F-test on seasonal dummies

The F-test on seasonal dummies checks for the presence of deterministic seasonality. The model used here uses seasonal dummies (mean effect and 11 seasonal dummies for monthly data, mean effect and 3 for quarterly data) to describe the (possibly transformed) time series behaviour. The test statistic checks if the seasonal dummies are jointly statistically not significant. When this hypothesis is rejected, it is assumed that the deterministic seasonality is present and the test results are displayed in green.

This test refers to Model-Based $\chi^{2}\ $and F-tests for Fixed Seasonal Effects proposed by LYTRAS, D.P., FELDPAUSCH, R.M., and BELL, W.R. (2007) that is based on the estimates of the regression dummy variables and the corresponding t-statistics of the RegARIMA model, in which the ARIMA part of the model has a form (0,1,1)(0,0,0). The consequences of a misspecification of a model are discussed in LYTRAS, D.P., FELDPAUSCH, R.M., and BELL, W.R. (2007).

For a monthly time series the RegARIMA model structure is as follows:

$\left( 1 - B \right)\left( y_{t} - \beta_{1}M_{1,t} - \ldots - \beta_{11}M_{11,t} - \gamma X_{t} \right) = \mu + (1 - B)a_{t}$ , [1]

where:

$M_{j,t} = \begin{cases} 1 & \text{ in month } j = 1, \ldots, 11 \\ - 1 & \text{ in December}\\ 0 & \text{ otherwise} \end{cases} \text{ - dummy variables;}$

$y_{t}$ – the original time series;

$B$ – a backshift operator;

$X_{t}$ – other regression variables used in the model (e.g. outliers, calendar effects, user-defined regression variables, intervention variables);

$\mu$ – a mean effect;

$a_{t}$ – a white-noise variable with mean zero and a constant variance.

In the case of a quarterly series the estimated model has a form:

$\left( 1 - B \right)\left( y_{t} - \beta_{1}M_{1,t} - \ldots - \beta_{3}M_{3,t} - \gamma X_{t} \right) = \mu + (1 - B)a_{t}$ , [2]

where:

$M_{j,t} = \begin{cases} 1 & \text{ in quarter} j = 1, \ldots, 3 \\ - 1 & \text{ in the fourth quarter}\\ 0 & \text{ otherwise} \end{cases} \text{ - dummy variables;}$

One can use the individual t-statistics to assess whether seasonality for a given month is significant, or a chi-squared test statistic if the null hypothesis is that the parameters are collectively all zero. The chi-squared test statistic is ${\widehat{\chi}}^{2} = {\widehat{\beta}}^{‘}{\lbrack Var(\widehat{\beta})}^{\ })^{- 1}\rbrack{\widehat{\beta}}^{\ }$ in this case compared to critical values from a $\chi^{2}\left( \text{df} \right)$-distribution, with degrees of freedom $df = 11\ $(monthly series) or $df = 3$ (quarterly series). Since the ${Var(\widehat{\beta})}^{\ }$ computed using the estimated variance of $\alpha_{t}$ may be very different from the actual variance in small samples, this test is corrected using the proposed $\text{F}$ statistic:

$F = \frac{ {\widehat{\chi}}^{2}}{s - 1} \times \frac{n - d - k}{n - d}$ , [3]

where $n$ is the sample size, $d$ is the degree of differencing, s is time series frequency (12 for a monthly series, 4 for a quarterly series) and $k$ is the total number of regressors in the RegARIMA model (including the seasonal dummies $\text{M}_{j,t}$ and the intercept).

This statistic follows a $F_{s - 1,n - d - k}$ distribution under the null hypothesis.