Consider the regression model: .

Which of the following is correct?

Question

Consider the regression model:  .

Which of the following is correct?

BlackTom AI · Accepted Answer

In the equation y = β0 + β1 x + u, x is the independent (explanatory) variable and y is the dependent (response) variable, so x is used to predict y.

1) Relevant knowledge points and concepts
- Regression expresses the dependent variable as a function of explanatory variables plus an error: y = f(x) + u.  
- β0 and β1 are parameters; β1 is the marginal effect of x on y (change in y per unit change in x).  
- u is the error term (unobserved factors, random disturbance), not a predictor.  
- A prediction uses the estimated function ŷ = β̂0 + β̂1 x.

2) Step-by-step reasoning
- The model writes y on the left-hand side and x on the right-hand side: this notation indicates y depends on x (and u).  
- Given a value of x, the model gives a predicted y: ŷ = β̂0 + β̂1 x. That is the operational prediction formula used in OLS or other estimation.  
- The parameter β1 quantifies how y changes when x changes, confirming x’s role as the predictor.

3) Intermediate expression / calculation
- After estimating parameters, prediction is explicit: ŷ = β̂0 + β̂1 x. No algebraic inversion or use of u is required or appropriate for prediction.

4) Why the correct answer is correct
- The functional form and predictive formula directly show x is used to produce estimates/predictions of y. The error term u is not observed and represents deviations, so it cannot be used as a predictor.

5) Why other options are incorrect
- “y is used to predict x”: the model does not express x as a function of y; that would require a different model x = ...  
- “u is used to predict y” or “u is used to predict x”: u is the disturbance (unobserved noise), not an explanatory variable used for prediction.

Conclusion: therefore the correct statement is “x is used to predict y,” which matches the provided answer.

Consider the regression model: . Which of the following is correct? 单项选择题

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