Question

1) Define and explain what is meant by the following:

a. Limited Dependent Variables

b. The linear Probability Model

c. Probit and Logit Models

d. Maximum Likelihood estimation

2) Explain whether, or under what circumstance Probit and Logit Models causes a problem for inference in Maximum Likelihood estimation?

Answer 1

1) a) Limited Dependent Variables -

A limited dependent variable means that there is a limit or boundary on the dependent variable and some of the observations hit the limit or boundary. It is a variable whose range of possible values is restricted in some or the other way. It is a continuous variable with a lot of repeated observations at the lower or upper limit.

In econometrics, this term is often used when the estimation of the relationship between the limited dependent variable of interest and other variables requires methods that take this restriction into account. The limited dependent variable models gives two issues which are censoring and truncation.

Example : Quantity of any product consumed, number of hours a person works.

b) The Linear Probability Model -

It is a regression model where the outcome variable is a binary variable, and one or more explanatory variables are used to predict the outcome. The explanatory variables can be binary or continuous. The Linear Probability Model is a special case of a binomial regression model. Here, the dependent variable for each observation takes values which are either 0 or 1

The Linear Probability Model allows the model to be fitted by simple linear regression. A drawback of this model is that, unless restrictions are placed on , the estimated coefficients can imply probabilities outside the unit interval [0,1] .

c) Probit and Logit Models -

Logit and Probit Models are appropriate while attempting to model a dichotomous dependent variable,for e.g., like/dislike, yes/no etc.

In the Probit Model, the inverse standard normal distribution of the probability is modelled as a linear combination of the predictors. It is a type of regression where the dependent variable can take only two values, for example, single or not single. The word comes from "probability + unit". The Probit Model is a popular specification for a binary response model.

The Logit Model is a statistical model that in its basic form uses a logistic function to model a binary dependent variable, although many more complex extensions exist. In regression analysis, logit regression is estimating the parameters of a logistic model.

Logit and Probit Models are used very extensively in the literature to capture the distribution functions of the outcome variable, which is the selection equation.

d) Maximum Likelihood Estimation -

It is a statistical method for estimating population parameters (mean, variance) from sample data that selects as estimates those parameter values maximizing the probability of obtaining the observed data. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference.

Maximum likelihood estimation is a method that determines values for the parameters of a model. The parameter values are found such that they maximize the likelihood that the process described by the model produced the data that were actually observed.

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