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⁵ The terms ‘contingent relationship’ and ‘categorical relationship’ have nothing to do with categorical (i.e., nominal-level) data and contingency tables. Rather, they refer to the nature of the influence of economic conditions upon voters and the results of elections.

⁶ Owens, John R. and Wade, L., ‘Economic Conditions and Constituency Voting in Great Britain’, Political Studies, 36 (1988), 33–51, at p. 33.CrossRefGoogle Scholar

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⁹ Several analyses of ‘popularity functions’ (as opposed to ‘vote functions’), however, do consider third parties. See Clarke, Harold D. and Zuk, Gary, ‘The Dynamics of Third-Party Support: The British Liberals, 1951–1979’ American Journal of Political Science, 33 (1989), 299–315CrossRefGoogle Scholar; and Monroe, Kristen R. and Erikson, Lynda, ‘The Economy and Political Support: The Canadian Case’, Journal of Politics, 48 (1986), 616–47.CrossRefGoogle Scholar

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¹⁶ Kiewiet, and Rivers, , ‘A Retrospective’, p. 372Google Scholar; Hibbs, , The American Political Economy, p. 393.Google Scholar

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⁴³ Each constituency in Australia consists of approximately thirty subdivisions. Each constituency in Canada consists of approximately 100 polling divisions, and each constituency in New Zealand consists, on average, of two counties.

⁴⁴ I choose these three third parties because they contested most elections between the Wars and because they were the most prominent third parties in rural parts of their respective countries. I choose these major parties because they existed continuously throughout the inter-war years and because they formed no pacts or coalitions with these third parties. Their principal opponents (namely, the Nationalist party and United Australia party in Australia; the Liberal party in Canada; and the Reform party, Liberal/United party and National party in New Zealand) do not fulfil these criteria.

⁴⁵ The social structural variables are ‘grouping variables’ which mitigate the incidence of aggregation bias in the model. See Langbein, Laura I. and Lichtman, Allan J., Sage University Series on Quantitative Applications in the Social Sciences, no. 07–001: Ecological Inference (Beverley Hills, Calif.: Sage Publications, 1976), pp. 11, 15–25.Google Scholar

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⁴⁷ Precise figures appear at the foot of Tables 2–5.

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⁴⁹ These inferences may be incorrect because one can observe a party's vote only in the subconstituency units in which candidates stood for election. Such inferences thus ignore the behaviour of voters who would have voted for these candidates if they had stood in the sub-constituency unit in which they resided. See Achen, Christopher, The Statistical Analysis of Quasi-Experiments (Berkeley: University of California Press, 1986), pp. 73–95Google Scholar; Heckman, John, ‘Sample Selection Bias as Specification Error’, Econometrica, 47 (1979), 153–61CrossRefGoogle Scholar; and Nelson, Forrest D., ‘Censored Regression Models with Unobserved Stochastic Censoring Thresholds’, Journal of Econometrics, 6 (1977), 309–25.CrossRefGoogle Scholar

⁵⁰ The formal model consists in an unobserved random variable, z*, which represents the candidate's propensity to contest a particular constituency. The observed counterpart of z*, the actual presence or absence of the candidate, is z. If one sets the unobserved threshold of z* to 0, then one can write z = 1 if z* > 0; z = 0 if z* ≤0. One thus observes the vote only when z = 1, and can express this vote as Y = 1/[1 + exp(–β''X)], z = 1/[1 + exp(– a′ V)]; z = 1, if z* > 0; z = 0 if z* ≤ 0. One mitigates the selection bias problem by applying the standard binary logit analysis to z in order to estimate the coefficients of a; by computing the quantity λ = φ(a′V)/Φ for each observation; and by appending λ to the matrix of independent variables, X, and performing the standard binomial logit analysis. One requires an identification condition in order to estimate models such as these. One variable, voter turnout in the sub-constituency unit, has been included in the V matrix and excluded from the X matrix.

⁵¹ Achen, , The Statistical Analysis of Quasi-Experiments, pp. 97–137.Google Scholar

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