# Multiple Regression & Auto Correlation

Part I: Actuarial Model (Questions 1-12)

An actuary wanted to develop a model to predict how long individuals will live. After consulting a number of physicians, she collected the age at death (y), the average number of hours of exercise per week (x1), the cholesterol level (x2), and the number of points that the individual's blood pressure exceeded the recommended value (x3). A random sample of 40 individuals was selected. The computer output of the multiple regression model is shown below.

THE REGRESSION EQUATION IS

y = 55.8 + 1.79x1 ? 0.021x2 ? 0.061x3

Predictor Coef StDev T

Constant 55.8 11.8 4.729

x1 1.79 0.44 4.068

x2 ?0.021 0.011 ?1.909

x3 ?0.016 0.014 ?1.143

S = 9.47 R?Sq = 22.5%

ANALYSIS OF VARIANCE

Source of Variation df SS MS F

Regression 3 936 312 3.477

Error 36 3230 89.722

Total 39 4166

1. How many degrees of freedom do we have for a t-test?

2. How many degrees of freedom (numerator and denominator) do we have for an F-test?

3. What is the relevant rejection region for an F test at the 5% significance level?

4. Is there enough evidence at the 5% significance level to infer that the model is useful in predicting length of life?

5. Is there enough evidence at the 1% significance level to infer that the average number of hours of exercise per week and the age at death are linearly related?

6. Is there enough evidence at the 5% significance level to infer that the cholesterol level and the age at death are negatively linearly related?

7. Is there sufficient evidence at the 5% significance level to infer that the number of points that the individual's blood pressure exceeded the recommended value and the age at death are negatively linearly related?

8. What is the coefficient of determination? What does this statistic tell you?

9. What is the adjusted coefficient of determination in this situation? What does this statistic tell you?

10. Interpret the coefficient b1.

11. Interpret the coefficient b2.

12. Interpret the coefficient b3.

Part 2: Multicollinearity (Questions 13-15)

Three predictor variables are being considered for use in a linear regression model. Use the correlation matrix below to identify where multicollinearity could be a problem.

x1 x2 x3

x1 1.000

x2 0.025 1.000

x3 0.968 0.897 1.000

13. Does the data in this table suggest a multicollinearity problem between X1 and X2?

14. Does the data in this table suggest a multicollinearity problem between X1 and X3?

15. Does the data in this table suggest a multicollinearity problem between X1 and Y?

Part 3: Autocorrelation (Questions 16-20)

Check for the presence of positive first-order autocorrelation, given that: Durbin-Watson Statistic d = 1.12, n = 45, k = 5, and ? = 0.05.

16. What are dL and dU?

17. What is the range in which we can reject the null hypothesis and conclude that positive autocorrelation is present?

18. What is the range in which we say that the test is inconclusive?

19. What is the range in which we can conclude that there is no evidence of positive autocorrelation?

20. Test the hypotheses H0: no first-order autocorrelation vs. H1: positive first-order autocorrelation.

#### Solution Summary

Multiple Regression & Auto Correlation