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Introduction to Econometrics, 5th edition Chapter 12: Autocorrelation

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1 Introduction to Econometrics, 5th edition Chapter 12: Autocorrelation
Type author name/s here Dougherty Introduction to Econometrics, 5th edition Chapter heading Chapter 12: Autocorrelation © Christopher Dougherty, All rights reserved.

2 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Dependent Variable: LGFOOD Method: Least Squares Sample: Included observations: 45 Variable Coefficient Std. Error t-Statistic Prob. C LGDPI LGPRFOOD R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn crite F-statistic Durbin-Watson stat Prob(F-statistic) The output shown in the table gives the result of a logarithmic regression of expenditure on food on disposable personal income and the relative price of food. 1

3 TESTS FOR AUTOCORRELATION III: EXAMPLES
Residuals, static logarithmic regression for FOOD The plot of the residuals is shown. All the tests indicate highly significant autocorrelation. 2

4 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Dependent Variable: RLGFOOD Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. RLGFOOD(-1) R-squared Mean dependent var 3.28E-05 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat RLGFOOD in the regression above is the residual from the LGFOOD regression. A simple regression of RLGFOOD on RLGFOOD(–1) yields a coefficient of 0.79 with standard error 0.11. 3

5 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Dependent Variable: RLGFOOD Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. RLGFOOD(-1) R-squared Mean dependent var 3.28E-05 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat Technical note for EViews users: EViews places the residuals from the most recent regression in a pseudo-variable called resid. resid cannot be used directly. So the residuals were saved as RLGFOOD using the genr command: genr RLGFOOD = resid 4

6 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Dependent Variable: RLGFOOD Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. C LGDPI E LGPRFOOD RLGFOOD(-1) R-squared Mean dependent var 3.28E-05 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Next, the Breusch‒Godfrey test. Adding an intercept, LGDPI and LGPRFOOD to the specification, the coefficient of the lagged residuals becomes 0.81 with standard error R2 is , so nR2 is 5

7 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Dependent Variable: RLGFOOD Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. C LGDPI E LGPRFOOD RLGFOOD(-1) R-squared Mean dependent var 3.28E-05 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) (Note that here n = 44. There are 45 observations in the regression in Table 12.1, and one fewer in the residuals regression.) The critical value of chi-squared with one degree of freedom at the 0.1 percent level is 6

8 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Breusch-Godfrey Serial Correlation LM Test: F-statistic Probability Obs*R-squared Probability Test Equation: Dependent Variable: RESID Method: Least Squares Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-Statistic Prob. C LGDPI E LGPRFOOD RESID(-1) R-squared Mean dependent var-1.85E-18 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Technical note for EViews users: one can perform the test simply by following the LGFOOD regression with the command auto(1). EViews allows itself to use resid directly. 7

9 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Breusch-Godfrey Serial Correlation LM Test: F-statistic Probability Obs*R-squared Probability Test Equation: Dependent Variable: RESID Method: Least Squares Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-Statistic Prob. C LGDPI E LGPRFOOD RESID(-1) R-squared Mean dependent var-1.85E-18 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) The argument in the auto command relates to the order of autocorrelation being tested. At the moment we are concerned only with first-order autocorrelation. This is why the command is auto(1). 8

10 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Breusch-Godfrey Serial Correlation LM Test: F-statistic Probability Obs*R-squared Probability Test Equation: Dependent Variable: RESID Method: Least Squares Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-Statistic Prob. C LGDPI E LGPRFOOD RESID(-1) R-squared Mean dependent var-1.85E-18 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) When we performed the test, resid(–1), and hence RLGFOOD(–1), were not defined for the first observation in the sample, so we had 44 observations from 1960 to 2003. 9

11 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Breusch-Godfrey Serial Correlation LM Test: F-statistic Probability Obs*R-squared Probability Test Equation: Dependent Variable: RESID Method: Least Squares Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-Statistic Prob. C LGDPI E LGPRFOOD RESID(-1) R-squared Mean dependent var-1.85E-18 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) EViews uses the first observation by assigning a value of zero to the first observation for resid(–1). Hence the test results are very slightly different. 10

12 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Dependent Variable: RLGFOOD Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. C LGDPI E LGPRFOOD RLGFOOD(-1) R-squared Mean dependent var 3.28E-05 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) We can also perform the test with a t test on the coefficient of the lagged variable. 11

13 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Breusch-Godfrey Serial Correlation LM Test: F-statistic Probability Obs*R-squared Probability Test Equation: Dependent Variable: RESID Method: Least Squares Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-Statistic Prob. C LGDPI E LGPRFOOD RESID(-1) R-squared Mean dependent var-1.85E-18 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Here is the corresponding output using the auto command built into EViews. The test is presented as an F statistic. Of course, when there is only one lagged residual, the F statistic is the square of the t statistic. 12

14 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Dependent Variable: LGFOOD Method: Least Squares Sample: Included observations: 45 Variable Coefficient Std. Error t-Statistic Prob. C LGDPI LGPRFOOD R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn crite F-statistic Durbin-Watson stat Prob(F-statistic) dL = 1.24 (1% level, 2 explanatory variables, 45 observations) The Durbin–Watson statistic is dL is 1.24 for a 1 percent significance test (2 explanatory variables, 45 observations). 13

15 TESTS FOR AUTOCORRELATION III: EXAMPLES
Breusch–Godfrey test Test statistic: nR2, distributed as c2(q) Alternatively, F test on the lagged residuals H0: r1 = ... = rq = 0, H1: not H0 The Breusch–Godfrey test for higher-order autocorrelation is a straightforward extension of the first-order test. If we are testing for order q, we add q lagged residuals to the right side of the residuals regression. We will perform the test for second-order autocorrelation. 14

16 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Dependent Variable: RLGFOOD Method: Least Squares Sample(adjusted): Included observations: 43 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. C LGDPI LGPRFOOD RLGFOOD(-1) RLGFOOD(-2) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Here is the regression for RLGFOOD with two lagged residuals. The Breusch–Godfrey test statistic is With two lagged residuals, the statistic has a chi-squared distribution with two degrees of freedom under the null hypothesis. It is significant at the 0.1 percent level 15

17 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Dependent Variable: RLGFOOD Method: Least Squares Sample(adjusted): Included observations: 43 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. C LGDPI LGPRFOOD RLGFOOD(-1) RLGFOOD(-2) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) We will also perform an F test, comparing the RSS with the RSS for the same regression without the lagged residuals. We know the result, because one of the t statistics is very high. 16

18 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Dependent Variable: RLGFOOD Method: Least Squares Sample: Included observations: 43 Variable Coefficient Std. Error t-Statistic Prob. C LGDPI LGPRFOOD R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Here is the regression for ELGFOOD without the lagged residuals. Note that the sample period has been adjusted to 1961 to 2003, to make RSS comparable with that for the previous regression. 17

19 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Dependent Variable: RLGFOOD Method: Least Squares Sample: Included observations: 43 Variable Coefficient Std. Error t-Statistic Prob. C LGDPI LGPRFOOD R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) The F statistic is This is significant at the 1% level. The critical value for F(2,35) is That for F(2,38) must be slightly lower. 18

20 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Breusch-Godfrey Serial Correlation LM Test: F-statistic Probability Obs*R-squared Probability Test Equation: Dependent Variable: RESID Method: Least Squares Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-Statistic Prob. C LGDPI LGPRFOOD RESID(-1) RESID(-2) R-squared Mean dependent var-1.85E-18 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Here is the output using the auto(2) command in EViews. The conclusions for the two tests are the same. 19

21 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Dependent Variable: LGFOOD Method: Least Squares Sample (adjusted): Included observations: 44 after adjustments Variable Coefficient Std. Error t-Statistic Prob. C LGDPI LGPRFOOD LGFOOD(-1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn crite F-statistic Durbin-Watson stat Prob(F-statistic) The output above gives the result of a parallel logarithmic regression with the addition of lagged expenditure on food as an explanatory variable. Again, there is strong evidence that the specification is subject to autocorrelation. 20

22 TESTS FOR AUTOCORRELATION III: EXAMPLES
Residuals, ADL(1,0) logarithmic regression for FOOD Here is a plot of the residuals. 21

23 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Dependent Variable: RLGFOOD Method: Least Squares Sample(adjusted): Included observations: 43 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. RLGFOOD(-1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat A simple regression of the residuals on the lagged residuals yields a coefficient of 0.43 with standard error We expect the estimate to be adversely affected by the presence of the lagged dependent variable in the regression for LGFOOD. 22

24 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Dependent Variable: RLGFOOD Method: Least Squares Sample(adjusted): Included observations: 43 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. C LGDPI LGPRFOOD LGFOOD(-1) RLGFOOD(-1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) With an intercept, LGDPI, LGPRFOOD, and LGFOOD(–1) added to the specification, the coefficient of the lagged residuals becomes 0.60 with standard error 0.17. 23

25 TESTS FOR AUTOCORRELATION III: EXAMPLES
============================================================ Dependent Variable: RLGFOOD Method: Least Squares Sample(adjusted): Included observations: 43 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. C LGDPI LGPRFOOD LGFOOD(-1) RLGFOOD(-1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) R2 is , so nR2 is 10.62, significant at the 1 percent level and nearly significant at the 0.1 percent level. (Note that here n = 43.) The t statistic for the coefficient of the lagged residual is also highly significant. 23

26 Copyright Christopher Dougherty 2016.
These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 12.2 of C. Dougherty, Introduction to Econometrics, fifth edition 2016, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre Individuals studying econometrics on their own who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics or the University of London International Programmes distance learning course 20 Elements of Econometrics

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