MF-852 Financial Econometrics

1. MF-852 Financial Econometrics Lecture 10 Serial Correlation and Heteroscedasticity Roy J. Epstein Fall 2003

2. Topics • Serial correlation • What is it? • Effect on hypothesis tests • Testing and correcting for serial correlation • Heteroscedasticity • Ditto. • ARCH (or how to win a Nobel prize)

3. Serial Correlation • The error terms in the regression should be independent, i.e., E(eiej) = 0 for any i and j. • If this assumption is not true then the errors are serially correlated. • Only a problem for time-series data.

4. Serial Correlation — Possible Causes • Omitted variables. • Wrong functional form. • “inertia” in economic data—error term composed of many small effects, each with a similar trend.

5. Correlated Error Terms • Suppose E(eiei-1)  0. Implies neighboring observations are correlated, not independent. • 1st-order process. Most common form of serial correlation. • Suppose E(eiei-4)  0. • 4th-order process. Often occurs with quarterly data.

6. Graph of Residuals from a Regression

7. Importance of Serial Correlation • Regression coefficients (the marginal effects) are unbiased. • BUT their standard errors are biased. • Bias generally understates the standard errors, so significance tests are biased against H0. • H0 is rejected too often.

8. Bias in Standard Errors • Standard errors for the coefficients depend on estimated variance of error term, s2e. • Regression program assumes independent errors with mean 0, so program calculates

9. Why the Standard Errors are Biased • Calculation ignores covariance when errors are NOT independent. • Covariance between errors, when it exists, is usually positive. • So s2e would be understated and standard errors would be biased downward.

10. Testing for Serial Correlation • Most common test is Durbin-Watson statistic • Only used for 1st order serial correlation • Calculated as

11. Durbin-Watson Stat. • When covariance between neighboring observations is zero then DW should be close to 2. High covariance implies DW —> 0. • H0 for no 1st order serial correlation: DW = 2 • Look up critical values in table (RR, p. 592) • See sample regression in xls file.

12. Model with Serial Correlation • Yt = 0 + 1Xt + et • Suppose et = et-1 + ut , where ut is another error with mean 0 that is serially independent and uncorrelated with e or X. • -1 <  < 1 (or the process is explosive) • ut is called the innovation in e because it is the new component of e each period. • Serially correlated: E(etet-1) = var(et).

13. How to find  • Estimate it as  = 1 – DW/2. • We can do this in Excel. • Fancier procedures: Cochrane-Orcutt and Hildreth-Liu and others. • A good regression program will calculate  automatically.

14. Fixing Serial Correlation • Suppose  is known. Then “difference” the model: Yt – Yt-1 = 0(1–) + 1(Xt–Xt-1) + (et – et-1) Or Yt – Yt-1 = 0(1–) + 1(Xt–Xt-1) + ut • ut is a “well behaved” error. • Differenced model yields unbiased coefficients and unbiased standard errors. • See example.

15. Heteroscedasticity • Strange name! Greek for “different variances.” • Violation of last assumption about residual: same variance for each error term. • Can occur with any kind of data.

16. Heteroscedasticity — Possible Causes • Wrong functional form. • Var(e) correlated with an included X variable on the right side of the regression. • E(var(e), X)  0, NOT E(e, X)  0

17. Heteroscedasticity — Importance • Regression coefficients (the marginal effects) are unbiased. • BUT their standard errors are biased. • Direction of bias not usually known. • Confidence levels, p-values, t statistics not reliable.

18. Model with Heteroscedasticity • Yt = 0 + 1Xt + et • Suppose var(et) = 2Xt2. • Var(e) is different for each observation.

19. Fixing Heteroscedasticity — Weighted Least Squares • Observations with smaller error variance are “better.” Give them more weight when estimating the model. • Weighted Least Squares (WLS): Multiply observations by weighting factors that equalize the variance. (1/Xt)Yt = (1/Xt)0 + (1/Xt)1Xt + (1/Xt)et • Var((1/Xt)et) = ((1/Xt2)2Xt2 = 2

20. Calculating WLS • Suppose form of heteroscedasticity is known, e.g., need to weight by Xt. • You just need to create new variables. (1/Xt)Yt = (1/Xt)0 + 1 + ut • Intercept in WLS is 1, slope on 1/X is 0. • “Well behaved” error term, yields unbiased coefficients and unbiased standard errors.

21. ARCH models • AutoRegressive Conditionally Heteroscedastic model • Regression model with serial correlation (“autoregressive”) AND heteroscedasticity. • Used to model volatility, i.e., variance, of returns.

22. ARCH models • Sometimes you want to model volatility itself (e.g., it’s an input to an option pricing model). • Volatility can change over time, periods of high and low volatility. • ARCH describes this process.

23. Formulation of ARCH model • Yt = 0 + 1Xt + et • Var(et) = 0 + 1et-1. • 1st order ARCH process. • Can estimate ’s and ’s and perform WLS.