Linear Predictor for AR(1) Process: Finding Optimal Prediction Function for Missing Values

Assignment Task

1. Suppose that I am dealing with Yt an AR(1) process. In addition, suppose that we have observed Y1, and Y3, and we would like to estimate the missing value Y2. What could be the best linear predictor of Y2 given Y1 and Y3? This is not just a conceptual question, please also list any necessary mathematical proofs, calculations, or steps to support your answers.

2. In your mathematical statistics class, your professor asks you to find a prediction function h(x) that minimizes a quantity named: H = E[(y − h(x))²] , where x and y are two random variables that are jointly distributed with density function f(x, y)

a) What could be the appropriate h(x) to minimize H?

b) Apply the above result to the model y = x2 + z where x and z are normal random variables with mean zero and variance 1. Show that H = 1.

c) Suppose that your professor asks you to restrict the choices for the function h(x) to linear functions of the form h(x)=ax+b and determine a and b that minimizes H. Show that a =1, and

b = E(xy)/E(x²) = 0

3. Consider a (weakly) stationary process Y with zero mean and autocovariance function denoted by acv(m).

a. Given Yt, what could be the best predictor Yˆt+1?

b. Given Yt and Yt−1, what could be the best linear predictor Y ̃t+1 of Yt+1.

c. Let us define a transformation named Difference of Mean Squared Errors (DOMSE) expressed as:

DOMSE = E{(Yt+1 − Yˆt+1)²} − E{(Yt+1 − Y ̃t+1)²}

Find an expression for DOMSE and evaluate it when Yt = cos(nU) when U is uniform on [-π, π], and when Yt is an AR model with acv(k) = α|k| where |α| < 1>

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