A SIMULTANEOUS EQUATIONS MODEL OF SPLIT TICKET VOTING IN ISRAELI ELECTIONS1 Martin S. Geisel, Carnegie -Mellon University Naomi E. Kies, Hebrew University Timothy W. McGuire, Carnegie -Mellon University Howard Rosenthal, Carnegie -Mellon University 1. Introduction An election in which citizens vote simul- Let ei = u21 + (uli and assume the u2i)RKi taneously for more than one office is a complex RKi's are given numbers. Then, if E(ei) = social process where two or more decisions are interrelated. Where more than one office is at and E(e.e.) a2 if i =j and zero otherwise, stake and where candidates are labeled as to political party voters can vote a "straight" ordinary least squares applied to party ticket or they can "split" their ballot, (1.2) voting for candidates for two or more parties. RMi ei Politicos have long felt that the two choices yields minimum variance linear unbiased estimates were mutually dependent. In this paper we pre- of and p2 via the relations + sent models of one such election, that of Israel o l' where and are the least in 1965. Although the choice of the Israeli 1965 P2 ßO' 0 data was largely one of circumstance, it was squares estimates from equation (1.2). For the fortunate in that the Israelis voted for only constant variance assumption to hold, either two offices, the national legislature (Knesset) uli or RKi2 Var(uli) + and local municipal councils. We thus avoid u2i effects 2 the contaminating of additional contests. (1 Var(u2i) + Cov(uli,u2i) is Furthermore, for the city of Jerusalem, much split- ticket voting was suggested by the presence constant for all i. Otherwise, procedures which of Teddy Kollek, an attractive candidate at the allow for heteroscedastic errors will be required local level representing the otherwise unsuccess- in order to obtain best linear unbiased estimates. ful Rafi party. Our data base consists of voting and census statistics for the sixty -four census A variant of the Goodman model was suggested tracts of (pre -1967) Jerusalem. by Rosenthal [12]. Instead of letting the pro- portion won from the remaining parties be a con - To describe these interrelated phenomena a stant, he suggested letting this proportion be a system of equations is required since a party's function of the Knesset vote proportion, for municipal vote depends on, among other things, example, p2 /(1 Then, its national vote, and vice versa. The statis- P2 tical analysis of systems of simultaneous linear ( u2i RKi).(I3) equations has received much attention from econo- /(1- metricians. The models with which we shall deal are somewhat more complicated since they involve Simple algebra will show that (1.3) also leads to both non -linearities and inequality constraints (1.2) except that the estimates are on the parameters of the system. p2 The reader can conceive of still further models that could be associated with (1.2). We can introduce our estimation problem Although the values of and may enable through an example involving only a two equation model. Let RKi denote the proportion of Rafi one to eliminate some of these models on logical grounds (see Rosenthal [12] and Goodman [4]), voters in the i -th district for the Knesset any regression poses difficult problems of inter- election in 1965; the corresponding munic- pretation. ipal proportion; p's, q's, ß's, and y's con- stants to be estimated; u's and v's stochastic If we regard R.. and both as endoge- terms in the model; and let i 1,2,...,n index nous (mutually determined by the electoral system) the census tracts. Let us first examine asingle- variables, then the least squares estimates of equation model originally suggested by Goodman the parameters of (1.2) will be biased and incon- [4] and frequently used in political science sistent since ei and will be correlated applications in the study of two temporally sepa- rated elections.2 For each census tract, assume in general (see, e.g., Johnston [6,pp.231-4]). that the Rafi municipal list maintained a pro- portion pl + u its Knesset vote while To analyze the simultaneous problem another relationship involving RKi is required. An winning a proportion p2 + of the Knesset u2i equation analogous to (1.2) with the roles of vote for all other parties (including absten- RMi and RKi interchanged will not do, for the tions). Then parameters of this two equation system are not RMi estimable or identifiable. The parameters can (1-RKi) (1.1) be identified if we complicate the equations by P2 u2i (uli adding exogenous (determined outside the system) variables to the model. To illustrate this, let 71 X and Y be two sociological variables (e.g. We assume that the columns of X are lin- educational level and proportion of Oriental early independent so that the rank of X is K.4 birth). Modifying (1.1) so as to make the pro- Further, we assume that so we can solve portions functions of the sociological variables for the reduced form leads, for example, to: Y = -XBr -1 + -1 = + V. (2.3) (p1 +uli)RKi Regarding the u's we assume (1.4) E(utg) 0, t= 1,...,T,g= 1,...,G, + (1-i) and = (q1 =ag if s=t, E( u u + (1.5) sg th) if s #t. the equations: This leads to structural If we let u'(t) [utl,...,utg] we may express = 53RKiXi the above as = + + + + ei, (1.6) E(u'(t))= [0,...,0], + gi, (1.7) RKi YO + + Y3RMiYi + E = h] if s =t, where the relations between the parameters of E(u(s)u'(t)) ' g (1.4) -(1.5) and (1.6) -(1.7) are immediate. = if s#t, An interesting feature of (1.6) is that where E is a positive definite symmetric G x G even in this simple case there are variables in matrix5 and is the null matrix of order G. the system which are products of endogenous and exogenous variables. This does not typically The reduced form (2.3) is similar to a multi- occur in models considered by econometricians. variate linear regression model. This suggests estimation of the reduced form by ordinary least In the next section of the paper we give a squares and then estimation of structural para- brief discussion of identification and estima- meters using the relation tion of systems of linear equations. In section n = -Br . 3 we present methods for dealing with the "adding up" constraint (that is, for each census dis- To see that this is impossible in general let us trict the sum of the Municipal proportions is post -multiply the structure (2.2) by an arbitrary unity, and similarly for the Knesset propor- non -singular matrix L to give tions). In section 4 we discuss the methods L + XBL UL employed for dealing with the product variables and parameter constraints. The results of the or * * * analysis for the 1965 Jerusalem elections are Yr + XB U (2.4) given in section 5. Concluding remarks are given in section 6. with reduced form * * 2. Identification and Estimation of Systems of Y +v , Linear Equations.3 where In this section we give a brief exposition * -1 = B *-1 = - BL(rL) = - Br -1 = , of methods of estimation of the parameters of systems of equations of the form: so that the two structures have the same reduced form.6 The two structures (2.2) and (2.4) are G K observationally equivalent in the sense that the E x =ut likelihood function for one structure is the same E + g m=1 g as that for the other. It is thus impossible in t= 1,...,T, g= 1,...,G. (2.1) general to estimate structural parameters from the reduced form without the imposition of some The y's are jointly dependent or endogenous vari- additional prior information about the structural ables and the x's are predetermined variables parameters. This is called the identification which are stochastically independent of the problem.7 errors, The equations (2.1) may be ex- pressed conveniently in matrix notation as It should be emphasized that identification is not a problem unique to simultaneous equation + XB = U, (2.2) systems nor is it related to sample size. For where Y is a T x G matrix of values of the example, suppose we are interested in estimating endogenous variables, X a T x K matrix of the weights of an apple and an orange. If we values of predetermined variables, U a T x G always weigh them together we can by repetitive matrix of unobservable stochastic disturbances, weighing obtain a very precise estimate of the and r and B are G x G and K x G matrices sum of the weights, but regardless of sample size of structural parameters. we never can gain any information about the 72 individual weights from this experiment. condition for identification. This rank can be G only if K Kg > Gg; that is, only if the We shall discuss only one kind of prior number of exogenous variables excluded from the information in relation to the identification gth relation is greater than or equal to the problem; namely, knowledge that certain elements number' of endogenous variables in the equation of and B are zero. We shall discuss con- minus one. This necessary condition is called ditions for identification of the parameters of the order condition. Both conditions may easily the gth equation. be extended to the case of homogeneous linear restrictions on the parameters of the gth equa- Let us write the gth equation of the system tion.