My previous post on aggregating time series data over time dealt with some of the consequences for unit roots. The next logical thing to consider is the effect of such aggregation on cointegration, and on testing for its presence.
As in the earlier discussion, we'll consider the situation where the aggregation is over "m" high-frequency periods. A lower case symbol will represent a high-frequency observation on a variable of interest; and an upper-case symbol will denote the aggregated series. So,
Yt = yt + yt - 1 + ......+ yt - m + 1 .
If we're aggregating quarterly (flow) data to annual data, then m = 4. In the case of aggregation from monthly to quarterly data, m = 3, and so on.
We know, from my earlier post, that if yt is integrated of order one (i.e., I(1)), then so is Yt.
Suppose that we also have a second temporally aggregated series:
Xt = xt + xt - 1 + ......+ xt - m + 1 .
If we're aggregating quarterly (flow) data to annual data, then m = 4. In the case of aggregation from monthly to quarterly data, m = 3, and so on.
We know, from my earlier post, that if yt is integrated of order one (i.e., I(1)), then so is Yt.
Suppose that we also have a second temporally aggregated series:
Xt = xt + xt - 1 + ......+ xt - m + 1 .
Again, if xt is I(1) then Xt is also I(1). There is the possibility that xt and yt are cointegrated. If they are, is the same true for the aggregated series, Xt and Yt?