The following ratio is regarded as a linear regression model for time series:
Xt = A0 + A1 * Y1t + … + Ak * Ykt + errt
Where:
Xt. Explained variable.
Y1, …, Ykt. Associated independent (explanatory) variables.
errt. Disturbance with zero mean of distribution and the variance δ2, which values are independent at different moments in time and uniformly distributed (otherwise, white noise).
This model has k+1 unknown parameters A0, A1, …, Ak, δ2, that should be estimated based on the available data on the studied process.
Features of building regression models for time series consist in probable linear dependency (high correlation) between explanatory variables.
If the dependency exists, calculation procedures are multicollinear and poorly conditioned, and the model's estimated coefficients have poor statistical properties.
To estimate system regression parameters, principal component analysis is used. This method is considered to be a robust (stable) method. It uses linear combinations of explanatory variables as new variables. The variables are selected so that the correlations between new variables are little or none, to avoid the problem mentioned above.
After the unknown parameters are estimated, the user can move to forecasting frequency of the analyzed series, which requires future or forecasted values of factorial series.
Xt+1 = A0 + A1 * Y1t+1 + … + Ak * Ykt+T
Where T – the lead period.
One of the regression model quality characteristics is the determination coefficient or adjusted determination coefficient, which is an unbiased estimation of true determination coefficient.
Model estimated coefficients follow the Student's distribution in the asymptotic form. This is used as the basis for checking the hypothesis of the coefficients equality to zero and for calculating confidence limits of the coefficients.
The Durbin-Watson coefficient is commonly used to test for autocorrelation of residuals (whether there is a time correlation in the system errors).
See also:
Library of Methods and Models | IEmLinearRegressionSettings | ISmLinearRegress