Consider the regression model:
𝑦
=
𝛽
0
+
𝛽
1
𝑥
+
𝑢
.

Which of the following is correct?

Question

Consider the regression model: 
𝑦
=
𝛽
0
+
𝛽
1
𝑥
+
𝑢
 .

Which of the following is correct?

BlackTom AI · Accepted Answer

In the linear model y = β0 + β1 x + u, x is the explanatory (independent) variable and is used to predict y.

1. Relevant knowledge points and concepts
- β0 is the intercept, β1 is the slope coefficient, and u is the error term capturing unobserved factors.
- The conditional expectation E[y | x] = β0 + β1 x under the usual assumption E[u | x] = 0. This conditional expectation is the prediction of y given x.
- Ordinary least squares (OLS) estimates β0 and β1 and yields predicted values ŷ = β̂0 + β̂1 x.

2. Complete reasoning process step by step
- Start from the model: y = β0 + β1 x + u.
- Take conditional expectation given x: E[y | x] = E[β0 + β1 x + u | x] = β0 + β1 x + E[u | x].
- Under the standard assumption E[u | x] = 0, this reduces to E[y | x] = β0 + β1 x. Thus the systematic part of y depends on x and provides the prediction function.
- In practice, estimate β0 and β1 by OLS: minimize Σ(y_i − β0 − β1 x_i)^2 to obtain β̂1 = Cov(x,y)/Var(x), β̂0 = ȳ − β̂1 x̄. Predicted values are ŷ_i = β̂0 + β̂1 x_i.

3. Intermediate calculation steps (estimator formulas)
- β̂1 = Σ(x_i − x̄)(y_i − ȳ) / Σ(x_i − x̄)^2
- β̂0 = ȳ − β̂1 x̄
- ŷ_i = β̂0 + β̂1 x_i

4. Why the correct answer is correct
- Because the deterministic component β0 + β1 x gives the expected/predicted value of y for any given x, x functions as the predictor or explanatory variable.

5. Why other possible statements would be incorrect
- Saying y predicts x reverses roles; the model is specified as y dependent on x.
- Claiming β1 is the intercept is false: β1 is the slope; β0 is the intercept.
- Suggesting u is used to predict y is wrong: u is the unobserved error, not a predictor.

6. Conclusion
Therefore, the correct statement is that x is used to predict y, which matches the provided answer.

Consider the regression model: 𝑦 = 𝛽 0 + 𝛽 1 𝑥 + 𝑢 . Which of the following is correct? Single choice

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