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Marietta College

Marietta College . Spring 2011 Econ 420: Applied Regression Analysis Dr. Jacqueline Khorassani. Week 14. Tuesday, April 12. Exam 3 : Monday, April 25, 12- 2:30PM Bring your laptops to class on Thursday too. Collect Asst 21. Use the data set FISH in Chapter 8 (P 274) to

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Marietta College

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  1. Marietta College Spring 2011 Econ 420: Applied Regression Analysis Dr. Jacqueline Khorassani Week 14

  2. Tuesday, April 12 Exam 3: Monday, April 25, 12- 2:30PM Bring your laptops to class on Thursday too

  3. Collect Asst 21 Use the data set FISH in Chapter 8 (P 274) to run the following regression equation: F = f (PF, PB, Yd, P, N) • Conduct all 3 tests of imperfect multicollinearity problem and report your results. • If you find an evidence for imperfect multicollinearity problem, suggest and implement a reasonable solution.

  4. Use EViews • Open FISH in Chapter 8 • Run P = f (PF, PB, Yd, N) • Click on view on regression output • Click on actual, fitted, residual • Click on residual graph • Do you suspect the residuals to be autocorrealted?

  5. This is what you should have got Positive residual is followed by positive residual possible positive autocorrelation

  6. Causes of Impure Serial Correlation • Wrong functional form • Example: effect of age of the house on its price • Omitted variables • Example: not including wealth in the consumption equation • Data error

  7. Cause of Pure Serial Correlation • Lingering shock over time • War • Natural disaster • Stock market crash

  8. Consequences of Pure Autocorrelation • Unbiased estimates but wrong standard errors • In case of positive autocorrelation standard error of the estimated coefficients drops • Consequences on the t-test of significance?

  9. Consequences of Impure Autocorrelation • Biased estimates • Plus wrong standard errors

  10. Let’s look at first order serial correlation єt = ρєt-1 + ut ρ (row) is first order autocorrelation coefficient It takes a value between -1 to +1 u2 is a normally distributed error with the mean of zero and constant variance

  11. A Formal Test For First Order Autocorrelation • Durbin-Watson test • Estimate the regression equation • Save the residuals, e • Then calculate the Durbin -Watson Stat (d stat) • d stat ~ 2 (1- ρ) • What is dstat under perfect positive correlation? ρ = +1  d = 0 • What is dstat under perfect negative correlation? ρ = -1  d = 4 • What is dstat under no autocorrelation? ρ = 0  d = 2 • What is the range of values for dstat? 0 to 4

  12. If 2>dstat>0 then suspect (test for) positive autocorrelation If 4>dstat>2 then suspect (test for) negative autocorrelation dstat=0 Perfect positive autocorrelation dstat=4 Perfect negative autocorrelation dstat=2 No autocorrelation

  13. EViews calculates d-stat automatically • It is included in your regression output • Run P = f (PF, PB, Yd, N) • Do you see the d-stat?

  14. What type of serial correlation shall we test for? Positive Dependent Variable: P Method: Least Squares Date: 04/12/11 Time: 08:59 Sample: 1946 1970 Included observations: 25 Variable Coefficient Std. Error t-Statistic Prob.   C -2.083188 0.271658 -7.668417 0.0000 PF 0.027143 0.017355 1.563934 0.1335 PB -0.012571 0.011620 -1.081865 0.2922 YD 0.001597 0.000387 4.132263 0.0005 N - 5.54E-05 1.27E-05 -4.376214 0.0003 R-squared 0.801154     Mean dependent var 0.160000 Adjusted R-squared 0.761384     S.D. dependent var 0.374166 S.E. of regression 0.182774     Akaike info criterion -0.384281 Sum squared resid 0.668123     Schwarz criterion -0.140506 Log likelihood 9.803514     Hannan-Quinn criter. -0.316668 F-statistic 20.14505     Durbin-Watson stat 1.498086 Prob(F-statistic) 0.000001

  15. If d stat<2, test for positive autocorrelation. • Null and alternative hypotheses • H0: ρ≤0 (no positive auto) • HA: ρ>0 (positive auto) • Choose the level of significance (say 5%) • Critical dstat (PP 591- 593) • Decision rule • If dstat< dL reject H0  there is significant positive first order autocorrelation • If dstat> dU  don’t reject H0  there is no evidence of a significant autocorrelation • if dstat is between dL and du  the test is inconclusive.

  16. N = 25, K = 4 At 5% level dL= 1.04, dU=1.77 dstat is between dL and du  the test is inconclusive Dependent Variable: P Method: Least Squares Date: 04/12/11 Time: 08:59 Sample: 1946 1970 Included observations: 25 Variable Coefficient Std. Error t-Statistic Prob.   C -2.083188 0.271658 -7.668417 0.0000 PF 0.027143 0.017355 1.563934 0.1335 PB -0.012571 0.011620 -1.081865 0.2922 YD 0.001597 0.000387 4.132263 0.0005 N - 5.54E-05 1.27E-05 -4.376214 0.0003 R-squared 0.801154     Mean dependent var 0.160000 Adjusted R-squared 0.761384     S.D. dependent var 0.374166 S.E. of regression 0.182774     Akaike info criterion -0.384281 Sum squared resid 0.668123     Schwarz criterion -0.140506 Log likelihood 9.803514     Hannan-Quinn criter. -0.316668 F-statistic 20.14505     Durbin-Watson stat 1.498086 Prob(F-statistic) 0.000001

  17. H0: ρ≤0 (no positive auto) • HA: ρ>0 (positive auto) • level of significance = 5% • Critical d-stat • dL =1.04 • dU = 1.77 • Decision • dstat is between dL and du  the test is inconclusive Fail to reject H0 Reject H0 inconclusive 1.5 DWstat=0 Perfect positive autocorrelation 1.04 1.77 DWstat=4 Perfect negative autocorrelation DWstat=2 No autocorrelation

  18. If dstat >2, you will to test for negative autocorrelation. • Null and alternative hypotheses • H0: ρ≥0 (no negative auto) • HA: ρ<0 (negative auto) • Choose the level of significance (1% or 5%) • Critical dstat (page 591- 593) • Decision rule • If dstat>4-dL reject H0  there is significant negative first order autocorrelation • If dstat< 4-dU  don’t reject H0  there is no evidence of a significant autocorrelation • if dstat is between 4 – dL and 4 – du  the test is inconclusive.

  19. Example d-sta >2  test for negative autocorrelation Dependent Variable: CONSUMPTION Method: Least Squares Date: 11/09/08 Time: 20:11 Sample: 1 30 Included observations: 30 Variable Coefficient Std. Error t-Statistic Prob.   C 16222.97 5436.061 2.984324 0.0060 INCOME 0.641166 0.166878 3.842131 0.0007 WEALTH 0.148788 0.041327 3.600281 0.0013 R-squared 0.847738      Mean dependent var 52347.37 Adjusted R-squared 0.836459      S.D. dependent var31306.54 S.E. of regression 12660.43Akaike info criterion 21.82499 Sum squared resid 4.33E+09   Schwarz criterion 21.96511 Log likelihood -324.3748  Hannan-Quinn criter. 21.86982 F-statistic 75.16274      Durbin-Watson stat 2.211726 Prob(F-statistic) 0.000000

  20. Let’s test for autocorrelation at 1% level in our example H0: ρ≥0 (no negative auto) HA: ρ<0 (negative auto) • 1% level of significance, k=2, n=30 • dL=1.07, du= 1.34 • 4- dL=2.93, 4- du= 2.66 • dstat <4- du, don’t reject H0

  21. Asst 22: Due Thursday • Use the data on Soviet Defense spending (Page 335– Data set: DEFEND Chapter 9) to regress SDH on SDL, UDS and NR only. • Conduct a Durbin-Watson test for serial correlation at 5% level of significance • If you find an evidence for autocorrelation, is it more likely to be pure or impure autocorrelation? Why?

  22. Thursday April 15 • Exam 3: Monday, April 25, 12- 2:30PM • Bring your laptops to class next Tuesday

  23. Collect Asst 22 • Use the data on Soviet Defense spending (Page 335– Data set: DEFEND Chapter 9) to regress SDH on SDL, USD and NR only. • Conduct a Durbin-Watson test for serial correlation at 5% level of significance • If you find an evidence for autocorrelation, is it more likely to be pure or impure autocorrelation? Why?

  24. Solutions for Autocorrelation Problem • If the D-W test indicates autocorrelation problem • What should you do?

  25. Adjust the functional form • Sometimes autocorrelation is because we use a linear form while we should have used a non-linear form With a linear line, errors have formed a pattern The first 3 observations have positive errors The last 2 observations have negative errors Revenue curve is not linear (It is bell shaped) What should we use? revenue 3 * * 4 2 * 5 * 1 * Price

  26. 2. Add other relevant (missing) variables 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 3 years Errors form a pattern We should include wealth in our model • Sometimes autocorrelation is caused by omitted variables. consumption * 5 4 3 * * 2 * * 1 Income

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

  28. After you make adjustments 1, 2 and 3 • Test for autocorrelation again • If autocorrelation is still a problem then suspect pure autocorrelation • Follow the Cochrane-Orcutt procedure • Say what?????

  29. Suppose our model is Yt = β0 + β1Xt + єt(1) And the error terms in Equation 1 are correlated єt = ρє t-1 + ut(2) Where ut is not auto-correlated. Rearranging 2 we get 3 єt - ρє t-1 = ut(3) Let’s lag Equation 1 Yt-1 = β0 + β1Xt-1 + єt-1 (4)

  30. Now multiply Equation 4 by ρ ρ Yt-1 = ρ β0 + ρβ1Xt-1 + ρ єt-1 (5) Now subtract 5 from 1 to get 6 Yt = β0 + β1 Xt + єt - ρ Yt-1 = - ( ρ β0 + ρβ1Xt-1 + ρ єt-1) ___________________________________ Yt - ρ Yt-1 = β0 - ρ β0 + β1 Xt - ρβ1Xt-1 + єt - ρ єt-1 (6) Note that the last two terms in Equation 6 are equal to Ut So 6 becomes Yt - ρ Yt-1 = β0 - ρ β0 + β1 (Xt - ρ Xt-1 ) + ut (7)

  31. Yt - ρ Yt-1 =β0- ρ β0 + β1 (Xt - ρ Xt-1 ) + ut (7) • What is so special about the error term in Equation 7? It is not auto-correlated • So, instead of equation 1 we can estimate equation 7 Define Zt = Yt – ρYt-1 & Wt = Xt – ρXt-1 Then 7 becomes Zt= M + β1 Wt + ut(8) Where M is a constant = β0 (1-ρ) Notice that the slope coefficient of Equation 8 is the same as the slope coefficient of our original equation 1.

  32. The Cochrane-Orcutt Method:So our job will be Step 1: Apply OLS to the original model (Equation 1) and find the residuals et Step 2: Use ets to estimate Equation 2 and find ρ^ (Note: this equation does not have an intercept.) Step 3: Multiply ρ^ by Yt-1 and Xt-1 & find Zt & Wt Step 4: Estimate Equation 8

  33. 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. • The estimated coefficient of AR(1) is ρ^ • Let’s apply this procedure to Asst 22

  34. Dependent Variable: SDH Variable Coefficient Std. Error t-Statistic Prob.   C -9.11 8.4 -1.08 0.2940 • SDL 1.38 0.17 8.10 0.0000 • USD 6.71E-05 0.013 0.005 0.9959 • NR 0.0005 0.0004 1.46 0.1608 • AR(1) 0.82 0.10 8.002 0.0000 • R-squared 0.997927      • Adjusted R-squared 0.997490      • Durbin-Watson stat 2.463339 Dependent Variable: SDH Variable Coefficient Std. Error t-Statistic Prob.   C 8.83 2.50 3.52 0.0020 SDL 0.97 0.04 22.18 0.0000 USD -0.005 0.008 -0.60 0.5553 NR 0.002 0.0002 9.30 0.0000 R-squared 0.996792      Adjusted R-squared 0.996334    Durbin-Watson stat 1.076364 What happened to standard errors as we corrected for serial correlation? They went up Positive autocorrelation standard error What is this? It is ρ^

  35. Return and discuss Asst 21 Use the data set FISH in Chapter 8 (P 274) to run the following regression equation: F = f (PF, PB, Yd, P, N) • Conduct all 3 tests of imperfect multicollinearity problem and report your results. • If you find an evidence for imperfect multicollinearity problem, suggest and implement a reasonable solution.

  36. Correlation Matrix First test PF is more correlated with PB than with F  PF is a problem Yd is more correlated with PB and PF than with F Yd is a problem N is more correlated with PB, PF and Yd than with F N is a problem PB is more correlated with PF than with F  PB is a problem P is more correlated with everything else than with F P is a problem F P PB PF YD N F 1 0.58 0.82 0.85 0.79 0.74 P 1 0.66 0.73 0.78 0.57 PB 1 0.96 0.82 0.78 PF 1 0.92 0.88 YD 1 0.93 N 1

  37. Correlation Matrix • Second test: • problem areas: • PF and PB • PF and Yd • PF and N • PB and Yd • Yd and N F P PB PF YD N F 1 0.58 0.82 0.85 0.79 0.74 P 1 0.66 0.73 0.78 0.57 PB 1 0.96 0.82 0.78 PF 1 0.920.88 YD 1 0.93 N 1 Note: F being highly correlated with independent variables is a good thing not a bad thing

  38. Test 3 • Need 5 regression equations • PF = f (P, Yd, PB, N) • P = f (PF, Yd, PB, N) • Yd = f (P, PF, PB, N) • PB = f (PF, Yd, P, N) • N = f (PF, Yd, PB, P) • For all find R2 then find VIF • For all VIF>5  Each independent variable is highly correlated with the rest

  39. Solutions • Increase sample size • Note: we want at least a df= 30, we have df=19 • Do we have an irrelevant variable? • Seth argued N is not needed? • What is N? (P 273) • Seth, what was your argument? • Generate a new variable that measures the ratio of prices • Makes sense but doesn’t solve the high correlation between Yd and N • Note: make sure your transformed variable makes sense • That is the estimated coefficient has a meaning that people can understand • The ratio PF/Yd makes no sense

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