Deriving Equation for $\gamma_0$

[roadworx]

Hi,

I have the following equation

$$\gamma_0 = \phi^2 \gamma_0 + (1+\Theta^2)\sigma^2 + 2\Theta\phi\sigma^2$$

The answer is

$$\gamma_0 = \frac{(1 + 2\Theta\phi+\Theta^2)}{1-\phi^2}$$

I get how to factorize the numerator. I'm not sure of the denominator. It looks like a geometric series formula, but does this mean $$\gamma_0$$ = $$(1+\Theta^2)\sigma^2 + 2\Theta\phi\sigma^2$$ ?

 

[arildno]

1. A factor of sigma squared is lacking from the numerator.

2. Subtract $\phi^{2}\gamma_{0}$ from both sides of the equation; factorize, and you'll see how the expression is arrived at.