David Byers
email: byersjd@cf.ac.uk

James Davidson

David Peel

C22 -Time-Series Models < C2 -Econometric Methods: Single Equation Models < C -Mathematical and Quantitative Methods

Keywords: fractional process, political support Fractional Integration, Long Memory, Opinion Polls, Partisanship, Political Support JEL Classification: C22, D72

 on long memory in support for the Conservative and Labour Parties in the UK using longer samples and additional poll series. It finds continuing support for the ARFIMA(0,d,0) model though with somewhat smaller values of the long memory parameter. We find that the move to telephone polling in the mid-1990s has no apparent effect on the estimated value of d for either party. Finally, we find that we cannot reject the hypotheses that the parties share a common long memory parameter which we estimate at around 0.65.

Introduction

Interest in the stochastic properties of time series measures of political support was stimulated by models of politico-economic interaction. [START_REF] Frey | A politico-economic model of the United Kingdom[END_REF][START_REF] Schneider | Politico-economic models of macroeconomic policy: a review of the empirical evidence[END_REF], Holden and Peel, 1985, Minford. and Peel 1982). Much of the early empirical work in this area relied upon the assumption that measures of party support or approval of the government were trendstationary, an assumption that appeared inconsistent with rational behaviour. Byers, Davidson andPeel (1997, 2002) -hereafter BDP -proposed a long memory model of aggregate support for political parties and estimated it using Gallup poll data for the Conservative and Labour parties in the UK. The approach relies upon explicit aggregation of individual support to derive an aggregate function in which the influence of events on party support is highly persistent. Analysis of data on a number of parties in other countries in [START_REF] Byers | The dynamics of aggregate political popularity: evidence from eight countries[END_REF] provided further support for the model. Appealingly, for almost all parties considered, a simple one parameter model with uncorrelated innovations captured the observed intertemporal dependence in political popularity. Formally, the series can be adequately modelled as pure fractionallyintegrated processes with long memory parameter, d. [START_REF] Box-Steffensmeier | The Dynamics of Aggregate Partisanship[END_REF] Thirdly, we can examine the effect of different polling methods. Partly as a response to the perceived failure to correctly forecast the outcome of the 1992 General Election, each of the polling organisations changed their sampling methods in the 1990s. ICM began telephone polling in November 1995 and Gallup in January 1997. MORI remained committed to quota sampling but changed its procedures. In addition to presenting results for each of the series, we use a multivariate approach on a common sample of Gallup and MORI data to test for equality of the long memory parameter across parties and organisations.

The Model

Let i t x be a variable which takes the value 1 in any time period, t, when individual i supports Party X and zero otherwise. Aggregate support for Party X in period t can be measured as the average value of

( ) 0 1 i i t i E y = If 0 i , ( ) 0 i i t E y . If 0 0 i i = = the
1 d t t s L z =
where t z is a stationary stochastic process and

1 d v =
. The restriction that i is strictly less than one, which is required to establish the result, implies that an individual's party support is mean reverting. Absent new shocks it would eventually return to some value, ( )

i t
E y , which is independent of the previous history of news. Individuals with 1 i = would simply stick with the political views which they had when the shock process was turned off. and so the restriction amounts to assuming that individuals exhibit a preference for some particular position on the political spectrum.

The Data and Estimation Procedures

The data analysed here are the log-odds ratios of monthly series on voting intentions carried out by Gallup, ICM and MORI. Each of these organisations asks a similar question to gauge support for the various parties. For instance, Gallup's question is 'If there were a General election tomorrow, which party would you vote for?' Those answering 'Don't Know' are asked to indicate which party they would be most inclined to vote for and the figures are then adjusted to add up to 100%. For our purposes the various technical issues which are used in the attempt to ensure that the sample is properly representative of the population as a whole are not immediately relevant though it should be noted that the published figures often include adjustments designed to improve the performance of a poll as a forecast of electoral outcomes and so are not the 'raw' numbers. The Gallup data is taken from King and Wybrow (2001). The continuous monthly series starts in January 1948 and ends in December 2000. The MORI series starts in August 1979 and continues to be available. The data is taken from the MORI website (http://www.mori.com/polls/trends.shtm). The ICM series runs on a continuous basis from December 1987 and was taken from the ICM polls archive (http://www.icmresearch.co.uk/reviews/polls-archive.asp). In the case of the latter two polls, the sample terminates at April 2005. Although we use the word 'continuous' there are, in fact, gaps in all of the polls in the form of data missing for particular months. We interpolate these by simply taking an average of the preceding and succeeding months 1 . A further issue is the choice of poll when there are several polls in a month, as happens close to General Elections.

When this occurs we use the poll which appears to have been carried out at the usual time of the month.

The estimation procedure which we use for the univariate analysis has two stages. Firstly, we use the Schwartz information criterion to select an appropriate model of the ARFIMA(p,d,q) class ( )( ) ( )

1 d t t L L s L u =
where the autoregressive component, ( ) L , is an lag polynomial of order p and the moving average component, ( ) L , is a lag polynomial of order q We compare models over a range of values for , 2, 2 p q p q + . For all of the series considered here, the SIC chooses the pure fractional process, ARFIMA (0,d,0). We than estimate the model using a maximum likelihood

Results

In Table 1 we present estimates of the long memory parameter for Conservative and Labour support as measured by Gallup, MORI and ICM. For the Gallup data, we estimate d for two samples, the period up to the adoption of telephone polling and for the complete sample running from 1948 to 2000. The period of telephone sampling is too short to provide a useable sub-sample. For ICM we estimate d for the sample as a whole and for sub-samples corresponding to the period before telephone sample and the period after. For comparison purposes we break the MORI sample at the same value and estimate d for these.

The estimates presented in Table 1 vary somewhat across polling organisations but this appears to be the result of different sample sizes. Leaving the 'telephonepolling' sample to one side, the estimated value of d falls as the sample size increases. In addition, there is evidence that estimated ds for Conservative and hoc manner by looking at local trends. Given the sample sizes we do not think that these procedures induce any noticeable bias in the estimates. In Table 2 we present results from combining the available data to produce a continuous series from 1948 onwards. We use Gallup to the end of 1996 and either MORI or ICM from then onwards. The former series is consistent in the sense that it uses quota sampling throughout. However, as we have seen, the introduction of telephone sampling seems to have little of no effect. The two combined series produce effectively identical estimates of d for the Conservatives and for Labour and also suggest that a single value of d can be used to characterise both processes.

To test for equality between the various ds we estimate a four variable Vector

Conclusion

In this paper we have used longer samples and additional poll series to extend the results of [START_REF] Byers | Modelling political popularity: an analysis of long range dependence in opinion poll series[END_REF] on long memory in support for the Conservative and Labour Parties in the UK. We find continuing support for the model though the estimates for our larger samples suggest somewhat smaller values of d than reported by BDP. For instance, the longer Gallup series produces estimates of 0.707 and 0.706 for Conservatives and Labour, respectively, compared with the earlier estimates of 0.779 and 0.726. We find that the move to telephone polling in the mid-1990s has no apparent effect on the estimated value of d for either party. Finally, we find that we cannot reject the hypotheses that the parties share a common long memory parameter which we estimate at around 0.65

An interesting question raised by these results is why we find smaller ds. 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000

  estimate a similar model for the USA and obtain similar results as do Dolado et al for Spain This paper extends the earlier results for the Conservative and Labour parties in the UK. Firstly, we use a longer sample period. The original paper used the Gallup 'Snapshot' series for the period September 1960 to May 1995. In this paper we use Gallup data from January 1948 to December 2000, the longest regularly sampled series of data available. Secondly, the passage of time means reasonable size are now available for the voting intentions surveys carried out by other survey organisations, in particular ICM and MORI.

  Labour get closer as the sample size increases. The other main feature of the tables is the dramatically lower value of d estimated for the 'telephone-polling' sample. However, since the results for MORI and ICM are similar, this would appear not to be the result of the change in sampling method. To investigate further, we estimate rolling regressions of sample size 100 using the 'Whittle' frequency domain estimator. The results are graphed in Figure1. Note that this estimates the d for the first differences of the series. The vertical line in the graphs marks the start of telephone sampling by ICM. It is clear that the introduction of telephone polling is not associated with a change in the estimated d. There is a fall in the d for ICM but it occurs much later, at the start of 2001.The estimated d for the MORI series shows no sharp change but does fall steadily. The reasons for these results merits further investigation.

ARFIMA( 0

 0 ,d,0) model using the Gallup and MORI series over a common sample from August 1978 to December 2000. This exploits the strong correlations between the contemporaneous values of the series. The results are given in Table

  top part of the Table reports the estimated value of d for the four series and the middle part reports all possible pairwise equality tests. The numbers above the principal diagonal are the Wald test statistic, distributed as the pair -and the prob-value -the lower figure. The numbers below the principal diagonal are the constrained estimates of d. None of the pairwise nulls is rejected. At the bottom of the Table we report the Wald test for equality of all the ds. This null also fails to be rejected. Imposing the constraint we estimate the common value of d as 0.654.

  Figure 1: Rolling Regressions

  estimator. Since there are quite a lot of apparent outliers in the data, suggesting that the underlying 'shock' process is fat-tailed, we assume that t u has Student's t distribution. This has the effect of giving less weight to observations which are relatively far from the centre of the distribution. Though it makes little difference to the estimates of d, we estimate the model in first differences, thereby obtaining an estimate of 1-d in the stationary process for t s . We add 1 to get the results reported below. The multivariate models are estimated by Least GeneralisedVariance. All estimation and testing was carried out using James Davidson's Time Series Modelling package. For details seeDavidson (2005.

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1 On rare occasions there are two successive missing values. These were adjusted in a rather ad Page 7 of 15 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 1 : Univariate Estimates

 1 One possibility is that one requires a very large sample to remove 'small sample' bias in the estimator. Another, suggested by the graphs of the rolling regression estimates, is that d, itself, evolves through time or,. pushing things back one step,

	that the distribution of the underlying AR(1) parameters is changing. This is a
	matter for further investigation.
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Table 2 : Estimation of Combined Series

 2

		Gallup/ICM	Gallup/MORI
		CON	LAB	CON	LAB
	Intercept	-0.010	-0.002	-0.009	0.000
		0.004	0.004	0.004	0.004
	d	0.710	0.710	0.731	0.733
		0.038	0.035	0.032	0.032
	AR(1)	0.677	1.011	0.417	0.391
	LBQ(12)	17.291	16.509	15.393	12.054
	ARCH(1) 6.9445*	2.532	11.2589*	7.8288*
	Sample: Jan 1948-April 2005		n = 688
	Notes: See Table 1.

Table 3 : Multivariate Estimates Unrestricted Vector-ARFIMAI

 3

		Gallup	MORI
		CON	LAB	CON	LAB
	d	0.648	0.626	0.690	0.654
	s.e.	0.047	0.046	0.044	0.045
			Pairwise Equality Tests
		CON	LAB	CON	LAB
	CON		0.195	0.802	0.009
	Gallup		0.658	0.370	0.924
	LAB	0.637		1.398	0.399
	MORI			0.236	0.521
	CON	0.670	0.656		0.490
					0.483
	LAB	0.651	0.641	0.670
		Restricted Vector-ARFIMA
	d	0.654
	Wald Test for equality:		chisq(3)	1.511
				probval	0.679
	Sample: Aug 1979 -Dec 2000

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript