## Friday, August 22, 2014

### The Econometrics of Temporal Aggregation - III - Unit Roots

In two previous posts, I've talked a bit about the effect that aggregating time series data can have on some standard econometric results. The first of those posts was about a talk that I gave last month at the 2014 Conference of the New Zealand Association of Economists. That talk was on the general topic of "The Econometrics of Temporal Aggregation". The second post looked more specifically of the consequences of such aggregation on the results of testing for Granger causality.

Here, I'll provide some information about the impact of temporal aggregation on the existence of, and testing for, unit roots in time-series data.
Let's think about a time series that is constructed by aggregating higher-frequency observations. For instance, we may have data that are recorded on a regular quarterly basis, but we add up these quarterly (flow) values to get a single annual (flow) value. The notation that I'll use is that yt is the higher-frequency value of the variable at time "t"; and Yt is the aggregated value.

If the aggregation is over "m" high-frequency periods, then

Yt = yt + yt - 1 + ......+ yt - m + 1 .

In our example of quarterly and annual data, m = 4. In the case of aggregation from monthly to quarterly data, m = 3, and so on.

Now, let's consider two situations - one where the yt data are integrated of order zero (i.e., I(0)); and one where they are integrated of order one (i.e., I(1)).

The first of these case is straightforward. If we add together two or more time series, each of which is stationary, then the resulting aggregate series is also stationary. So, in our notation, Yt will be stationary.

Next, suppose that the yt data are I(1). This implies that the series, wt = Δyt = (yt - yt - 1), is I(0).

Correspondingly, consider the first-difference of the aggregated series, and define

Wτ = ΔYτ = (Yτ - Yτ - 1 ) = [Σyτ - j - Σyτ - 1 - j] ,

where the summations are each for j = 0 to (m - 1).

So,
Wτ = (yτ - yτ - 1) + (yτ - 1 - yτ - 2) + ....... + (yτ - m + 1 - yτ - m) = (wτ + wτ - 1 + .... + wτ - m + 1) .

Because each of the "w" series are I(0), we see that "W" is the sum of "m" I(0) series, and so it is I(0) itself. So, aggregating an I(0) series results in another I(0) series.

In general, by a similar argument, aggregating an I(d) series results in another I(d) series, for d ≥ 0.

This is all very well, but just because the order of integration of a time series is unaltered by temporal aggregation, this doesn't necessarily mean that our ability to detect the order of integration is unaffected by this manipulation of the data. In other words, how are standard tests for (non-) stationarity impacted by temporal aggregation of the data?

There's actually quite a large literature addressing this important question. However, this is something that I've discussed previously, here, so I won't go over it again.