Consider the version of Fair's model in Example 10.6 . Now,...

Consider the version of Fair's model in Example $10.6 .$ Now, rather than predicting the proportion of the two-party vote received by the Democrat, estimate a linear probability model for whether or not the Democrat wins. $\begin{array}{l}{\text { (i) Use the binary variable demwins in place of demvote in }(10.23) \text { and report the results in }} \\ {\text { standard form. Which factors affect the probability of winning? Use the data only through } 1992 \text { 2 }}\end{array}$ $\begin{array}{l}{\text { (ii) How many fitted values are less than zero? How many are greater than one? }} \\ {\text { (iii) Use the following prediction rule: if demwins }>.5, \text { you predict the Democrat wins; otherwise }} \\ {\text { the Republican wins. Using this rule, determine how many of the } 20 \text { elections are correctly }} \\ {\text { predicted by the model. }}\end{array}$ $\begin{array}{l}{\text { (iv) Plug in the values of the explanatory variables for } 1996 . \text { What is the predicted probability that }} \\ {\text { Clinton would win the election? Clinton did win; did you get the correct prediction? }} \\ {\text { Clse a heteroskedasticity-robust } t \text { test for } A R(1) \text { serial correlation in the errors. What do you }} \\ {\text { find? }}\end{array}$ $\begin{array}{l}{\text { (v) Use a heteroskedasticity-robust test for AR (1) serial correlation in the errors. What do you }} \\ {\text { find? }} \\ {\text { (vi) Obtain the heteroskedasticity-robust standard errors for the estimates in part (i). Are there }} \\ {\text { notable changes in any } t \text { statistics? }}\end{array}$

Consider the version of Fair's model in Example $10.6 .$ Now, rather than predicting the proportion of
the two-party vote received by the Democrat, estimate a linear probability model for whether or not the
Democrat wins.
$\begin{array}{l}{\text { (i) Use the binary variable demwins in place of demvote in }(10.23) \text { and report the results in }} \\ {\text { standard form. Which factors affect the probability of winning? Use the data only through } 1992 \text { 2 }}\end{array}$
$\begin{array}{l}{\text { (ii) How many fitted values are less than zero? How many are greater than one? }} \\ {\text { (iii) Use the following prediction rule: if demwins }>.5, \text { you predict the Democrat wins; otherwise }} \\ {\text { the Republican wins. Using this rule, determine how many of the } 20 \text { elections are correctly }} \\ {\text { predicted by the model. }}\end{array}$
$\begin{array}{l}{\text { (iv) Plug in the values of the explanatory variables for } 1996 . \text { What is the predicted probability that }} \\ {\text { Clinton would win the election? Clinton did win; did you get the correct prediction? }} \\ {\text { Clse a heteroskedasticity-robust } t \text { test for } A R(1) \text { serial correlation in the errors. What do you }} \\ {\text { find? }}\end{array}$
$\begin{array}{l}{\text { (v) Use a heteroskedasticity-robust test for AR (1) serial correlation in the errors. What do you }} \\ {\text { find? }} \\ {\text { (vi) Obtain the heteroskedasticity-robust standard errors for the estimates in part (i). Are there }} \\ {\text { notable changes in any } t \text { statistics? }}\end{array}$

Key Concepts

Hypothesis Testing for Model Parameters

Hypothesis testing, including the use of t-tests, is used to assess whether individual explanatory variables significantly affect the dependent variable in a model. In the context of models with potential heteroskedasticity and serial correlation, adjusting the test statistics using robust methods is essential to obtain valid inferences about the significance of the predictors.

Prediction Rule and Classification

A prediction rule in binary outcome models involves classifying observations based on their predicted probability of the event occurring, often using a threshold (commonly 0.5) as the decision boundary. This rule helps in evaluating the binary prediction performance of the model by comparing predicted outcomes to actual outcomes, providing a measure of predictive accuracy.

Robust Standard Errors

Robust standard errors provide a way to correct for heteroskedasticity by adjusting the estimated standard errors so that they are valid even when the error variance is non-constant. In applications such as the LPM, using heteroskedasticity?robust standard errors is crucial to ensuring that hypothesis tests and confidence intervals for the coefficients are reliable.

Serial Correlation

Serial correlation, or autocorrelation, occurs when the error terms in a regression model are correlated across time or observations. Detecting and correcting for serial correlation is important, especially in time series data, because it can lead to underestimated standard errors, making test statistics unreliable. Techniques such as AR(1) tests are used to assess whether present autocorrelation exists in the error structure.

Fitted Values Outside [0,1]

In the context of a linear probability model, the linear nature of the regression function means that some fitted values may fall below 0 or above 1, which are not interpretable as probabilities. This issue highlights one of the key challenges of using the LPM and motivates the use of alternative modeling approaches or adjustments when strict probability bounds are required.

Binary Outcome Modeling

Binary outcome modeling involves statistical methods used when the dependent variable takes on only two possible outcomes. While models like logit and probit are specifically designed to handle the dichotomous nature of the data, the linear probability model is a simpler alternative that estimates the probability of an event occurring using a linear specification, acknowledging that some theoretical limitations exist.

Linear Probability Model

A linear probability model (LPM) is an OLS regression framework where the dependent variable is binary, and the estimated coefficients are interpreted as the marginal effects on the probability of the outcome. Despite its simplicity and ease of estimation, it often produces predicted probabilities that can fall outside the interval [0,1] and suffers from heteroskedasticity because the variance of the error term depends on the predicted value.

Heteroskedasticity

Heteroskedasticity refers to the circumstance in which the variance of the error term is not constant across observations. In the case of LPMs, this is almost inherent because the variance of a binary variable depends on its probability. This violation of the constant variance assumption can lead to inefficiencies in the estimates and biases in the standard errors if not properly addressed.

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