Transform the following ARMA(1,1) model into a MA(∞) model. What is the requiredcondition on the parameters 𝜙𝜙 and 𝜃𝜃 for the transformation to work? (8 points)𝑟𝑟𝑡𝑡 = 𝜙𝜙𝑟𝑟𝑡𝑡−1 + 𝑎𝑎𝑡𝑡 + 𝜃𝜃𝑎𝑎𝑡𝑡−1.

The ARMA(1,1) model is given by:

rt = φrt-1 + at + θat-1

We want to transform this into a MA(∞) model. The MA(∞) model is a moving average model of infinite order, which can be represented as:

rt = Σ (ψi * at-i) from i=0 to ∞

Step 1: Substitute the AR part of the ARMA model into the model itself iteratively.

rt = φrt-1 + at + θat-1 = φ(φrt-2 + at-1 + θat-2) + at + θat-1 = φ^2rt-2 + φat-1 + φθat-2 + at + θat-1 = φ^3rt-3 + φ^2at-2 + φ^2θat-3 + φat-1 + φθat-2 + at + θat-1 = ...

Step 2: Continue this process indefinitely.

rt = Σ (φ^i * at-i) + Σ (φ^(i-1) * θ * at-i) from i=1 to ∞

Step 3: Combine the two summations.

rt = Σ [φ^i * at-i + φ^(i-1) * θ * at-i] from i=1 to ∞

Step 4: Recognize this as a MA(∞) model.

rt = Σ (ψi * at-i) from i=0 to ∞

where ψi = φ^i + φ^(i-1) * θ

The required condition on the parameters φ and θ for the transformation to work is that the absolute value of φ is less than 1. This is because for the infinite series to converge (which is necessary for the model to make sense), the terms φ^i * at-i and φ^(i-1) * θ * at-i must get smaller and smaller as i increases. This will only happen if -1 < φ < 1.