To determine the order of AR and MA models, Autocorrelation and Partial auto correlation function provide better idea, while the spectrum analysis is not so conclusive. So while developing a model (ARMA or ARIMA) for generating monthly streamflows, can periodicity analysis be neglected?
ISSUE: Is periodicity test a requirement for ARMA or ARIMA? Or autoregressive modeling in general?
PERIODICITY & STOCHASTIC DATA: Periocidity in the general literature is called “cycle.” It is not an obstruction to AR-family of modeling. In fact, business cycles are also modeled under AR. This is a case of using AR for nonlinear data set.
Does the existence of periodicity or cycle vitiate stochastic condition of data? Stochastic means random. Random means without pattern. Cycle or periodicity, by common sense, would tell us that it breaks this stochastic condition. The logic seems to follow thus.
In general, autoregressive model states that the output variable depends linearly on the previous values. However, in real modeling, we need to put AR in order: (i) AR(0) manifests no dependence between terms. Only the error contributes to the output of the process; (ii) AR(1), only the previous term and the noise contributes to the output; (iii) AR(2) says that the previous 2 terms and the noise contributes to the output. As on moves to a higher order of AR, say AR(2) the stricture of process stationarity may also be relaxed. The idea of periodicity (cycle) may turn into regime shifting or changing which may be accommodated by smoothing function; this is a case of autoregressive modeling of nonlinear data set, i.e. Smoothed Transition Autoregressive (STAR).
AUTOREGRESSIVE MODEL: A simple case of linear data set model my autoregressive is given by:
(1) Xt = c + sum(kXt-1) + e
… where Xt = output; c = constant; k = parameter; Xt-i is a observation at time t minus i; and e = white noise. In this simple case of AR(p) or autoregressive to order of p, strict linearity is required.
However, in some case, we start seeing data set not being strictly linear beyond one previous observation; there might be regime shift or change, i.e. business cycle, hence periodicity. In such a case, we need to go to a higher order of AR and introduce smoothing functions. Without smooth transition, there will appear a erupt change and thus disrupt the model.
Smooth Transition Autoregressive (STAR) is introduced as:
(2) Yt = p10 + p1’wt + (p20 + p2’wt) F(yt-d) + ut
… where u = nid(0, sig2); pj = pj1, pj2, …, pjp) that j = 1, 2, …; and wt = (yt-1, yt-2, …, yt-p).
Incorporating into this general smoothed function STAR, two smoothing functions may be used:
(3) F(yt-d) = 1 + exp[-g(yt-d – c)]-1
… where g > 0; another smoothing function is given by:
(4) F(yt-d) = 1 – exp(-g(yt-d – c)2)
… where g>0. Note that F(yt-d) is between 0 and 1. this allows smooth transition in regime switching (periodicity or business cycle); c = threshold value; and g = the speed and smoothness of the transition.
IF g goes to infinity, yt-d c then F(yt-d) = 1, the model becomes a threshold autoregressive (TAR).
IF g goes to zero, F(yt-d) becomes linear AR(p) in (1) above.
Model (2) with transition function (3) becomes Logistic Smooth Transition Autoregressive (LSTAR). If model (1) uses transition function (3), it is called exponential Smooth Transition Autoregressive (ESTAR).
REFERENCES: See links below.
http://www.bportugal.pt/en-US/BdP%20Publications%20Research/WP200309.pdf
http://www.nber.org/chapters/c7196.pdf
http://ageconsearch.umn.edu/bitstream/21365/1/sp06ra02.pdf
Periodicity can be considered by developing seasonal ARIMA models. Use of correlograms is very subjective, and it gives an idea about the number of parameters under the model. So better to go with the grid search approach and choose an adequate model as suggested in Box-Jenkins methodology. Or build some non-linear or data driven models for the seasonal data.
Thank you so much for your valuable replies. Ya I shall incorporate STAR as well as SARIMA and see the effect.