Concept Comprehension: Predicted Probabilities

Here are the logit coefficients for a model of home ownership. \(\texttt{realhhinc}\) is household income in 1000s of dollars and \(\texttt{city}\) is a binary variable indicating whether the respondent lives in a city.

\[ \begin{aligned} \beta_{0} &= -.388 \\ \beta_{realhhinc} &= .019 \\ \beta_{city} &= -.719 \end{aligned} \]

1. What is \(\mathbf{x}\mathbf{\beta}\) for a respondent with a household income of $70,000 per year who lives in a city? [1]

2. What is the predicted probability of home ownership for this respondent? [1]

3. What are the odds of home ownership for this respondent? [1]

4. In your own words, explain the difference between the mean predicted change and the predicted change at the mean. [2]

5. As the baseline probability of an outcome changes from 0 to 1, how does the magnitude of the marginal change in \(\Pr(y=1)\) for an explanatory variable vary? (For example, when is the marginal change the biggest? When is it the smallest?) [2]

6. If \(\beta_{x} = .8\), what is the maximum marginal change in the outcome probability for a change in \(x\)? [1]