Introduction
This example estimates a VARMA(3,0,0) model following Example 4.1 found on page 86 of Reinsel, G.C. Elements of Multivariate Time Series Analysis. The dataset "mink.csv" contains the data.
Step 1: Load data
This example loads the data using the GAUSS function loadd. The function loadd utilizes the GAUSS formula string syntax and allows users to load and transform specific variables directly from the dataset.
new;
library tsmt;
// Load data
// Create file name with full path
fname = getGAUSSHome() $+ "pkgs/tsmt/examples/mink.csv";
// Load two variables from dataset
y = loadd(fname, "LogMink + LogMusk");
// Difference the data
y = vmdiffmt(y, 1);Step 2: Estimate The Model
Because this model is strictly a VAR model, the only additional input after the data matrix is the AR order.
// Estimate the parameters of the VAR(3) model
// and print diagnostic information
call varmaFit(y, 3);
Step 3: Output
The coefficient estimates from varmaFit read
Phi Plane [1,.,.] -0.065819 0.56348 -0.55990 0.25347 Plane [2,.,.] 0.16426 -0.27348 -0.31914 -0.30793 Plane [3,.,.] -0.21124 0.34758 -0.45904 0.32457
In addition to estimating coefficients, varmaFit provides a number of diagnostic tests. The tests include unit root and cointegration tests:
Augmented Dickey-Fuller UNIT ROOT Test for Y1
Critical Values
ADF Stat 1% 5% 10% 90% 95% 99%
No Intercept -5.2278 -2.5328 -1.9498 -1.6266 0.9152 1.3168 2.1179
Intercept -5.1774 -3.5663 -2.9370 -2.6152 -0.4393 -0.0499 0.6942
Intercept and Time Trend -5.2981 -4.0892 -3.4615 -3.1709 -1.2584 -0.9195 -0.2986
Augmented Dickey-Fuller UNIT ROOT Test for Y2
Critical Values
ADF Stat 1% 5% 10% 90% 95% 99%
No Intercept -4.7720 -2.5328 -1.9498 -1.6266 0.9152 1.3168 2.1179
Intercept -4.7352 -3.5663 -2.9370 -2.6152 -0.4393 -0.0499 0.6942
Intercept and Time Trend -4.6155 -4.0892 -3.4615 -3.1709 -1.2584 -0.9195 -0.2986
Phillips-Perron UNIT ROOT Test for Y1
PPt 1% 5%
No Intercept -7.2621 -2.5328 -1.9498
Intercept -6.9186 -3.5663 -2.9370
Intercept and Time Trend -6.9497 -4.0892 -3.4615
Phillips-Perron UNIT ROOT Test for Y2
PPt 1% 5%
No Intercept -6.3499 -2.5328 -1.9498
Intercept -5.9127 -3.5663 -2.9370
Intercept and Time Trend -5.8505 -4.0892 -3.4615
Augmented Dickey-Fuller COINTEGRATION Test for Y1 Y2
Critical Values
ADF Stat 1% 5% 10% 90% 95% 99%
No Intercept -5.0448 -3.4003 -2.8198 -2.4901 -0.2841 0.1628 0.9912
Intercept -5.0375 -4.0246 -3.4040 -3.0890 -0.9988 -0.6383 0.0929
Intercept and Time Trend -5.1691 -4.5041 -3.9157 -3.6062 -1.6464 -1.3413 -0.6750
Johansen's Trace and Maximum Eigenvalue Statistics. r = # of CI Equations
Critical Values
r Trace Max. Eig 1% 5% 10% 90%
No Intercept 0 65.5906 46.3981
1 19.1925 19.1925 1.0524 1.7046 2.1927 9.3918
Intercept 0 65.6102 46.4140
1 19.1962 19.1962 2.2515 3.3599 4.0975 12.8635
Intercept and Time Trend 0 66.2392 46.3986
1 19.8406 19.8406 4.0389 5.3796 6.1879 16.1762