In the following regression equation, y is a binary variable:

y=β0+β1x1+...+βkxk+u

Notice that P(y=1|x)=E(y|x) where P(X) stands for probability of event X occurring
So, P(y=1|x)=
ˆ
β
0+
ˆ
β
x1+...+
ˆ
β
xk

In this case, the estimated slope coefficient
^
β1
measures _____.

Question

In the following regression equation, y is a binary variable:

y=β0+β1x1+...+βkxk+u

Notice that P(y=1|x)=E(y|x) where P(X) stands for probability of event X occurring
So, P(y=1|x)=
ˆ
β
0+
ˆ
β
x1+...+
ˆ
β
xk

In this case, the estimated slope coefficient 
^
β1
 measures _____.

BlackTom AI · Accepted Answer

The model given is a linear probability model where the dependent variable y is binary (0 or 1) and the expectation E(y|x) is interpreted as the predicted probability P(y=1|x). In this setup, the coefficient on x1, denoted β1 hat, represents the average change in the predicted probability of the event y=1 associated with a one-unit increase in x1, holding all other regressors constant. This interpretation comes directly from the linear form y = β0 + β1 x1 + ... + βk xk + u, where the partial derivative of E(y|x) with respect to x1 is ∂E(y|x)/∂x1 = β1.

Therefore, the statement that the estimated slope coefficient β1 hat measures the predicted change in the probability of success (y=1) when x1 increases by one unit, everything else remaining constant, is the correct interpretation in the context of a linear probability model.

Note: In practice, the linear probability model can produce predicted probabilities outside the [0,1] interval and standard errors can be biased due to heteroskedasticity. In contrast, nonlinear models like logit or probit interpret coefficients differently (as changes in the latent utility or as marginal effects at certain values of x), which is why those contexts require different interpretations. However, within this regression specification, the described interpretation for β1 holds true.

AS.440.606.80.SP25 M07: Assignment

In the following regression equation, y is a binary variable: y=β0+β1x1+...+βkxk+u Notice that P(y=1|x)=E(y|x) where P(X) stands for probability of event X occurring So, P(y=1|x)= ˆ β 0+ ˆ β x1+...+ ˆ β xk In this case, the estimated slope coefficient ^ β1 measures _____.

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