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

6.4.4.10.

Box-Jenkins Analysis on Seasonal Data

Example with the SEMPLOT Software for a Seasonal Time Series

A computer software package is needed to do a Box-Jenkins time series analysis for seasonal data. The computer output on this page will illustrate sample output from a Box-Jenkins analysis using the SEMSTAT statisical software program. It analyzes the series G 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.

Model Identification Section

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

? bookg.bj 



MAX	MIN	MEAN	VARIANCE	NO. DATA
622.0000	104.0000	280.2986	14391.9170	144

Do you wish to make transformations? y/n	y
The following transformations are available:   1 Square root  2 Cube root  3 Natural log  4 Natural log log  5 Common log  6 Exponentiation  7 Reciprocal  8 Square root of Reciprocal  9 Normalizing (X-Xbar)/Standard deviation  10 Coding (X-Constant 1)/Constant 2

Enter your selection, by number:	3
Statistics of Transformed series:
Mean:     5.542	Variance     0.195
Input order of difference or 0:	1
Input period of seasonality (2-12) or 0:	12
Input order of seasonal difference or 0:	0
Statistics of Differenced series:
Mean:     0.009	Variance     0.011
Time Series: bookg.bj.
Regular difference: 1	Seasonal Difference: 0

Autocorrelation Function for the first 36 lags 

 1   0.19975     13   0.21509      25   0.19726
 2  -0.12010     14  -0.13955      26  -0.12388
 3  -0.15077     15  -0.11600      27  -0.10270
 4  -0.32207     16  -0.27894      28  -0.21099
 5  -0.08397     17  -0.05171      29  -0.06536
 6   0.02578     18   0.01246      30   0.01573
 7  -0.11096     19  -0.11436      31  -0.11537
 8  -0.33672     20  -0.33717      32  -0.28926
 9  -0.11559     21  -0.10739      33  -0.12688
10  -0.10927     22  -0.07521      34  -0.04071
11   0.20585     23   0.19948      35   0.14741
12   0.84143     24   0.73692      36   0.65744
   

Analyzing Autocorrelation Plot for Seasonality

If you observe very large autocorrelations at lags spaced n periods apart, for example at lags 12 and 24, then there is evidence of periodicity. That effect should be removed, since the objective of the identification stage is to reduce the autocorrelations throughout. So if simple differencing was not enough, try seasonal differencing at a selected period. In the above case, the period is 12. It could, of course, be any value, such as 4 or 6.

The number of seasonal terms is rarely more than 1. If you know the shape of your forecast function, or you wish to assign a particular shape to the forecast function, you can select the appropriate number of terms for seasonal AR or seasonal MA models.

The book by Box and Jenkins, Time Series Analysis Forecasting and Control (the later edition is Box, Jenkins and Reinsel, 1994) has a discussion on these forecast functions on pages 326 - 328. Again, if you have only a faint notion, but you do know that there was a trend upwards before differencing, pick a seasonal MA term and see what comes out in the diagnostics.

The results after taking a seasonal difference look good!

Model Fitting Section

Now we can proceed to the estimation, diagnostics and forecasting routines. The following program is again executed from a menu and issues the following flow of output:

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

? bookg.bj 

MAX	MIN	MEAN	VARIANCE	NO. DATA
622.0000	104.0000	280.2986	14391.9170	144

Do you wish to make transformations? y/n	y (we selected a square root transformation because a closer inspection of the plot revealed increasing variances over time)
Statistics of Transformed series:
Mean:     5.542	Variance     0.195
Input order of difference or 0:	1
Input NUMBER of AR terms:	Blank defaults to 0
Input NUMBER of MA terms:	1
Input period of seasonality (2-12) or 0:	12
Input order of seasonal difference or 0:	1
Input NUMBER of seasonal AR terms:	Blank defaults to 0
Input NUMBER of seasonal MA terms:	1
Statistics of Differenced series:
Mean:     0.000	Variance     0.002
Pass 1 SS:	0.1894
Pass 2 SS:	0.1821
Pass 3 SS:	0.1819

Estimation is finished after 3 Marquardt iterations. 

Output Section

MA estimates with Standard Errors
Theta 1 :    0.3765    0.0811 

Seasonal MA estimates with Standard Errors
Theta 1 :    0.5677    0.0775 

Original Variance            :             0.0021
Residual Variance   (MSE)    :      0.0014 
Coefficient of Determination :     33.9383 

AIC criteria ln(SSE)+2k/n    :      -1.4959 
BIC criteria ln(SSE)+ln(n)k/n:     -1.1865 

k = p + q + P + Q + d + sD = number of estimates + order of regular difference + product of period of seasonality and seasonal difference. 

n is the total number of observations.
In this problem k and n are:  15   144

***** Test on randomness of Residuals *****
The Box-Ljung  value      =   28.4219
The Box-Pierce value      =   24.0967
with degrees of freedom  =   30 
The 95th percentile           =   43.76809 

Hypothesis of randomness accepted. 

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%.

Next Period	Lower	Forecast	Upper
145	423.4257	450.1975	478.6620
146	382.9274	411.6180	442.4583
147	407.2839	441.9742	479.6191
148	437.8781	479.2293	524.4855
149	444.3902	490.1471	540.6153
150	491.0981	545.5740	606.0927
151	583.6627	652.7856	730.0948
152	553.5620	623.0632	701.2905
153	458.0291	518.6510	587.2965
154	417.4242	475.3956	541.4181
155	350.7556	401.6725	459.9805
156	382.3264	440.1473	506.7128