Note: This page has been translated by MathWorks. Click here to see

To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

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.

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 (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.

Define the restriction function, and calculate its Jacobian matrix.

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

$$r(c,{\varphi}_{1},{\varphi}_{2},{\sigma}_{\epsilon}^{2})={\varphi}_{2}-0=0.$$

The Jacobian of the restriction function is

$$\left[\begin{array}{cccc}\frac{\partial r}{\partial c}& \frac{\partial r}{\partial {\varphi}_{1}}& \frac{\partial r}{\partial {\varphi}_{2}}& \frac{\partial r}{\partial {\sigma}_{\epsilon}^{2}}\end{array}\right]=\left[\begin{array}{cccc}0& 0& 1& 0\end{array}\right]$$

Evaluate the restriction function and Jacobian at the unrestricted MLEs.

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

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`

).