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The STATESPACE Procedure

Example 18.1: Series J from Box and Jenkins

This example analyzes the gas furnace data (series J) from Box and Jenkins. (The data are not shown. Refer to 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 18.1.1.

Output 18.1.1: 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 > 2*std error,  - is < -2*std error,  . is between

 
Gas Furnace Data
Box & Jenkins Series J
Automatically Selected Model

The STATESPACE Procedure

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

 
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
 
x(T;T) y(T;T) x(T+1;T) y(T+1;T) Information Criterion Chi Square DF
1 1 0.906681 0.607529 122.3358 134.7237 7
 
x(T;T) y(T;T) x(T+1;T) y(T+1;T) x(T+2;T) Information Criterion Chi Square DF
1 1 0.909434 0.610278 0.186274 -1.54701 10.34705 6
 
x(T;T) y(T;T) x(T+1;T) y(T+1;T) y(T+2;T) Information Criterion Chi Square DF
1 1 0.91014 0.618937 0.206823 0.940392 12.80924 6
 
x(T;T) y(T;T) x(T+1;T) y(T+1;T) y(T+2;T) y(T+3;T) Information Criterion Chi Square DF
1 1 0.912963 0.628785 0.226598 0.083258 -7.94103 2.041584 5

 
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

 
Gas Furnace Data
Box & Jenkins Series J
Automatically Selected Model

The STATESPACE Procedure
Selected Statespace Form and Preliminary Estimates

Variance Matrix for Innovation
0.035274 -0.00734
-0.00734 0.097569

 
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

 
Gas Furnace Data
Box & Jenkins Series J
Automatically Selected Model

The STATESPACE Procedure
Selected Statespace Form and Fitted 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 CO2 concentration (variable Y) as the output. Since the parameter estimates shown in Output 18.1.1 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 2 pages of the output are shown in Output 18.1.2.

Output 18.1.2: 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

 
Gas Furnace Data
Box & Jenkins Series J
Transfer Function Model

The STATESPACE Procedure
Selected Statespace Form and Fitted 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

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