PROC VARMAX: Vector Autoregressive Process with Exogenous Variables

The VARMAX Procedure

Vector Autoregressive Process with Exogenous Variables

A VAR process can be affected by other observable variables that are determined outside the system of interest. Such variables are called exogenous (independent) variables. Exogenous variables can be stochastic or nonstochastic. The process can also be affected by the lags of exogenous variables. A model used to describe this process is called a VARX( $\text{[math]}$ , $\text{[math]}$ ) model.

The VARX( $\text{[math]}$ , $\text{[math]}$ ) model is written as

$\text{[math]}$

where $\text{[math]}$ is an $\text{[math]}$ -dimensional time series vector and $\text{[math]}$ is a $\text{[math]}$ matrix.

For example, a VARX(1,0) model is

$\text{[math]}$

where $\text{[math]}$ and $\text{[math]}$ .

The following statements fit the VARX(1,0) model to the given data:

   data grunfeld;
      input year y1 y2 y3 x1 x2 x3;
      label y1='Gross Investment GE'
            y2='Capital Stock Lagged GE'
            y3='Value of Outstanding Shares GE Lagged'
            x1='Gross Investment W'
            x2='Capital Stock Lagged W'
            x3='Value of Outstanding Shares Lagged W';
   datalines;
   1935  33.1 1170.6  97.8 12.93  191.5   1.8
   1936  45.0 2015.8 104.4 25.90  516.0    .8
   1937  77.2 2803.3 118.0 35.05  729.0   7.4
   
   ... more lines ...

   /*--- Vector Autoregressive Process with Exogenous Variables ---*/
   
   proc varmax data=grunfeld;
      model y1-y3 = x1 x2 / p=1 lagmax=5
                            printform=univariate
                            print=(impulsx=(all) estimates);
   run;

The VARMAX procedure output is shown in Figure 30.18 through Figure 30.20.

Figure 30.18 shows the descriptive statistics for the dependent (endogenous) and independent (exogenous) variables with labels.

Figure 30.18 Descriptive Statistics for the VARX(1, 0) Model

The VARMAX Procedure

Number of Observations	20
Number of Pairwise Missing	0

Simple Summary Statistics
Variable	Type	N	Mean	Standard Deviation	Min	Max	Label
y1	Dependent	20	102.29000	48.58450	33.10000	189.60000	Gross Investment GE
y2	Dependent	20	1941.32500	413.84329	1170.60000	2803.30000	Capital Stock Lagged GE
y3	Dependent	20	400.16000	250.61885	97.80000	888.90000	Value of Outstanding Shares GE Lagged
x1	Independent	20	42.89150	19.11019	12.93000	90.08000	Gross Investment W
x2	Independent	20	670.91000	222.39193	191.50000	1193.50000	Capital Stock Lagged W

Figure 30.19 shows the parameter estimates for the constant, the lag zero coefficients of exogenous variables, and the lag one AR coefficients. From the schematic representation of parameter estimates, the significance of the parameter estimates can be easily verified. The symbol "C" means the constant and "XL0" means the lag zero coefficients of exogenous variables.

Figure 30.19 Parameter Estimates for the VARX(1, 0) Model

The VARMAX Procedure

Type of Model	VARX(1,0)
Estimation Method	Least Squares Estimation

Constant
Variable	Constant
y1	-12.01279
y2	702.08673
y3	-22.42110

XLag
Lag	Variable	x1	x2
0	y1	1.69281	-0.00859
	y2	-6.09850	2.57980
	y3	-0.02317	-0.01274

AR
Lag	Variable	y1	y2	y3
1	y1	0.23699	0.00763	0.02941
	y2	-2.46656	0.16379	-0.84090
	y3	0.95116	0.00224	0.93801

Schematic Representation
Variable/Lag	C	XL0	AR1
y1	.	+.	...
y2	+	.+	...
y3	-	..	+.+
+ is > 2std error, - is < -2std error, . is between, * is N/A

Figure 30.20 shows the parameter estimates and their significance.

Figure 30.20 Parameter Estimates for the VARX(1, 0) Model Continued

Model Parameter Estimates
Equation	Parameter	Estimate	Standard Error	t Value	Pr > \|t\|	Variable
y1	CONST1	-12.01279	27.47108	-0.44	0.6691	1
	XL0_1_1	1.69281	0.54395	3.11	0.0083	x1(t)
	XL0_1_2	-0.00859	0.05361	-0.16	0.8752	x2(t)
	AR1_1_1	0.23699	0.20668	1.15	0.2722	y1(t-1)
	AR1_1_2	0.00763	0.01627	0.47	0.6470	y2(t-1)
	AR1_1_3	0.02941	0.04852	0.61	0.5548	y3(t-1)
y2	CONST2	702.08673	256.48046	2.74	0.0169	1
	XL0_2_1	-6.09850	5.07849	-1.20	0.2512	x1(t)
	XL0_2_2	2.57980	0.50056	5.15	0.0002	x2(t)
	AR1_2_1	-2.46656	1.92967	-1.28	0.2235	y1(t-1)
	AR1_2_2	0.16379	0.15193	1.08	0.3006	y2(t-1)
	AR1_2_3	-0.84090	0.45304	-1.86	0.0862	y3(t-1)
y3	CONST3	-22.42110	10.31166	-2.17	0.0487	1
	XL0_3_1	-0.02317	0.20418	-0.11	0.9114	x1(t)
	XL0_3_2	-0.01274	0.02012	-0.63	0.5377	x2(t)
	AR1_3_1	0.95116	0.07758	12.26	0.0001	y1(t-1)
	AR1_3_2	0.00224	0.00611	0.37	0.7201	y2(t-1)
	AR1_3_3	0.93801	0.01821	51.50	0.0001	y3(t-1)

The fitted model is given as

	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

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