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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,y0) = (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,ϕ1,ϕ2,σε2)=ϕ2-0=0.

The Jacobian of the restriction function is

[∂r∂c∂r∂ϕ1∂r∂ϕ2∂r∂σε2]=[0010]

Evaluate the restriction function and Jacobian at the unrestricted MLEs.

r = EstMdl.AR{2};

R = [0 0 1 0];

Conduct 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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