Serial Correlation and the Housing price function

1. Serial Correlation and the Housing price function Aka “Autocorrelation”

2. More on House prices • We return to the house price function to illustrate the issue of serial correlation • It turns out that OLS may NOT give us the best estimate of the price function • The reason is that one of the assumptions of the GM theorem is probably violated in the consumption model • The data is probably serially correlated • E[utut-1] ≠ 0

3. The Issue • In time series we can think of the residual as representing an economic shock • This is literally true in a statistical sense but may also be true in an economic sense • If the residual really is an economic shock it is a sort of omitted variable • It is likely that the effect of the shock persist across calender time periods • Think of the current economic situation: crisis will not stop on 31st of dec • Cant happen in cross section data

4. The Impact • This implies that the residuals will be correlated across time • Violates the GM theorem • Specifically if we estimate the model Yt=b1+b2Xt+ut • GM requires: • var[ut] =E[ut2]= 2(homoskedasticty) • E[utut-1] = 0 (no autocorrelation) • The second is likely violated in time series data

5. Characteristics of Serial Correlation • Systematic pattern exists in residuals • First order serial correlation:ut =  ut-1 + vt • effect from error in previous period ut-1, • random error vtwhich satisfies the OLS assumptions • Think of what happens if rho is positive: • a positive shock tends to be followed by another positive shock • The shock persists • Aside: This is known as first order serial correlation. You can have serial correlation of higher order but we wont deal with it here. An example would be • ut= 1 ut-1 + 2 ut-2+ …….. + kut-k +vt

6. Serial Correlation vs Het • Systematic pattern exists in the residuals • Similar idea to heteroscedasticity but crucially different • Recall Het is a pattern in variance of residuals • Serial correlation is correlation between the draws from the same distribution • Think of the dice or roulette wheel example • Intuition: if random bit come from roll of dice then homo is with same dice and hetero is with different dice • Rolls of same dice for different people but rolls are linked • Note to confuse the issue: the two phenomena can occur together (GARCH) • We will treat them as separate • Evident in time series and not cross section because no natural ordering of data

7. Consequences • OLS is unbiased • OLS is consistent • OLS is no longer efficient • Variance formula used previously is incorrect • significance test, confidence intervals etc. cannot be used • Aside: a corrected formula can be used • Stata: regress y x, robust  • We don’t bother with this because can do better with alternative estimator • Same consequences as Het – which can lead to confusion

8. Testing for AC • Plot of residuals against time • Stata: scatter u year • Plot residuals against lagged value • Scatter u L.u • Not a formal test but can give an idea of what's going on • Graphs are from housing data • Looks like there is positive serial correlation • Not surprising given the bubble

9. Residual vs Year

10. Residual vs Lagged Residual

11. The Durbin Watson Test • Formal test of AC • Most complicated hypothesis test we have encountered • Wont work with all AC • Test requires • Testing for first order only • model must include intercept • model cannot include a lagged dependent variable (this is a big problem)

12. Formal Structure of the test • H0: =0 H1a: <0 H1b: >0, • Form the test statistic: Stata command: dwstat • Find the critical values from DW tables • N,K and SL • Each reading will produce two value: du and dL • Compare the test statistic and the critical values using the chart (over) • State Conclusion

13. Chart for DW test

14. Comments • The test statistic is (sort of) the coefficient of regression of residual on its lagged value • It is approximately equal to 2(1-rho)

15. This explains the boundaries on the chart • We have this slightly weird set-up because we don’t actually know the critical values for this test with certainty • All we know is min and max values for the true critical value: du and dL • This hypothesis test is unusual in that there is a zone of indecision where the test produces no result • This is different from all the other hypothesis tests that we have encountered • Don’t try this manually use the stata command: dwstat

16. The housing model regress price inc_pchstock_pc if year<=1997 Source | SS df MS Number of obs = 28 -------------+------------------------------ F( 2, 25) = 88.31 Model | 1.1008e+10 2 5.5042e+09 Prob > F = 0.0000 Residual | 1.5581e+09 25 62324995.9 R-squared = 0.8760 -------------+------------------------------ Adj R-squared = 0.8661 Total | 1.2566e+10 27 465423464 Root MSE = 7894.6 ------------------------------------------------------------------------------ price | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- inc_pc | 10.39438 1.288239 8.07 0.000 7.741204 13.04756 hstock_pc | -637054.1 174578.5 -3.65 0.001 -996605.3 -277503 _cons | 135276.6 35433.83 3.82 0.001 62299.24 208253.9 ------------------------------------------------------------------------------ . dwstat Durbin-Watson d-statistic( 3, 28) = .746281

17. Comparing with Critical Values • Critical values dL = 1.18, du = 1.65 • Place on chart: 4-du =2.42, 4-dL= 2.67 • Locate the test statistic on chart • dw=0.71<dL is in the zone of positive serial correlation

18. We can reject the null of no serial correlation and cannot reject the alternative of positive correlation at 5% significance level • This is exactly what we would expect in a model of house prices • We expect shocks to persist across time boundaries • Postively correlated residuals

19. Efficient Estimation • If we find AC we know that OLS will be inefficient • Remember why this might be a problem (see over) • Can we do better? • Yes. There is an efficient estimator called Generalised Least Squares (GLS) • Two steps • Remove the AC from the data • Do OLS on the transformed data • Aside: this is similar to het but different in part 1

20. Prob of error is lower for efficient estimator at any sample size Same sample size, different estimator

21. The GLS Procedure • Assume that ris known: • Basic model: Yt= 1 + 2Xt+ ut ut =  ut-1 + vt • Create new data with each observations weighted by the rho

22. The GLS Procedure • Then run the regression on the transformed data • This regression doesn’t have AC • The slope estimates are the BLUE of the coefficients of the original model • Note the intercept term is slightly different

23. How it Works • The model: Yt=1+2Xt+ut ut=  ut-1 + vt • This implies: Yt=1+2Xt+ ut-1 + vt • We also have: ut= Yt-1-2Xt => ut-1 = Yt-1-1-2Xt-1 • substitute for ut-1 • Yt=1+2Xt+ (Yt-1-1-2Xt-1 )+vt • Collect terms: • Yt - Yt-1 =1(1-  )+2(Xt- Xt-1 )+vt • This is the equation we had earlier and the residual v does not have serial correlation • So the est of 2 from the transformed model will be the BLUE of the coefficient from the original model

24. FGLS • In reality we wont know rho • We can make a guess from the DW statistic • dw=2(1-rho) • We can start with any value, retest for ac and if its there repeat the whole process until it is eliminate • and iterate until convergence

25. CorchraneOrcutt • Estimate the basic model using OLS:Yt=1+2Xt+ut => Yt=b1+b2Xt+ut • Calculate the residuals:ut= Yt- b1- b2Xt • Use OLS to get initial estimate of rho from the regression: ut=put-1 + wt • Transform model using the estimated rho as outlined before • Use OLS on the transformed data to get estimates 1and 2 These will be different from those of step 1 • Generate residual by applying these second estimates to the original (not transformed) data These will be a different set of residuals than those from step 2 • Get a new estimate of rho by applying OLS to the equation in step 3 but using the new residual series • Transform the original data by this new estimate of rho • Get new GLS estimates of the betas by applying OLS to the second set of transformed data • Repeat steps 6-9 until successive estimates of rho are very close