PACF Lag Two for MA(1)

Problem

Derive the partial autocorrelation at lag two for an MA(1) process. The lag-one autocorrelation is r1 = theta / (1 + theta^2), the lag-two autocorrelation is zero, and the PACF at lag two follows from these as -r1^2 / (1 - r1^2).

Return 0.0 if the denominator is zero. This nonzero PACF is why an MA(1) model has a slowly decaying partial autocorrelation.

Hint

The PACF at lag two is the coefficient on x[t-2] after removing the effect of x[t-1].

Test cases

case 1

in: [0.5]

expected: -0.19047619047619047

case 2

in: [1]

expected: -0.3333333333333333

case 3

in: [0]

expected: 0

case 4

in: [2]

expected: -0.19047619047619047

Starter codePython

def pacf_lag2_ma1(theta):
    # Your code here
    pass

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