Methods and formulas for ARIMA

Select the method or formula of your choice.

Coefficients

The coefficients are estimated using an iterative algorithm that calculates least squares estimates. At each iteration, the back forecasts are computed and SSE is calculated. For more details, see Box and Jenkins1.

The ARIMA algorithm is based on the fitting routine in the TSERIES package written by Professor William Q. Meeker, Jr., of Iowa State University2. We are grateful to Professor Meeker for his help in the adaptation of his routine to Minitab.

Back forecasts

Back forecasts are calculated using the specified model and the current iteration's parameter estimates. For more details, see Cryer3.

SSE

Formula

Notation

TermDescription
n total number of observations
residuals using that iteration's parameter estimates, including back forecasts

SS for residuals

Formula

Notation

TermDescription
n total number of observations
at residuals using the final parameter estimates, excluding back forecasts

DF for residuals

Formula

For a model with a constant term:

(nd) – pq – 1

For a model without a constant term:

(nd) – pq

Notation

TermDescription
n total number of observations
d number of differences
p number of autoregressive parameters included in the model
q number of moving average parameters included in the model

MS for residuals

Formula

SS / DF

Chi-square statistic

Formula

Notation

TermDescription
n total number of observations
d number of differences
K 12, 24, 36, 48
k lag
autocorrelation of the residuals for the k th lag

DF for chi-square statistic

Formula

For a model with a constant term:

Kpq – 1

For a model without a constant term:

Kpq

Notation

TermDescription
K 12, 24, 36, 48
p number of autoregressive parameters included in the model
q number of moving average parameters included in the model

P-value for chi-square statistic

Formula

P(X < χ 2)

Notation

TermDescription
X distributed as χ 2 (DF)

Forecasts

Formula

Forecasts are calculated recursively, based on the model and the parameter estimates. For example, if an ARIMA model is fit with 1 autoregressive term (AR(1)) and one seasonal differencing term with a seasonal period of 12, this model is fit:

Yt Yt–12 = γ + Φ(Yt–1Yt–12–1)

To estimate , the first forecast, where k is the origin, find:

Then, you find , in the same manner, and so on.

Notation

TermDescription
Yt actual value at time t
Φ autoregressive term
estimated autoregressive term
γ constant term
estimated constant term
1 G.E.P. Box and G.M. Jenkins (1994). Time Series Analysis: Forecasting and Control, 3rd Edition. Prentice Hall.
2 W. Q. Meeker (1978). "TSERIES-A User-Oriented Computer Program for Time Series Analysis" , The American Statistician, 32, 111-112.
3 J.D. Cryer (1986). Time Series Analysis. Duxbury Press.
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