Paper 235-2009 Stationarity Testing in High-Frequency Seasonal Time Series D. A. Dickey, N. C. State University, Raleigh NC Abstract Deciding whether seasonality is of a stochastic nature, and thus slowly changing over time, or deterministic and thus repeating in the same way each season can have a substantial impact on forecast accuracy. Tests for stochastic seasonality, called seasonal unit root tests, have been developed for certain common seasonal periods, like 12 (monthly data) 4 and 2, but until now have not been available for high frequency (like daily data over years or minute by minute over days). This paper fills the gap, arriving at a simpler distributional result than is usually the case with unit roots. An example using natural gas supply is used to illustrate. Key Words: Nonstationarity, unit roots, asymptotics Introduction Time series quite often show patterns that repeat periodically. Monthly retail sales provide a good example. If the seasonality is very regular, seasonal dummy variables can be used to give, for example, additive monthly effects. With this approach, the January effect is assumed to be the same regardless of the year. Seasonal ARMA error terms can be added to make some local modifications. An alternate model that is useful when the seasonality changes over the years is the seasonal unit root model. Motivated by Box and Jenkins’ approach to modeling international airline ticket sales, this method takes a span d difference for seasonality d, e.g. d=12 for monthly data, and analyzes these seasonal span differences. Using the backshift operator B, the polynomial (1-B d ) represents the span d difference. Tables of percentiles for testing that the polynomial has unit roots (as does 1-B d ) are available (Dickey, Hasza, Fuller, 1984, henceforth “DHF”) for seasonal periods d=2, 4, and 12. As with ordinary (d=1) unit root tests, these are nonstandard distributions that shift when typical deterministic inputs like seasonal means are included in the model. It is possible that a user may want to test for unit roots at a longer lag, for example one might suspect periodicity 24 or 7x24=168 in hourly data and hence might ask if unit roots at those lags give an appropriate model. This paper deals with large d results for unit root tests. Some features emerge that are nicer than those of the shorter period cases. Simulations using SAS 1 software show the fidelity of finite sample behavior to the limit theory. The Lag d Model Let Y t denote data at time t, d denote the period of seasonality and B the standard backshift operator so B d Y t = Y t-d . A simple model relating Y t to Y t-d is Y t – f(t) = α( Y t-d – f(t-d)) + e t where e t is white noise and f(t) represents deterministic terms such as a constant mean, seasonal means, sinusoid, and trends. In line with nonseasonal unit root testing, a user might be interested in testing the null hypothesis that α=1 and as usual this would entail assumptions about starting values. For simplicity, we begin with the mean 0 assumption, f(t)=μ=0, known starting values Y -j =μ=0 for j=0,1,2,…,-d+1 and n=md, that is, complete seasons. The results carry over into more realistic scenarios. As usual, we base a test on the least squares estimator obtained by regressing Y t on Y t-d for t=1,2,…,n=md with no intercept. This maximizes the conditional (on Y -j ) likelihood giving the estimator ˆ α . The usual algebra of least squares holds here. The algebra does not depend on any distributional assumptions. We find that 1 2 1 ( 1) 1 ( 1) ( 1) 1 1 1 1 1 ˆ ( ) (1 / ) /[ d m d m di s di s di s s i s i m d d m Y e m d Y α α − − − 2 ] − +− − + − +− = = = = − = ∑∑ ∑∑ , a ratio of two normalized sums. In this expression s is the period (or season) within a seasonal cycle of d time periods. For monthly data d=12 and s=1 is the January index. Here i represents the cycle (the year for example) so the time subscript t is t=d(i-1)+s when i-1 cycles have passed and we are in period s of the i th cycle. 1 SAS is the registered trademark of SAS Institute, Cary, NC. Statistics and Data Analysis SAS Global Forum 2009