Dear all, I am certain that one of the variables in my vector can affect the rest only if it is lagged. For example, lets imagine an agriculture product which is planted in a year and harvest in the following year. For its development, it is critical the rain during the year where planting took place. So, I want to investigate cointegration between the rain values in the previous year (Xt-1) and the harvest in the following year (Yt). I have read many papers on time series analysis and I always have seen the investigation of cointegration using values of the variables in same periods. In the following link,
https://stats.stackexchange.com/a/285589/234696
a professor provides a clear explanation indicating the validity of the framework, but I haven't found any published paper applying it. Indeed, that link is the only reference I have found in this topic. So, I worry about the considerations of potential reviewers. Therefore, I would like to gather suggestions of published papers using this. In addition, suggestions of literature on this topic are welcomed.
Finally, I want to apply this in a panel data framework and using both Engle and Granger residual based framework and Johansen Fisher panel cointegration tests. Is the framework correct for both panel and time series?
Very much thank you in advance,
Dear Professor Thomas Dimpfl,
Very much thank you for your response and help. It is great to get a recomended paper as a bonus. In reality, I am not working with agriculture and rainfall. I just brought the example because it was a clear case where in case of cointegration, the long run relationship is not contemporaneous.
Hi, it is ok to have cointegration between variables that are not contemporaneous. The simplest case is the random walk: Y(t+1)=c+Y(t)+e(t+1). In this extremely simple case Y(t+1) and Y(t) are cointegrated with cointegration vector given by (1,-1).
More generally you can put forward a similar argument in ARIMA(p,1,q) model.
Hope it helps!
Yes, it is possible.
My contribution is also in line with Pablo Pincheira above.
If x(t) and y(t) are I(1) and cointegrated then the exists an a and
x(t) - a y(t) is stationary.
Now as y(t) is I(1) y(t) - y(t-1) or a( y(t) - y(t-1) ) is stationary.
The sum of two stationary variables is stationary. Therefore
x(t) - a y(t) + a( y(t) - y(t-1) ) = x(t) -a y(t-1) is stationary and x(t) and y(t-1) are cointegrated. Continuing this logic you may fid that if x(t) and y(t) are I(1) and cointegrated then x(t-i) and y(t-j) Where i and j are any two finite integers are cointegrated. Any lags or leads to be included in a cointegration test should be determined by the economics of the situation. They can not be determined by cointegration tests.
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