I want to fit a SARIMA model and have daily sales data with a weekly seasonal pattern (frequency = 7) with this auto correlation function (ACF):
Clearly, there are seasonal effects as the spikes of the ACF indicate. As explained in (Shumway, Stoffer (2011): Time Series Analysis and its applications. Example 3.46, Figure 3.23.) the ACF of my time series also indicates seasonal non-stationarity and therefore a unit root in the seasonal component.
After differencing at lag $k=7$ ($y_t = y_t – y_{t-7}$) the ACF looks like this:
Since I wanted to automate this process I wanted to test for a unit root with the R package uroot using the Canova and Hansen test. This test assumes no unit root in $H_0$
However, when I execute:
z <- ts(data[,1], frequency=7) res <- ch.test(z, type = "dummy", sid = 7) res I get the result:
Canova and Hansen test for seasonal stability data: z statistic pvalue [1,] 0.1584 0.4785 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Test type: seasonal dummies NW covariance matrix lag order: 12 First order lag: no Other regressors: no P-values: interpolation in original tables Am I right assuming that the test gives me a $text{p-value}=0.4785$ and therefore I cannot reject $H_0$ on a 95% confidence level and therefore have to assume there is no unit root in the seasonal component? And if yes, how does this fit the ACF indicating non-stationarity in the seasonal component? Is there a difference between unit root in the seasonal component and non-stationary seasonal component?
Am I missing/confusing something here?
Exemplary plot of the first 30 days:
All data:
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5223, 5558, 4665, 4797, 0, 4359, 3650, 3797, 3897, 3808, 3530, 0, 5054, 5042, 4767, 4427, 4852, 4406, 0, 4395, 3558, 3464, 3769, 3706, 4364, 0, 6102, 5011, 4782, 5020, 5263] EDIT:
I edited the question a couple of times in order to feedback the comments below.
EDIT:
Differencing like suggested above seems the wrong way to go (see comments). Furthermore the statement there is a unit root could not be backed further (see comments). I will look into the OCSB test before writing an answer just to be sure.
Best Answer
Just because the ACF shows seasonal spikes with relatively slow decay it can not automatically be concluded that there is a unit-root in the seasonal componend.
Actually (Ghysels, Osborn (2001): The Econometic Analysis of Seasonal Time Series, p.29) states:
"[…] for the sample sizes often observed in practice, it may be difficult to discriminate between deterministic seasonality and a seasonal unit root process."
Furthermore on page 42 they write:
"It is now well known that a series generated by a unit root process can wander widely and smoothly over time without any inherent tendency to return to its underlying mean value […]. In the seasonal context, there are S unit root processes, none of which has an inherent tendency to return to a deterministic pattern. As a result, the values for the seasons can wander widely and smoothly in relation to each other […]"
Therefore it is useful to visualize the time series of the different seasons $S$ and see if they look non-stationary. In our case we have daily data and model each day as a season. So we get seasons $S=7$. Plotting gives:
None of the single seasonal time series looks particularly non-stationary.
Therefore the Canova and Hansen test statistics is perfectly fine. The same result is obtained when using the forecast::nsdiffs() function or the OCSBtest() function from the R forecast package source code on github.
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