6.4.4.9. Example of Univariate Box-Jenkins Analysis

6. Process or Product Monitoring and Control
6.4. Introduction to Time Series Analysis
6.4.4. Univariate Time Series Models

6.4.4.9. Example of Univariate Box-Jenkins Analysis

Example with the SEMPLOT Software

A computer software package is needed to do a Box-Jenkins time series analysis. The computer output on this page will illustrate sample output from a Box-Jenkins analysis using the SEMSTAT statistical software program. It analyzes the series F data set in the Box, Jenkins and Reinsel text.

The graph of the data and the resulting forecasts after fitting a model are portrayed below.

Output from other software programs will be similar, but not identical.

Model Identification Section

With the SEMSTAT program, you start by entering a valid file name or you can select a file extension to search for files of particular interest. In this program, if you press the enter key, ALL file names in the directory are displayed.

Enter FILESPEC or EXTENSION (1-3 letters): To quit, press F10.

MAX	MIN	MEAN	VARIANCE	NO. DATA
80.0000	23.0000	51.7086	141.8238	70

Do you wish to make transformations? y/n	n
Input order of difference or 0:	0
Input period of seasonality (2-12) or 0:	0

Time Series: bookf.bj. Regular difference: 0 Seasonal Difference: 0

Autocorrelation Function for the first 35 lags

    0   1.0000       12   -0.0688       24  -0.0731
    1  -0.3899       13    0.1480       25  -0.0195
    2   0.3044       14    0.0358       26   0.0415
    3  -0.1656       15   -0.0067       27  -0.0221
    4   0.0707       16    0.1730       28   0.0889
    5  -0.0970       17   -0.7013       29   0.0162
    6  -0.0471       18    0.0200       30   0.0039
    7   0.0354       19   -0.0473       31   0.0046
    8  -0.0435       20    0.0161       32  -0.0248
    9  -0.0048       21    0.0223       33  -0.0259
   10   0.0144       22   -0.0787       34  -0.0629
   11   0.1099       23   -0.0096       35   0.0261

Model Fitting Section

Enter FILESPEC or EXTENSION (1-3 letters): To quit, press F10.

MAX	MIN	MEAN	VARIANCE	NO. DATA
80.0000	23.0000	51.7086	141.8238	70

Do you wish to make transformations? y/n	n
Input order of difference or 0:	0
Input NUMBER of AR terms:	2
Input NUMBER of MA terms:	0
Input period of seasonality (2-12) or 0:	0

*********** OUTPUT SECTION ***********

    AR estimates with Standard Errors
    Phi 1   :   -0.3397    0.1224
    Phi 2   :    0.1904    0.1223

    Original Variance           :    141.8238
    Residual Variance           :    110.8236
    Coefficient of Determination:     21.8582

    ***** Test on randomness of Residuals *****
    The Chi-Square value       =  11.7034
    with degrees of freedom    =  23
    The 95th percentile        =  35.16596

    Hypothesis of randomness accepted.
    Press any key to proceed to the forecasting section.

Forecasting Section

   ---------------------------------------------------
             FORECASTING SECTION
   ---------------------------------------------------

   Defaults are obtained by pressing the enter key, without input.
   Default for number of periods ahead from last period = 6.
   Default for the confidence band around the forecast  = 90%.

   How many periods ahead to forecast? (9999 to quit...):
   Enter confidence level for the forecast limits      :

   90 Percent Confidence limits
   Next        Lower     Forecast         Upper
     71      43.8734      61.1930       78.5706
     72      24.0239      42.3156       60.6074
     73      36.9575      56.0006       75.0438
     74      28.4916      47.7573       67.0229
     75      33.7942      53.1634       72.5326
     76      30.3487      49.7573       69.1658