## Wednesday, 21 December 2011

### Autoregressive (AR) Models Time Series (Part C) - Quantitative Methods Level II

Mean Reversion.

A time series shows mean reversion if it tends to move towards its mean i.e. decrease when its current value is above its mean and increase when its current value is below its mean.When a time series equals its mean-reverting value, then the model predicts that the value of the time series will be the same in the next period.

Multi-period Forecasts and the Chain Rule of Forecasting.

The chain rule of forecasting is a process in which a predicted value two periods ahead is estimated by first predicting the next period’s value and substituting it into the equation of a predicted value two periods ahead. It is important to note that the Multi-period forecast is more uncertain than single-period forecasts because the uncertainty increases when number of periods in the forecast increase.

Comparing Forecasting Model Performance.

The accuracy of the model depends on its forecast error variance. The smaller the forecast error variance, the more accurate the model will be.

In-sample forecast errors: These are the residuals from the fitted time series model i.e. residuals within a sample period.
Out-of-sample forecast errors: These are the residuals outside the sample period. It is more important to have smaller forecast error variance for out-of-sample forecasts because the predicted values are always out of sample. In order to evaluate out-of-sample forecasting accuracy of the model. Root Mean Squared Error (RMSE) is used. RMSE is the square root of average squared error.
Decision Rule: The smaller the RMSE, the more accurate the model will be.

The RMSE (Root Mean Squared Error) is used as a criterion for comparing forecasting performance of different forecasting models. To accurately evaluate uncertainty of forecast, both the uncertainty related to the error term and the uncertainty related to the estimated parameters in the time-series model must be considered.

NOTE: If the model has the lowest RMSE for in-sample data, it does not guarantee that the model will have the lowest RMSE for out-of-sample data as well.

Instability of Regression Coefficients.

When the estimated regression coefficients in one period are quite different from those estimated during another period, this problem is known as instability or nonstationarity. The estimates of regression coefficients of the timeseries model can be different across different sample periods i.e. the estimates of regression coefficients using shorter sample period will be different from using longer sample periods. Thus, sample period selection is one of the important decisions in time series regression analysis.

· Using longer time periods increases statistical reliability but estimates are not stable.
· Using shorter time periods increase stability of the estimates but statistical reliability is decreased.

NOTE: We cannot select the correct sample period for the regression analysis by simply analyzing the autocorrelations of the residuals from a time-series model. In order to select the correct sample, it is necessary that data should be Covariance Stationary.