Suggested Steps in Time-Series Forecasting (Part B) - CFA Level II

4. When serial correlation is detected in the model, AR model should be used. However, before using AR model, time series must be tested for Covariance Stationarity.
· If time series has a linear trend and covariance nonstationary; it can be transformed into covariance stationary by taking the first difference of the data.
· If time series has exponential trend and covariance nonstationary; it can be transformed into covariance stationary by taking natural log of the time series and then taking the first difference.
· If the time series exhibits structural change, two different time-series model (i.e. before & after the shift) must be estimated.
· When time series exhibits seasonality, seasonal lags must be included in the AR model.

5. When time series is converted into Covariance Stationarity, AR model can be used i.e.
· Estimate AR (1) model;
· Test serial correlation in the regression errors; if no serial correlation is found only then AR (1) model can be used. When serial correlation is detected in AR (1), then AR (2) should be used and tested for serial correlation. When no serial correlation is found, AR (2) can be used. If serial correlation is still present, order of AR is kept on increasing until all serial correlation is removed.

6. Plot the data and detect any seasonality. When seasonality is present, add seasonal lags in the model.

7. Test the presence of autoregressive conditional heteroskedasticity in the residuals of the model i.e. using ARCH (1) model.

8. In order to determine the better forecasting model, calculate out-of-sample RMSE of each model and select the model with the lowest out of-sample RMSE.

Suggested Steps in Time-Series Forecasting (Part A) - CFA Level II

Following is a guideline to determine an accurate model to predict a time series.

1. Select the model on the basis of objective i.e. if the objective is to predict the future behavior of a variable based on the past behavior of the same variable, use Time series model and if the objective is to predict the future behavior of a variable based on assumed casual relationship with other variables Cross sectional model should be used.

2. When time-series model is used, plot the series to detect Covariance Stationarity in the data. Trends in the time series data include:
· A linear trend
· An exponential trend
· Seasonality
· Structural change i.e. a significant shift in mean or variance of the time series during the sample period.

3. When there is no seasonality or structural change found in the data, linear trend or exponential trend is appropriate to use i.e.
i. Use linear trend model when the data plot on a straight line with an upward or downward slope.
ii. Use log-linear trend model when the plot of the data exhibits a curve.
iii. Estimate the regression model.
iv. Compute the residuals
v. Use Durbin-Watson statistic to test serial correlation in the residual.

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.