1. ANAREGWEEK 14 AUTOCORRELATION IN TIME SRIES DATA  Problems of autocorrelation  First-order autoregressive error model  Durbin-Watson test for autocorrelation  Remedial measures for autocorrelation

2. P ROBLEMS OF A UTOCORRELATION When there is an autocorrelation: 1. The estimated regression coefficient are still unbiased, but they no longer have the minimum variance property and may be quite inefficient. 2. MSE may seriously underestimate the variance of the error terms. 3. S(b k ) calculated according to ordinary least squares procedures may seriously under- estimate the true standard deviation of the estimated regression coefficient. 4. The confidence intervals and tests using the t and F distributions, discussed earlier, are no longer strictly applicable.

3. F IRST - ORDER A UTOREGRESSIVE E RROR M ODEL Suppose in multiple linear regression model with the random error terms following a first-order autoregressive process is given by: Where ρ is a parameter such that | ρ| < 1 u i are independent N(0,σ 2 ) It can be shown that the mean and variance of ε i for the first-order autoregressive error models are as follows:

4. D URBIN -W ATSON T ESTS FOR A UTOCORRELATION Hypotheses statements: H o : ρ = 0 H a : ρ ǂ 0 The test statistics D is calculated by: Where n is the number of cases. The decision rule: 0 4 – d u d l d u 4 – d l 4

5. R EMEDIAL M EASURES FOR A UTOCORRELATION Two principal remedial measures when autocorrelated error terms exist are: 1. To add one or more independent variables to the regression model, or 2. To use transformed variables There are three methods on transformed variables: Cochrane-Orcutt procedure Hildreth-Lu procedure First-difference procedure

6. R EMEDIAL M EASURES FOR A UTOCORRELATION (2) Cochrane-Orcutt Procedure 1. Estimation of ρ, by calculating : 2. Fitting of transformed model Y t ’ 3. Test for need to iterate by using the Durbin- Watson test First-Difference procedure: 1. Transformed 2. Regress Y t ’ on X t ’ 3. Use Durbin-Watson test to examine whether the first-diffrence procedure has removed the autocorrelations