In Chapter 9, the focus of study is the Dichotomous Variable. Briefly construct a model (example) to predict a dichotomous variable outcome. It can be something that you use at your

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If we assume the data-generating process for Y to be:

Y1 = α + β1X1i + … + βKXKi + Ui

Y is a dichotomous dependent variable

THEN:

Β1, …, βK represent the change in the probability of Y equaling one with a one unit increase in X1, … ,XK (respectively), holding all other Xs constant. For example, we can express β1 as: β1= Pr(Y = 1|X1 + 1, X2, …, XK) – Pr(Y = 1|X1 + 1, X2, …, XK)

The Linear Probability Model

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Merits

Imposes no restrictions on the associated regression analysis, so all methods discussed earlier (use of dummy variables, selecting controls, instrumental variables, panel data methods) seamlessly apply.

Shortcomings

It ignores the limitation of the dependent variable

The lack of restrictions on the range of predicted values of the outcome

Merits and Shortcomings of the Linear Probability Model

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Limit-violating prediction is a predicted value for a limited dependent variable that does not fall within that variable’s limits

For many applications, limit-violating predictions may not be a problem in practice

Could engineer the Xs in such a way as to preclude predictions of Y outside of 0-1

Merits and Shortcomings of the Linear Probability Model

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To overcome the limitations of the linear probability model, probit and logit models are used

The choice between a linear probability model and the alternative models is not obvious

There is no universally ”right” model

Probit and Logit Models

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The key difference the linear probability model and the alternative models (logit and probit models) is the connection between the determining function, the unobservables, and the dependent variables.

Purchase = α + βSubFee

Two Shortcomings of the linear probability model:

It is hard to believe that the determining function and unobservables always add up to exactly 0 or 1

Predictions about the effect of subscription fee (SubFee) on purchases may be unrealistic

Probit and Logit Models

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Rather than setting the dependent variable equal to the sum of the determining function and the unobservables, let the value of the dependent variable depend on this sum but in a coarse way

The sum of the determining function and the unobservables equal a latent variable

A latent variable is a variable that cannot be observed, but information about it can be inferred from other observed variables

Probit and Logit Models

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We define the sum of defining function ( α + βSubFeei) and the unobservables (Ui) to be a latent variable

If we call latent variable Utility, then:

Utilityi = α + βSubFeei + Ui

We assume a purchase occurs if utility is positive (> 0) and a purchase does not occur if utility is not positive (≤ 0)

We can express the purchase decision as:

Purchasei =

Probit and Logit Models

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Examples of Dichotomous Dependent Variables Coupled with Latent Variables