This example uses the Series J data from Box and Jenkins (1976). First, the input series X
is modeled with a univariate ARMA model. Next, the dependent series Y
is cross-correlated with the input series. Since a model has been fit to X
, both Y
and X
are prewhitened by this model before the sample cross-correlations are computed. Next, a transfer function model is fit with
no structure on the noise term. The residuals from this model are analyzed; then, the full model, transfer function and noise,
is fit to the data.
The following statements read Input Gas Rate
and Output CO
from a gas furnace. (Data values are not shown. The full example including data is in the SAS/ETS sample library.)
title1 'Gas Furnace Data'; title2 '(Box and Jenkins, Series J)'; data seriesj; input x y @@; label x = 'Input Gas Rate' y = 'Output CO2'; datalines; -0.109 53.8 0.000 53.6 0.178 53.5 0.339 53.5 ... more lines ...
The following statements produce Output 7.3.1 through Output 7.3.11:
proc arima data=seriesj; /*--- Look at the input process ----------------------------*/ identify var=x; run; /*--- Fit a model for the input ----------------------------*/ estimate p=3 plot; run; /*--- Crosscorrelation of prewhitened series ---------------*/ identify var=y crosscorr=(x) nlag=12; run; /*--- Fit a simple transfer function - look at residuals ---*/ estimate input=( 3 $ (1,2)/(1) x ); run; /*--- Final Model - look at residuals ----------------------*/ estimate p=2 input=( 3 $ (1,2)/(1) x ); run; quit;
The results of the first IDENTIFY statement for the input series X
are shown in Output 7.3.1. The correlation analysis suggests an AR(3) model.
Output 7.3.1: IDENTIFY Statement Results for X
Gas Furnace Data |
(Box and Jenkins, Series J) |
Name of Variable = x | |
---|---|
Mean of Working Series | -0.05683 |
Standard Deviation | 1.070952 |
Number of Observations | 296 |
Output 7.3.2: IDENTIFY Statement Results for X: Trend and Correlation
The ESTIMATE statement results for the AR(3) model for the input series X
are shown in Output 7.3.3.
Output 7.3.3: Estimates of the AR(3) Model for X
Conditional Least Squares Estimation | |||||
---|---|---|---|---|---|
Parameter | Estimate | Standard Error | t Value | Approx Pr > |t| |
Lag |
MU | -0.12280 | 0.10902 | -1.13 | 0.2609 | 0 |
AR1,1 | 1.97607 | 0.05499 | 35.94 | <.0001 | 1 |
AR1,2 | -1.37499 | 0.09967 | -13.80 | <.0001 | 2 |
AR1,3 | 0.34336 | 0.05502 | 6.24 | <.0001 | 3 |
Constant Estimate | -0.00682 |
---|---|
Variance Estimate | 0.035797 |
Std Error Estimate | 0.1892 |
AIC | -141.667 |
SBC | -126.906 |
Number of Residuals | 296 |
Model for variable x | |
---|---|
Estimated Mean | -0.1228 |
Autoregressive Factors | |
---|---|
Factor 1: | 1 - 1.97607 B**(1) + 1.37499 B**(2) - 0.34336 B**(3) |
The IDENTIFY statement results for the dependent series Y cross-correlated with the input series X
are shown in Output 7.3.4, Output 7.3.5, Output 7.3.6, and Output 7.3.7. Since a model has been fit to X
, both Y
and X
are prewhitened by this model before the sample cross-correlations are computed.
Output 7.3.4: Summary Table: Y Cross-Correlated with X
Correlation of y and x | |
---|---|
Number of Observations | 296 |
Variance of transformed series y | 0.131438 |
Variance of transformed series x | 0.035357 |
Output 7.3.5: Prewhitening Filter
Autoregressive Factors | |
---|---|
Factor 1: | 1 - 1.97607 B**(1) + 1.37499 B**(2) - 0.34336 B**(3) |
Output 7.3.6: IDENTIFY Statement Results for Y: Trend and Correlation
Output 7.3.7: IDENTIFY Statement for Y Cross-Correlated with X
The ESTIMATE statement results for the transfer function model with no structure on the noise term are shown in Output 7.3.8, Output 7.3.9, and Output 7.3.10.
Output 7.3.8: Estimation Output of the First Transfer Function Model
Conditional Least Squares Estimation | |||||||
---|---|---|---|---|---|---|---|
Parameter | Estimate | Standard Error | t Value | Approx Pr > |t| |
Lag | Variable | Shift |
MU | 53.32256 | 0.04926 | 1082.51 | <.0001 | 0 | y | 0 |
NUM1 | -0.56467 | 0.22405 | -2.52 | 0.0123 | 0 | x | 3 |
NUM1,1 | 0.42623 | 0.46472 | 0.92 | 0.3598 | 1 | x | 3 |
NUM1,2 | 0.29914 | 0.35506 | 0.84 | 0.4002 | 2 | x | 3 |
DEN1,1 | 0.60073 | 0.04101 | 14.65 | <.0001 | 1 | x | 3 |
Constant Estimate | 53.32256 |
---|---|
Variance Estimate | 0.702625 |
Std Error Estimate | 0.838227 |
AIC | 728.0754 |
SBC | 746.442 |
Number of Residuals | 291 |
Output 7.3.9: Model Summary: First Transfer Function Model
Model for variable y | |
---|---|
Estimated Intercept | 53.32256 |
Input Number 1 | |
---|---|
Input Variable | x |
Shift | 3 |
Numerator Factors | |
---|---|
Factor 1: | -0.5647 - 0.42623 B**(1) - 0.29914 B**(2) |
Denominator Factors | |
---|---|
Factor 1: | 1 - 0.60073 B**(1) |
Output 7.3.10: Residual Analysis: First Transfer Function Model
The residual correlation analysis suggests an AR(2) model for the noise part of the model. The ESTIMATE statement results for the final transfer function model with AR(2) noise are shown in Output 7.3.11.
Output 7.3.11: Estimation Output of the Final Model
Conditional Least Squares Estimation | |||||||
---|---|---|---|---|---|---|---|
Parameter | Estimate | Standard Error | t Value | Approx Pr > |t| |
Lag | Variable | Shift |
MU | 53.26304 | 0.11929 | 446.48 | <.0001 | 0 | y | 0 |
AR1,1 | 1.53291 | 0.04754 | 32.25 | <.0001 | 1 | y | 0 |
AR1,2 | -0.63297 | 0.05006 | -12.64 | <.0001 | 2 | y | 0 |
NUM1 | -0.53522 | 0.07482 | -7.15 | <.0001 | 0 | x | 3 |
NUM1,1 | 0.37603 | 0.10287 | 3.66 | 0.0003 | 1 | x | 3 |
NUM1,2 | 0.51895 | 0.10783 | 4.81 | <.0001 | 2 | x | 3 |
DEN1,1 | 0.54841 | 0.03822 | 14.35 | <.0001 | 1 | x | 3 |
Constant Estimate | 5.329425 |
---|---|
Variance Estimate | 0.058828 |
Std Error Estimate | 0.242544 |
AIC | 8.292809 |
SBC | 34.00607 |
Number of Residuals | 291 |
Output 7.3.12: Residual Analysis of the Final Model
Output 7.3.13: Model Summary of the Final Model
Model for variable y | |
---|---|
Estimated Intercept | 53.26304 |
Autoregressive Factors | |
---|---|
Factor 1: | 1 - 1.53291 B**(1) + 0.63297 B**(2) |
Input Number 1 | |
---|---|
Input Variable | x |
Shift | 3 |
Numerator Factors | |
---|---|
Factor 1: | -0.5352 - 0.37603 B**(1) - 0.51895 B**(2) |
Denominator Factors | |
---|---|
Factor 1: | 1 - 0.54841 B**(1) |