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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 ( $y_{-1},y_0$ ) = (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];

model = arima(2,0,0);
[fit,V] = estimate(model,Y,'Y0',Y0);
 
    ARIMA(2,0,0) Model:
    --------------------
    Conditional Probability Distribution: Gaussian

                                  Standard          t     
     Parameter       Value          Error       Statistic 
    -----------   -----------   ------------   -----------
     Constant        2.88021       2.52387        1.14119
        AR{1}       0.606229       0.40372        1.50161
        AR{2}       0.106308      0.292833       0.363034
     Variance       0.123855     0.0425975        2.90756

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

$$r(c,\phi_1,\phi_2,\sigma_\varepsilon^2) = \phi_2 - 0 = 0.$$

The Jacobian of the restriction function is

$$\left[\matrix{\frac{\partial r}{\partial c} & \frac{\partial r}{\partial \phi_1} & \frac{\partial r}{\partial \phi_2} & \frac{\partial r}{\partial \sigma_\varepsilon^2}}\right] = \left[\matrix{0 &0 &1 &0}\right]$$

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 =

     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).

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