Unit Root Testing: Sample Size vs. Sample Span

The more the merrier when it comes to the number of observations we have for our economic time-series data - right? Well, not necessarily. 

There are several reasons to be cautious, not the least of which include the possibility of structural breaks or regime-switching in the data-generating process. However, these are topics for future posts. Here, I want to discuss a different issue - namely, the impact of data frequency on the properties of tests for the stationarity of economic time-series.

To be specific, let's consider the following question: "Which is better when I'm applying the (augmented) Dickey-Fuller test - 20 annual observations for the series, or 80 quarterly observations?"

The short answer is that it really doesn't matter.

However, a couple of caveats are in order. I'm going to assume that any unit roots that may be present in the series are at the zero frequency. That is, I'm not going to consider unit roots at the seasonal frequencies. Clearly, we can't test for these using annual data. Moreover, if seasonal unit roots are present, then this should be taken into account when testing for unit roots at the zero frequency. For example, see Hylleberg et al. (1990).

The question that I've posed has received quite a bit of consideration in the time-series econometrics literature. A slightly more general way of posing the question is to ask if tests for unit roots are affected when we either temporally aggregate or selectively sample the data. The former process arises with flow variables when, for example, we add up monthly flows to get a quarterly or annual flow. The second process is associated with stock variables where we use a particular higher-frequency observation to represent the value of the lower-frequency variable. An example would be where we use the March-month unemployment rate as a measure of the rate for the March quarter.

In general, let's let "m" denote the frequency at which we selectively sample; or the number of periods that are aggregated in the case of a flow variable, As a detail, note that if, in the case of a stock variable, we average some higher-frequency values to get a low frequency value, then this constitutes m-period temporal aggregation, with a scaling of ( 1 / m). An example would be using the average of four end-of-quarter values for the CPI to get an annual CPI measure (m = 4).

Now, suppose that we have a time series generated according to:

                          y_t = ρ y_t-1 + u_t    ;  t = 1, 2, ...., T

where u_t follows a finite-order stationary ARMA process whose maximum AR modulus is less than ρ.

We want to test H₀: ρ = 1 against a sequence of local alternatives. That is,  against  H_A: ρ = exp(-c / T), where c > 0.

In this context, perhaps the most important result for you to aware of is the following, from Pierce and Snell (1995, p.336):

“Any test that is asymptotically independent of nuisance parameters under both H₀ and H_A has  a limiting distribution under both H₀ and H_A that is independent of m.”

So, this means that, asymptotically, temporal aggregation or selective sampling have no consequences in terms of size distortion, or loss of power, for the ADF, Phillips-Perron test, or Hall's (1994) IV-based unit root test. The same is true for several other related tests.

Pierce and Snell also show that even in finite samples, this result holds quite well; and it also applies to tests of non-cointegration, such as those of Engle and Granger, and Johansen.

Let's look at a simple, illustrative, example.The data I'll use are for imports of merchandise (c.i.f. basis) into New Zealand. The data are in nominal values, and I've seasonally adjusted them using the Census X-13 method, assuming a multiplicative decomposition of the time-series. The data were downloaded using the excellent, free, Infoshare facility provided by Statistics New Zealand.

(Snide remark: I hate to say it, but Statistics Canada could learn a few things from Statistics N.Z., and from plenty of other such agencies, when it comes to providing easy access to long-term economic time-series data.)

The fact that the data have been seasonally adjusted using the ratio-to-moving average method, prior to our unit root testing, can raise some issues of its own. Again, this is something that's best left for a separate post.

The data that I've used are available on this blog's data page, and the EViews workfile is on the code page. This is what the monthly, quarterly and annual data look like:

Now let's apply some unit root tests. I've used both the ADF test (where the null hypothesis is that the series is I(1), and the alternative is that it is I(0)); and th KPSS test (where the null hypothesis is that the series is I(0), and the alternative is that it is I(1)). 

(None of the series were found to be I(2).)

The ADF tests have been applied with an allowance for a drift and trend in the data, and the SIC was used to select degree of augmentation, k. For the  KPSS tests the bandwidth, b, was selected using the Newey-West method, with the Bartlett kernel.

Here are the results:

Sample                               T             k             ADF                   b          KPSS    

                                                                      (p-val.)

------------------------------------------------------------------------------------------------

1960M1 - 2014M3              651           2            -2.050                21          0.709*

                                                                      (0.572)

1960Q1 - 2014Q1              217            0            -2.027               11          0.456*

                                                                      (0.583)

1960 - 2013                        54            4            -1.409                5          0.252*

                                                                      (0.846)

-----------------------------------------------------------------------------------------------

* Significant at the 1% level, based on asymptotic critical values.

As you can see, results strongly support the presence of a unit root in the imports time-series, regardless of the degree of aggregation of the data (and hence the sample size, T) over the 54-year time-span under consideration here.

References

Hall, A., 1992. Testing for a unit root in time series using instrumental variable estimators with pretest data based model selection. Journal of Econometrics, 54, 223-250.

Hylleberg, S., R. F. Engle, C. W. J. Granger, and B. S. Yoo, 1990. Seasonal integration and cointegration. Journal of Econometrics, 44, 215-238.     

Pierse, R. G. and J. Snell, 1995. Temporal aggregation and the power of tests for a unit root. Journal of Econometrics, 65, 333-345.

© 2014, David E. Giles