ARIMA (p,d,q) forecasting equation:

Auto-Regressive

Integrated Moving Average models are efficient class of models used for forecasting time

series data which can be transformed to be “stationary” by differencing. A

random variable is said to be stationary if the statistical properties of that

variable remains constant over the whole time series. A series that is

stationary in nature has a negligible trend and the fluctuations around its

mean have similar constant amplitude and it squirm in a regular patterns

i.e. its short term random patterns of time always look similar. The stationary

series Autocorrelation factor also remain constant over that particular period

of time and the power spectrum remain constant over the entire series. A random

variable composed of Autocorrelation factor and power spectrum series and

contains both signal and noise, the signal pattern could be fast or slow

reversion or a sinusoid oscillation or it could be a rapid alteration in sign

containing a seasonal component. ARIMA model is considered as “filter” that

segregates the signal from noise to gain valuable information for the future

movements.

ARIMA forecasting

mathematical equation for stationary time series is linear like regression

equation in that the predictors/variables consists of lags of dependent

variables and the lags of forecast errors. The mathematical formulation of

model is,

Forecasted value of Y = constant/weighted sum of single or more

values of Y and/ weighted sum of single or more values of errors.

Non-Seasonal ARIMA model

is defined as ARIMA (p,q,d) where,

p is the number of

autoregressive terms,

q is the number of lagged

forecast errors in the prediction equation

d is the number of non-seasonal

differences needed for stationary, and

The forecasting equation

is constructed as follows.

First, let y denote

the dth difference of Y, which means:

If d=0: yt = Yt

If d=1: yt = Yt – Yt-1

If d=2: yt = (Yt – Yt-1) – (Yt-1 – Yt-2) = Yt –

2Yt-1 + Yt-2

Note that the second

difference of Y (the d=2 case) is not the difference from two periods

ago. Rather, it is the first-difference-of-the-first

difference, which is the discrete analog of a second derivative, i.e., the

local acceleration of the series rather than its local trend.

In order to identify the

appropriate ARIMA model for Y, the first step involves the determining of the

differencing order (d) needed to stationary the series and also to remove the

gross features of seasonality in conjunction with a variance-stabilizing

transformation such as logging or deflating the stationary series also may

contains auto correlated errors representing some number of AR terms (p>1)

and some number MA terms (q>1) would also be need to formulate the

forecasting equation.