I have a question in time series analysis. While we are testing for autocorrelation of the residuals, it is mentioned we check for correlation between the error term and the kth lagged variable. However, shouldn’t it be only the previous value?Since autocorrelation is the correlation of error terms over successive period…Moreover, in the example in schweser, page 218, it mentions different lags. I am not getting what it means. For example lag 2, is it the correlation of the residuals of yt with 2 periods before it? or the correlation of the residuals between lag 1 and lag 2?
Yes, a 2-period lag means that you’re checking the correlation of the time series with the same series from 2 periods earlier. So you’re comparing values from times 4, 5, 6, 7, 8, 9, 10 with values from times 2, 3, 4, 5, 6, 7, 8 respectively.
Im sorry I might sound a bit dumb but this concept is confusing me. So does it mean when the autocorrelation at Lag 2 is = to X, the correlation of the residuals between the value at Y4 and Y2 is equal to X? this is making me confused as I thought that serial correlation is the exhibition of correlation in residuals from one period to the next…not 2 or 12 periods before.
When you build AR models they couldn’t be correctly specified (the error terms of the model describes a non-stationary behavior), so this violates OLS stimation assumptions and makes the model useless for interpretation and forecasting. Happilly, there is one way to check for incorrect specification which is calculating autocorrelation of the model error terms.
Lag 1 : Correation [e(t) , e(t-1)] must be statistically equal to zero, this implies low T-values (less than 2.0 approx for 5% level of significance)
The same for Lag 2 which is Correlation [e(t) , e(t-2)] and Lag 3, Lag 4, etc. Note that lags are always (t) vs (t - i) where i goes from 1 to n
What happens when one autocorrelation is statistically significant different from zero? The model is not correctly specified.
When you use seasonal data, sales of company for example, the 4Q quarters or 12th month of year commonly are highly correlated with previous 4Q quarters or 12th months respectively, so the lag 4 or lag 12 of autocorrelation errors display significance, this implies our model is no correctly specified and must be corrected adding one extra independent variable to the model of the same time location of the detected significant autocorrelation. Doing this, the model absorbs that bad error behavior and error now is more stationary.
This is the idea behind using autocorrelation of errors. It is to know what AR independent variables to keep adding until errors autocorrelation are not significant.
Hope this helps
It does a lot actually…Thanks a lot 