Let denote a dimensional time series vector of random variables of interest. The thorder VAR process is written as

where the is a vector white noise process with such that , , and for ; is a constant vector and is a matrix.
Analyzing and modeling the series jointly enables you to understand the dynamic relationships over time among the series and to improve the accuracy of forecasts for individual series by using the additional information available from the related series and their forecasts.
Consider the firstorder stationary bivariate vector autoregressive model

The following IML procedure statements simulate a bivariate vector time series from this model to provide test data for the VARMAX procedure:
proc iml; sig = {1.0 0.5, 0.5 1.25}; phi = {1.2 0.5, 0.6 0.3}; /* simulate the vector time series */ call varmasim(y,phi) sigma = sig n = 100 seed = 34657; cn = {'y1' 'y2'}; create simul1 from y[colname=cn]; append from y; quit;
The following statements plot the simulated vector time series shown in Figure 36.1:
data simul1; set simul1; date = intnx( 'year', '01jan1900'd, _n_1 ); format date year4.; run; ods graphics on; proc timeseries data=simul1 vectorplot=series; id date interval=year; var y1 y2; run;
Figure 36.1: Plot of Generated Data Process
The following statements fit a VAR(1) model to the simulated data. First, you specify the input data set in the PROC VARMAX statement. Then, you use the MODEL statement to designate the dependent variables, and . To estimate a VAR model with mean zero, you specify the order of the autoregressive model with the P= option and the NOINT option. The MODEL statement fits the model to the data and prints parameter estimates and their significance. The PRINT=ESTIMATES option prints the matrix form of parameter estimates, and the PRINT=DIAGNOSE option prints various diagnostic tests. The LAGMAX=3 option is used to print the output for the residual diagnostic checks.
To output the forecasts to a data set, you specify the OUTPUT statement with the OUT= option. If you want to forecast five steps ahead, you use the LEAD=5 option. The ID statement specifies the yearly interval between observations and provides the Time column in the forecast output.
The VARMAX procedure output is shown in Figure 36.2 through Figure 36.10.
/* Vector Autoregressive Model */ proc varmax data=simul1; id date interval=year; model y1 y2 / p=1 noint lagmax=3 print=(estimates diagnose); output out=for lead=5; run;
Figure 36.2: Descriptive Statistics
Number of Observations  100 

Number of Pairwise Missing  0 
Simple Summary Statistics  

Variable  Type  N  Mean  Standard Deviation 
Min  Max 
y1  Dependent  100  0.21653  2.78210  4.75826  8.37032 
y2  Dependent  100  0.16905  2.58184  6.04718  9.58487 
The VARMAX procedure first displays descriptive statistics. The Type column specifies that the variables are dependent variables. The column N stands for the number of nonmissing observations.
Figure 36.3 shows the type and the estimation method of the fitted model for the simulated data. It also shows the AR coefficient matrix in terms of lag 1, the parameter estimates, and their significance, which can indicate how well the model fits the data.
The second table schematically represents the parameter estimates and allows for easy verification of their significance in matrix form.
In the last table, the first column gives the lefthandside variable of the equation; the second column is the parameter name AR, which indicates the ()th element of the lag autoregressive coefficient; the last column is the regressor that corresponds to the displayed parameter.
Figure 36.3: Model Type and Parameter Estimates
Type of Model  VAR(1) 

Estimation Method  Least Squares Estimation 
AR  

Lag  Variable  y1  y2 
1  y1  1.15977  0.51058 
y2  0.54634  0.38499 
Schematic Representation 


Variable/Lag  AR1 
y1  + 
y2  ++ 
+ is > 2*std error,  is < 2*std error, . is between, * is N/A 
Model Parameter Estimates  

Equation  Parameter  Estimate  Standard Error 
t Value  Pr > t  Variable 
y1  AR1_1_1  1.15977  0.05508  21.06  0.0001  y1(t1) 
AR1_1_2  0.51058  0.05898  8.66  0.0001  y2(t1)  
y2  AR1_2_1  0.54634  0.05779  9.45  0.0001  y1(t1) 
AR1_2_2  0.38499  0.06188  6.22  0.0001  y2(t1) 
The fitted VAR(1) model with estimated standard errors in parentheses is given as

Clearly, all parameter estimates in the coefficient matrix are significant.
The model can also be written as two univariate regression equations.






The table in Figure 36.4 shows the innovation covariance matrix estimates and the various information criteria results. The smaller value of information criteria fits the data better when it is compared to other models. The variable names in the covariance matrix are printed for convenience; means the innovation for , and means the innovation for .
Figure 36.4: Innovation Covariance Estimates and Information Criteria
Covariances of Innovations  

Variable  y1  y2 
y1  1.28875  0.39751 
y2  0.39751  1.41839 
Information Criteria  

AICC  0.554443 
HQC  0.595201 
AIC  0.552777 
SBC  0.65763 
FPEC  1.738092 
Figure 36.5 shows the cross covariances of the residuals. The values of the lag zero are slightly different from Figure 36.4 due to the different degrees of freedom.
Figure 36.5: Multivariate Diagnostic Checks
Cross Covariances of Residuals  

Lag  Variable  y1  y2 
0  y1  1.26271  0.38948 
y2  0.38948  1.38974  
1  y1  0.03121  0.05675 
y2  0.04646  0.05398  
2  y1  0.08134  0.10599 
y2  0.03482  0.01549  
3  y1  0.01644  0.11734 
y2  0.00609  0.11414 
Figure 36.6 and Figure 36.7 show tests for white noise residuals. The output shows that you cannot reject the null hypothesis that the residuals are uncorrelated.
Figure 36.6: Multivariate Diagnostic Checks Continued
Cross Correlations of Residuals  

Lag  Variable  y1  y2 
0  y1  1.00000  0.29401 
y2  0.29401  1.00000  
1  y1  0.02472  0.04284 
y2  0.03507  0.03884  
2  y1  0.06442  0.08001 
y2  0.02628  0.01115  
3  y1  0.01302  0.08858 
y2  0.00460  0.08213 
Schematic Representation of Cross Correlations of Residuals 


Variable/Lag  0  1  2  3 
y1  ++  ..  ..  .. 
y2  ++  ..  ..  .. 
+ is > 2*std error,  is < 2*std error, . is between 
Figure 36.7: Multivariate Diagnostic Checks Continued
Portmanteau Test for Cross Correlations of Residuals 


Up To Lag  DF  ChiSquare  Pr > ChiSq 
2  4  1.58  0.8124 
3  8  2.78  0.9473 
The VARMAX procedure provides diagnostic checks for the univariate form of the equations. The table in Figure 36.8 describes how well each univariate equation fits the data. From two univariate regression equations in Figure 36.3, the values of in the second column are 0.84 and 0.80 for each equation. The standard deviations in the third column are the square roots of the diagonal elements of the covariance matrix from Figure 36.4. The statistics are in the fourth column for hypotheses to test and , respectively, where is the th element of the matrix . The last column shows the values of the statistics. The results show that each univariate model is significant.
Figure 36.8: Univariate Diagnostic Checks
Univariate Model ANOVA Diagnostics  

Variable  RSquare  Standard Deviation 
F Value  Pr > F 
y1  0.8351  1.13523  491.25  <.0001 
y2  0.7906  1.19096  366.29  <.0001 
The check for white noise residuals in terms of the univariate equation is shown in Figure 36.9. This output contains information that indicates whether the residuals are correlated and heteroscedastic. In the first table, the second column contains the DurbinWatson test statistics to test the null hypothesis that the residuals are uncorrelated. The third and fourth columns show the JarqueBera normality test statistics and their values to test the null hypothesis that the residuals have normality. The last two columns show statistics and their values for ARCH(1) disturbances to test the null hypothesis that the residuals have equal covariances. The second table includes statistics and their values for AR(1), AR(1,2), AR(1,2,3) and AR(1,2,3,4) models of residuals to test the null hypothesis that the residuals are uncorrelated.
Figure 36.9: Univariate Diagnostic Checks Continued
Univariate Model White Noise Diagnostics  

Variable  Durbin Watson 
Normality  ARCH  
ChiSquare  Pr > ChiSq  F Value  Pr > F  
y1  1.94534  3.56  0.1686  0.13  0.7199 
y2  2.06276  5.42  0.0667  2.10  0.1503 
Univariate Model AR Diagnostics  

Variable  AR1  AR2  AR3  AR4  
F Value  Pr > F  F Value  Pr > F  F Value  Pr > F  F Value  Pr > F  
y1  0.02  0.8980  0.14  0.8662  0.09  0.9629  0.82  0.5164 
y2  0.52  0.4709  0.41  0.6650  0.32  0.8136  0.32  0.8664 
The table in Figure 36.10 gives forecasts, their prediction errors, and 95% confidence limits. See the section Forecasting for details.
Figure 36.10: Forecasts
Forecasts  

Variable  Obs  Time  Forecast  Standard Error 
95% Confidence Limits  
y1  101  2000  3.59212  1.13523  5.81713  1.36711 
102  2001  3.09448  1.70915  6.44435  0.25539  
103  2002  2.17433  2.14472  6.37792  2.02925  
104  2003  1.11395  2.43166  5.87992  3.65203  
105  2004  0.14342  2.58740  5.21463  4.92779  
y2  101  2000  2.09873  1.19096  4.43298  0.23551 
102  2001  2.77050  1.47666  5.66469  0.12369  
103  2002  2.75724  1.74212  6.17173  0.65725  
104  2003  2.24943  2.01925  6.20709  1.70823  
105  2004  1.47460  2.25169  5.88782  2.93863 