Fractional ( Co ) Integration and Governing Party Support in Britain

FRACTIONAL (CO)INTEGRATION AND GOVERNING PARTY SUPPORT IN BRITAIN: A METHODOLOGICAL RESEARCH NOTE Recent developments in the analysis of long-memoried processes provide important leverage for analyzing time series variables of interest to political scientists. This paper provides an accessible exposition of these methods, and illustrates their utility for addressing protracted controversies regarding the political economy of party support in Britain. Estimates of the fractionally differencing parameter, d, reveal that governing party support, prime ministerial approval and economic evaluations are longmemoried and nonstationary, and that governing party support and prime ministerial approval are fractionally cointegrated. Pace conventional wisdom that party leader images matter little, if at all, analyses of multivariate fractional error correction models show that prime ministerial approval has important shortand long-run effects on party support. Prospective and retrospective personal economic evaluations are influential but, contrary to a longstanding claim, national economic evaluations are not significant. The paper concludes by suggesting that individual-level heterogeneity is a likely source of the observed aggregate-level fractional integration in governing party support and its determinants. Specifying parsimonious models that incorporate theoretically meaningful heterogeneity is a challenging topic for future research. 1 FRACTIONAL INTEGRATION AND GOVERNING PARTY SUPPORT IN BRITAIN: A METHODOLOGICAL RESEARCH NOTE The history of science testifies that methodological innovation can provide important leverage for addressing theoretical controversies. In this paper, we provide an accessible exposition of leadingedge statistical methods for analyzing long-memoried time series, and employ these methods to address protracted debates concerning the political economy of public support for parties and party leaders in Great Britain. After reviewing the controversies at issue, we introduce the concepts of long-memory and fractional integration, and consider why one might expect key variables in party support models to be fractionally integrated. Next, we discuss the concepts of fractional cointegration and fractionally integrated error correction mechanisms, and analyze the memory properties of political and economic variables of interest. Multivariate fractional error correction models are used to investigate the determinants of governing party support, and encompassing tests are employed to address disputes about the explanatory efficacy of various kinds of economic evaluations. In the conclusion, we summarize major findings and discuss methodological and theoretical implications for future studies of party support in Britain and elsewhere. Continuing Controversies Following pioneering studies in the 1970s (Goodhart and Bhansali, 1970; Miller and Mackie, 1973) most studies of the dynamics of party support in Britain have focused exclusively on the effects of objective economic indicators such as inflation and unemployment rates. However, during the past decade, research has demonstrated the utility of employing measures of voters' judgments about their own and their country's financial well-being (e.g., Clarke and Stewart 1995; Sanders, 1991). Although there is now consensus that economic evaluations provide substantial explanatory power, analysts continue to disagree which type of evaluation drives party support. These debates have a lengthy intellectual lineage. Over three decades ago V.O. Key (1968) articulated his famous two-fold argument that voters are rational actors whose electoral decisions are 2 guided by retrospective rather than prospective judgments of government performance. Key’s thesis was consistent with a frequently articulated conventional wisdom that incumbents are rewarded or punished for the state of voters’ pocketbooks in the recent past (Kiewiet, 1983:5-7). According to this hypothesis, personal retrospective economic evaluations are what matter for party support. However, claims also have been advanced on behalf of personal prospective judgments. In Britain, these latter arguments gained considerable credence when Sanders (1991) used personal economic expectations as the linchpin in his two-stage forecasting model (the "Essex model") that predicted the widely unanticipated Conservative victory in the 1992 British general election an impressive 16 months before that event occurred. Consistent with Sanders' analysis, Figure 1A shows, personal economic expectations have a substantial positive correlation (r = +.49) with governing party support over the 1979-1996 period studied in this paper. (Figure 1 about here) There also are disputes between scholars such as Sanders who emphasize the importance of personal, or "egocentric," evaluations and those such as Kinder and Kiewiet (1981), MacKuen, Erikson, and Stimson (1992, 1996), Erikson, MacKuen and Stimson (2000) and Norpoth (e.g., 1996a, 1996b) who favor national, or "sociotropic," evaluations. The sociotropic camp, in turn, is deeply divided. In research on presidential approval in the United States, MacKuen et al. argue that national expectations outperform national retrospections, whereas Kinder and Kiewiet and Norpoth make strong claims on behalf of the latter. Still others (e.g., Clarke and Stewart, 1994; Lewis-Beck, 1988) find that both national prospections and national retrospections are influential. In their British study, Clarke and Stewart (1995) report that national expectations perform as well as personal ones in analyses of governing party support, but that national retrospections outperform other types of economic evaluations as determinants of prime ministerial approval. An additional controversy in British studies concerns the explanatory role of leader images in 3 party support models (see, e.g., Butler and Stokes 1976; Clarke and Stewart, 1995; Clarke, Stewart and Whiteley, 1997; Crewe and King, 1994). Party leader image variables typically are closely related to measures of party support in analyses of aggregate time series data. For example, over the 1979-1996 period, prime ministerial approval and governing party support track one other very closely, the correlation between the two series being fully +.91 (see Figure 1B). Impressive statistical relationships such as this have led some analysts to argue that including prime ministerial approval ratings in models of governing party support is tautological (see, e.g., Sanders et al., 1987), thus providing support for the traditional claim that party leader images matter little, if at all, as determinants of party support in Britain. Clarke and Stewart (1995) respond that strong correlations do not necessarily indicate that two variables are measuring the same thing, and that powerful (causal) relationships between conceptually distinct variables are exactly what political scientists wish to discover. Additionally, they note that multivariate analyses of cross-sectional survey data from the British election studies (BES) also consistently indicate that voters' feelings about party leaders have sizable effects on electoral choice. This latter controversy has motivated interest in the concept of cointegration and the specification of error correction models that enable analysts to capture possible long-run relationships between party and leader support (e.g., Clarke and Stewart, 1995; Clarke, Stewart and Whiteley, 1997). Below, we show that the traditional cointegration assumption undergirding these models is too restrictive, and that the relationship between prime ministerial approval and governing party support is best described as fractionally cointegrated. Before doing so, we discuss what fractional dynamics are, how they can arise, and why one might expect the British time series data to exhibit this property. Fractional Dynamics and Party Support The concept of fractional integration generalizes the crucial distinction between stationary and non-stationary data. Until recently, time-series analysts have made a "knife-edged" decision that their data are either stationary I(0) or non-stationary I(1), i.e., unit-root, processes (Box-Steffensmeier and 4 Smith, 1996). This decision is theoretically consequential. Choosing stationarity implies that one believes a series has a constant mean -stochastic innovations (shocks) are gradually "forgotten," and the series is mean reverting. In contrast, the unit-root choice entails the belief that a series has “perfect memory” -absent shocks, its value at time t is the same as its value at the previous period, t-1. However, because the data generating process (DGP) is stochastic, shocks do occur and are not discounted over time. As shocks accumulate, the unit-root series can meander widely, and is said to exhibit a stochastic trend. In the case of party support, such behavior contradicts theories and empirical evidence concerning the re-equilibriating tendencies of party systems in mature democracies (see, e.g., Bartoloni and Mair, 1990; Butler and Stokes, 1976; Stokes and Iversen, 1966). The stationarity-non-stationarity decision also has important consequences for analyses of factors affecting party support. As is well known, the possibility of spurious regressions -concluding that there is a statistically significant relationship between variables that are actually independent of one another -increases dramatically when nonstationary variables are analyzed in level form (Granger and Newbold, 1974). Recent research indicates that similar threats to inference obtain for near-integrated series, i.e., series for which the DGP is very strongly autoregressive (DeBoef and Granato, 1997, 2000). Assuming nonstationarity also can cause problems. When diagnostics indicate the presence of a unit root, the conventional remedy is to difference the series. However, differencing is not a cost-free option because one will not be able to detect long-run relationships among the variables being analyzed (Smith, 1993). Analysts concerned about this possibility typically include error correction mechanisms (ECMs) in their models, asserting that the ECM specification is warranted by diagnostics indicating that variables of interest travel together in the long run. Such variables are said to cointegrate (Engle and Granger, 1987). In recent years, researchers have challenged the assumption that it is necessary for time series variables to have integer orders of integration. Influenced by pioneering econometric research by Granger and Joyeux (1980) and Hosking (1981), analyses of American time series data by Box5 Steffensmeier and Smith (1996, 1998) and Lebo et al. (2000) indicate that theoretically important variables such as presidential approval and party identification are fractionally integrated. Series exhibiting fractional dynamics may be analyzed by generalizing the familiar ARIMA models developed by Box and Jenkins (1970). The DGP of a (fractionally) integrated series Y is: φ(B)(1-B)Yt = θ(B)εt where εt ~N(0,σ ) (1) where B is the backshift operator such that Bεt = εt-k, φ(B) represents a stationary autoregressive (AR) process, θ(B) represents an invertible moving-average (MA) process, and d, the fractional differencing parameter, measures the degree of integration of Y. If one assigns integer values to d , the traditional ARIMA framework is maintained. Fractionally integrated processes relax the assumption that d is an integer and permit its value to lie on the real number line between 0 and 1. Series in this middle ground possess long memory because they exhibit statistically significant correlations across long time lags, and they cease to be variance and covariance stationary when 0.5 < d < 1 (Baillie 1996; Beran, 1992). It often is assumed that when d > .5, a fractionally integrated series remains mean reverting. Phillips and Xiao (1999:33-34) recently have pointed out that the latter is not true in the conventional sense, although shocks do eventually decay if d < 1. As in the unit-root case, a fractionally integrated series can be rendered I(0) by differencing. For a fractionally integrated series, fractionalrather than integerdifferencing is employed. To illustrate the defining long-memory property imparted by fractional integration, we simulate a series with d value of .85, and calculate its autocorrelation function across 60 lags. We also calculate autocorrelation functions for a simulated random walk (a nonstationary series) and a simulated stationary, first-order autoregressive series. As Figure 2 shows, although the stationary series is specified to have a large AR1 coefficient (β1 = .85), its autocorrelation function dampens quickly. In contrast, the random walk's autocorrelation function has extremely large values at low lags, and statistical significance is maintained over the first 56 consecutive lags. As intuition would suggest, the fractionally 6 integrated series has intermediate (but decidedly nontrivial) persistence -its autocorrelations are greater (in absolute value) than those for the stationary series through the first 36 successive lags, but they always are less than those for the random walk. Indicative of its long memory, the fractional series autocorrelations remain statistically significant through 33 consecutive lags -nearly three years, if data were gathered on a monthly basis. (Figure 2 about here) Granger (1980) (see also Box-Steffensmeier and Smith, 1997; Wlezien, 2000) discusses the statistical basis by which a fractionally integrated series can be created, and demonstrates how aggregating individual-level heterogeneous behavior produces fractional dynamics. This heterogeneity exists with respect to varying autoregressive and moving average processes at the micro level. There are many political science time series variables that would seem to fit this characterization. Of particular interest here are British time series measures of party support, prime ministerial approval and subjective evaluations of national and personal economic conditions. These series are constructed by aggregating individual-level data from monthly national surveys. Recent analyses of public opinion formation (e.g., Sniderman, Brody and Tetlock, 1991; Zaller, 1992; see also Krause, 1998, 2000; Krause and Granato, 1988) suggest that in each such survey there will be individuals with a diverse array of autoregressive tendencies. Although much remains to be learned about the sources of these differences, it may be hypothesized that they are products of varying levels of political information and cognitive capacity, as well as variation in partisan attachments and other core political beliefs and values. For example, Zaller's (1992) contention that political knowledge is normally distributed across the electorate implies that voters respond differently to political and economic events. Poorly informed people may repeatedly change their opinions based on very little new information, whereas the opinions of more politically knowledgeable individuals will vary less, barring some major event (e.g., the Falklands War, the ERM crisis). 7 It also may be hypothesized that the degree of persistence manifested by different aggregate series will vary. For example, a series such as prime ministerial approval may exhibit unit-root behavior because its dynamics are dominated by the presence of a sizable group of voters who neither "forgive nor forget." Perhaps because the relationship between voters and prime minister has a specificity and interpersonal quality that is absent when more amorphous and impersonal entities such as political parties or the economy are considered, some voters' images of a prime minister will constitute a simple accumulation of salient information about that person's behavior in office. In contrast, aggregate subjective economic evaluation series are likely to be somewhat shorter memoried as a vast majority of voters update their views of the state of the economy and their personal economic circumstances, while simultaneously discounting previously acquired information. However, it is unlikely that these processes are homogeneous because voters have different locations in the economy, and "media, markets and mates" supply differing types and quantities of economic information. As demonstrated by Granger (1980), the result will be that aggregate economic evaluation series will exhibit long, but imperfect, memory. More generally, it is plausible that fractional dynamics (of varying degree) may characterize many of the series studied by political economists. It bears emphasis that such series should not be treated as if they were "garden variety" stationary (i.e., I(0)) processes. In cases where d > .5, failing to account for fractional dynamics will lead to problems in regression analyses including biased standard errors and Durbin-Watson test statistics, and an enhanced possibility of spurious regressions (Lebo et al. 2000; see also Box-Steffensmeier and Tomlinson, 2000). How should analysts concerned about the possibility of fractional dynamics in their data proceed? Prior to developing and testing multivariate models, two steps analogous to those employed in a traditional ARIMA modeling exercise should be followed. First, rather than assuming variables are integrated at integer levels, e.g., I(0) or I(1), the memory characteristics of each series should be identified empirically. The identification process 8 involves estimating the fractional differencing parameter d and its standard error for each series. Second, before employing the data for estimating the parameters in multivariate models, each series should be differenced by its estimated d value. In the next section, we describe the British data and estimate d for each series. Data and d We employ data from 210 consecutive monthly British Gallup polls spanning the July 1979 December 1996 period. The variables are: governing (Conservative) party support, prime ministerial approval, and four subjective economic evaluations -personal expectations, personal retrospections, national expectations and national retrospections. Each variable is bounded (between 0 and 100 in the cases of governing party support and prime ministerial approval, and between -100 and +100 in the case of the economic evaluations) and each is an aggregation of individual-level data. These characteristics suggest that the series are fractional integrated. Estimates of d for each series are consistent with this hypothesis. Although several techniques may be used for this purpose, we employ Robinson’s semi-parametric estimator and Sowell’s maximum likelihood estimator. To determine the ARFIMA (autoregressive, fractionally integrated, moving average) model that best accounts for both shortand long-term dynamics of each series, d is estimated for each variable in a (0, d, 0) model and in (p, d, q) models containing up to three autoregressive and three moving average parameters (at lags 1, 2 and 3). The Akaike Information Criterion and the Schwartz Bayesian Criterion (see, e.g., Franses, 1998:59) are employed to distinguish which of the 16 rival models per series is superior. According to both criteria, a (0, d, 0) model is best for each of the six variables. Table 1 presents the values of d computed using Robinson’s and Sowell’s estimators for these (0, d, 0) models. The two procedures yield very similar results -d does not equal zero for any series (p < .05), and it does not equal 1.0 for any series except possibly prime ministerial satisfaction. Although the 9 point estimates for d of .96 and .91 for the latter variable indicate fractional integration, each is within two standard errors of 1.0. Thus, one cannot reject the unit-root hypothesis (p = .05). Based on these results, we conclude that fractional integration characterizes at least five of the six series. In the next section, we use these findings to help specify error correction models of the determinants of governing party support. (Table 1 about here) Fractional Cointegration and Fractional Error Correction Models Just as the likelihood of finding fractional dynamics in political variables motivates a rethinking of the knife-edged distinction between stationarity and nonstationarity, the methodology of cointegration must likewise be rethought in light of the possibility of long-memory equilibrium relationships between variables. Among the advantages of this new methodology is the ability to specify and estimate more precisely equilibrium relationships among variables of interest. Thus, along with ARFIMA modeling procedures, fractional cointegration techniques improve the chances of resolving longstanding controversies regarding rival models of the dynamics of important variables such as governing party support. The concept of fractional cointegration is most easily understood as a more general form of traditional cointegration. Cointegration involves a long-term relationship between two or more variables (Granger 1981; Engle and Granger 1987, 1991). This relationship is one such that shocks to one variable at any time t are re-equilibrated at a constant rate in subsequent time periods. Standard operating procedure for diagnosing cointegration between two variables involves, first, determining if the original series are nonstationary (I(1)) and, second, determining if the residuals from a regression of one of the variables on the other are stationary (I(0)). Unit-root tests are employed to assess the order of integration of the individual variables and the regression residuals. As noted above, the problem with this strategy is that unit-root tests have low power, making it 10 difficult to reject the null of nonstationarity when d is large but less than 1. This, and the possibility that many political and economic series may have such d values, erodes confidence in the conventional cointegration testing methodology (Barkoulas and Baum 1997; Dueker and Startz 1998). However, as Cheung and Lai (1993) observe, the requirement that the residuals of a cointegrating regression are I(0) is unnecessary. It is sufficient that these equilibrium errors are not infinitely persistent. As long as the residuals produced by regressing one fractionally integrated series on another one are of a lower order of integration than are either of the fractionally integrated series, those series can be said to be in dynamic equilibrium and to fractionally cointegrate. Hence, one may fuse the methodology of cointegration with the concept of fractional integration. Extending the methodology of cointegration to include fractional cases enables one to generalize the concept of an error correction mechanism (ECM). Following Engle and Granger (1987), standard practice is to specify an error correction mechanism when two series cointegrate in the traditional sense, thereby enabling one to study long-run relationships among non-stationary variables. This methodology assumes that the residuals of the cointegrating regression, and hence, the ECM, are I(0). Relaxing this assumption as described above to allow for the possibility of fractional cointegration is a natural extension of this methodology. Rather than using (non)stationarity tests to decide whether the ECM is I(0) or I(1), estimating the d parameter for the ECM permits a more nuanced understanding of the nature of the equilibrium relationship. The range of possible behavior for the ECM is greatly increased -defining an ECM as possessing long memory simply implies that shocks to one variable are reequilibrated at a slower rate than the exponential decay of a conventional ARMA process. The existence of a fractionally integrated ECM does present a problem to the multivariate modeler, however. If one wishes to include a fractionally integrated ECM as an independent variable in a larger equation, one should be wary of threats to inference generated by its long-memory characteristics. Monte Carlo analyses indicate that even when a dependent variable and other 11 independent variables are I(0), a single fractionally integrated independent variable can threaten inference, as both its coefficient estimates and standard errors are biased downwards (Lebo, 2000). Interpretation of an ECM’s coefficient as a measure of the strength and significance of the adjustment mechanism will be flawed if one fails to account for fractional cointegration. Hence, it is important to ensure that ECMs are I(0) before including them in one's model. As discussed below in the analysis of British governing party support, this can be done by fractionally differencing the residuals of the cointegrating regression by their estimated value of d. A full multivariate ARFIMA model with an ECM specification thereby can be estimated without worrying about threats to inference posed by the presence of a fractionally integrated component. Prior to estimating such models with British data, prime ministerial approval and Conservative party support are tested for fractional cointegration. As noted above, previous studies of the relationship between public support for British parties and their leaders have employed an ECM to capture a long-run equilibrium relationship that exists between the two variables. However, given the possibility of fractional cointegration, these studies may have over-estimated the rate at which leader approval and party support return to equilibrium following a shock. To investigate this possibility, a cointegrating regression is performed between Conservative party vote intentions and prime ministerial approval and the residuals are tested for stationarity. The d estimate for the residuals is .59 (s.e. = .059). This is over four standard errors below the d values for either Conservative vote intentions or prime ministerial approval (see Table 1). Also, one cannot reject (p = .05) the hypothesis that d for the residual series is less that .5, i.e., .5 (.059 x 2) = .472. Thus, recalling that series with d's less .5 indicate stationarity (see p. 7 above), one cannot reject the possibility that the residuals constitute a stationary series. Taken together, these results are consistent with the hypothesis that the two series are fractionally cointegrated. As discussed above (p. 7, see also Figure 2), fractionally integrated series are said to be longmemoried because they manifest statistically significant correlations at long lags. The presence of such 12 long-memory in the residuals of the governing party support prime ministerial approval regression tells us that a shock to one of these variables will persist longer than the geometric decay of a standard cointegrating system. Hence, assuming a standard cointegrating (d = 0) relationship between prime ministerial approval and governing party support may over-estimate the speed with which the two variables return to their long-term equilibrium following a shock. The implication is that, despite the strong correlation between the two variables, voters' opinions about the governing party and the prime minister can diverge for long periods before a shock dissipates and the variables return to equilibrium. Next, we will show how accounting for this long-run relationship in a multivariate context can improve model specification and enhance confidence in empirical findings. Fractional Error Correction Models of Governing Party Support We now specify and estimate fractional error correction models of governing party support. To ensure that threats to inference are minimized, governing party support (Conservative vote intentions), prime ministerial approval, and the four subjective measures are rendered I(0) by fractionally differencing each by its own value of d (see Table 1). This procedure is analogous to conventional (full) differencing in ARIMA modelling, and is easily implemented using specialized filtering programs. To capture the long-term equilibrium relationship between Conservative support and prime ministerial approval, the error correction term (lagged one period) is included in the multivariate models. Before doing so, this variable (the residuals of the cointegrating regression of Conservative support on prime ministerial approval) is fractionally differenced by its d estimate (.59). Other independent variables in the party support models include interventions for the Falklands War, the introduction of the poll tax, the replacement of Margaret Thatcher by John Major as prime minister, Major's reselection as Conservative leader, the ERM crisis and a summary variable to capture miscellaneous political events. The error correction model of governing party support is: GOVFt = β0 +β1PMFt + β2EVALFt αFECMt-1 + Gβ3-kSHOCKSt-i + εt (2) 13 where: GOVF = fractionally differenced governing party support; PMF = fractionally differenced prime ministerial approval; EVALF = fractionally differenced economic evaluations (personal prospective, personal retrospective, national prospective, national retrospective); FECM = fractional error correction mechanism estimated from first-stage regression of GOVF on PMF; SHOCKS = interventions (Falklands War, etc.); β, α = model parameters (to be estimated); ε = stochastic error term (~N(0, σ). Four separate analyses, using each of the subjective economic evaluations in turn, are performed. Column 1 of Table 2 shows prime ministerial approval, personal economic expectations , the error correction mechanism, and each of the intervention variables have statistically significant effects (p < .05). Column 2 shows a nearly identical pattern with personal retrospective economic evaluations being significant. However, columns 3 and 4 show that prospective and retrospective judgments of national economic conditions fail to achieve statistical significance (t ratios = 1.16, and 0.49,respectively). In the four models, the estimated coefficient for the error correction mechanism is properly (negatively) signed, and varies by only .01 (from -.27 to -.28), thereby indicating that shocks to governing party support are slowly eroded by the (fractionally) cointegrating relationship with prime ministerial approval. A variety of such shocks were at work in the period under consideration. As one would anticipate, the Falklands War, the replacement of Thatcher by Major, and Major's reselection as party leader boosted the Conservatives' vote intention share, whereas the imposition of the poll tax and the currency crisis diminished it. Miscellaneous events also were influential. (Table 2 about here) The vote intention analyses indicate that both personal economic expectations and personal economic retrospections exercise significant effects on governing party (Conservative) support. This finding, and failure of prospective or retrospective evaluations of the national economy to achieve statistical significance, are consistent with the Essex model's emphasis on the importance of personal 14 economic judgments. It also contradicts -for the British case at least -the arguments of the many analysts who have advanced claims on behalf of national prospective or, more frequently, national retrospective economic judgments. But is one of the personal economic evaluation models preferable to the other? Similar to models employing standard cointegration methodology, one may address this question by performing encompassing tests to compare the performance of rival non-nested models (Charemza and Deadman, 1997:ch. 8). Empirically, a battery of encompassing tests (Davidson and MacKinnon, 1981; Mizon and Richard, 1986) show that neither personal economic evaluation model encompasses the other (Table 3). This result, in turn, suggests that although the fractional error correction models indicate that personal economic evaluations perform better than national ones, the time horizon over which the former operate remains ambiguous. (Table 3 about here) One-Step Estimation: Some analysts have argued that considerations of statistical efficiency make it preferable to estimate error correction model parameters using a one-step, rather than a two-step, procedure (e.g., DeBoef, 2001; see also Beck, 1992, 1993). The implicit assumption is that all variables of interest, including those in the error correction mechanism, are stationary -either as "supplied by nature," or by the application of suitable differencing procedures. As noted, when variables are fractionally integrated, stationarity can be achieved by filtering them using their estimated d values. Once governing party support, prime ministerial approval and the economic evaluations have been filtered by their d values, we may specify the one-step fractional error correction model of governing party support as: GOVFt = β0 +β1PMFt + β2EVALFt α(GOVFt-1 γPMFt-1) + Gβ3-kSHOCKSt-i + εt (3) where GOVF, PMF, EVALF, SHOCKS and ε are as described above (p. 13), and β, α,and γ are parameters to be estimated. Parameter estimates (see Table 4) for different variants of this model using the four economic 15 evaluation variables are very similar to those for the two-step models. Personal economic expectations and personal economic retrospections have significant effects ( t ratios = 2.17 and 1.75, respectively), but neither national economic expectations nor national economic retrospections achieve significance (t ratios = 1.09 and 0.35, respectively). Once more, encompassing tests fail to indicate that the personal expectations model is preferable to its personal retrospective rival or vice versa (data not shown). Also as in the two-step models, the adjustment parameters (α) are significant in each model (p < .001), as are various intervention variables. The γ coefficients capturing the fractional cointegrating relationship between governing party support and prime ministerial approval in the one-step models are significant as well. (Table 4 about here) The several coefficients in the one-step models all carry the same signs and are very similar in magnitude to their two-step counterparts. Adjustment parameters for the one-step models are slightly smaller than their two-step counterparts (by .045 on average), but Wald tests show that the differences are not statistically significant (e.g., χ = 0.38, df = 1, p = .54 for the personal expectations model). This also is true for the coefficients (γ's) measuring the long-run fractional cointegrating relationships between governing party support and prime ministerial approval -these coefficients are smaller (mean difference = .096) than that (.648) estimated in the two-step procedure, but the differences are not significant (e.g., χ = 0.37, df = 1, p = .54 for the personal expectations model). A more general test constraining all parameters in the one-step models to be equal to their counterparts in the two-step ones reveals that the differences are not significant (e.g., χ = 1.53, df = 11, p = 1.00 for the personal expectations model). Thus, alternative estimation procedures for rival models incorporating the fractionally integrated variables tell essentially the same story about the determinants of governing party support in Britain. In the conclusion, we will reprise the highlights of this story, and suggest avenues for future research. 16 Conclusion: New Methods, New Research Agendas The past decade has witnessed substantial theoretical and methodological innovation in studies of the dynamics of party support in Britain and other mature democracies. Sensitive to the serious threats to inference posed by nonstationary data, researchers have tested their data for unit-roots, and specified error correction models of the impact of economic evaluations and party leader images on vote intentions. Some critics have been slow to accept the results of these analyses, issuing caveats that unitroot tests have low statistical power, and that variables such as voting intentions, prime ministerial approval and economic evaluations are bounded (either by nature or construction) and, hence, cannot be truly nonstationary at the I(1) level. In the present paper, we have argued that the concepts of fractional integration and fractional cointegration enable one to generalize the conventional integrationcointegration-error correction methodology to take account of these criticisms. Estimating the fractional differencing parameter d for key variables in a model of the dynamics of British governing party support reveals that threats to inference would arise if these variables were treated as stationary I(0) processes. The d parameters indicate that all of the variables are fractionally integrated and nonstationary. Governing party support and prime ministerial approval are fractionally cointegrated. These findings warrant the specification of models that generalize the error correction approach used in previous studies by relaxing the assumption of integer orders of integration and cointegration. Estimating the parameters in these new fractional error correction models using either twoor one-step procedures documents the existence of a governing party vote intention-prime ministerial approval (fractional) error correction mechanism. The size of the ECM coefficient indicates that although these two variables travel in tandem in the long run, shocks caused by economic or political interventions can drive them apart for lengthy periods of time. Parameter estimates also reveal that evaluations of personal economic conditions are significant predictors of governing party support, but evaluations of the national economy are not significant. 17 The fractional error correction models thus help to resolve two longstanding controversies in the literature on party support in Britain. Pace the (perhaps mythical) era when social class "was everything" and "all else was embellishment and detail" (Pulzer, 1967), these models forcefully testify that party leader images have been important elements in the calculus of party support in Britain since at least the late 1970s. Although the looseness of the (fractional) cointegrating relationship between prime ministerial approval and governing party support indicates that various economic and political interventions can have sizable short-term effects, it is equally clear that leader images matter a great deal. Model estimates also testify that personal, not national, economic evaluations matter – at least in terms of direct effects on party support. However, this latter finding is not an unequivocal endorsement of the widely cited Essex model. Encompassing tests pitting personal expectations against personal retrospections are inconclusive, so therefore we caution that additional inquiry is in order. When researchers conduct future studies of the dynamics of party support, fractional (co)integration methods will prove helpful. As demonstrated in this paper, these methods are easily implemented in the context of generalized versions of familiar ARIMA and error correction models, and they enable analysts to avoid the methodologically contentious and often theoretically unattractive assumption that important political and economic variables evolve through time with integer orders of integration. The concepts of fractional integration and fractional cointegration also have important heuristic value in that they encourage researchers to offer conjectures concerning micro-level processes responsible for fractional aggregate-level phenomena. Individual-level heterogeneity is a likely source of aggregate-level fractional integration. Specifying such heterogeneity in the context of parsimonious, theoretically attractive, models of party support poses an interesting and important challenge to political economists in Britain and elsewhere. 18 ENDNOTES 1. For literature reviews, see Lewis-Beck (1988); Norpoth, Lewis-Beck and Lafay (1991). 2. A weak or wide-sense stationary series has a constant mean, a constant variance and a constant covariance for any lag k, i.e., E(yt) = μ; E[(yt μ)] = γ0; E[(yt μ)(yt-k μ)] = γk for all t and k (Franses, 1998:68). The simplest unit-root series is the random walk: Yt = 1.0*Yt-1 + εt, where εt ~ N(0,σ). Recursive substitution shows that the random walk may be expressed as Yt = Y0 + Σεi where Y0 is the initial value of Y and Σεi is the sum of all shocks from time 1 through time t. 3. Thus, a near-integrated process is: Yt = β1Yt-1 +εt where εt ~ N(0,σ), β1 = 1.0 + ζ, and ζ is a small negative number, e.g., -.01. 4. The invertibility condition implies that |θ| < 1 so that an MA process can be represented as an infinite autoregression with the sum of the parameters being less than 4. See Mills (1993:14). 5. When d = 0, a series exhibits mean reversion, finite variance, and covariance stationarity (see note 2 above), and it can be modeled using combinations of moving-average and autoregressive parameters in an ARMA (p,0,q) specification. When d = 1, the variable is a unit-root process characterized by mean, variance, and covariance nonstationarity. In this case, the series can be rendered stationary by first differencing and then modeled using AR and MA parameters in an ARIMA (p,1,q) format. 6. The simplest fractionally integrated process is fractional Gaussian noise, i.e., (1 B)Yt = εt, where εt ~ N(0,σ). (1 B) can be expanded (infinitely) as 1 dL (1/2)d(1 d)B (1/6)d(1 d)(2 d)B ...(1/j!)d(1 d)(2 d)...((j 1) d)B -.... See Franses (1998:79). 7. The three series were simulated using RATS 4.3, with the fractional series being created using the ARFSIM.PRG procedure. 8. For example, variation in extent of concern with different aspects of the macroeconomy, e.g., inflation versus unemployment for middleand working-class persons respectively, is a possible source of heterogeneity in economic evaluations. See, e.g., Hibbs (1982a, 1982b). Similarly, as Sanders (1991) observes, one would expect concern with interest rates to vary according to whether a voter is a mortgage holder or a tenant. 9. Estimates of the d parameter with aggregate time series data will not enable one to determine the proportions of voters manifesting various patterns of stability and change in economic and political evaluations. Individual-level analyses of multi-wave panel data are required for this purpose. Promising in this regard are mixed Markov latent class models which allow for multiple "mover" and "stayer" chains and take account of random measurement error in observed variables. See, e.g., Clarke, Stewart and Whiteley (1999). 10. In the run-up to the 1997 general election (January 1997), Gallup changed its survey methodology from in-person interviews with quota samples to telephone interviews with random digit dialing sample selection procedures. To avoid possible mode effects artefacts, we terminate the time series in December 1996. The decision is innocuous – only four of a possible 214 observations are lost. 11. The Gallup vote intention questions are: (a) "If there were a General Election tomorrow, which party 19 would you support?" (b) [If "don't know"] "Which party would you be most inclined to vote for?" Conservative vote intentions are calculated as the sum of the percentages of respondents answering "Conservative" to (a) or (b). 12. The Gallup prime ministerial approval question is: "Are you satisfied or dissatisfied with [name] as Prime Minister?" Prime ministerial approval is measured as the percentage saying they are "satisfied." 13. The subjective economic evaluation questions are: (a) personal prospections "How do you think the financial situation of your household will change over the next 12 months?" (b) personal retrospections "How does the financial situation of your household now compare with what it was 12 months ago?" (c) national prospections "How do you think the general economic situation in this country will develop over the next 12 months?" (d) national retrospections "How do you think the general economic situation in this country has changed over the last 12 months?" The response categories are: "get(got) a lot better," "get a little better," "stay the same," "get a little worse," "get a lot worse." The economic evaluation variables are constructed by subtracting the percentage offering negative responses from the percentage offering positive ones. 14. Our diagnoses of the dynamic properties of the time series also included standard unit-root tests. As Maddala and Kim (1998) emphasize, these tests have limited power in the face of fractional alternatives, and different tests can produce contradictory results. For example, the Dickey-Fuller test (Dickey and Fuller, 1979) indicates that party support is nonstationary. However, the variance ratio test (Cochrane, 1988) rejects the hypothesis that d equals 1.0, and the KPSS test (Kwiatkowski et al., 1992) rejects the hypothesis that d equals 0.0. The tests speak more consistently when prime ministerial approval and the economic evaluation series are analyzed. For prime ministerial approval, the Dickey-Fuller and variance ratio tests fails to reject the null hypothesis of nonstationarity, and the KPSS test rejects the null of stationarity. For economic evaluations, Dickey-Fuller and variance ratio test reject the null hypothesis of nonstationarity, and the KPSS test fails to reject the null of stationarity. 15. Procedures for estimating d include, inter alia, semiparametric estimation (Geweke and Porter-Hudak 1983; Robinson 1995), approximate maximum likelihood estimation in the frequency domain (Fox and Taqqu 1986), exact maximum likelihood estimation in the time domain (Sowell 1992), Bayesian techniques (Koop, Ley, Osiewalski, and Steel 1995), bootstrapping (Andersson and Gredenhoff 1998), and autoregression-based estimation (Franses, 1998: 78-80). 16. Robinson's estimator is calculated using the RATS RGSE.SRC procedure; Sowell's is calculated using OX. RATS and OX ARFIMA software may be downloaded from the following web sites: www.estima.com (RATS); www.nuff.ox.ac.uk/Users/Doornik (OX). 17. For example, if X and Y are cointegrated and the re-equilibration rate is .5, ceteris paribus, a 10-point shock to Y at time t would be eroded by its cointegrating relationship with X by 50% in each subsequent time period, i.e., to 5 points at t+1, to 2.5 points at t+2, to 1.25 points at t+3, etc.. 18. Technically, a bivariate fractional cointegrating relationship may be defined as follows: two variables, Zt and Xt where Zt is I(d) and Xt is I(b) are fractionally cointegrated if there is a function g(.) such that ηt = Zt +g(Xt) is I(s) and s < d. Thus, the existence of some function g(Xt) that reduces the order of integration for Zt establishes that Zt and Xt are fractionally cointegrated (Abadir and Taylor 1998:6). 20 19. The RATS FIF.SRC procedure is used to filter the series by their d values. The filtering operation produces a series of residuals which represent the difference between the original series and the values of the series predicted by their (0, d, 0) ARFIMA models. 20. The Falklands war, replacement of Thatcher by Major, the ERM crisis and Major reselection interventions are coded as 1 for the month in which the intervention occurred and 0 otherwise. Since the poll tax implementation is hypothesized to exert a temporary one-month effect, it is scored 1 for the month it occurred, -1 for the next month, and 0 otherwise. The miscellaneous intervention variable is scored +1 for months in which there was an intervention hypothesized to increase Conservative support, 1 for months in which there was an intervention hypothesized to decrease Conservative support, and 0 for other months. A list of interventions is available upon request. 21. The models are estimated using the OLS regression program in MICROFIT 4.0. MICROFIT 4.0 provides an extensive battery of regression diagnostics, as well as encompassing tests for rival nonnested models. 22. With the exception of normality of residuals, all regression diagnostic tests are statistically insignificant. Thus, there is no evidence of residual autocorrelation (first-order or general), heteroscedasticity (ARCH or general) or functional form misspecification. Also, CUSUM and CUSUM squared tests (not shown) are consistent with the hypothesis of parameter constancy. 23. Since prime ministerial approval enters the Conservative support model without a lag, the possibility of simultaneity bias is investigated. Test results for both the two-step and one-step estimates of the party support models are consistent with the hypothesis that prime ministerial approval is weakly exogeneous to Conservative vote intentions and, hence, it is possible to perform single-equation estimation of the parameters in the Conservative support model. For example, for the two-step estimates, adding the ECM to a model of the determinants of prime ministerial approval (personal expectations variant) shows that the ECM is statistically insignificant (t = 1.096). Adding the residuals of the prime ministerial model to the Conservative support model (personal expectations variant) also yields an insignificant coefficient (t = 1.295). On weak exogeneity tests in the context of error-correction models, see Charemza and Deadman (1997:ch.7). Details re: the prime ministerial approval models are available upon request. 24. 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