1. Welcome to Econ 420 Applied Regression Analysis Study Guide Week Thirteen

2. Asst 10 The answer key is included in the next 3 slides. Study the key even if you received full point on your assignment as the answers to these questions should be treated as independent lessons

3. 1. # 5, Page 156 a) D.W. = 1.27, N = 40, k = 2 Because D.W. < 2, we want to test for positive autocorrelation Null Hypothesis: H0: ρ ≤ 0 Alt. Hypothesis: HA: ρ > 0 for N = 40, k = 2, at 5% significance: dL = 1.39 Because our D.W., 1.27 is less than d-lower, (1.27 < 1.39) we reject the null hypothesis of no autocorrelation and assume positive autocorrelation. b) D.W. = 1.27, N = 25, k = 2 Because D.W. < 2, we want to test for positive autocorrelation Null Hypothesis: H0: ρ ≤ 0 Alt. Hypothesis: HA: ρ > 0 for N = 25, k = 2, at 5% significance: dL = 1.21 and dU = 1.55 Because our D.W., 1.27 lies between d-lower and d-upper, (1.21 < 1.27 < 1.55) the test is inconclusive. c) D.W. = 2.45, N = 80, k = 5 Because D.W. > 2, we want to test for negative autocorrelation Null Hypothesis: H0: ρ ≥ 0 Alt. Hypothesis: HA: ρ < 0 for N = 80, k = 5, at 5% significance: dL = 1.51 and dU = 1.77 4 – dL = 4 – 1.51 = 2.49 and 4 – dU = 4 – 1.77 = 2.23 Because our D.W., 2.45 lies between 4 – dU and 4 – dL, (2.23 < 2.45 < 2.49) the test is inconclusive. d) 4-dl=4-1.61=2.39, 4-du=4-1.66=2.34. Since 2.45 is so far away from the center value of 2 that it exceeds 4-dl, reject the null hypothesis of no autocorrelation.

4. 2. #6 Page 156 a.The sample size is smaller in part b than in part a, but the given D.W. statistic and the number of independent variables stays the same. With the smaller sample size, the outcome of the test changes from rejecting the null hypothesis to an inconclusive result. In general, if you have a smaller sample size, the distance between dl and du becomes larger, and so there is a bigger chance you will have an inconclusive result. b.The number of independent variables falls from part c to part d, but the given D.W. statistic and the sample size stay the same. In part c, the test is inconclusive. In part d, the null hypothesis of no autocorrelation is rejected. In general, if the number of independent variables in the model falls, the distance between dl and du becomes smaller, so you are less likely to get an inconclusive result.

5. 2. #9 Page 156 No. When the first 12 observations are using in the original order, the Durbin- Watson statistic is 2.26. If the data are ordered as shown in the table in Question 7, the Durbin-Watson statistic is 1.37. Note: If you change the order of the data, the Durbin-Watson statistic will come out differently, unless the order is exactly reversed.

6. Solutions for Autocorrelation Problem If the D-W test indicate autocorrelation problem What should you do?

7. 1.Adjust the functional form Sometimes autocorrelation is caused because we use a linear form while we should have used a non-linear form revenue Price * * * * * With a linear line, errors have formed a pattern The first 3 observations have a positive error The last 2 observations have a negative errors Revenue curve is no-linear (bell shaped) 1 2 3 4 5

8. 2. Add other relevant (missing) variables Sometimes autocorrelation is caused because of omitted variables. consumption Income * * * * * 1 2 3 4 5 We forget to include wealth in our model In year one (obs. 1) wealth goes up drastically  big positive error The effect of the increase in wealth in year 1 lingers for takes 3 -4 years Errors form a pattern

9. 3. Examine the data Any systematic error in the collection or recording of data may result in autocorrelation.

10. After you make adjustments 1, 2 and 3 Test for auto again If auto correlation is still a problem –Follow the Cochrane-Orcutt procedure

11. Suppose our model is And the error terms in Equation 7- 6 are correlated Let’s lag Equation 7-6 Where u t is not auto-correlated. Rearranging 7-6B we get 7-7 e t = ρ e t-1 + u t (7-6B)

12. Now multiply Equation 7-8 by ρ Now subtract 7-9 from 7-6 to get 7-10 Note that the last two terms in Equation 7-10 are u t in 7-7 So 7-10 becomes Define Z t = Y t – ρY t-1 & W t = X t – ρX t-1 Then 7-11 becomes Z t = M + B 1 W t + u t (7-12) Where M is a constant = B 0 (1- ρ) (

13. Notice that the slope coefficient of Equation 7-12 is the same as the slope coefficient of our original equation 7-6. But Equation 7-12 does not have autocorrelation problem. This means that if we estimate Equation 7- 12, we have solved the autocorrelation problem.

14. The Cochrane-Orcutt Method: So our job will be Step 1: Apply OLS to the original model (7-6) and find the residuals e t ^ Step 2: Use e t ^s to estimate Equation 7- 6B and find ρ^ (Note: this equation does not have an intercept.) Step 3: Multiply ρ^ by Y t-1 and X t-1 & find Z t & W t Step 4: Estimate Equation 7-12

15. Luckily EViews does this (steps 1- 4) automatically All you need to do is to add AR(1) to the set of your independent variables.

16. Asst 11 (40 points) Due: Before 10 PM Sunday, November 25 # 10, 11, 12 and 13 page 157