Question

Question 5 (20p): Discrete choice models a. What is a binary choice model? What do we eatimate? Discribe the linear, probit and logit models. b. Derive the formula for the probit model for a consumer buying/not buying a product , based on product characteristics. c. In the linear and probit models, how would you calculate the predicted value at the value Xmean d. In estimating the linear binary choice model, what can we say about heteroscedasticity? given estimated coefficients? Provide a formula and explain the components. od luck!

Answer 1

Answer a & b.

Binary Choice Models: Binary Choice model is one of the discrete Choice Models. A binary decision helps choose between two alternatives available. Binary Choice Model is a statistical model whose primary objective is to know and understand the outcome of the binary decision taken.

Primarily, the estimation under Maximum Likelihood aims at measuring the "fit"of the model to the data. The ultimate objective is to predict the choice amongst the two options available.

Linear Model: We take help of Linear regression to create a linear model. It helps us denote a continuous response variable as a function of one or more predictor variables.

Probit Model: We take help of Probit regression to create a linear model. Probit regression is used to model binary outcome variables.In simple terms, the dependent variable under study can only take two values.

Logit Model: We take help of Logistic regression to create a linear model. It uses a logistic function to model a binary dependent variable.

Answer c.

Linear Regression formula:-

The below equation helps to derive the line of best fit.

y = a + bx

where

b = Sum { xi yi - nMean(x) Mean(y) } / { xi*xi - n Mean(x) * Mean(x) }

i = 1 to n

Probit regression:-

It is a special type of Generalised Linear Model. The bivariate outcome Y has a bernoulli distribution with paramter p.

EY = p.

The probit Link Function is:-

Probit(EY) = X $\beta$ , Where $\beta$ is a vector of unknown parameters

Answer d.

In estimating the linear choice model, what can we say about heteroscedasticity?

We refer to heteroscedasticity as a circumstance wherein the variability of a variable is unequal across the range of values of a second variable that predicts it. In simple terms , it means unequal scatter. It is referred in the context of the residuals or error term. OLS regression gives equal weight to all the observations , but when heteroscedasticity is present, the cases with larger disturbances have more pull than other observations.