# When would I use a trig function in a Regression/Time Model Series

I was casually flipping through some older journals in my university's library and found that some authors employ trig functions to forecast social phenomena.

I'd like to know under what circumstances would I consider using trig functions as a tool in forecasting or cross sectional models? Would I ever use a trig function in time series analysis if I used seasonal dummies in a given ARIMA,VAR or VECM model?

Trigonometric functions fulfill the same roles as seasonal components in exponential smoothing: they are periodic and tie together time points one cycle apart.

The advantage of trigonometrics is that they are nice and smooth. If you just treat the, say, 12 seasonal indices in monthly exponential smoothing as free parameters, they do not hang together in any way. The seasonal index for January could be -10, for February +15, for March -5, for April +27, without rhyme or reason - this "models" discontinuous jumps between months, which usually does not make a lot of sense. Trigonometric functions impose a certain continuity between neighboring points.

Alternatives to trigonometric functions would be periodic splines, or periodic "bump functions".

One example for using trigonometric functions in time series modeling and forecasting is the model. The "T" actually stands for "trigonometric". And there are people who work in the frequency domain, using Fourier transforms, which of course are also trigonometry.

Note that seasonal ARIMA will typically not use seasonal dummies, but seasonal differences. (Of course, you can run a regression with ARIMA errors or an ARIMAX model, where you account for the seasonality using trigonometric or other periodic functions. This requires a few degrees of freedom, but it doesn't throw away an entire year's worth of observations by seasonal differencing.)

• Excellent! can you share your thoughts on this thred as well? economics.stackexchange.com/questions/19172/… Nov 10, 2017 at 1:23
• Good points made there. Of course the focus (economic and econometric modeling) is somewhat different from mine - my point about dummies making for discontinuities is more relevant for long seasonal cycles, like weekly or daily data, than for short ones, like the quarterly data relevant in econometrics. Plus, harmonics are of course not very useful for cycles with stochastic length, like business cycle (which Alecos seems to be referring to), so it seems Tsay's comment is slightly off-base Nov 10, 2017 at 7:47
• While I feel utterly unqualified to critique a Professor of Economics at Chicago, I suspected Alecos is right about that. Perhaps Tsay was simplifying for his audience - even though it is a graduate-level text, its emphasis is for practitioners. At any rate I was uncomfortable enough with the claim I felt I should quote it verbatim and let others decide how much weight to put on it; in the context of cycles of non-stochastic length, Tsay's comment is still correct. Nov 10, 2017 at 10:20