How can I check if my time series data is zero mean, stationary and independent identically distributed?

Question

I have time series data which is international monthly tourist arrival to Malaysia (N=264).

My objective is to forecast tourist arrival for 6 months ahead. After analyzing my data pattern, I found that using Box-Jenkins method gives me the best forecasting model.

The question is that, at such preliminary stage, when using Box-Jenkins methodology time series data must follow certain criteria: mean is zero, variance is constant and the data is stationary.

I used graphical methods to check if my time series data is stationary, has zero mean and constant variance.

However, how to check whether my time series data is independent and identically distributed (iid)? What method should i used?

Answer 1

The errors from the model should have a zero mean or a mean that is not significantly different from zero everywhere. (1) In practice this means no Pulses, no Level/Step shifts , no seasonal pulses and no local time trends. (2) The variance of the errors from the final model should be constant which means no structural shift in error variance or dependency on the level of the original series. (3) The parameters of the model must be invariant over time. The appropriate tests for (1) is available via Intervention Detection Tests ( Tiao , Tsay and others ). The appropriate test for (2) is both the Tsay Test for constant error variance AND the Box-Cox test for transformations. The test for (3) is the Chow test . In case you don't want to program these tests yourself as they are not freely available you might want to look at AUTOBOX. All of these have been seamlessly integrated into a piece of software that I helped write , available from http://www.autobox.com. Hope this helps.

Answer 2

Box and Jenkins suggest used the autocorrelation and partial autocorrelation functions to identify the model. The general Box Jenkins models are seasonal ARIMA modles which allow for nonstationary components (periodic components and polynomial trend). The rule for testing for nonstationarity is to compute the autocorrelation function and if the correlations are large and drop off slowly that is an indication of nonstationarity. This is somewhat subjective but IrishStat does this in a more formal automated way with AUTOBOX. As he mentioned level shifts and pulses can also be indications of nonstationary behavior which the AUTOBOX procedures will detect and incorporate into the model. Once you fit the model the diagnostic checking looks at the residuals to determine if they are independent with 0 mean and constant variance. Independence cannot be tested for but you can test for significant autocorrelation in the residuals and such correlation is an indication of lack of independence. The Ljung-Box test is a general test for correlation in the residual series. If zero autocorrelation is rejected higher order AR and or MA terms may be needed. If the residual series also looks nonstationarity in terms of slowly decaying autocorrelation then differencing of the series 1st 2nd or 3rd for linear quadratic or cubic trends respectively can be tried. If the appear to be periodic components remaining in the residual series then seasonal differencing can be used.