Logit Estimation Results
Model Estimation by the Method of Maximum Likelihood
The coefficients of the logit model are estimated by the
method of maximum likelihood estimation. The likelihood
function L gives the joint probability density function,
or likelihood, of observing the sample
Solution is obtained by a numerical optimization algorithm.
The first step is to set some starting values for the coefficients.
A sensible approach is to set all slope coefficients to zero.
In this case, the probability that
The next step (iteration 1) is to apply an updating rule to update the starting values to a new set of coefficient estimates. In successive iterations, coefficient estimates are updated from the previous iteration. The iterations stop when there is little improvement. This is called convergence.
For the logit model, convergence is usually achieved in 4 or 5 iterations. Furthermore, the form of the log-likelihood function for the logit model guarantees that different starting values will lead to unique maximum likelihood estimates. This attractive property makes the logit model an estimation method that can be used with some success. A general warning is that if the number of iterations exceeds 10 or 15 then this may be a signal of multicollinearity in the data set. A result may be relatively high estimated standard errors on the estimation output.
For the first model in the school budget voting study, the SHAZAM output below shows that the estimation converged in 4 iterations.
The first part of the above output states that, of 95 observations
in the sample, there were 59 yes votes
The value of the log-likelihood function when all slope coefficients
are zero is
The estimation output for the
school budget voting study also reports the results for a second model
estimation where income and property taxes are log-transformed.
For this model, the estimation output shows that the value
of the log-likelihood function when all slope coefficients
Interpretation of the Results
Logit estimation results for the first model in the school budget voting study are below:
Rubinfeld (1977, p. 35) states:
The income variable serves as a measure of the capacity of households to consume both private and public goods. On the assumption that local school education is a normal good, we expected, other things equal, that income and the demand for public schools would be positively correlated. In the context of our voting model, this suggests a positive relationship between income and the probability of a yes vote.
The logit estimation results can be used to test the above hypothesis. The statistical properties for the maximum likelihood estimator are established for "large" samples (asymptotically). This means that the asymptotic t statistic has an approximate normal distribution in "large" samples. In other words, the results of any hypothesis testing exercise can be considered as approximate when applied to "small" samples.
The sign of an estimated coefficient gives the direction of the effect of a change in the explanatory variable on the probability of a success (an observation at one). The positive estimated coefficient on income suggests that an increase in income gives a higher probability of a yes vote. A test of the null hypothesis that the income coefficient is zero against the alternative hypothesis that the income coefficient is positive has a t-test statistic of 1.83. For a one-sided test, the approximate 5% critical value is 1.645. The calculated test statistic exceeds the critical value and therefore the null hypothesis is rejected in favour of the alternative.
Elasticities and Marginal Effects for Roughly Continuous Variables
For the logit model, the estimated coefficients do not have a direct economic interpretation. Measures that are familiar to economists are marginal effects and elasticities. First consider the calculation of elasticities. An elasticity gives the percentage change in the probability of a success in response to a one percentage change in the explanatory variable. For the kth explanatory variable this is obtained using partial derivatives as:
Since the elasticities vary for every observation it is desirable
to report a summary measure. A convenient summary measure is to
evaluate the elasticity at the sample means of the explanatory
variables. This measure is reported as
To address this limitation, Hensher and Johnson (1981, p. 59)
propose evaluating the elasticities
at every observation and then constructing a weighted average
where the predicted probabilities are the weights.
This measure is reported on the SHAZAM output as the
Using the weighted aggregate elasticity measure, the SHAZAM output for the first model in the school budget voting study shows that, on average, a one percent increase in income gives a 0.48 percent increase in the probability of a yes vote, holding all else constant.
The marginal effect of the kth explanatory variable on the response probability is obtained from:
That is, estimates of the marginal effects are calculated
by rescaling the estimated coefficients. The scale factor
varies with the observed values of X. For reporting purposes,
the scale factor can be evaluated at the sample means
of the explanatory variables. This value is reported as
Greene (2000, p. 816) notes that a preferred method may be to obtain a scale factor by evaluating the expression at every observation and then taking the average. He comments that in "large" samples, by applying the Slutsky theorem, this result will be similar to the result obtained by evaluating the scale factor at the sample means.
By inspecting the marginal effects reported on the SHAZAM output for the first model in the school budget voting study it is found that, on average, a $1000 increase in income leads to a 0.015 increase in the probability of a yes vote, holding all else constant.
Some cautionary notes are required when interpreting the elasticities and marginal effects discussed above. The formula derived from a partial derivative is meaningful when the explanatory variable of interest is roughly continuous, such as income. The formula do not apply to determining the magnitude of the partial effect from changing a dummy variable from zero to one. That is, the estimated elasticities reported on the SHAZAM output may not have any useful interpretation for dummy explanatory variables. The calculation of marginal effects for dummy explanatory variables is discussed in the next section.
It should also be noted that special treatment is required for evaluating elasticities and marginal effects when the explanatory variables include quadratic terms or interaction variables (see the discussion in Wooldridge (2000, p. 533)).
A common application is to work with log-transformed variables.
The LOG option on the LOGIT command assumes observations
are in the form
Marginal Effects for Dummy Variables
Suppose the kth explanatory variable is a 0-1 dummy variable. The change in the probability of a success (Y=1) that results from changing Xk from zero to one, holding all other variables at some fixed values, denoted by X*, is given by the difference:
P(Y=1 | Xk=1 , X*)
Values must be set for X*. An approach is to set values to represent a "typical case". A "typical case" can be defined by setting all dummy variables to their modal values and all other variables to their mean values.
The SHAZAM output reports marginal effects for all dummy explanatory
variables in the section labelled
P(Y=1 | Xk=0 , X*)
and the column labelled
P(Y=1 | Xk=1 , X*)
The final column labelled
For the school budget voting study, the variable
To get an estimate of the magnitude of the effect,
set the explanatory variables to values that represent a
"typical voter" in the sample. The definitions
of the variables in the data set can be reviewed.
The dummy variables
An individual that is not a school teacher, with "typical" characteristics on all variables in the model, has a probability of a yes vote of 0.439. If the individual is a school teacher the probability increases to 0.836. The marginal effect is the difference 0.397.
Note that the marginal effect obtained using the less precise method of taking the partial derivative with respect to the school dummy variable was calculated to be 0.426.
Overall Significance and Goodness of Fit Measures
For the first model in the school budget voting study the SHAZAM output reports the statistics below:
Denote the value of the
Approximate critical values are obtained from the chi-square distribution with degrees of freedom equal to the number of slope coefficients. The SHAZAM output reports a p-value for the test statistic. Inspection of the above output shows that, at a 10% significance level, the null hypothesis is rejected.
The final part of the SHAZAM logit estimation output reports a variety of goodness-of-fit measures that have been proposed by various researchers. Formula and references are given in the SHAZAM User's Reference Manual. A useful comment from Wooldridge (2000, p. 536) is: "goodness-of-fit is not usually as important as statistical and economical significance of the explanatory variables."
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