Chapter 5 Chapter 5 Heterogeneity in Issue Salience in the American Electorate
5.1 Homogeneity, Distributions, and Random Pa- rameters Logit
Recall the "ideal" model of voter choice from Chapter 1. This model represents the probability that voter i votes for candidate c as:
(5.1)
With the survey data commonly available in political science this model cannot be estimated. Some restrictions must be placed on the estimated issue weights in order to estimate this model. By far the most common assumption about issue weights employed in empirical research is the homogeneity assumption. Rather than attempt to estimate a separate set of issue weights for each voter, we assume that all voters place the same weight on a given issue, meaning f3i = (3 for all i. If we maintain the assumption that the unobserved portions of utility are distributed independently and identically as Type I Extreme Value distributions then this leads to the familiar conditional logit specification.
(5.2) This is an easy model to estimate, and most software packages include commands that will estimate models of this type. Thus conditional logit and its variants are popular
among researchers because they are simple models that yield some information about the impact of candidate issue positions on vote choice. Of course, the drawback to this type of model is that it assumes that all voters behave in exactly the same way.
Thus, models of this type are of no use in studying heterogeneity in the impact of issues on vote choice, as all heterogeneity was assumed away in order to make the model tractable.
What is needed is an assumption about /3i across voters that allows for hetero- geneity in issue weights, while still leading to a tractable model. One possibility is to assume that the weight placed on a particular issue by voters is random, but is distributed among voters by some known probability distribution. That is, for issue j,
/Ji
rv f(/3jl/Jii O'f3J· Assuming a probability distribution f, our goal will be to esti- mate the mean/Ji
and the standard deviation O'f:Ji of this distribution. The mean gives us a measure of the average weight placed on the issue in the electorate, while the standard deviation will give us a measure of the heterogeneity in the weight placed on the issue. This assumption strikes the middle ground between estimating a unique set of weights for each voter, as I did in Chapter 4, and assuming all voters are the same, as I did in Chapter 2. This distributional assumption is the motivation behind random parameters logit.Consider equation 5.1 again. This is the probability that voter i chooses candidate c conditional on f3i· If we knew /3i for each voter the solution to equation 5.1 for each individual would be trivial. However, without this information estimation of equation 5.1 will generally not be possible. If /3i represents the weights on Q issues, then the choice probability is conditional on Q random variables. Without some information about these random variables it is not possible to determine the choice probability in equation 5.1.
However, if the distribution of the Q random variables is known then it is possible for us to determine the unconditional choice probability. I assume throughout this
chapter that these Q random variables are independently distributed. To obtain the unconditional choice probability of individual i choosing candidate c we must integrate equation 5.1 with respect to the Q independent random variables.
(5.3) The intuition behind this estimator is actually quite simple. Let us assume for a moment that there is only one random parameter,
/3j,
and that parameter follows a known probability distributionf.
Essentially, for each individual we are calculating the probability that that individual selects candidate c for every possible value of/3j,
and weight the contribution of each particular calculated probability to the likelihood function by the density
f.
Thus, values of/3j
far out into the tails of a continuous distributionf
may produce very different calculated probabilities of selecting can- didate c for a particular voter, but they will not carry much weight in the overall likelihood function. However, as a~j increases more weight is assigned to values of/3i
that are further from the mean. This might improve the calculated likelihood for some individuals, since their particular values of/3j
may be far from the mean. If enough individuals are far from the mean then this estimator will estimate a signifi- cant standard deviation on/3j -
this is how random parameters logit determines if there is significant heterogeneity in the weight placed on a particular issue. Thus, random parameters logit offers a clear statistical test for heterogeneity in the impact of issues on vote choice.Another useful property of random parameters logit is that it is not burdened with the IIA property that plagues conditional logit. Examination of equation 5.3 reveals that the ratio of Pie to Pid for c
:f.
d depends on all candidates in the choice set. Because we are estimating individual-specific unobserved portions of the utility function, we are also estimating correlation in the unobserved portion of utility overcandidates.1 Random parameters logit can approximate any substitution pattern that any other discrete choice model can estimate, depending on the specific specification of the random parameters logit model (McFadden and Train 1998). For instance, a random parameters logit model can be specified that will approximate the substitution patterns that could be estimated with a multinomial probit model (Brownstone and Train 1999). However, in the specific context of this chapter it is unclear how much of the correlation between choices is accounted for. Correlation between choices in the empirical application below is solely a function of variation in the estimated issue weights, and not of all portions of unobserved utility as it is in many multinomial probit models utilized in political science (e.g., Alvarez and Nagler 1998).
To estimate the random parameters logit model we set up the following log- likelihood function:
where I is the set of all voters, C is the set of all candidates, and
{
1 if i chooses c
Yic =
0 otherwise
(5.4)
The log-likelihood function in equation 5.4 involves estimation of a Q-dimensional integral, where Q is the number of random parameters. This integral cannot be evaluated analytically since it does not have a closed-form solution. If Q is equal to one or two the integration may be performed numerically. However, when Q is greater than two numerical techniques cannot compute the integral with sufficient precision
1To see this, rewrite individual i's utility for voting for candidate c as Uic
=
(3' Xie + 8iXic + C:ic·The unobserved portion of utility is 8iXic +c:ic, which is correlated across choices due to the common influence of 8i.
(Revelt and Train 1997).
Since we cannot compute the integral numerically when Q 2 we must apply Monte Carlo simulation techniques to approximate the choice probabilities in the log-likelihood function of equation 5.4. We then maximize the resulting simulated log-likelihood function.
The Monte Carlo simulation technique I employ here approximates the choice probabilities in equation 5.4 by computing the integrand in equation 5.4 at randomly chosen values for each
/3iq·
For each individual, for each random parameter, I first draw a random variable from the distribution I have assumed the random parameter follows. Since I have assumed that the random deviations from the mean weight placed on an issue are independent across individuals and issues, I generate a matrix of I x Q independent random variables drawn from the appropriate distributions (f(/3q) for the qth random parameter). I then compute the corresponding choice probabilities for a given value of the parameter vector B. I then repeat this process R times for the given value of the parameter vector. Let P[c(B) be the realization of the choice probability for the rth draw. I approximate the choice probabilities for a given parameter vectore
by averaging over the values of Pi~:R
A 1 ~A
Pic(B) = R Li P[c(B)
r=l
(5.5) Pic(B) is the simulated choice probability of individual i choosing candidate c given
e.
This simulated choice probability is an unbiased estimator of the actual probability Pic(B), with a variance that decreases as R increases. It is also twice differentiable and strictly positive for any realization of the finite R draws. These properties are especially appealing because they imply that log-likelihood functions constructed with Pie( B) are always defined and can be maximized with conventional gradient-based optimization methods.Thus, we construct a simulated log-likelihood function:
I C
SC= LLYiclog
(Pic(e)J
(5.6)i=l c=l
The parameter vector
e
is the vector that maximizes the simulated log-likelihood function. Under weak conditions this estimator is consistent, asymptotically efficient, and asymptotically normal (Lee 1992). However, this estimator does display some bias at low values of R, which decreases as R increases. The bias is exceedingly small when R = 250 (Brownstone and Train 1999); most empirical work uses R equal to 500 or 1000.The next section applies a random parameters logit model to the 1996 presidential election and reveals a significant degree of heterogeneity in the weights that voters place on issues. A conditional logit model is also estimated in order to show that random parameters logit reveals all of the information that conditional logit does, plus information on heterogeneity in issue weights.