¹ Cycles of party dominance are only one among many important kinds of cyclic patterns we might find in politics, e.g., we may have cycles in the structure of ideological competition within a country, but in this article we will limit ourselves to cycles in party dominance.

² There are at least two key reasons that more seems to have been written about and theorized about party realignments and the possible cycling of party dominance in the United States than about these phenomena in the rest of the democratic world put together. First, Lipset and Rokkan’s ‘frozen cleavages’ thesis held a grip on theorizing about European party systems for several decades (see Lipset, S. M. and Rokkan, S., ‘Cleavage Structures, Party Systems and Voter Alignments: An Introduction’, in S. M. Lipset and S. Rokkan, eds, Party Systems and Voter Alignments (New York: The Free Press, 1967), pp. 1–64Google Scholar). If cleavages are frozen, then the kinds of ‘critical elections’ described by V. O. Key, which introduce new issue dimensions to restructure political competition, seemed irrelevant for understanding contemporary European politics (see Key, V. O. Jr, ‘A Theory of Critical Elections’, Journal of Politics, 17 (1955), 3–18CrossRefGoogle Scholar, and ‘Secular Realignment and the Party System’, Journal of Politics, 21 (1959), 198–210). Secondly, just as the literature on party identification was thought by many European scholars to have the label ‘made in the USA, not intended for the export market’ (see various essays in Budge, Ian, Crewe, Ivor and Farlie, Dennis, eds, Party Identification and Beyond (Chichester: Sussex: Wiley, 1976)Google Scholar), so, too, with the American literature on realignment. In particular, arguably the key ideas in this literature – the notion of regular alternation of two parties in power (exemplified in Samuel Lubell’s notion of one party as the sun and the other as the moon) with one dominant both in terms of votes and seats and in terms of defining the ideational structure of political competition – were seen as limited to the peculiarly American case of two-party competition.

³ We are deliberately not using the Laakso–Taagepera index to count how many parties there are, because it understates the importance of the third party for understanding outcomes in a first-past-the-post system (see, e.g., Taylor, P., Gudgin, G. and Johnston, R. J., ‘The Geography of Representation: A Review of Recent Findings’, in Bernard Grofman and Arend Lijphard, eds, Electoral Laws and their Political Consequences (New York: Agathon Press, 2003), pp. 183–192Google Scholar.).

⁴ Data for the United Kingdom include England, Scotland, Wales and the whole of Ireland (through 1918), and include only Northern Ireland (beginning with 1922). Data are taken from Rallings, Colin and Thrasher, Michael, eds, British Electoral Facts 1832–2006 (Farnham, Surrey: Ashgate, 2007)Google Scholar, and Leeke, Matthew, UK Election Statistics 1945–2003 (Research Paper 03/59, House of Commons Library, 2003)Google Scholar.

⁵ Lebo, Matthew and Norpoth, Helmut, ‘The PM and the Pendulum: Dynamic Forecasting of British Elections’, British Journal of Political Science, 37 (2007), 71–87CrossRefGoogle Scholar.

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⁷ In the nineteenth century, the interval between some elections was six years. In the twentieth century, the only exception to the five-year rule is the period 1935–45, when elections were suspended because of the Second World War.

⁸ Shumway, Robert and Stoffer, David, Time Series Analysis and Its Applications (New York: Springer, 2000)CrossRefGoogle Scholar.

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¹⁰ Periodograms were constructed in both JMP (see SAS Institute, JMP Start Statistics (Belmont, Calif.: Brooks-Cole, 2005)) and S-PLUS (see S-PLUS 6 for Windows Guide to Statistics (Seattle, Wash.: Insightful Corporation, 2001)) using detrended data and employing a 10 per cent split cosine bell taper. Tapering is used to reduce leakage, i.e., overestimated or irregular amplitudes in the vicinity of an amplitude peak (see S-PLUS, Vol. 2, p. 274, and Shumway and Stoffer, Time Series Analysis, pp. 247–8). Varying the moving-average frequency span from 3 to 7 resulted in variation in cycle-length estimates of no more than 3 years.

¹¹ Tests must be performed on unsmoothed data to obtain accurate significance levels.

¹² Fuller, Wayne, Introduction to Statistical Time Series, 2nd edn (New York: Wiley, 1996)Google Scholar.

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¹⁴ Udny Yule, George, ‘On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer’s Sunspot Numbers’, in Statistical Papers of George Udny Yule (New York: Hafner, 1971 [1927]), pp. 389–420Google Scholar.

¹⁵ Lebo and Norpoth, ‘Dynamic Forecasting of British Elections’.

¹⁶ Lebo and Norpoth (‘Dynamic Forecasting of British Elections’) performed the Yule calculations for the period 1929 to the present using vote rather than seat shares and without obtaining equally-spaced time points by interpolation; they obtain a cycle length of five elections, or an average of about 19 or 20 years. Our calculation using the Yule method for the same period for vote share yields a cycle length of 22 years without interpolation; 26 years, with interpolation. Since the method is intended for equally-spaced time points, the reliability of estimates based on raw data (without interpolation) is difficult to interpret. The interpolation-based estimate of 26 years, however, is similar to those in Table 1A.

¹⁷ Spectral analysis applied to 1928–2004 yields an estimate of twenty-seven years and both Fisher’s and Bartlett’s tests are significant for this era.

¹⁸ See Alesina, Alberto, Londregan, John and Rosenthal, Howard, ‘A Model of the Political Economy of the United States’, American Political Science Review, 87 (1993), 12–33CrossRefGoogle Scholar; Alesina, Alberto and Rosenthal, Howard, ‘Partisan Cycles in Congressional Elections and the Macroeconomy’, American Political Science Review, 83 (1989), 373–398CrossRefGoogle Scholar, and Partisan Politics: Divided Government, and the Economy (Cambridge: Cambridge University Press, 1995); Erikson, Robert, ‘Economic Conditions and the Congressional Vote: A Review of the Macrolevel Evidence’, American Journal of Political Science, 34 (1990), 373–399CrossRefGoogle Scholar.

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²⁴ Several studies on British politics have focused on forecasting rather than cycling patterns (see, e.g., Mughan, Anthony, ‘General Election Forecasting in Britain: A Comparison of Three Simple Models’, Electoral Studies, 6 (1987), 195–207CrossRefGoogle Scholar; Norpoth, Helmut, ‘Forecasting British Elections: A Dynamic Perspective’, Electoral Studies, 23 (2004), 297–305CrossRefGoogle Scholar; Lebo and Norpoth, ‘Dynamic Forecasting of British Elections’). Models of party support and turnout are addressed in Clarke, Harold D., Sanders, David, Stewart, Marianne C. and Whiteley, Paul, Political Choice in Britain (Oxford: Oxford University Press, 2004)CrossRefGoogle Scholar. In a Europe-wide study, Jerôme, Jerôme-Speziari and Lewis-Beck report evidence for joint cycles in 15 European nations, based on economic variables and the politics of economic integration (see Jerôme, Bruno, Jerôme-Speziari, Véronique and Lewis-Beck, Michael, ‘Partisan Dynamics in the European “Nation”’, presented at the First World Meeting of The Public Choice Societies, Amsterdam, 2007)Google Scholar. The latter study, however, covered a span of only 28 years.

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²⁸ Wittman, Donald, ‘Candidate Motivation: A Synthesis of Alternatives’, American Political Science Review, 77 (1983), 142–157CrossRefGoogle Scholar.

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³² Note that assumption 3, which leads to strengthening of the in-party effect, is likely to be counter-balanced in its effects by the forces identified in our fourth assumption. In fact, our model predicts that, on the average, incumbent parties lose about 2 percentage points in seat share each election.

³³ Schlesinger, Arthur M. Jr, The Cycles of American History (Boston, Mass.: Houghton Mifflin, 1986), p. 28Google Scholar.

³⁴ Stokes, Donald E. and Iversen, G. R., ‘On the Existence of Forces Restoring Party Competition’, Public Opinion Quarterly, 26 (1962), 159–171CrossRefGoogle Scholar.

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³⁷ Wlezien, Christopher, ‘The Public as Thermostat: Dynamics of Preferences for Spending’, American Journal of Political Science, 39 (1995), 981–1000CrossRefGoogle Scholar, and ‘Patterns of Representation: Dynamics of Public Preferences and Policy’, Journal of Politics, 66 (2004), 1–24.

³⁸ Bartle, John, Dellepiane, Sebastian and Stimson, James A., ‘The Moving Centre: Preferences for Government Activity in Britain, 1945–2005’ (unpublished paper, University of Essex, 2009)Google Scholar.

³⁹ Bartle, Dellepiane and Stimson’s measure of policy preferences applies the Dyad Ratios algorithm (see Stimson, James, Public Opinion in America: Moods, Cycles, and Swings, 2nd edn (Boulder, Colo.: Westview, 1998)Google Scholar) to the percentage of ‘Left’ responses (out of those classified as either ‘Left’ or ‘Right’) in all the domestic policy preference data that were available in the Gallup Political Index, British Election Studies (BES), NOP, ICM, British Social Attitudes (BSA), British Household Panel Study (BHPS), The European Social Survey, Eurobarometer and YouGov (based on a total of some 349 items asked in 2,482 separate administrations). We have inverted the scale to represent the percentage of ‘Right’ responses. We thank John Bartle for sharing the time series of voter preferences with us.

⁴⁰ See Budge, Ian, Klingemann, Hans-Dieter, Volkens, Andrea, Tannenbaum, Eric and Bara, Judith, eds, Mapping Policy Preferences: Estimates for Parties, Electors, and Governments 1945–1998 (Oxford: Oxford University Press, 2001)Google Scholar.

⁴¹ Cf. Grofman, Bernard, ‘The Neglected Role of the Status Quo in Models of Issue Voting’, Journal of Politics, 47 (1985), 231–237CrossRefGoogle Scholar.

⁴² See Wittman, ‘Candidate Motivation’.

⁴³ For simplicity of exposition, we will speak of movements relative to the median voter to denote movements relative to the expected value of the median voter distribution.

⁴⁴ We assume that the standard deviation of the voter distribution is σ_V = 0.5, so that the preferred positions of the parties are located at +/−2 standard deviations from the centre of the scale, which without loss of generality, is taken to be 0.

⁴⁵ Smoothed values for both the Conservative seat share and the model estimates were obtained by replacing each value s_t with a centre-weighted moving average , where s_t is the value in year t. We have used smoothed values because we wish to focus on long-term cycles. Calculations were performed with a time increment of four years.

⁴⁶ In succession, each parameter estimate was selected by a search procedure to generate the smallest sum of squared error for that parameter with other parameters temporarily fixed, and the procedure was repeated with each parameter until no change was observed in the estimated parameters to three decimal places.

⁴⁷ An alternative would be to fit the model by least square deviations between the actual election results and interpolated model projections for the same actual election years. But this approach renders the model projections dependent on each individual actual election time point and not just the model parameters.

⁴⁸ Since the model parameters reflect the effect of the timing advantage, the governing party’s seat and vote strength may be biased (over-predicted) by the model in years in which the government chose not to hold an election (such as years when it deemed that its electoral prospects were poor). The existence of cycles and their regularity as predicted by the model, however, should not be greatly affected by over-estimates of governing party strength (and hence under-estimates of opposition party strength) between elections.

⁴⁹ The projected and empirical plots for all-party seat share are presented in Figure A1 in the web appendix (see journals.cambridge.org/jps).

⁵⁰ The estimated cycle length is computed from the model projection by dividing the time duration (either of the full study or of a portion thereof) by the number of projected cycles. Truncating the study era to 1928–2005 (essentially the era studied by Lebo and Norpoth, ‘Dynamic Forecasting of British Elections’) yields estimates for both model parameters and cycle length that are substantially the same as those for the full study era 1832–2005. Vote share, which we have argued is not as reliable a measure of party strength as seat share, is irregular and is not fitted well by the model. This lack of fit of vote share (not shown) is particularly poor during the nineteenth century when aggregate party vote share was less reliable.

⁵¹ Statistics for fitting the model without the dummy transition variable are provided in Table 2; model projections are provided in the web appendix (Figure A2). Note that for this reduced model the correlations between projected and observed values are 0.56 and 0.66 for the two-party and all-party proportions, respectively – values that are again significantly positive at the 0.01 level. The weaker, although significant, fit underscores the value of introducing a dummy variable to account for a portion of Conservative strength during the Liberal/Labour transition but at the same time shows that the basic cycling pattern is present even without the dummy variable.

⁵² It is possible that the policy-motivation parameter may be larger directly after an election while the median convergence parameter may be larger as the next election approaches, as an anonymous referee suggested. Testing this possibility is, however, beyond the scope of this article as it would require measures of party and voter positions between elections.

⁵³ Bartle et al., ‘The Moving Centre’.

⁵⁴ First, we generated 44 random data points for party seat shares, according to a normal distribution with a mean of 0.5 and a standard deviation of 0.13 (approximately that observed for the real data). The data were smoothed as in the model (but without the transition dummy variable, which is not involved for random data). For the ten runs performed, the average correlation coefficient between smoothed random data and fitted model projections was 0.38. Indeed, this correlation is positive, but nowhere near the 0.87 obtained for the fit with real data. Visually, the fits for random data were generally very poor. Secondly, we returned to the real data and reduced the number of parameters of the full model by setting the median-convergence and party policy motivation parameters to 0. This reduced the correlation only from 0.87 to 0.76, still gave a moderately good visual fit, and projected a period of 28 years, similar to that provided by the full model.

⁵⁵ Other studies have related cycling to the Kondratiev long wave that involves the rise and fall of dominant technologies with a duration of about 55 years and with Kuznets growth cycles that have a duration of about 25 years (see Berry et al., The Rhythms of American Politics).

⁵⁶ Lewis-Beck, Michael, ‘Comparative Economic Voting: Britain, France, Germany, Italy’, American Journal of Political Science, 30 (1986), 315–346CrossRefGoogle Scholar, and Economics and Elections: The Major Western Democracies (Ann Arbor: University of Michigan Press, 1988); Lewis-Beck, Michael and Paldam, Martin, ‘Economic Voting: An Introduction’, Electoral Studies, 19 (2000), 113–121CrossRefGoogle Scholar.

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⁶⁰ Evans, Geoffrey and Anderson, Robert, ‘The Political Conditioning of Economic Perceptions’, Journal of Politics, 68 (2006), 194–207CrossRefGoogle Scholar.

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⁶⁴ In a number of instances, electoral results actually ran counter to economic expectations. For example, the incumbent Conservatives were defeated in 1964 and 1997 – although 1964 was the second year in a row with the GDP growth rate well above the median for the period, while 1997 was the fourth above the median. Yet the Conservatives were re-elected in 1992 despite an especially weak economy (1992 was the fourth year in a row below the median).

⁶⁵ Merrill et al., ‘Cycles in American Politics’. As noted earlier, in order to lessen the endogenous effects of election timing by the governing party, the empirical data are interpolated to represent the state of the system at four-year intervals; model projections are computed for these same time points.

⁶⁶ If, say, the model parameters are estimated from data through only 1945, the model projection rather accurately predicts the Conservative surge starting about 1950, peaking about 1960, and falling off in the late 1960s.

⁶⁷ Merrill et al., ‘Cycles in American Politics’. However, although cycles in Britain are well supported by the data, the pattern of alternation between Conservatives and Liberal/Labour strength is less regular than that found in Merrill et al., ‘Cycles in American Politics’ for American politics.

⁶⁸ See Brunell, Thomas, Grofman, Bernard and Merrill, Samuel III, ‘What if We Had a Realignment and Nobody Noticed? Putting Critical Elections in the US House and Senate in Historical Context, 1854–2006’, presented at the annual meeting of the Midwest Political Science Association, Chicago, 2009Google Scholar.

⁶⁹ For example, the largely centrist Liberal Democrats have had their effect on major-party positioning, tending to push the major parties (particularly the Conservatives) further apart. See Adams, James and Merrill, Samuel III, ‘Why Small, Centrist Third Parties Motivate Policy Divergence by Major Parties’, American Political Science Review, 100 (2006), 403–417CrossRefGoogle Scholar, and Nagel, Jack and Wlezien, Christopher, ‘Centre-Party Strength and Major-Party Divergence in Britain, 1945–2005’, British Journal of Political Science, 40 (2010), 279–304CrossRefGoogle Scholar.