Autocorrelation- Remedial Measures
- 2. Remedial Measures (Generalized Least Squares GLS) If we know ρ If we don’t know ρ Assume ρ=1 First Difference Method Estimate ρ from DW d-statistic Estimate ρ from OLS residuals (et)
- 3. If the regression holds at time t, it should hold at time t-1, i.e. (2) Multiplying the second equation by gives (3) Subtracting (3) from (1) gives t 11211 ttt uXY 11211 ttt uXY )()1( 1211 tttt XXpYY ttt uXY 21 11 AR(1) scheme Follows all CLRM assumptions. So the TRICK is to REPLACE ut with this term in the regression (1)
- 4. The equation can be re-written as: The error term satisfies all the OLS assumptions When we apply OLS to transformed models, the estimators thus obtained are called generalized least squares (GLS) estimators The estimators thus obtained will have the desirable BLUE property
- 5. Note that when we use GLS, we loose the first observation. Eg. If n=50, now the GLS regression will be run for 49 observations. t To avoid this loss of one observation, the first observation of Y and X is transformed as follows: Not generally used in large samples In small samples, results may be sensitive if we exclude first observation. )()1( 1211 tttt XXpYY
- 6. Method 1: Assume that = +1 (First difference Method) The generalized Least squares equation reduces to the first difference equation t Putting = 1 , we get OR (First difference) This assumption is appropriate if the coefficient of autocorrelation is very high, say in excess of 0.8, or the Durbin-Watson d is quite low. Feature :There is no intercept. Thus, we have to use the regression through the origin. )()( 112 tttttt uuXXYY ttt XY 2 )()1( 1211 tttt XXpYY
- 7. Method 2: Estimate from Durbin-Watson d statistic We know that , i.e. Obtain rho from this equation and use it to transform the data as shown in the GLS. t Good for reasonably large samples d-statistic is reported along with regression results by the software. Eg. If computed d=1.755, we can calculate ρ-hat. 2 1 d ˆ12 d )()1( 1211 tttt XXpYY
- 8. Method 3: Estimate from OLS residuals The sample counterpart of the following regression is given by Estimate the above equation using OLS residuals and obtain estimated-ρ. Use it to run GLS. t In small samples, this procedure gives biased estimate of ρ. ttt uu 11 ttt vuu 1 ˆ.ˆˆ )()1( 1211 tttt XXpYY
- 9. Describe how can the problem of autocorrelation be remedied (Assuming disturbance term follows AR(1) scheme) if P=0.5. The GLS equation is: t Put ρ=0.5, we get i.e. ttt uXY 21 )()1( 1211 tttt XXpYY
- 10. Thank you