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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.

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.

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 = EstMdl.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`

).