# Documentation

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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 () = (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} 0.106309 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

The Jacobian of the restriction function is

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`).

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