Hi Kingdom Nwuju

I use the R statistical language. Andrew Jebb's 2015 paper has a very useful approach to coding Fourier terms. Here's the study: Article Time series analysis for psychological research: examining a...

Article Time series analysis for psychological research: examining a...

sin1 = sin(2 * pi * 1 * time/12)

cos1 = cos(2 * pi * 1 * time/12)

sin2 = sin(2 * pi * 2 * time/12)

cos2 = cos(2 * pi * 2 * time/12)

…

sin6 = sin(2 * pi * 6 * time/12)

cos6 = cos(2 * pi * 6 * time/12)

Where "time" is your running variable. This is based on a monthly model, with six sine/cosine pairs. If you were to use a quarterly model, it would be two pairs of sine/cosine pairs, i.e., sin1, cos1, sin2, cos2, and instead of dividing by 12 you divide by 4. As an example of the analytical code, I use the following, with the nlme r package:

model

@Tyler J Lane

Thank you for your kind answer. Let me take my time to explore the links very well. I will get back to you should I encounter any challenge.

I don't know of a Fourier-ARIMA combined model. I know that an ARIMA model is a time-domain model. Another approach to time series analysis is the use of a Fourier series model the series. The second approach is called the frequency domain approach in which the series is a linear combination of sinusoidal functions. It is a resolution of the series into frequencies of different amplitudes. I doubt if you'll find a software to give you a ready-mades solution but if you use reviews you can estimate the equation as a linear combination of sine and cosine functions of the time points.

Fourier-ARMA models are not new at all. They were proposed by Bovas Abraham and George Box, "Deterministic and forecast‐adaptive time‐dependent models", J. Roy. Statist. Soc. Series C Applied Statist. 27(2), 1978, 120-130). Note that the paper is not cited by Jebb et al. (2015). A long time ago I have experimented with it and it is mentioned in my paper with Rajae Azrak on Exact maximum likelihood estimation for extended ARIMA models, in the book Developments in Time Series Analysis, edited by T. Subba Rao and published in 1993. That paper is available on ResearchGate. Deterministic seasonality is included in my program ANSECH developed in Fortran and included in Time Series Expert.

@Guy Mélard

Thanks for your answer.

Can you kindly drop the line to your work on Fourier Time Series, so that I can review it.

The paper I mentioned is here Article Exact maximum likelihood estimation for extended ARIMA models