AR(2) Forecast Error Variances to Horizon Two

Problem

Compute the forecast error variances for horizons zero, one, and two of a stationary AR(2) with innovation variance sigma2 and coefficients phi1 and phi2. The psi weights are 1, phi1, and phi1^2 + phi2, so the variances are sigma2 times 1, 1 + phi1^2, and 1 + phi1^2 + (phi1^2 + phi2)^2.

Return the three variances as a list in horizon order.

Hint

Each psi weight comes from the recursion psi[k] = phi1 * psi[k-1] + phi2 * psi[k-2] with psi[0] = 1.

Test cases

case 1

in: [1,0.5,0.2]

expected: [1,1.25,1.4525]

case 2

in: [2,0,0]

expected: [2,2,2]

case 3

in: [1,-0.5,0.25]

expected: [1,1.25,1.5]

case 4

in: [0,1,1]

expected: [0,0,0]

Starter codePython

def ar2_error_variances(sigma2, phi1, phi2):
    # Your code here
    pass

4 test cases · run your code against them in the editor