I’m confused with covariance stationary. In order to be covariance stationary, the expected value must be constant and finite. Does that means that all the independent variable should be like a horizontal line when plotted? Thanks!
I think you are right… If anyone thinks I am wrong let me know!
No, it does not meant this at all. It means that when you plot the independent variables, they do not display any trends. For example…plot S&P 500 returns against time. You will see that you certainly don’t have a constant line. You will notice that the expected value is constant (random walk hypothesis states that expected returns will be zero). However, you will see periods were the volatility displays clusters. High volatility periods tend to be followed by high vol, and vice-a-versa. Therefore, you have a constant and finite mean, but nonconstant volatility. Another example would be plotting GDP against time. In this case there is an obvious trend upwards when plotting GDP at its level. Take a period from 1940-1950, and then another from 1980-1990. You will see that the average value of the plots is certainly not the same. Therefore, you have finite expected value, but it is not constant throughout time. However, the CHANGE in GDP each period is stationary.
Covariance-stationary process has to have a constant mean and variance. White noise is a good example of a covariance-stationary process (notice that there is no clear evidence that mean or variance are changing): http://en.wikipedia.org/wiki/File:White-noise.png Most time series in finance are non-stationary though.
No, the values can move around, but when you fit a regression to the values it should run horizontally through the data. Furthermore, you want to make sure that the deviations from the line are approximately constant over the entire span for which data is available.
here’s my take: the time series is mean reverting : the expected value is constant *over time* the variance is constant over time autoregression should not show ‘trends’: no seasonality etc
Look at P/E over the long term http://www0.gsb.columbia.edu/students/organizations/cima/files/CIMA_2009_montier_presentation.pdf Page 6
Covariance stationary is required for only AR models for the forecasts to make sense. It is not required for trend models that do not have a lagged variable. For e.g linear trending model, trending upwards or dowwards, does not exhibit covariance stationary, yet it is a dependable model.