Autocorrelation Function Nature and Characterstics ppt 2.ppt
- 1. Autocorrelation • Recall that when estimating the parameters of a regression model using the OLS formulas, it is assumed that the population error terms for the different observations are independently and identically distributed • An independently distributed error-term means that when the observations are organized in a particular order (by time), the sign and size of the error term of an observation from a particular period is not related to (i.e. it is independent of ) the sign and size of the error term of the previous period
- 2. Autocorrelation/Introduction • This is equivalent to saying that the value taken by the dependent variable in particular observation is not related to (i.e. is independent of ) the value(s) taken by the dependent variable for the previous period • Autocorrelation occurs when this error term (i.e. dependent variable) independence assumption is violated
- 3. Autocorrelation/Introduction • It is most common in (but not unique to) time- series data; for example prices are often positively correlated through time: there are several year periods of “good” prices and of “depressed” prices
- 4. Autocorrelation Yi = β0 + β1Xt + ut First order autocorrelation: ut =ρ ut-1 + et , For positive autocorrelation: 0< ρ <1 For negative autocorrelation: -1< ρ <0 et are assumed to be random variables et are independent and have identical normal dist. with E(ei) =0 and σ (ei ) =σe All the ut have normal probability distributions with E(ut) =0 and σ (ut ) =σu = σe / 1- ρ2
- 5. Autocorrelation/Diagnostics • A visual inspection of a plot of the OLS residuals versus time can be used as a preliminary diagnostic tool for autocorrelation • Groups of several positive residuals followed by several negative residuals indicate possible positive (versus negative) autocorrelation, which is most common; residuals with clearly alternating signs indicate negative autocorrelation
- 6. Autocorrelation • ρ is the correlation between one period’s disturbance term and the next • If OLS is applied, estimators will be unbiased and consistent, but it will underestimate standard errors, t test procedure is invalid
- 7. Autocorrelation/Diagnostics • The best known formal test for autocorrelation is the Durbin-Watson statistic • The basic Durbin-Watson statistic tests for first order autocorrelation; i.e. correlation between the previous and the present period residuals (i.e. dependent variable) values • It is calculated as T t t T t t t e e e d 2 2 2 2 1 *
- 8. Autocorrelation/Diagnostics • If there is no first order autocorrelation the value of d* will be close to 2 • It would be smaller than 2 is there is positive first order autocorrelation and greater than 2 if there is negative first order autocorrelation
- 9. Autocorrelation/Diagnostics • More precisely, let H0: no first order autocorrelation Ha: there is either a positive or a negative first order autocorrelation ( a two tailed alternative) • Compare the calculated d* with the “significance points” in the Durbin-Watson statistic table at the desired α, for the appropriate sample size (n) and number of independent variables in the model (k)
- 11. Autocorrelation/Diagnostics • If 4-du>d*>du conclude H0 • If 4-dl<d*<dl conclude Ha • Otherwise the test is inconclusive
- 12. Autocorrelation/Diagnostics • Recall the Durbin-Watson test only detects first order autocorrelation when no lagged dependent variable has been included in the model • If a lagged dependent variable is included, d* would tend to be close to 2, even if there is first-order autocorrelation present in the model
- 13. Correcting for Autocorrelation • The solution for autocorrelation is to transform the original autoregressive error term into one with a non-correlated error term so as to permit the use of the OLS procedures; let: • Yt = B1 + B2X2t + ….+BkXkt + et, t=1…T
- 14. Correcting for Autocorrelation • where both et and Vt have zero expected values and constant variances through time, et is autocorrelated but Vt is not • The former defines a standard first-order autoregressive model: ρ is the correlation coefficient between errors in time-period t and errors in time period t-1 1 ρ 0 V ρ 1 t t t e e
- 15. Correcting for Autocorrelation • If ρ were known, the following procedure could be used to correct for autocorrelation • Notice that the former model implies: ρYt-1= ρB1+ ρB2X2t-1+….+ ρBkXkt-1+ ρet-1;thus
- 16. Correcting for Autocorrelation • Yt-ρYt-1=(1-ρ)B1+ B2(X2t-ρX2t-1)+….+ Bk(Xkt- ρXkt-1)+ (et-ρet-1); or • Y*t= (1-ρ)B1+ B2X2 * t+….+ BkXk * t+vt • Notice that the error term (vt) on the last equation, which is a transformed model, satisfies the OLS assumptions
- 17. The Cochrane-Orcutt Procedure • In a first step OLS is used to estimate the original model and its residuals, which are used to obtain a first estimate of ρ through the following regression: • et = ρet-1 + vt; • The estimated value of ρ is used to estimate the transformed et
- 18. The Cochrane-Orcutt Procedure • This procedure involves estimating the previously discussed transformed model in several iterations, each of which produces a better estimate of ρ