AUTOCORRELATION. Phung Thanh Binh

Transcription

1 AUTOCORRELATION Phung Thanh Binh

2 OUTLINE Time series Gauss-Markov conditions The nature of autocorrelation Causes of autocorrelation Consequences of autocorrelation Detecting autocorrelation Remedial measures

3 Time series Gauss Markov conditions 1) Linear: Y t = X 2t + 3 X 3t + u t 2) Zero conditional mean of error: E[u t X jk ] = 0 Unbiased 3) No perfect collinearity 4) Homoscedasticity: Var(u t X jk ) = 2 if violated => ARCH family models: Lecture 16 (for T.S data) 5) No serial correlation: Cov(u t,u s X jk ) = 0 BLUE violated Lecture 13

4 The nature of autocorrelation Terminological issue: Common practice: the terms autocorrelation and serial correlation are the same. Some authors prefer to distinguish the two terms as follows: Autocorrelation: Lag correlation of a given series with itself, lagged by a number of time units. Serial correlation: Lag correlation between two different series.

5 The nature of autocorrelation

6 The nature of autocorrelation

7 The nature of autocorrelation

8 Positive autocorrelation

9 Negative autocorrelation No autocorrelation

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11 The meaning of ρ: The error term u t at time t is a linear combination of the current and past disturbance.

12 The causes of autocorrelation The possible strong correlation between the observation i with the observation j could be due to: Inertia Cobweb phenomenon Lags Nonstationarity Pure autocorrelation Specification bias: Excluded variables case Specification bias: Incorrect functional form etc [see Gujarati (2009). Basic Econometrics] Impure autocorrelation

13 The consequences of autocorrelation The estimated coefficients are still unbiased. The variance of the is no longer the smallest. The standard error of the estimated coefficient, becomes large.

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15 The consequences of autocorrelation Table6_1.dta (in Econometrics by examples)

16 Graphical method Detecting autocorrelation Plot the values of the residuals, e t, chronologically If discernible pattern exists, autocorrelation likely a problem. Durbin-Watson test: Durbin-Watson s d statistic Durbin s h statistic Breusch-Godfrey (BG) test

17 predict s1, resid. gen s1_100=100*s1. label var s1_100 "Residuals". predict s2, rstandard. twoway (line s1_100 time) (line s2 time) Time Residuals Standardized residuals

18 Assumptions are: 1. The regression model includes an intercept term. 2. The regressors are fixed in repeated sampling. 3. The error term follows the first-order autoregressive (AR1) scheme: Durbin-Watson d Statistic 4. The regressors do not include the lagged value(s) of the dependent variable, Y t. 5. No missing observation. u u v t t 1 t

19 Durbin-Watson d Statistic d (e i e i 1 ) 2 e i 2, for n and K -1 d.f. Positive Zone of No Autocorrelation Zone of Negative autocorrelation indecision indecision autocorrelation 0 d-lower d-upper 2 4-d-upper 4-d-lower 4 Autocorrelation is clearly evident Ambiguous cannot rule out autocorrelation Autocorrelation in not evident

20 Durbin-Watson d Statistic

21 This test allows for: Breusch-Godfrey (BG) test (1) Lagged values of the dependent variables to be included as regressors (2) Higher-order autoregressive schemes, such as AR(2), AR(3), etc. (3) Moving average terms of the error term, such as u t-1, u t-2, etc. The error term in the main equation follows the following AR(p) autoregressive structure: u u u u v t 1 t 1 2 t 2... p t p t The null hypothesis of no serial correlation is: p 0

22 Breusch-Godfrey (BG) test The BG test involves the following steps: Regress e t, the residuals from our main regression, on the regressors in the model and the p autoregressive terms given in the equation on the previous slide, and obtain R 2 from this auxiliary regression. If the sample size is large, BG have shown that: (n p)r 2 ~ X 2 p That is, in large samples, (n p) times R 2 follows the chi-square distribution with p degrees of freedom. Rejection of the null hypothesis implies evidence of autocorrelation. As an alternative, we can use the F value obtained from the auxiliary regression. This F value has (p, n-k-p) degrees of freedom in the numerator and denominator, respectively, where k represents the number of parameters in the auxiliary regression (including the intercept term).

23 Breusch-Godfrey (BG) test

24 First-Difference Transformation If autocorrelation is of AR(1) type, we have: Assume ρ=1 and run first-difference model (taking first difference of dependent variable and all regressors) Generalized Transformation Estimate value of ρ through regression of residual on lagged residual and use value to run transformed regression Newey-West Method Generates HAC (heteroscedasticity and autocorrelation consistent) standard errors Model Evaluation Remedial Measures u u v t t 1 t

25 First-Difference Method

26 This outcome could be due to the wrong value of ρ (ρ = 1) chosen for transformation. Notes: There if no intercept in the first-difference model. If there is an intercept term, what does it stand for? Rule of thumb: First-Difference Method Use the first-difference form whenever d < R 2 (Maddala). Use the first-difference form when u t is nonstationary (or differently, Y t and X t are not cointegrated).

27 Feasible Generalized Least Squares (FGLS)

28 Feasible Generalized Least Squares (FGLS)

29 FGLS: Prais-Winsten Transformation

30 FGLS: Cochrane-Orcutt Transformation One of the iterative methods of estimating ρ

31 FGLS: Cochrane-Orcutt Transformation

32 How does the Cochrane-Orcutt procedure work?

33 How does the Cochrane-Orcutt procedure work?

34 How does the Cochrane-Orcutt procedure work? Prais-Winsten procedure is similar, but it transforms the first observation differently.

35 The Newey-West Method This method still uses OLS, but corrects the standard errors for autocorrelation. This is an extension of White s heteroscedasticityconsistent standard errors. The corrected standard errors are known as HAC (heteroscedasticity- and autocorrelation-consistent) standard errors or simply Newey-West standard errors. This is strictly valid in large samples.

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37 The Newey-West Method

38 The Newey-West Method