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Conduct a Wald Test

This example shows how to calculate the required inputs for conducting a Wald test with waldtest. The Wald test compares the fit of a restricted model against an unrestricted model by testing whether the restriction function, evaluated at the unrestricted maximum likelihood estimates (MLEs), is significantly different from zero.

The required inputs for waldtest are a restriction function, the Jacobian of the restriction function evaluated at the unrestricted MLEs, and an estimate of the variance-covariance matrix evaluated at the unrestricted MLEs. This example compares the fit of an AR(1) model against an AR(2) model.

Step 1. Compute the unrestricted MLE.

Obtain the unrestricted MLEs by fitting an AR(2) model (with a Gaussian innovation distribution) to the given data. Assume you have presample observations (y-1,y0) = (9.6249,9.6396)

Y = [10.1591; 10.1675; 10.1957; 10.6558; 10.2243; 10.4429;
     10.5965; 10.3848; 10.3972;  9.9478;  9.6402;  9.7761;
     10.0357; 10.8202; 10.3668; 10.3980; 10.2892;  9.6310;
      9.6318;  9.1378;  9.6318;  9.1378];
Y0 = [9.6249; 9.6396];

model = arima(2,0,0);
[fit,V] = estimate(model,Y,'Y0',Y0);
 
    ARIMA(2,0,0) Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic     PValue  
                _______    _____________    __________    _________

    Constant     2.8802        2.5239         1.1412        0.25379
    AR{1}       0.60623       0.40372         1.5016         0.1332
    AR{2}       0.10631       0.29283        0.36303        0.71658
    Variance    0.12386      0.042598         2.9076      0.0036425

When conducting a Wald test, only the unrestricted model needs to be fit. estimate returns the estimated variance-covariance matrix as an optional output.

Step 2. Compute the Jacobian matrix.

Define the restriction function, and calculate its Jacobian matrix.

For comparing an AR(1) model to an AR(2) model, the restriction function is

r(c,ϕ1,ϕ2,σε2)=ϕ2-0=0.

The Jacobian of the restriction function is

[rcrϕ1rϕ2rσε2]=[0010]

Evaluate the restriction function and Jacobian at the unrestricted MLEs.

r = fit.AR{2};
R = [0 0 1 0];

Step 3. Conduct a Wald test.

Conduct a Wald test to compare the restricted AR(1) model against the unrestricted AR(2) model.

[h,p,Wstat,crit] = waldtest(r,R,V)
h = logical
   0

p = 0.7166
Wstat = 0.1318
crit = 3.8415

The restricted AR(1) model is not rejected in favor of the AR(2) model (h = 0).

See Also

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