9.1. Box-Jenkins ARIMA
9.1.0. Overview
ARIMA stands for Auto Regressive Integrated Moving Average model. So called, because the model fits autoregressive and moving average parameters to a transformed (differenced) time series and integrates back to the original scale before forecasts are generated. The differencing transformation makes use of B, the backshift operator, which shifts the subscript of a time series observation backwards in time by one period.
Select the column to analyse by clicking on [Dependent]. This column should not contain any missing values.
9.1.1. Differencing Input Options
The Differencing Input Options dialogue is for entering the parameter values used in transforming the original data. The data can be transformed by differencing, taking logs, raising to a power and adding an offset to it.
Nonseasonal Differencing: The degree of differencing across whole series (d).
Seasonal Differencing: The degree of differencing between points with seasonal period units apart in the series (D).
Seasonal Period: The number of time units per season (s).
Lambda: This is a coded value which determines any logarithmic or power transformation. This is performed before any differencing:
= 1 No Effect 
= 0 Log of series 
else 
Offset: The value of offset is added to every value in the time series. This is used to allow taking logarithms of a series including negative values.
Maximum Lag: This is the maximum lag to calculate in correlation displays.
You can apply any transformation to the series during the data preparation phase. In this case, forecasts will be generated in the transformed scale. Transformations made within ARIMA are reversed to give forecasts in the original scale.
The program then displays a dialogue with two options. The Fit Model option proceeds with the next step in the analysis (where the ARIMA model is selected.) and the Differencing Output Options gives access to the intermediate results.
The Fit Model option should not be selected until the data series has been transformed into a stationary series.
9.1.2. Differencing Output Options
Differencing Output Options should be used to help set the transformation values before the model is estimated. You may move between this dialogue and the second dialogue a number of times before the series is transformed appropriately. The input series for the ARIMA model must be stationary. A stationary series has a constant mean, variance and autocorrelation. Also, the Autocorrelation Function and Partial Autocorrelation Function should give an idea about the number of parameters to fit to the model.
Character Plot of Series: The transformed series is displayed in the form of a character plot together with the values.
Autocorrelation Function: The Autocorrelation Function (ACF) displays the autocorrelations of the transformed series. The autocorrelation represents the correlation between points in the series at displacement lag. The number of autocorrelations displayed is controlled in the Differencing Input Options dialogue.
In ARIMA modelling the series is required to be stationary. If the ACF either cuts off fairly quickly or dies down fairly quickly, then the time series values should be considered stationary. If the ACF dies down extremely slowly, then the time series values should be considered non stationary and some differencing will be required.
The ACF should be examined to decide which model to fit in the final stages.
Partial Autocorrelation Function: The Partial Autocorrelation Function (PACF) displays the partial autocorrelations of the transformed series. The partial autocorrelations represent the correlation between points in the series at displacement lag, with the effects of the intervening observations eliminated. Hence the partial autocorrelation at lag 1 is equivalent to the autocorrelation at lag 1. The number of autocorrelations displayed is controlled in the Differencing Input Options dialogue.
The PACF should be examined to decide which model to fit in the final stages.
Hi-Res Plot of Series: This displays a graphical view of the transformed series.
Example
Table 3.1 on p. 83 from Bowerman, Bruce L. & Richard T. O’Connell (1987). Data on monthly Hotel Room Averages for 1973-1986 are given.
Open TIMESER, select Statistics 2 → ARIMA and Room Averages (C1) as [Variable]. At the Differencing Input Options dialogue enter:
· 0 Nonseasonal Differencing
· 1 Seasonal Differencing
· 12 Seasonal Period
· 0 Lambda (0 Log(X), else X^Lambda)
· 0 Offset (minimum 480)
· 25 Maximum Lag
Check all differencing output option boxes to obtain the following results:
ARIMA
Character Plot of Series: Room Averages
|
Row |
X(t) |
-0.0263 0.0831 |
|
1 |
0.0334 |
* |
|
2 |
0.0020 |
* |
|
3 |
0.0465 |
* |
|
4 |
0.0357 |
* |
|
5 |
0.0484 |
* |
|
… |
… |
… |
Autocorrelations: Room Averages
|
Lag |
Correlation |
Standard Error |
-1.0000 1.0000 |
|
1 |
0.1933 |
0.0801 |
( *****)* |
|
2 |
0.0244 |
0.0830 |
( ** ) |
|
3 |
-0.2442 |
0.0830 |
***(**** ) |
|
4 |
-0.1515 |
0.0875 |
(***** ) |
|
5 |
-0.2119 |
0.0892 |
*(***** ) |
|
… |
… |
… |
… |
Partial Autocorrelations: Room Averages
|
Lag |
Correlation |
Standard Error |
-1.0000 1.0000 |
|
1 |
0.1933 |
0.0801 |
( *****)* |
|
2 |
-0.0135 |
0.0801 |
( * ) |
|
3 |
-0.2560 |
0.0801 |
***(**** ) |
|
4 |
-0.0622 |
0.0801 |
( ** ) |
|
5 |
-0.1760 |
0.0801 |
*(**** ) |
|
… |
… |
… |
… |
9.1.3. Model Fitting
The Model Fitting dialogue should only be used when the time series is considered stationary. The following guidelines can be used to help choose an ARIMA model to fit. It is often possible to try different models on the same data.
9.1.3.1. Seasonal and Nonseasonal Operators
Nonseasonal Operators (p and q)
The ACF has spikes at lags 1, 2, ..., r and cuts off after lag r, and the PACF dies down; use q = r and p = 0.
The ACF dies down and the PACF has spikes at lags 1, 2, ..., r and cuts off after lag r; use q = 0 and p = r.
The ACF has spikes at lags 1, 2, ...,r and cuts off after lag r, and the PACF has spikes at lags 1, 2, ... ,s and cuts off after lag s; use q = r and p = s.
The ACF contains small autocorrelations at all lags and the PACF contains small autocorrelations at all lags; use q = 0 and p = 0.
The ACF dies down and the PACF dies down; use p = 1 and q = 1.
Seasonal Operators (P and Q)
The previous guidelines apply to P and Q, but only consider autocorrelations at s, 2s, 3s, ... where s is the seasonal period.
9.1.3.2. Model Fitting Parameters
The Model Fitting dialogue requires inputting the following parameters:
Overall Constant: If the overall constant value is non
zero, then Nonseasonal AR Parameters: The nonseasonal AR
Nonseasonal MA Parameters: The nonseasonal MA
Seasonal AR Parameters: The seasonal AR parameter
Seasonal MA Parameters: The seasonal MA parameter
Backforecasts: This is the number of backforecasts
generated before the model is fitted. If this value is zero then no
backforecasts are generated. Maximum Number of Iterations: The maximum number of
iterations is the number of iterations allowed before the model declares non
convergence. The model fitted is given by the following equations: where: ·
·
·
·
·
·
·
The model is fitted by an iterative least squares method.
The output options are accessed by selecting the ARIMA Results from the following dialogue. When a model has been fitted, you will have the following
output options: Model Results: The number of iterations made and the
transformation used are displayed. For each fitted parameter the estimated
value, the standard error and the t‑value are displayed. Parameter Covariance Matrix: A table of the covariance between each fitted parameter is displayed. Parameter Correlation Matrix: A table of the correlation between each fitted parameter is displayed. Plot of Residuals: This displays the residual
values in a table and allows them to be saved back to the data matrix. Residual Autocorrelation: This displays the Autocorrelation Function of the residuals. The residuals should be
unrelated because the model should account for the relationship in the time
series data. If the residuals are unrelated then the autocorrelations of the residuals should be small. The Ljung-Box statistic (see below) is a test of the residual autocorrelations. Ljung-Box Statistic: This displays the
Ljung-Box statistic, the degrees of freedom and the associated chi-square
probabilities at various values up to the lag. The Ljung-Box statistic is a test of the relationship between the residuals. A large value shows the residuals to
be related, and hence the model being inadequate. Example Following the example in Differencing Output
Options, select Fit
Model to select an ARIMA model. On the Model Fitting dialogue enter: ·
1 Overall Constant (0 No, Else Yes) ·
3 Nonseasonal AR Parameters (P) ·
0 Nonseasonal MA Parameters (Q) ·
0 Seasonal AR Parameters (Ps) ·
1 Seasonal MA Parameters (Qs) ·
0 Backforecasts ·
200 Maximum Number of Iterations ·
0.0001 Tolerance On the Model Output Options dialogue check only the Model
Results box to obtain the following results: ARIMA: Fit Model Model Results Transformation: X(t) = (1-B^12) log(Room
Averages) Parameter Estimate Std error t ratio Overall Constant 0.02699 0.01690 1.5976 (AR) P(1) 0.26089 0.06798 3.8380 (AR) P(2) 0.15688 0.06293 2.4929 (AR) P(3) -0.23467 0.07074 -3.3175 (SMA) Qs(1) 0.51389 0.07311 7.0293 Number of Iterations = 148 (Converged) Seasonal Period = 12 The numbers obtained here are not identical to the ones
given in the book, though the general characteristics of the fitted models are
similar. This is due to the highly iterative nature of the estimation process
and the results may differ from one implementation to the other. This dialogue requests the following parameters: Number of Forecasts: This is the number of forecasts
to be generated. Forecast Origin (<0, Offset): The forecast origin
determines the location in the series at which the forecasts will start. If you
enter a positive value, this is used as the forecast origin. If 0 is entered,
then the last point in the series is used as the origin. If a negative value is
entered, then this value is used as an offset from the last point. For
instance, -1 represents the penultimate point as the forecast origin. Confidence Level: The forecasts will be given with
confidence intervals at this level. The value must be greater than 0 and less
than 1. Typical values are 0.95, 0.99 or 0.9. When the above parameters have been specified, the following
output options will be available: Forecast Table: A table of forecasts and confidence
intervals from the given forecast origin is displayed. Character Forecast Plot: A character plot of the original data, lead 1 forecasts and forecasts from the forecast origin are
displayed. Plot of Forecast: A graphical display of the original
data, lead 1 forecasts and forecasts from the forecast origin is generated.
These are the same values as the Character Forecast Plot. Example Following the example in Model Output Options dialogue select Forecasting. On the Forecasting Input
Options dialogue enter: ·
24 Number of Forecasts ·
168 Forecast Origin (<0, Offset from last) ·
0.95 Confidence Level Select the Forecast Table
output option to obtain the following results: ARIMA: Forecasting Forecast Table: Room Averages Forecasts with Origin at
168 Row Forecast Lower 95% Upper 95% 169 840.0521 808.0632 873.3075 170 771.1056 741.7421 801.6315 171 777.0039 747.4158 807.7633 172 872.1992 838.9860 906.7271 173 858.4281 825.7393 892.4109 174 982.0313 944.6357 1020.9071 175 1154.9294 1110.9500 1200.6498 176 1181.2955 1136.3121 1228.0598 177 902.9520 868.5678 938.6974 178 903.5714 869.1636 939.3412 179 783.1983 753.3743 814.2029 180 892.2127 858.2375 927.5330 181 860.6177 823.8597 899.0157 182 794.2411 760.3182 829.6776 183 804.2494 769.8990 840.1324 184 903.2180 864.6406 943.5167 185 888.6314 850.6769 928.2792 186 1015.3943 972.0257 1060.6980 187 1193.5981 1142.6182 1246.8526 188 1220.5764 1168.4441 1275.0345 189 933.1099 893.2557 974.7422 190 933.8563 893.9703 975.5220 191 809.5330 774.9569 845.6517 192 922.2238 882.8345 963.3704
is the overall
constant. 
is the AR operator. 
is the MA operator. 
is the seasonal AR
operator. 
is the seasonal MA
operator. 
is the seasonal white
noise. 
is the white noise and
assumed 
9.1.4.
Model Output Options
9.1.5. Forecasting
9.1.5.1. Forecasting Input
Options
9.1.5.2. Forecasting Output Options
