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.

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

2. **Adjusted R-squared:** The adjusted $R^2$ 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 $R^2$. If removing the variable increases the adjusted $R^2$, 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 $R^2$ increases after removing the Time variable. However, the adjusted $R^2$ 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 $R^2$, but $R^2$ alone isn’t the best criterion for model selection when adjusted $R^2$ is available.

3. **Yes, Time should be deleted because we could not prove its significance even for $\alpha = 0.10$:** 
   - This is a valid point because the Time variable is not significant at $\alpha = 0.10$, 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 $R^2$, then keeping it would be better despite its insignificance. However, the impact on the adjusted $R^2$ 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 $\alpha = 0.10$.**

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

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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 $R^2$ and adjusted $R^2$?
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 $R^2$ 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 $R^2$ over $R^2$, especially when the number of predictors varies.

Explore the significance of variables in regression models using a practical example. Learn when to remove a variable based on statistical tests and adjusted R-squared. Detailed analysis includes a case study on loan balance prediction with time and loan type variables.

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