The STATESPACE Procedure

Example 28.1 Series J from Box and Jenkins

This example analyzes the gas furnace data (series J) from Box and Jenkins. (The data are not shown; see Box and Jenkins (1976) for the data.)

First, a model is selected and fit automatically using the following statements.

title1 'Gas Furnace Data';
title2 'Box & Jenkins Series J';
title3 'Automatically Selected Model';

proc statespace data=seriesj cancorr;
   var x y;
run;

The results for the automatically selected model are shown in Output 28.1.1.

Output 28.1.1: Results for Automatically Selected Model

Gas Furnace Data

Box & Jenkins Series J

Automatically Selected Model

The STATESPACE Procedure

Number of Observations	296

Variable	Mean	Standard Error
x	-0.05683	1.072766
y	53.50912	3.202121

Gas Furnace Data

Box & Jenkins Series J

Automatically Selected Model

The STATESPACE Procedure

Information Criterion for Autoregressive Models
Lag=0	Lag=1	Lag=2	Lag=3	Lag=4	Lag=5	Lag=6	Lag=7	Lag=8	Lag=9	Lag=10
651.3862	-1033.57	-1632.96	-1645.12	-1651.52	-1648.91	-1649.34	-1643.15	-1638.56	-1634.8	-1633.59

Schematic Representation of Correlations
Name/Lag	0	1	2	3	4	5	6	7	8	9	10
x	+-	+-	+-	+-	+-	+-	+-	+-	+-	+-	+-
y	-+	-+	-+	-+	-+	-+	-+	-+	-+	-+	-+
+ is > 2std error, - is < -2std error, . is between

Output 28.1.2: Results for Automatically Selected Model

Schematic Representation of Partial Autocorrelations
Name/Lag	1	2	3	4	5	6	7	8	9	10
x	+.	-.	+.	..	..	-.	..	..	..	..
y	-+	--	-.	.+	..	..	..	..	..	.+
+ is > 2std error, - is < -2std error, . is between

Yule-Walker Estimates for Minimum AIC
	Lag=1		Lag=2		Lag=3		Lag=4
	x	y	x	y	x	y	x	y
x	1.925887	-0.00124	-1.20166	0.004224	0.116918	-0.00867	0.104236	0.003268
y	0.050496	1.299793	-0.02046	-0.3277	-0.71182	-0.25701	0.195411	0.133417

Output 28.1.3: Results for Automatically Selected Model

Gas Furnace Data

Box & Jenkins Series J

Automatically Selected Model

The STATESPACE Procedure

Canonical Correlations Analysis

x(T;T)	y(T;T)	x(T+1;T)	Information Criterion	Chi-Square	DF
1	1	0.804883	292.9228	304.7481	8

Output 28.1.4: Results for Automatically Selected Model

Gas Furnace Data

Box & Jenkins Series J

Automatically Selected Model

The STATESPACE Procedure

Selected Statespace Form and Preliminary Estimates

State Vector
x(T;T)	y(T;T)	x(T+1;T)	y(T+1;T)	y(T+2;T)

Estimate of Transition Matrix
0	0	1	0	0
0	0	0	1	0
-0.84718	0.026794	1.711715	-0.05019	0
0	0	0	0	1
-0.19785	0.334274	-0.18174	-1.23557	1.787475

Input Matrix for Innovation
1	0
0	1
1.925887	-0.00124
0.050496	1.299793
0.142421	1.361696

Output 28.1.5: Results for Automatically Selected Model

Variance Matrix for Innovation
0.035274	-0.00734
-0.00734	0.097569

Output 28.1.6: Results for Automatically Selected Model

Gas Furnace Data

Box & Jenkins Series J

Automatically Selected Model

The STATESPACE Procedure

Selected Statespace Form and Fitted Model

State Vector
x(T;T)	y(T;T)	x(T+1;T)	y(T+1;T)	y(T+2;T)

Estimate of Transition Matrix
0	0	1	0	0
0	0	0	1	0
-0.86192	0.030609	1.724235	-0.05483	0
0	0	0	0	1
-0.34839	0.292124	-0.09435	-1.09823	1.671418

Input Matrix for Innovation
1	0
0	1
1.92442	-0.00416
0.015621	1.258495
0.08058	1.353204

Output 28.1.7: Results for Automatically Selected Model

Variance Matrix for Innovation
0.035579	-0.00728
-0.00728	0.095577

Parameter Estimates
Parameter	Estimate	Standard Error	t Value
F(3,1)	-0.86192	0.072961	-11.81
F(3,2)	0.030609	0.026167	1.17
F(3,3)	1.724235	0.061599	27.99
F(3,4)	-0.05483	0.030169	-1.82
F(5,1)	-0.34839	0.135253	-2.58
F(5,2)	0.292124	0.046299	6.31
F(5,3)	-0.09435	0.096527	-0.98
F(5,4)	-1.09823	0.109525	-10.03
F(5,5)	1.671418	0.083737	19.96
G(3,1)	1.924420	0.058162	33.09
G(3,2)	-0.00416	0.035255	-0.12
G(4,1)	0.015621	0.095771	0.16
G(4,2)	1.258495	0.055742	22.58
G(5,1)	0.080580	0.151622	0.53
G(5,2)	1.353204	0.091388	14.81

The two series are believed to have a transfer function relation with the gas rate (variable X) as the input and the CO $_{2}$ concentration (variable Y) as the output. Since the parameter estimates shown in Output 28.1.1 support this kind of model, the model is reestimated with the feedback parameters restricted to 0. The following statements fit the transfer function (no feedback) model.

title3 'Transfer Function Model';
proc statespace data=seriesj printout=none;
   var x y;
   restrict f(3,2)=0 f(3,4)=0
            g(3,2)=0 g(4,1)=0 g(5,1)=0;
run;

The last two pages of the output are shown in Output 28.1.8.

Output 28.1.8: STATESPACE Output for Transfer Function Model

Gas Furnace Data

Box & Jenkins Series J

Transfer Function Model

The STATESPACE Procedure

Selected Statespace Form and Fitted Model

State Vector
x(T;T)	y(T;T)	x(T+1;T)	y(T+1;T)	y(T+2;T)

Estimate of Transition Matrix
0	0	1	0	0
0	0	0	1	0
-0.68882	0	1.598717	0	0
0	0	0	0	1
-0.35944	0.284179	-0.0963	-1.07313	1.650047

Input Matrix for Innovation
1	0
0	1
1.923446	0
0	1.260856
0	1.346332

Output 28.1.9: STATESPACE Output for Transfer Function Model

Variance Matrix for Innovation
0.036995	-0.0072
-0.0072	0.095712

Parameter Estimates
Parameter	Estimate	Standard Error	t Value
F(3,1)	-0.68882	0.050549	-13.63
F(3,3)	1.598717	0.050924	31.39
F(5,1)	-0.35944	0.229044	-1.57
F(5,2)	0.284179	0.096944	2.93
F(5,3)	-0.09630	0.140876	-0.68
F(5,4)	-1.07313	0.250385	-4.29
F(5,5)	1.650047	0.188533	8.75
G(3,1)	1.923446	0.056328	34.15
G(4,2)	1.260856	0.056464	22.33
G(5,2)	1.346332	0.091086	14.78