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

First, let me restate the question and options for clarity:
- The regression equation is y = β0 + β1 x1 + ... + βk xk + u, where y is a binary variable and P(y = 1 | x) = E(y | x) (i.e., the probability of y = 1 given x).
- The question asks: In this case, the estimated slope coefficient β1 hat measures _____.
- Answer options:
  1) the predicted change in the value of y when x1 increases by one unit, everything else remaining constant
  2) the predicted change in the value of y when x1 increases by one unit, everything else remaining constant
  3) the predicted change in the probability of success when x1 increases by one unit, everything else remaining constant
  4) the predicted change in the probability of failure when x1 increases by one unit, everything else remaining constant

Now, let’s evaluate each option with careful language and logic.

Option 1: the predicted change in the value of y when x1 increases by one unit, everything else remaining constant
- Why this is not correct: Although y is binary (0 or 1), the model as described is interpreted through its conditional expectation E(y|x), which equals P(y = 1 | x). The coefficient β1 hat captures how the expected value of y changes with x1, which, in this context, is the change in the probability that y = 1. Saying it measures a change in the raw y value disregards the probabilistic interpretation; the focus is on probability, not a direct change in the observed binary outcome y itself. In short, the slope is not about a direct shift in the binary outcome, but about a shift in the expected probability of the outcome being 1.

Option 2: the predicted change in the value of y when x1 increases by one unit, everything else remaining constant
- Why this is not correct: This is essentially a restatement of option 1 with slightly different wording. It still treats the slope as the change in a raw binary y, rather than the change in the probability representation P(y = 1 | x). Since y takes only 0 or 1, talking about its change ignores the probabilistic interpretation and is misleading for a regression understood through E(y|x).

Option 3: the predicted change in the probability of success when x1 increases by one unit, everything else remaining constant
- Why this is correct: Given P(y = 1 | x) = E(y | x) and the linear specification y = β0 + β1 x1 + ... + βk xk + u, the estimated slope β1 hat describes how the expected value of y, i.e., the probability that y equals 1, changes as x1 increases by one unit, keeping other predictors fixed. In other words, β1 hat represents the change in P(y = 1 | x) for a one-unit increase in x1, which is precisely the change in the probability of success under this formulation.

Option 4: the predicted change in the probability of failure when x1 increases by one unit, everything else remaining constant
- Why this is not correct: The probability of failure is P(y = 0 | x) = 1 - P(y = 1 | x). While there is a relationship between the two probabilities, the slope β1 hat in this linear probability model directly governs P(y = 1 | x), not P(y = 0 | x). Therefore, saying the slope measures the change in the probability of failure would be the complement of what β1 hat actually measures, unless you explicitly transform the interpretation, which is not what the model as stated implies.

In summary, the slope coefficient in this linear representation of a binary outcome is interpreted as the change in the probability of the event y = 1 occurring (success) when x1 increases by one unit, holding other variables constant. The other options misinterpret whether the coefficient refers to a direct change in the binary outcome, the direction of change, or the probability of the complementary event (y = 0).

AS.440.606.80.SP25 M07: Assignment

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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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