Explanation

For an MA(1) model, the relationship between the first-order autocorrelation coefficient (ρ₁) and the MA parameter (θ₁) is given by:

$$\rho_1 = \frac{\theta_1}{1 + \theta_1^2}$$

Given ρ₁ = -0.45, we can set up the equation:

$$-0.45 = \frac{\theta_1}{1 + \theta_1^2}$$

Cross-multiplying:

$$-0.45(1 + \theta_1^2) = \theta_1$$ $$-0.45 - 0.45\theta_1^2 = \theta_1$$

Rearranging terms:

$$0.45\theta_1^2 + \theta_1 + 0.45 = 0$$

This is a quadratic equation in the form $a\theta_1^2 + b\theta_1 + c = 0$ where:

• $a = 0.45$
• $b = 1$
• $c = 0.45$

Using the quadratic formula:

$$\theta_1 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ $$\theta_1 = \frac{-1 \pm \sqrt{1^2 - 4(0.45)(0.45)}}{2(0.45)}$$ $$\theta_1 = \frac{-1 \pm \sqrt{1 - 0.81}}{0.9}$$ $$\theta_1 = \frac{-1 \pm \sqrt{0.19}}{0.9}$$ $$\theta_1 = \frac{-1 \pm 0.43589}{0.9}$$

This gives two possible solutions:

1. $\theta_1 = \frac{-1 + 0.43589}{0.9} = \frac{-0.56411}{0.9} = -0.6268$
2. $\theta_1 = \frac{-1 - 0.43589}{0.9} = \frac{-1.43589}{0.9} = -1.5954$

Since the problem states that θ₁ lies between -1 and +1, we select the first solution:

$$\theta_1 = -0.6268$$

Therefore, the Yule-Walker estimate for θ₁ is -0.6268.