Question

Question 1

(a) (4 points) What are they key advantages of the Logit model over the Linear Probability Model?

(b) (15 points) In class we saw that efficient estimates of the coefficients from a linear regression model can be obtained under the presence of heteroskedasticity using Generalized Least Squares (GLS). How does GLS work? That is, describe the mechanism through which GLS addresses non-constant error variances to achieve efficient estimation.

(c) (5 points) Let Zi be the log-odds ratio in the context of the Logit model. Show that if P(Yi = 1|X) > P(Yi = 0|X) then it must be that Zi > 0, and that if P(Yi = 1|X) < P(Yi = 0|X) then it must be that Zi < 0.

(d) (6 points) In class we saw that the linear probability model is inherently heteroskedastic. Why is this the case? That is, show why the variance of the error term in the LPM will necessarily be non-constant.

Answer 1

(a) Advantages of Logit Model ;

1.The logistic model is unavoidable if it fits the data much better than the linear model.

2.The logistic model is less prone to over fitting, but it can overfit in high dimensional datasets . You should consider Re ularization (L1 and L2) techniques to avoid over - fitting to these scenario.

3. Logistic model is easier to implement , interpret and very efficient to train.

4. Logistic model is not only give a measure of how relevent a predictor (coefficient size)is, but also its direction of association ( positive or negative).

(b)

Generalized Least Squres (GLS) is technique for estimating the unknown parameters in a linear regression model whwn there is a certain degree of correlation between the residuals in a regression model .

Heteroskedasticity produces a simple example. To produce observations with equal variances, each data point is divided by the standard deviation

This corresponds to choosing A equal to a diagonal matrix with the reciprocals of these standard deviations arrayed along its diagonal. The estimation criterion function is

which is a weighted sum of squared residuals. For this reason, in this special case GLS is often called weighted least squares (WLS). WLS puts most weight on the observations with the smallest variances, showing how GLS improves upon OLS, which puts equal weight on all observations. Those n for which σ n is relatively small tend to be closest to the mean of y n and, hence, more informative about β.

Faced with AR(1) serial correlation in a time series, the appropriate choice of A transforms each data point (except the first) into differences:

y͂n = yn - ρyn-1,

x͂nk = Xnk - ρxn-1, k, k = 1, …,K.

This transformed y˜ n display zero covariances:

(c)

THE LOGISTIC REGRESSION MODEL (LRM). The logistic regression model (LRM) (also known as the logit model) can then be written as Odds X Zi P Yi P Yi K k = i = + k ik = − = = ∑=1 ln( ) 1 ( 1) ( 1) ln α β The above is referred to as the log odds and also as the logit. Zi is used as a convenient shorthand for α + ΣβkXik. By taking the antilogs of both sides, the model can also be expressed in odds rather than log odds, i.e. ∏ ∏ ∑ = = + ∑ = = = = = + = = − = = = = K k X K k X X Z K k i k ik k k k k

(d)

The error term in an LPM is heteroskedastic because the variance isn't constant. Instead, the variance of an LPM error term depends on the value of the independent variable(s). Because the variance of the error depends on the value of X, it exhibits heteroskedasticity rather than homoskedasticity.