Methods for Forecast with Best ARIMA Model

Forecast with Best ARIMA Model compares many models and selects a final model with a criterion in the specifications of the analysis. For information on the results for the final ARIMA model, go to Methods and formulas for ARIMA. The following sections contain details that are unique to Forecast with Best ARIMA Model.

Model selection

The model selection uses the following steps:

Estimate the model parameters for every model. If a model includes a constant and the estimation of the parameters fails, try to estimate the parameters without the constant term.
Calculate the information criterion for each model. The default criterion is the corrected Akaike Information Criterion (AICc).
Produce results for the model with the best value of the information criterion.

The following sections describe details that differ in the selection of non-seasonal and seasonal models.

Non-seasonal models

The specifications for the analysis contain the order of differencing. For the specified order, the process evaluates all combinations of the autoregressive and moving average orders in the specifications for the analysis with the following restrictions:

When you fit models with a constant term, candidate models have p + q ≤ 9.
When you fit models without a constant term, candidate models have p + q ≤ 10.
Models with d = 2 never include a constant term.
The model evaluates ARIMA(0, d, 0) only when d = 1.

Seasonal models

The specifications for the analysis contain the orders of nonseasonal and seasonal differencing. For the specified orders, the process evaluates all of the combinations of seasonal and non-seasonal autoregressive and moving average orders with the following restrictions:

When you fit models with a constant term, candidate models have p + q + P + Q ≤ 9.
When you fit models without a constant term, candidate models have p + q + P + Q ≤ 10.
Models with d + D > 1 never include a constant term.
The search for a seasonal model requires the order of at least one of the seasonal parameters to be able to be greater than 0. The search includes non-seasonal models if the specifications for the search include models where all the seasonal parameters have orders of 0.
At least 1 of p, q, P, and Q is non-zero in every model.

Criteria

To evaluate ARIMA models with the same orders of differencing, the analysis uses 1 of 3 information criteria:

Akaike Information Criterion (AIC)
Corrected Akaike Information Criterion (AICc)
Bayesian Information Criterion (BIC)

The calculation of the information criteria for a model uses the log-likelihood value for the model. The calculation of the log-likelihood value uses a recursive algorithm. For more information, see section 8.6 of Brockwell & Davis (1991)¹.

Notation

Term	Description
k	The number of parameters in the model For a seasonal model with a constant, k = p + q + 2 For a seasonal model without a constant, k = p + q + 1 For a seasonal model with a constant, k = p + q + P + Q + 2 For a seasonal model without a constant k = p + q + P + Q + 1
L_c	the log-likelihood of the current model
n	the sample size of the time series

Box-Cox transformation

The analysis allows a Box-Cox transformation of the data. The transformation of the data happens before the model selection. For information on the Box-Cox transformation for time series data, go to Methods and formulas for Box-Cox Transformation for Time Series.

The results of the analysis include the back-transformed forecasts and the probability limits of the forecasts. The t^th value of the transformed time series depends on the value of λ for the transformation:

for λ > 0
for λ = 0
for λ < 0

where is the t^th value of the original time series and t = 1, …, n.

Let be the l^th forecast value starting from the origin, t, for the transformed data. Let be the l-step forecast variance from the transformed data. Then, the l^th forecast value from t for the original series depends on the value of λ:

The transformation of the probability limit for a forecast uses the inverse of the Box-Cox transformation. For details on the calculations of the probability limits, go to Methods and formulas for ARIMA. The inverse transformation for the upper probability limit is the same as the inverse transformation for the lower probability limit. The inverse transformation depends on the value of λ.

where is the limit in the original scale and is the limit in the transformed scale.

Random walk model

The ARIMA(0, 1, 0) model, with or without a constant term, is the random walk model. In Minitab Statistical Software, Forecast with Best ARIMA Model fits the random walk model. The command Stat > Time Series > ARIMA requires at least one autoregressive or moving average parameter. The estimation and probability limits for the random walk model have specific forms. The calculations for the loglikelihood, the forecast limits, and the probability limits for the forecasts depend on whether the model includes a constant term.

Definitions

Term	Description
	the observations for a time series with t = 1, …, n
	the first differenced data from the original time series,

Use the following equations to represent the model without a constant:

where are independently and identically distributed and follow the normal distribution with mean 0 and variance σ², t = 2, …, n.

Equations that represent the model with a constant are similar:

The likelihood value calculations make use of the following equation:

Model without a constant term

The loglikelihood has the following form:

Loglikelihood

where

The forecast value at t + l, l = 1, …, 150, starting from the time order, t has the following form:

The 100 × (1 – α) probability limit for the forecast value has the following form:

where represents the 100 × (1 – α/2)^th percentile from the standard normal distribution.

Model with a constant term

For a model with a constant, the calculations for the loglikelihood require the estimation of the constant, C. First, difference the data from the original series for t = 2, …, n. The constant is the sample mean of and has the following form: