Prediction for Dichotomous Variable
Chapter 9
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Learning Objectives
Identify a limited dependent variable and its applications
Describe the linear probability model
Identify merits and shortcomings of the linear probability model
Model probit and logit models as determined by the realization of latent variable
Calculate marginal effects for logit and probit models
Execute estimation of a probit and logit model via maximum likelihood
Identify the merits and shortcomings of the probit and logit models in practice

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Limited Dependent Variable
Limited dependent variable
A dependent variable whose range of possible values has consequential constraints
Some constraints include upper and/or lower bounds or the ability to take on only discrete values

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IN THIS TABLE, A RANDOM VARIABLE SPENTit REPRESENTING THE AMOUNT OF MONEY SPENT BUYING PRODUCTS ONLINE BY HOUSEHOLD i IN WEEK k.
PRODUCTS ESSENTIALLY ALWAYS HAVE NON-NEGATIVE PRICES, SO THIS RANDOM VARIABLE IS CONSTRAINED TO BE AT LEAST ZERO.
IN THIS TABLE, WE SEE SEVERAL OBSERVATIONS WITH SPENTit VALUES EXACTLY EQUALS THE CONSTRAINT OF ZERO.
Limited Dependent Variables

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Dichotomous (or binary) dependent variable
A limited dependent variable that can take on just two values, typically recorded as 0 and 1
Measure many different types of outcomes: purchase/don’t purchase, project success/project failure, employed/unemployed, approve/disapprove
Limited Dependent Variables

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Linear probability model defined as regression analysis applied to a dichotomous dependent variable
Widely used model
The act of fitting the equation Purchase = α + βSubFee to the data by solving the moment condition is an application of a linear probability model.
The Linear Probability Model

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Data on Subscription Fees and Purchase Decisions for SaferContent

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The following table shows regression estimates that fit the function Purchase = α + βSubFee to the data:
Based on the estimates in the above table the determining function would be: Purchasei = 1.65 – 0.05 × SubFeei
The Linear Probability Model

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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

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Define the latent variable, Y*, as the sum of the determining function and the unobservables:
Y*i = α + β1X1i + … + βKXKi + Ui
Then define the dependent variable, Yi , to be 1 if the latent variable exceeds 0, and otherwise:
Yi =
Notice we do not need the determining function and unobservables to add up exactly to 0 or 1
Probit and Logit Models

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The latent variable formation for Y also prevents unreasonable predictions about the probability of Y equaling 1
Pr(Yi = 1|X1i, …, XKi) = Pr(Y*i > 0|X1i , …, XKi)
This equation states that the probability the outcome (Y) equals 1, given the values for the Xs, is equal to the probability that the latent variable (Y*) is greater than 0, given the values for the Xs
Probit and Logit Models

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Probit and Logit Models
Pr(Yi = 1|X1i, …, XKi) = Pr(α + β1X1i + … + βKXKi + Ui > 0|X1i , …, XKi)
Uncertainty about Y is due to uncertainty about U.
Pr(Yi = 1|X1i, …, XKi) = Pr(Ui > ‒ α ‒ β1X1i ‒ … ‒ βKXKi|X1i , …, XKi)
While the determining function is unconstrained, the probability that Y equals 1 is explicitly defined to be a probability in terms of U, so constrained to be between 0 and 1

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Probit model a latent variable formulation for a dichotomous dependent variable that assumes a standard normal distribution for the unobservables
The probability that Y equals 1 for given values of the Xs using formula:
Pr(Yi = 1|X1i, …, XKi) = Pr(Ui > ‒ α ‒ β1X1i ‒ … ‒ βKXKi|X1i , …, XKi)
Probit and Logit Models

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Probit and Logit Models
Probability Y Equals 1 for Given Xs, Assuming Standard Normal Distribution for U

IN THE GRAPH ϕ(U) IS THE PROBABILITY DENSITY FUNCTION (pdf) FOR THE STANDARD NORMAL DISTRIBUTION.

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Probit and Logit Models
Since we know Ui is a standard normal random variable, simplify the expression for Pr(Yi = 1|X1i, …, XKi)
Normal distribution is symmetric around its mean (which is 0 for U)
Pr(Ui > ‒ α ‒ β1X1i ‒ … ‒ βKXKi|X1i , …, XKi) =
Pr(Ui < α + β1X1i + … + βKXKi|X1i , …, XKi)
Define ɸ(.) as the cumulative distribution function (cdf) for a standard normal random variable U, where ɸ(m) = Pr(U < m)
Pr(Yi = 1|X1i, …, XKi) = ɸ(α + β1X1i + … + βKXKi)

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Probit and Logit Models
Probability Y Equals 1 for Given Xs, Assuming Standard Normal Distribution for U and Using cdf for U

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Logit model is a latent variable formulation for a dichotomous dependent variable that assumes a Logistic(0,1) distribution for the unobservables
The logistic distribution generates a simple formula for the probability of Y equaling 1 for a given set of Xs
When we assume that Ui~Logistic(0,1), the probability that Y equals 1 for given values of the Xs can be expressed as:
Pr(Yi = 1|X1i, …, XKi) =
Probit and Logit Models

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Marginal effect is the rate of change in the probability of a dichotomous dependent variable equaling 1 with one-unit increase in an independent variable (holding all other independent variables constant)
For the linear probability model, the βs in the determining function measure marginal effects
Marginal Effects

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Marginal Effects
Consider the following general latent variable model:
Y*i = α + β1X1i + … + βKXKi + Ui
The marginal effect of Xj is:
MargEffxj = Pr(Yi = 1|X1i, …, Xji +1,…, XK) ‒ Pr(Yi = 1|X1i, …, Xji , …, XKi)

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Marginal Effects
For Probit:
MargEffxj = ɸ(α + β1X1i + … βj(Xji +1) + … + βKXKi) ‒ ɸ(α + β1X1i + … + βjXji + … + βKXKi)
For Logit:
MargEffxj = ‒

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Marginal Effects
Probit and logit marginal effects generally depend on the magnitude of the change in the independent variable
Probit and logit marginal effects generally differ depending on the level of X from which a change is being considered
Because the marginal effects we measure depend on the starting point of X, there is not an obvious, single number as the marginal effect of X
In practice, it is common to attempt to summarize the marginal effect of x for a probit or logit model using a single number

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We have taken the parameters (e.g., α, β) as given, in practice we get estimates for these parameters using the data
For the linear probability model, solve for the parameters using the sample moment equations
Maximum likelihood estimation (MLE) using this approach population level parameters are estimated using values that make the observed outcomes as likely as possible for a given model
Estimation and Interpretation

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Maximum Likelihood Estimation
Consider the following general latent variable model:
Y*i = α + β1X1i + … + βKXKi + Ui
Let Yi be 1 if the latent variable exceeds 0, and 0 if otherwise
Assuming a probit model:
Pr(Yi = 1|X1i, …, XKi) = ɸ(α + β1X1i + … + βKXKi)
To get estimates for our parameters , collect a sample of Ys and Xs of size N

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Maximum Likelihood Estimation
The probability of this observation is:
1 ‒ ɸ(α + β1X1i + … + βKXKi)
Assuming the logit model:
Everything is the same as in the probit example, except the probabilty formulas

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Probit Results for SaferContent Data

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Logit Results for SaferContent Data

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Merits and Shortcomings
Merits of probit and logit models
Help overcome shortcomings of the linear probability model
The latent variable formation places no restrictions per se on the relationship between the determining function and unobservables
Both models predict probabilities rather than the actual value (0 or 1) for the dependent variable

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Merits and Shortcomings
Shortcomings of probit and logit models
The probabilities implied by the probit and logit models directly depend on the assumption of a normal or logistic distribution for the unobservables
Added complexity of calculating marginal effects, relative to the linear probability model
Use of instrumental variables and fixed effects

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