as well as incorrect standard errors; failing to account forcross-sectional problems in the data is usually less serious. In anyevent, users of our preferred methods have no need to resort toautocorrelation consistent standard errors.

ables in one country impact other countries.Models of the political causes of economic per-formance, for example, must take into accountthat the economic performance of any countryis a function of the economic performance ofits trading partners. These issues have been dis-cussed in the context of TSCS data elsewhere(Beck et al. 2006, Franzese & Hayes 2007), andhere we simply point out that our preferredapproach to dynamics makes it easy for ana-lysts to deal with this critical cross-sectionalissue.

Another cross-sectional feature that shouldbe considered (see Beck & Katz 2007 and thereferences cited there) is that parameters mayvary randomly by country, and possibly as afunction of country-level covariates. It is easyto allow for this by using the “random coeffi-cients model” (which is equivalent to a “mixed”or “hierarchical” or “multilevel” model) if theerror process is iid. Note that one of the ran-domly varying parameters can be that of thelagged dependent variable, the parameter thatcontrols the speed of adjustment in the model.Perhaps countries differ in that speed of adjust-ment. As we see in Section 5.1, this issue is easyto examine when errors are iid.


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3.2. Fixed Effects and LaggedDependent Variables

Perhaps the most common cross-sectional is-sue is heterogeneity of the intercepts. In theTSCS context, this is usually dealt with byadding “fixed effects” (country-specific inter-cepts) to the specification. We would adjointhese country-specific intercepts to the pre-ferred ADL specification. But here we get intopotential trouble, since it is well known that au-toregressive models with fixed effects lead to bi-ased parameter estimates (Nickell 1981). Thisbias is induced because centering all variablesby country, which eliminates the heterogeneityof the constant term, induces a correlation be-tween the centered lagged dependent variableand the centered error term.

It is also well known that this bias is of order1T , and almost all of the work on this prob-lem has been in the context of small-T “pan-els.” When T is 2 or 3, the bias is indeed severe(50% or so). But when T is 20 or more, the biasbecomes small.

Various corrections for this bias are wellknown. Most of them involve the use of in-strumental variables, building on the work ofAnderson & Hsiao (1982). As is often the case,it is hard to find good instruments, and sothe instrumental variable corrections often ob-tain consistency at the price of rather poor fi-nite sample properties. Other estimators (Kiviet1995) are hard to combine with other methodsand hard to generalize to even non-rectangulardata sets.

We ran Monte Carlo experiments to com-pare OLS estimation of a simple LDV modelwith fixed effects to the Kiviet and Anderson-Hsiao estimators (Beck & Katz 2009). For theT ’s seen typically in TSCS analysis (20 ormore), OLS performs about as well as Kivietand much better than Anderson-Hsiao. Giventhe advantages of the OLS method discussedin the previous subsection, we do not hesitateto recommend OLS when country-specific in-tercepts must be adjoined to the specificationof a TSCS model. Judson & Owen (1999) give

similar advice following a similar discussion ofthis issue.

4. NON-STATIONARITY INPOLITICAL ECONOMYTSCS DATA

Before we look at some examples, one topicremains: what to do with non-stationary data.During the past two decades, with the pioneer-ing work of Engle & Granger (1987), time-series econometrics has been dominated by thestudy of non-stationary series. There are manyways to violate the assumptions of stationar-ity presented in Equation 2, but most of thework has focused on the issue of unit rootsor integrated series in which shocks to the se-ries accumulate forever. These series are long-memoried; even distant shocks persist to thepresent. The key question is how to estimatemodels where the data are integrated (we re-strict ourselves to integration of order one withno loss of generality). Such data, denoted I(1),are not stationary but their first difference is sta-tionary. The simplest example of such an I(1)process is a random walk, where

yi,t = yi,t−1 + νi,t, 15.

with νi,t being stationary by definition. Inte-grated data look very different from data gener-ated by a stationary process. Most importantly,they do not have equilibria (because there is nomean reversion), and the best prediction of anintegrated series many periods ahead is the cur-rent value of that series.

There is a huge literature on estimatingmodels with integrated data. Such methodsmust take into account that standard asymptotictheory does not apply, and also that

limt→∞

Var(yi,t) = ∞. 16.

Thus, if we wait long enough, any integratedseries will wander “infinitely” far from its mean.Much work on both diagnosing and estimatingmodels with integrated series builds on boththese issues. Our interest is not in the estimation

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of single time series but rather TSCS politicaleconomy data.5

Political economy data is typically observedannually for relatively short periods of time(often 20–40 years). Of most relevance, dur-ing that time period we often observe very fewcycles. Thus, although the series may be verypersistent, we have no idea if a longer timeperiod would show the series to be stationary(though with a slow speed of adjustment) ornon-stationary. These annual observations on,for example, GDP or left political control ofthe economy are very different from the dailyobservations we may have on financial rates. Soalthough it may appear from an autoregressionthat some political economy series have unitroots, is this the right characterization of theseseries? For example, using Huber & Stephens’(2001) data, an autoregression of social secu-rity on its lag yields a point estimate of theautoregressive coefficient of 1.003 with a stan-dard error of 0.009; a similar autoregression ofChristian Democratic party cabinet participa-tion yields 1.03 with a standard error of 0.001.It does not take heavy-duty statistical testing tosee we cannot reject the null (that the autore-gressive coefficient is one) in favor of the alter-native (that it is less than one). But does thismean that we think the series might be I(1)?

If these series had unit roots, they wouldtend to wander far from their means, and thevariance of the observations would grow largerand larger over time (a similar argument is madeby Alvarez & Katz 2000). But by definition boththe proportion of the budget spent on social se-curity and Christian Democratic cabinet par-ticipation are bounded between 0% and 100%,which then bounds how large their variancescan become. Further, if either series were I(1),then we would be equally likely to see an in-crease or decrease in either variable regardlessof its present value. Do we really believe thatthere is no tendency for social security spend-ing to be more likely to rise when it is low

5There is a literature on panel unit roots (Im et al. 2003, Levinet al. 2002), but at this point the literature is still largely abouttesting for unit roots.

and to fall when high, or for Christian Demo-cratic cabinet strength to exhibit a similar ten-dency? In the Huber & Stephens data, socialsecurity spending ranges only between 3% and33% of the budget, and Christian Democraticcabinet strength ranges between 0% and 34%.Even though these series are very persistent,they simply cannot be I(1). The impressive ap-paratus built over the past two decades to esti-mate models with I(1) series does not providethe tools needed for many, if not most, politicaleconomy TSCS datasets.

One possibility is to induce stationarity byfirst differencing all slowly changing variables,leading to a model that explains changes in y bychanges in x. In practice, first-difference mod-els often perform poorly (at least from the per-spective of the researcher, for whom changes inx appear unrelated to changes in y). Modelingfirst differences also throws out any long-runinformation about y and x, so the effect of achange in x is the same regardless of whether yis high or low by historical standards.

Fortunately, the modeling issue is not reallyabout the univariate properties of any seriesbut the properties of the stochastic processthat generated the y’s conditional on theobserved covariates. Even with data similarto Huber & Stephens’, the errors may appearstationary and so the methods of the previoussection can be used. In particular, whether theseries are integrated or stationary but slowlymoving, they may be well modeled by the ECspecification (Equation 8), which, as we haveseen, is just an alternative parameterization ofthe ADL model. The EC form is nice becauseit combines the short-run first-differencesmodel with the long-run tendency for series tobe in equilibrium. If the estimated λ in the ECspecification is zero, that indicates that y and xhave no long-run equilibrium relationship. Wehave already seen that if x and y are stationary,they always have a long-run relationship, so thisis only a problem if the series are integrated.In other words, if the series are stationary butadjust very slowly, the EC (or equivalent ADL)model is a good place to start, and if the seriesare integrated, either the EC model will work


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(the series are said to be co-integrated) or theresiduals will appear highly correlated. Becauseour preferred methodology chooses specifi-cations with almost uncorrelated residuals, itshould never lead to choosing an incorrect EC(or ADL) specification.

Why do we propose ignoring much of whathas dominated econometric argument for twodecades? First, economists study many series(such as interest or exchange rates) that inher-ently are in levels, and so are likely to be inte-grated; variables in political economy are oftenexpressed as a proportion of GDP or the gov-ernment budget and hence are much less likelyto be integrated. Other political variables, suchas party control of government, may be per-sistent, but cannot possibly be integrated (theytake on values of zero and one only, and so haveneither infinite variance nor no tendency to re-vert back toward the mean). A second differ-ence is that economists have no theory aboutwhether one short-run exchange rate adjusts toa second rate, or the second rate adjusts to thefirst, or both; this leads to complicated estima-tion issues. In many political economy mod-els, it is clear that y adjusts to x but not viceversa. We think that left governments increasespending but we do not think that low spendingleads directly to a right-wing government (Beck1992). Thus, even with highly persistent data,the EC (or ADL) model, estimated by OLS, willquite often work well, and, when it fails, simpletests and a rigorous methodology will indicatethat failure.6

5. EXAMPLES

In this section we consider two examplesto explore the practical issues in estimating

6There is a slight technical problem in that the distributionof the estimated λ is not normal if the series are not co-integrated. Instead, they have a Dickey-Fuller type distribu-tion, which has fatter tails. Thus, there may be some caseswhere a standard test of the null hypothesis that λ = 0 yieldsincorrect conclusions. But given the large n and T of TSCSdata, in many cases it is clear that the EC model is adequateor not, and if we incorrectly assume stationarity, consistentapplication of appropriate standard methods will indicate theproblem.

dynamics in political economy TSCS datasets.The first example, presented in some detail,looks at the political determinants of capitaltaxation rates where adjustment speeds arefairly slow. The second example, presentedmore cursorily, looks at the impact of politicalvariables on the growth of GDP. In the GDPexample, where the dynamics are quite fast, thespecification choice has fewer consequences.All computations were done using Stata 11.1,with data kindly provided by Geoff Garrett.Although our analysis is different from thoseof Garrett & Mitchell, we began by easilyreplicating their results.

5.1. Capital Taxation Rates

Our first example models capital taxation ratesin 16 OECD nations from 1961 through 1993using the data and specification of Garrett &Mitchell (2001).7 Obviously tax rates moverelatively slowly over time; the autoregressivecoefficient of tax rates is 0.77. Thus, althoughtax rates are clearly stationary, it will take somenumber of years for the system to get close tofully adjusting; it takes about 2.65 years for anyshock to dissipate.

Before estimation, one should examine thedata to see whether there is sufficient within-country heterogeneity to make TSCS analy-sis meaningful, to see whether there appears tobe very much inter-country heterogeneity thatmight need to be modeled, and to see whetherthere are any temporal pattens, such as trends,that need to be modeled. For the first two issuesa standard country-specific box plot of tax rates,shown in Figure 2, is appropriate; for the thirdquestion time-series plots by country, shown inFigure 3, are more useful.

7The data set is not rectangular; some countries only reporttax rates for a portion of the period under study. In total therewere 322 observations after one drops missing data at thebeginning of a period (and omits observations with missinglagged data so that all results pertain to the same data). Theextra lag in the ADLLDV2 model leads to the loss of thefirst observation for each country, yielding 306 observationsfor that estimation. This loss of data points is an additionalreason why analysts may not prefer this specification.

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Australia Austria Belgium Canada

Denmark Finland France Germany

Italy Japan Netherlands Norway

Sweden Switzerland UK US

Capital T

axation R

ate

Figure 2Box plots of capital taxation rates by country, 1967–1992.

Whereas some countries (Austria,Denmark, the Netherlands, and New Zealand)show little if any variation in their tax rates, theother countries show enough variation overtime to make a TSCS analysis of interest. Thereis also some inter-country heterogeneity, withFrance, Sweden, and the United Kingdomhaving generally higher rates. Figure 3shows that taxation rates in some countriesare strongly trending whereas others showlittle trend; this figure also clearly shows thebeginning of period missingness pattern in thedata. A regression of tax rates on time shows atrend of ∼0.33% (with a small standard error)per annum in those rates. Thus, it appearsas though a TSCS analysis of these data issensible, and it may be the case that there will

be unexplained temporal and cross-sectionalheterogeneity. Following Garrett & Mitchell,we mean-centered all observations by countryand year, which is equivalent to allowing foryear- and country-specific intercepts.8

8Of course one can only decide whether these year- andcountry-specific intercepts are needed after a specificationis chosen, and because these intercepts are atheoretical, oneshould attempt to find specifications where they are not nec-essary. Alas, this is often impossible. Here the intercepts weresignificant in all specifications. We might have preferred amodel with a time trend instead of year-specific intercepts,but the difference between the two specifications was neg-ligible, and we preferred to stay consistent with Garrett &Mitchell. Obviously in actual research such decisions shouldbe made with care, and researchers should not simply do whatothers have done before.


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1970 1975 1980 1985 1990 1970 1975 1980 1985 1990 1970 1975 1980 1985 1990 1970 1975 1980 1985 1990

Australia Austria Belgium Canada

Denmark Finland France Germany

Italy Japan Netherlands Norway

Sweden Switzerland UK USCapital T

axation R

ate

Year

Figure 3Time-series plots of capital taxation rates by country.

Garrett & Mitchell (2001) wish to explaincapital taxation rates (this is only one of theiranalyses) by variables related to the economy,the demand for services, and political factors.We work more or less with the Garrett &Mitchell specification, dropping a few vari-ables that were not substantively interestingnor statistically significant in any specification.We thus regress the capital tax rate (CAP-TAX ) on unemployment (UNEM), economicgrowth (GDPPC), the proportion of the pop-ulation that is elderly (AGED), vulnerability ofthe workforce as measured by low wage imports(VULN), foreign direct investment (FDI), andtwo political variables: the proportion of thecabinet portfolios held by the left (LEFT ) andthe proportion held by Christian Democrats

(CDEM). Because we mean-centered all vari-ables, there are no intercepts in the model.Table 1 reports the results of the various dy-namic estimations. All standard errors are ourrecommended panel-corrected standard errors(Beck & Katz 1995), which are easy to computewith our recommended methodology.

The static model (not shown) is clearly in-adequate; a Lagrange multiplier test for se-rial correlation of the errors strongly rejectsthe null hypothesis of serially independent er-rors. Because the static model is nested in-side both the LDV and SC models, standardWald tests (a t-test of either H0:ρ = 0 or H0:βTAX L = 0) clearly show that the static modelcan be rejected in favor of either of these twomodels.

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Table 1 Comparison of SC, LDV, ADL, and ADLLDV2 estimates of Garrett & Mitchell’s (2001)model of capital taxation in 16 OECD nations, 1967–1992 (country and year centered)

SC LDV ADL ADLLDV2

Variable β̂ PCSE β̂ PCSE β̂ PCSE β̂ PCSEVULN −0.22 0.12 −0.10 0.07 −0.28 0.13 −0.33 0.14FDI 0.51 0.26 0.37 0.21 0.59 0.26 0.48 0.28UNEM −0.18 0.22 −0.34 0.14 −0.68 0.27 −0.68 0.30AGED 1.42 0.51 0.35 0.24 0.26 0.71 −0.27 0.87GDPPC −0.69 0.11 −0.62 0.12 −0.80 0.13 −0.81 0.14LEFT 0.004 0.012 0.006 0.009 0.003 0.013 0.002 0.014CDEM 0.018 0.022 0.015 0.012 0.015 0.025 0.031 0.024TAXL 0.70 0.06 0.76 0.07 0.93 0.10VULNL 0.21 0.14 0.24 0.15FDIL −0.55 0.29 −0.56 0.31UNEML 0.48 0.26 0.62 0.28AGEDL 0.24 0.76 0.98 0.94GDPPCL 0.29 0.12 0.36 0.14LEFTL 0.005 0.013 0.004 0.014CDEML 0.005 0.024 −0.010 0.025TAXL2 −0.26 0.09ρ 0.66

Abbreviations: ADL, autoregressive distributed lag; ADLLDV2, the ADL specification with a second lag of the dependentvariable; LDV, lagged dependent variable; PCSE, panel corrected standard errors; SC, first-order serially correlated errormodel. See text for definitions of variables in the left-hand column.

But we must compare both the LDV andSC specifications to the more general ADLspecification. Again, since both these specifi-cations are nested inside the ADL specificationwe can use standard Wald tests (in this case anF-test of the null hypothesis that the coefficientson all the lagged x’s are zero); that null is deci-sively rejected, so the more general ADL spec-ification is preferred.

The ADL specification still shows serial cor-relation of the errors; a Lagrange multiplier testof the null hypothesis of independent errorsshows we can reject that null of iid errors. (Aregression of the residuals from the ADL spec-ification on the lagged residuals and all the otherindependent variables has an R2 of 0.046, which,multiplied by the number of observations inthat regression, yields a statistic of 14; sincethis statistic is distributed χ2

1 , the null hypoth-esis of independent errors is clearly rejected.)As discussed in Section 2.5, we added a second

lag of capital taxation to the specification; re-sults of estimating this specification are in theADLLDV2 columns. We cannot reject the nullhypothesis of independent errors for this re-gression (χ2

1 = 1.1). The ADLLDV2 specifica-tion is both statistically superior to the simplerspecifications and shows iid errors. There are,of course, many other specifications that a sub-stantive article would test (multiple speeds ofadjustment, for example), but we do not pursuethese here.

All the models show that a one-time unitshock to the error process dies out exponentially(or nearly exponentially for the ADLLDV2model) with similar decay rates ranging from24% to 34% per annum for the first three mod-els; for the ADLLDV2 model, the initial de-cay rate is only 7% in the first year but in-creases to 33% (one minus the sum of the coef-ficients on the lagged dependent variable) afterthe first year. Given the standard errors on these


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coefficients, the decay rates are quite similar.Thus, for example, a classical confidence inter-val for the decay rate in the ADL model is (11%,38%), and in the ADLLDV2 model after thefirst year it is (17%, 49%).

Turning to the estimated effect of the vari-ous independent variables (omitting the two po-litical variables, which show almost no effect buthuge standard errors), recall that the SC speci-fication assumes that the effect of the variablesis only instantaneous, the LDV model assumesthe effect decays geometrically, and the ADLand ADLLDV2 models allow us to test thoseassumptions. In those latter specifications, thecoefficients on the current and lagged valuesof VULN and FDI are close in absolute valueand of opposite sign. Thus, the impact of thosevariables on capital taxation rates is more or lessonly instantaneous, and the ADL coefficient es-timates of this instantaneous effect are similarto the SC estimates but different from the LDVestimates. Of course, the ADL specifications al-low us to study the speed of adjustment, whereasthe SC specification just assumes instantaneousadjustment.

The coefficients on UNEM and GDPPC andtheir lags are of opposite sign but do not quiteoffset each other. Here the ADL estimates are,as we would expect, much closer to the LDVestimates than to the SC estimates. But again,we need not make the assumptions about de-cay rates that the LDV specification imposes;instead, we can examine what the decay pro-cess looks like. Interestingly, and contrary toAchen’s notion of the lagged dependent vari-able “dominating” a regression, the coefficientsof all four of these substantive coefficients areas large as or larger than the similar coefficientsin the SC specification.

The variable AGED determines tax rates inthe SC specification but fails to show any impactin any of the other specifications. Intuitively,although AGED perhaps “ought” to affect taxrates, its coefficient in the SC specification“seems” a bit large; would a one-point increasein the aged population be expected to lead toa more-than-one-point increase in capital taxa-tion rates? Perhaps it is not so simple to discuss

which results “make sense,” and making senseis hardly a statistical criterion. Note also thatAGED is itself highly trending (its autoregres-sion has a coefficient of 0.93 with a standard er-ror of 0.01). Although we can reject the null thatAGED has a unit root, it, like the capital tax rate,changes very slowly. Thus, we might suspectthat the simple contemporaneous relationshipbetween the two variables is spurious (in thesense of Granger & Newbold 1974). Of coursewe cannot know the “truth” here, but it is notobvious that the ADL (or LDV) results on theimpact of AGED are somehow foolish or wrong.

The moral so far is that researchers shouldestimate a model flexible enough to accountfor various types of dynamics; they should alsotry hard to make sure that the error process isclose to iid. The ADLLDV2 model performsvery well here, both in terms of its passing var-ious tests and its interpretability (with the sim-pler ADL model being easier to interpret butnot quite passing the statistical tests). While nospecification will be good in all situations, it isclear that researchers should not consider moregeneral specifications before accepting highlyconstrained ones such as either the SC or LDVmodel.

Our focus is on dynamics, but no TSCSanalysis is complete without a final assessmentof heterogeneity over countries. Rememberthat our analysis uses country-centered data, sothere can be no heterogeneity in the variouscentered means. But we can see if the model failsto work for some subset of countries by cross-validation (Stone 1974), leaving out one coun-try at a time. Thus we reran the ADLLDV2specification, leaving out one country at a timeand then using the estimated values to pre-dict capital tax rates in the omitted country.The mean absolute prediction error was thencomputed for each country. For all observa-tions, the absolute forecast error was about 2.3.Four countries—Japan, Norway, Sweden, andthe United Kingdom—had mean absolute fore-cast errors above 3.5, indicating at least somelack of homogeneity. We do not pursue this is-sue further here, but clearly this issue would bepursued in a more complete analysis. (We also

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do not present other post-estimation analysesthat should be standard, such as residual plotsby countries.)

We also assessed heterogeneity by test-ing for parameter heterogeneity (by country).Here, since we focus on dynamics, we fit theADL specification allowing for the coefficientof the lagged dependent variable (θ ) for eachcountry to be a random draw from a normaldistribution with zero mean. This allows us tosee whether the general speed of adjustmentvaries by country. Results of this estimation re-veal no statistically (or substantively) significantparameter heterogeneity (on the lagged depen-dent variable); the estimated standard deviationon the normal from which the coefficients weredrawn was only 0.09.

The standard error of the estimated stan-dard deviation was 0.07. A test of the null hy-pothesis that θ does not vary by country yieldsa statistic of 0.70; this statistic is χ2

1 , so far fromthe critical value for rejection of the null. Wecan look at the individual country estimates ofθ . Most are within 0.01 of the overall estimateof θ , with only the coefficient for the UnitedKingdom really differing; the estimated θ forthe United Kingdom is 0.11 under the over-all estimate for θ , though with a standard errorof ∼0.07. Given all this, we prefer not to pur-sue whether further investigation of the speedof adjustment in tax rates in the United King-dom is needed, but clearly this type of analy-sis in other situations might prove extremelyuseful.

5.2. The Growth of GrossDomestic Product

Our second example relates to political econ-omy explanations of the growth of GDP in 14OECD nations observed from 1966 through1990 (yielding 336 observations), using datafrom Garrett (1998). Our treatment is cur-sory for reasons of space. We use one of hismodels, taking the growth in GDP as a lin-ear additive function of political factors andeconomic controls. The political variables arethe proportion of cabinet posts occupied by

left parties (LEFT ), the degree of central-ized labor bargaining as a measure of corpo-ratism (CORP), and the product of the lat-ter two variables (LEFT x CORP). The eco-nomic and control variables are a dummy mark-ing the relatively prosperous period through1973 (PER73), overall OECD GDP growth(DEMAND), trade openness (TRADE), capitalmobility (CAPMOB), and a measure of oil im-ports (OILD). All variables, following Garrett’suse of country fixed effects, were mean centeredby country. As before, all standard errors arepanel corrected.

GDP growth appears stationary, with an au-toregressive coefficient of 0.32. Thus, all spec-ifications are expected to show relatively fastdynamics, with quick returns to equilibrium.Turning to models with explanatory variables,results of estimating various specifications arein Table 2.

The static model showed modest serial cor-relation of the errors; a Lagrange multipliertest showed we could clearly reject the null ofserially independent errors (χ2

1 = 8.6). Sub-stantively, the serial correlation of the errorsis small (0.12), so the OLS results are similarto the slightly more correct results in the twodynamic specifications.

Given the rapid speed of adjustment (the co-efficient on the LDV is 0.16), it is not surprisingthat all three specifications show similar esti-mates. Very few coefficients are significant inany of the specifications, but the two variablesthat show a strong impact in the static specifi-cation continue to show a strong impact in thetwo dynamic specifications.

The similarity of the SC and LDV estimatesis not surprising; because of the fast dynamics,the two models are not really very different. Af-ter one period, the various independent vari-ables in the LDV specification have only 3%of their original impact; the long-run effects inthe LDV specification are only 18% larger thanthe immediate impacts. Thus, the two specifi-cations are saying more or less the same things,and the estimated coefficients are quite simi-lar. Substantively, it appears as though GDPgrowth in a country is largely determined by


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Table 2 Comparison of SC and LDV estimates of Garrett’s (1998) model of economic growth in14 OECD nations, 1966–1990 (country centered)

Static SC LDV

Variable β̂ PCSE β̂ PCSE β̂ PCSEDEMAND 0.007 0.0012 0.007 0.002 0.007 0.001TRADE −0.018 0.019 −0.021 0.021 −0.019 0.019CAPMOB −0.20 0.21 −0.25 0.23 −0.24 0.21OILD −7.86 7.34 −6.69 7.89 −5.85 7.08PER73 1.76 0.42 1.76 0.45 1.45 0.43CORP 0.54 0.56 0.43 0.61 0.30 0.56LEFT −0.075 0.17 −0.076 0.18 −0.062 0.17LEFTxCORP 0.10 0.53 0.10 0.56 0.17 0.52GDP_L 0.16 0.07ρ 0.12

Abbreviations: LDV, lagged dependent variable; PCSE, panel corrected standard errors; SC, first-order serially correlatederror model. See text for definitions of variables in the left-hand column.

GDP growth in its trading partners, and poli-tics appears to play little if any role.

Both specifications were tested against thefull ADL specification that contained all theone-year lags of the independent variables.Standard hypothesis tests do not come close torejection of the simpler SC or LDV models infavor of the ADL model. Since the LDV andAIC specifications are not nested, discriminat-ing between them is not so simple. Because bothspecifications have the same number of parame-ters, discrimination using standard informationcriteria (AIC or BIC) simplifies to comparisonsof goodness of fit, on which criterion both spec-ifications perform almost equally well. This willoften be the case, since both the SC and LDVspecifications imply very quick adjustments toequilibrium when the dynamic parameters arenear zero.

In summary, the data are consistent withvery short-run impacts, and it does not muchmatter how we exactly specify those dynam-ics. In terms of the Achen critique, there aretwo predictors of GDP that are strong in theSC model; they remain about equally strong inthe LDV model. As we have argued, there isnothing about lagged dependent variables thatmakes them “dominate a regression” or makes“real” effects disappear. Given the nature of dy-

namics, this will always be the case when vari-ables adjust quickly.

6. CONCLUSION

There is no cookbook for modeling the dy-namics of TSCS models. Instead, careful ex-amination of the specifications, and what theyentail substantively, can allow TSCS analyststo think about how to model these dynamics.Well-known econometric tests help in this pro-cess, and standard methods make it easy to es-timate the appropriate dynamic model. Mod-eling decisions are less critical where variablesequilibrate quickly; as the adjustment processslows, the various specifications imply more andmore different characteristics of the data. An-alysts should take advantage of this to choosethe appropriate model, namely, one that impliesdynamics consistent with theoretical concerns.The specification chosen should of course beflexible enough to allow for testing against al-ternative dynamic specifications.

Being more specific, we have provided evi-dence that, contra Achen, there is nothing per-nicious in the use of a model with a lagged de-pendent variable. Obviously attention to issuesof testing and specification are as importanthere as anywhere, but there is nothing about

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lagged dependent variables that makes themgenerically harmful. As we have seen, there isa variety of generic dynamic specifications, andresearchers should choose among them by us-ing the same general methodology they usein other cases. The ADL model (or its ADL2complication) is a good place to start; at thatpoint, various specializations of the model canbe tested against this general specification. An-alysts, should, of course, interpret the dynamicresults in substantive terms, focusing on long-as well as short-run effects.

Instead of pushing dynamics into a compli-cated error process that then must be “fixedup” to allow for estimation, it is much better tomodel the dynamics directly—that is, in termsof observable variables. There are both theoret-

ical and practical advantages to this. The theo-retical advantage is that dynamic issues becomemuch more than nuisances for estimation. Thepractical advantage is that it is easy to estimatemodels with (approximately) independent er-ror processes via OLS, and easy then to esti-mate these models with additional complicatingcross-sectional features.

There are many important features of TSCSdata, both in the temporal and spatial realms.Both sets of features are interesting, and nei-ther should be swept under the rug. Fortu-nately, the econometrics involved with goodTSCS modeling are not difficult, and a clear eyeon specification and testing allows researchersto find substantively interesting features of thedata.

DISCLOSURE STATEMENT

The authors are not aware of any affiliations, memberships, funding, or financial holdings thatmight be perceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS

For research support we thank the National Science Foundation. An earlier version was presentedat the Annual Meeting of the Society for Political Methodology, Stanford University, Stanford,California, July 29–31, 2004. We thank Geoff Garrett, Evelyne Huber, and John Stephens forproviding data. We also thank the many colleagues who have discussed TSCS issues with us andallowed us to present in various forums, among whom Chris Achen and Simon Jackman (for effortswell beyond the call of duty) must be singled out.

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PL14-frontmatter ARI 11 April 2011 17:22

Annual Review ofPolitical Science

Volume 14, 2011Contents

A Life in Political ScienceSidney Verba � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � i

Leadership: What It Means, What It Does, and What We Want toKnow About ItJohn S. Ahlquist and Margaret Levi � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 1

Examining the Electoral Connection Across TimeJamie L. Carson and Jeffery A. Jenkins � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �25

Presidential Appointments and PersonnelDavid E. Lewis � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �47

Understanding the 2007–2008 Global Financial Crisis: Lessons forScholars of International Political EconomyEric Helleiner � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �67

Presidential Power in WarWilliam G. Howell � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �89

The Politics of Regulation: From New Institutionalism to NewGovernanceChristopher Carrigan and Cary Coglianese � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 107

The New Judicial Politics of Legal DoctrineJeffrey R. Lax � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 131

The Rhetoric Revival in Political TheoryBryan Garsten � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 159

The Rhetoric of the Economy and the PolityDeirdre Nansen McCloskey � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 181

The Contribution of Behavioral Economics to Political ScienceRick K. Wilson � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 201

The Causes of Nuclear Weapons ProliferationScott D. Sagan � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 225

Network Analysis and Political ScienceMichael D. Ward, Katherine Stovel, and Audrey Sacks � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 245

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PL14-frontmatter ARI 11 April 2011 17:22

The Big Five Personality Traits in the Political ArenaAlan S. Gerber, Gregory A. Huber, David Doherty, and Conor M. Dowling � � � � � � � � � � 265

ClientelismAllen Hicken � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 289

Political Economy Models of ElectionsTorun Dewan and Kenneth A. Shepsle � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 311

Modeling Dynamics in Time-Series–Cross-Section PoliticalEconomy DataNathaniel Beck and Jonathan N. Katz � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 331

Voting TechnologiesCharles Stewart III � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 353

Indexes

Cumulative Index of Contributing Authors, Volumes 10–14 � � � � � � � � � � � � � � � � � � � � � � � � � � � 379

Cumulative Index of Chapter Titles, Volumes 10–14 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 381

Errata

An online log of corrections to Annual Review of Political Science articles may be foundat http://polisci.annualreviews.org/

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Documents

Modeling Dynamics in Time-Series Cross-Section Political ...users.cla.umn.edu/~uggen/beck_arps_11.pdfPL14CH16-Beck ARI 14 April 2011 17:11 Modeling Dynamics in Time-Series–Cross-Section