Math Problem Statement

A researcher wants to examine how the remaining balance on $100,000 loans taken 10 to 20 years ago depends on whether the loan was a prime or subprime loan. He collected a sample of 25 prime loans and 25 subprime loans and recorded the data in the following variables:

Balance = the remaining amount of loan to be paid off (in $),

Time = the time elapsed from taking the loan,

Prime = a dummy variable assuming 1 for prime loans, and 0 for subprime loans.

Variable Model B Intercept 90,269 (t = 24.35) Time −148 (t = −0.64) Prime −28,493 (t = −5.36) Time × Prime 662 (t = 2.03) SSE 1,369,126,091 R2R-squared 0.7547 Adjusted R2R-squared 0.7388

Note: The values of relevant test statistics are shown in parentheses below the estimated coefficients.

Suppose that at a 10% significance level, you do not reject the null hypothesis, H0: β1 = 0, when testing the individual significance of Time in Model B. Would you delete Time from Model B?Multiple Choice

Yes, removing Time from Model B results in Model C which has a higher adjusted R2R-squared.

No, Model B has the highest R2R-squared, so it should be used for making predictions.

Yes, Time should be deleted because we could not prove its significance even for α = 0.10.

No, removing Time from Model B results in Model C which has a with lower R2R-squared.

Solution

To answer this question, let's consider the key details:

Key Points:

  1. Time variable significance: The t-statistic for the Time variable in Model B is 0.64-0.64. The corresponding p-value is not provided, but the question states that at a 10% significance level (α=0.10\alpha = 0.10), we do not reject the null hypothesis H0:β1=0H_0: \beta_1 = 0. This indicates that the Time variable is not statistically significant at this level.

  2. Adjusted R-squared: The adjusted R2R^2 of Model B is given as 0.7388.

  3. Decision criteria: When considering removing a variable from a model, we should consider:

    • Whether the variable is statistically significant.
    • The impact on the adjusted R2R^2. If removing the variable increases the adjusted R2R^2, it suggests a better-fitting model without that variable.

Answer Options Analysis:

  1. Yes, removing Time from Model B results in Model C which has a higher adjusted R-squared:

    • This might be true if the adjusted R2R^2 increases after removing the Time variable. However, the adjusted R2R^2 value after removal isn't provided directly. The question asks whether you would delete Time based on its significance.
  2. No, Model B has the highest R-squared, so it should be used for making predictions:

    • This option focuses on the regular R2R^2, but R2R^2 alone isn’t the best criterion for model selection when adjusted R2R^2 is available.
  3. Yes, Time should be deleted because we could not prove its significance even for α=0.10\alpha = 0.10:

    • This is a valid point because the Time variable is not significant at α=0.10\alpha = 0.10, suggesting it may not contribute meaningfully to the model.
  4. No, removing Time from Model B results in Model C which has a lower adjusted R-squared:

    • If removing Time reduces the adjusted R2R^2, then keeping it would be better despite its insignificance. However, the impact on the adjusted R2R^2 is not directly provided.

Conclusion:

Given that the Time variable is not statistically significant at the 10% level, the best response, based on the information provided, is:

Yes, Time should be deleted because we could not prove its significance even for α=0.10\alpha = 0.10.

This option acknowledges the statistical insignificance of the Time variable, which generally justifies its removal.


Would you like more details on this topic or have any questions? Here are some related questions to deepen your understanding:

  1. What is the difference between R2R^2 and adjusted R2R^2?
  2. How do you interpret the t-statistic in the context of regression analysis?
  3. What is the purpose of the dummy variable in a regression model?
  4. How do we calculate the adjusted R2R^2 in a regression model?
  5. What is the implication of a negative coefficient in regression analysis?
  6. When should you consider removing a variable from a regression model?
  7. How does the interaction term (Time × Prime) affect the interpretation of the model?
  8. What could be the impact on model predictions if an insignificant variable is kept in the model?

Tip: When comparing models, always prioritize the adjusted R2R^2 over R2R^2, especially when the number of predictors varies.