Conduct Wald Test

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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.

Compute 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 ( $y_{- 1}, y_{0}$ ) = (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];

Mdl = arima(2,0,0);
[EstMdl,V] = estimate(Mdl,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.

Compute 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

$r (c, ϕ_{1}, ϕ_{2}, σ_{ε}^{2}) = ϕ_{2} - 0 = 0 .$

The Jacobian of the restriction function is

$[\begin{array}{c} \frac{\partial r}{\partial c} & \frac{\partial r}{\partial ϕ_{1}} & \frac{\partial r}{\partial ϕ_{2}} & \frac{\partial r}{\partial σ_{ε}^{2}} \end{array}] = [\begin{array}{c} 0 & 0 & 1 & 0 \end{array}]$

Evaluate the restriction function and Jacobian at the unrestricted MLEs.

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