The mean of a time series is suppose to give a measurement of a typical value (average) of the data over the time horizon. In a stationary time series the mean is a value around which the time series fluctuates over and above as it continuously approaches this value. In fact one would expect that the value of this time series would remain around this typical value. In a non stationary time series there is no constant mean. The data does not revolve around a particular value and although the mean can be calculated it does no give an idea of the typical value which the time series is approaching.
This is just one possibility as shown in the figure.
Since in the model it is implied that u t is a linear combination of two non-stationary time series then this term would also be non-stationary
The variables might be related as suggested by the underlying theory to be investigate but since the data is non-stationary a spurious regression might be obtained.
If cointegration is established then this implies a long run equilibrium (non spurious) relationship exist among the set of variables as expressed by the OLS equation.
where: Γi are known as short run parameters, α is the speed of adjustment parameter, ν t is a random disturbance term
The Ai matrices contains all the coefficients to be estimated
Econometric Modelling using Eviews Edward Bahaw March19 th 2008 Natural Gas Institute of the Americas
Residuals (u t ) arise as the regression line might not pass through all the points
Ordinary least squares – minimizes the square of such residuals
Non stationary data Time Non - Stationary Time Series Mean does not represent the value which the time series approaches Time Mean represents the value which the series approaches over time Stationary Time Series
Spurious Regression If the residual term u t is non-stationary about a mean of zero then t he regression equation would be spurious or unreliable
If X 1t and X 2t are two variables
OLS regression would give:
X 1t = μ + β 2 X 2t + u t ,
Spurious Regression Residuals Time u t is non-stationary u t corresponding to a spurious regression u t Mean
Spurious Regression Using X 1t = μ + β 2 X 2t + u t , Then u t = X 1t – β 2 X 2t – μ If X 1t and X 2t are non-stationary a spurious regression may be obtained
Cointegration and Non-Stationary Variables In the model: X 1t = μ + β 2 X 2t + u t , Or u t = X 1t – β 2 X 2t – μ If u t is stationary about a mean of zero then cointegration exists.