Wednesday, July 16, 2014

The Econometrics of Temporal Aggregation - II - Causality Testing

In my recent post about my presentation at the recent conference of the N.Z. Association of Economists, I promised some follow-up posts of a more specific nature. I thought I'd begin with some brief comments about the effects of temporal aggregation on Granger causality.

By temporal aggregation, I'm referring to the situation where the economic activity takes place at some frequency (say daily), but our time-series data are recorded at only a lower frequency (say monthly). My use of the word "aggregation' tells you that we're "adding up" the data over time - so, at least implicitly, I'm thinking about flow data.

How does this sort of aggregation impact on Granger causality?

A number of authors have addressed this question from different perspectives. Some useful references include the papers by Sims (1971), Wei (1982), Christiano & Eichenbaum (1987), Marcellino (1999), Breitung and Swanson (2002), and Gulasekaran and Abeysinghe (2002).

I'll use yt  and xt to refer to the high-frequency data; and Yt and Xt to refer to the (aggregated) low-frequency data. When I write xt → yt, this will mean "x Granger-causes y". Similarly, yt ↔ xt will denote bi-directional Granger causality between y and x. On the other hand, Xt ↛ Yt will mean "X does not Granger-cause Y", etc.

First, let's look at a summary of a few situations that can arise when time-series data are temporally aggregated:
  1. If xt  ↛ yt, then  Xt ↛ Yt.
  2. If  xt → yor yt  → xt , then we can find Xt ↔ Yt; or Xt ↛ Yt; and / or  Yt ↛ Xt.
  3. If yt ↔ xt, then we can find only Yt  → Xt, or vice versa.
Next, let's look a simple illustrative empirical example. It involves the use of daily data for the price of crude oil, PC, (on the Edmonton hub), and the wholesale (rack) price, PW, of gasoline in Vancouver. As well as using the daily data, I've also aggregated the numbers to get end-of-week prices, and end-of-month prices.The prices are stocks, not flows, but we'll see that this type of temporal aggregation can also impact on Granger causality testing.

The testing that I've undertaken allows for the unit roots and cointegration in the data by using the Toda-Yamamoto methodology. I've discussed the latter in detail in several earlier posts - most notably here. Here are the results, with "p" denoting the p-value,  and "v" denoting the degrees of freedom, for the Wald test of the hypothesis of "no Granger causality":

As you can see, the upper part of the results table illustrates situation #2 noted above. The daily data indicate that there is Granger causality from crude oil prices to gasoline prices; whereas the aggregated data suggest the opposite (unlikely) result.


Breitung, J. and N. R. Swanson, 2002. Temporal aggregation and spurious instantaneous causality in multiple time series models. Journal of Time Series Analysis, 23, 651-665.

Christriano, L. J. and M. Eichenbaum, 1987. Temporal aggregation and structural inference in macroeconomics. Carnegie-Rochester Conference Series on Public Policy, 26, 63-130.

Gulasekaran, R. and T. Abeysinghe, 2002. The distortionary effects of temporal aggregation on Granger causality. Working Paper No. 0204, Department of Economics, National University of Singapore.

Marcellino, M., 1999. Some consequences of temporal aggregation in empirical analysis. Journal of Business and Economic Statistics, 17, 129-136.

Sims, C. A., 1971. Discrete approximations to continuous time distributed lags in econometrics. Econometrica, 39, 545-563.

Toda, H. Y. and T. Yamamoto, 1995. Statistical inferences in vector autoregressions with possibly integrated processes. Journal of Econometrics, 66, 225-250

Wei, W. W. S., 1982. The effect of systematic sampling and temporal aggregation on causality – A cautionary note. Journal of the American Statistical Association, 77, 316-319.

© 2014, David E. Giles


  1. Interesting post.

    Trying to think of an example: Suppose x is a forecast of y. Like Vancouver rainfall. Suppose rainfall follows a random walk at daily frequencies, so that yesterday's rainfall causes today's forecast of tomorrow's rainfall. But at annual frequencies, weather forecasters can watch what is happening in the Pacific, for signs of an approaching El Nino.

    (I don't think stocks or flows makes any difference, and the price of gas has the units $/litre, which is a stock divided by a stock, with no time dimension.)

    1. Nick - Given that Granger causality is all about predictability, conditioned by an information set, anything that changes the latter may impact on GC. This fits in with your example. Re. stocks and flows - the units are not the issue, With flows we temporally aggregate, with stocks we generally "selectively sample" (unless we average over the higher frequency values). In general, this distinction is crucial, as the effects on the MA properties of the data are different in each case. See my presentation slides - - especially slides 10, 11, and 20.