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

𝑦
=
𝛽
0
+
𝛽
1
𝑥
1
+
.
.
.
+
𝛽
𝑘
𝑥
𝑘
+
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
)
=
𝛽
^
0
+
𝛽
^
𝑥
1
+
.
.
.
+
𝛽
^
𝑥
𝑘

In this case, the estimated slope coefficient
𝛽
1
^
measures _____.

Question

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

𝑦
=
𝛽
0
+
𝛽
1
𝑥
1
+
.
.
.
+
𝛽
𝑘
𝑥
𝑘
+
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
)
=
𝛽
^
0
+
𝛽
^
𝑥
1
+
.
.
.
+
𝛽
^
𝑥
𝑘

In this case, the estimated slope coefficient 
𝛽
1
^
 measures _____.

BlackTom AI · Accepted Answer

The question presents a regression setup where y is a binary outcome and the model is written as a linear probability model: y = β0 + β1 x1 + ... + βk xk + u, with P(y = 1 | x) = E(y | x). In this context, the coefficient β1 hat represents the amount by which the predicted probability P(y = 1 | x) changes when x1 increases by one unit, holding all other predictors constant.

Option analysis:
- The sole option states: 'the predicted change in the probability of success when x1 increases by one unit, everything else remaining constant'. This aligns exactly with the interpretation of a slope coefficient in a linear probability model, where the dependent variable is binary and the link between x and the probability is linear. It captures the magnitude of the effect on P(y=1|x) per unit increase in x1, while other covariates stay fixed.

Potential misconceptions addressed:
- If one were thinking in terms of a nonlinear model like logistic or probit, the coefficient β1 would not equal a simple probability change; instead, it would relate to log-odds (logit) or z-scores (probit), and the marginal effect would depend on x and would require transformation to interpret as a change in probability. However, the equation given explicitly uses P(y=1|x) = E(y|x) and presents a linear form for y, which justifies interpreting β1 as a probability change.

Why other interpretations would be incorrect in this setup:
- Interpreting β1 as the change in any outcome other than the probability of y=1 would be inconsistent with P(y=1|x) = E(y|x) and the linear specification, since the dependent variable y itself is the indicator of the event. 
- Thinking β1 captures a change in the expected value of y without tying it to a probability would ignore that E(y|x) in this model is a probability (a value between 0 and 1). 
- Assuming the coefficient measures a change in the log-odds or require nonlinear transformation would be appropriate only if the model were specified as logit or probit; that is not the case here.

In summary, within the given linear probability framework, the coefficient β1 hat quantifies how much the predicted probability of y = 1 changes with a one-unit increase in x1, with all other predictors held constant.

AS.440.606.50.SP25 M07: Assignment

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Similar Questions

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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