## Monday, 2 January 2012

### Regression with more than One Time Series - CFA Level II Quantitative Methods

1. When neither of the time series (dependent & independent) has a unit root, linear regression can be used.
2. One of the two time series (i.e. either dependent or independent but not both) has a unit root, we should not use linear regression because error term in the regression would not be covariance stationary.
3. If both time series have a unit root, and the time series are not cointegrated, we cannot use linear regression.
4. If both time series have a unit root, and the time series is cointegrated, linear regression can be used. Because, when two time series are cointegrated, the error term of the regression is covariance stationary and the t-tests are reliable.

Cointegration: Two time series are cointegrated if
· A long term financial or economic relationship exists between them.
· They share a common trend i.e. two or more variables move together through time.

Detecting Cointegration: The Engle-Granger Dickey- Fuller test can be used to determine if time series are cointegrated.
Engle and Granger Test:
1. Estimate the regression
2. Unit root in the error term is tested using Dickeyfuller test but the critical values of the Engle- Granger are used.
3. If test fails to reject the null hypothesis that the error term has a unit root, then error term in theregression is not covariance stationary. This implies that two time series are not cointegrated and regression relation is spurious.
4. If test rejects the null hypothesis that the error term has a unit root, then error term in the regression is covariance stationary. This implies that two time series are cointegrated and regression results and parameters will be consistent.

NOTE:
· When the first difference is stationary, series has a single unit root. When further differences are required to make series stationary, series is referred to have multiple unit roots.
· For multiple regression model, rules and procedures for unit root and stationarity are the same as that of  single regression.