Calculate the best linear predictor of x3 its mean square

Question

Question 1. Suppose that we observe n=100 observations from a stationary time series Xt , and that we fit an AR(1) model to the data (whether it is or is not in fact an AR(1)), using -.4 as an estimate of φ, resulting in residuals:

et = Xt - (-.4)Xt-1 , t=2, 3, ..., n

Suppose that the sample ACF and PACF from the residuals are:

Lag #	1	2	3	4	5	6	7	8	9	10	11	12
ACF	.799	.412	.025	-.228	-.316	-.287	-.198	-.111	-.056	-.009	.048	.133
PACF	.799	-.625	-.044	.038	-.020	-.077	-.007	-.061	-.042	.089	.052	.125

(a) Are these values compatible with "whiteness" (i.e., white noise) of the residuals? Give an argument based upon the sample ACF and PACF. If the residuals do not appear to be "noise", what ARMA model, for the residuals, is suggested by the ACF and PACF for the residuals?

Give an argument justifying your assessment.

(b) Now go back and apply the result of 5 (Box-Jenkins) to (a), and argue what is the new model for Xt?

2, The following is designed to solidify the procedure by which all forecasting is done. The procedure can be implemented by various algorithms: by solving an equation via matrix inversion (φ = Γ-1γ), or by the Durbin-Levinson algorithm. You will use the Durbin-Levinson algorithm to forecast a future value (part a), and the matrix inversion equation to interpolate a missing data (part b). These are two very common scenarios. Suppose that the process is an ARMA(1,1) with the following parameters:

Xt -.8Xt-1 = Zt +.9Zt-1 {Zt}, WN(0,σz² ),σz² =1

(a) Suppose that the first two observed values are: x1=1.5, x2=1.8. Calculate the best linear predictor of x3, its mean square prediction error, and calculate a 95% prediction interval. (Hint: use Durbin-Levinson)

(b) Suppose that you have observed the values at times 1 and 3, x1=1.5, x3=1.8, but that the value at time 2, x², predictors: W = [X3 X1] and Y be the Predicted: Y = X2; Γ is the 2x2 covariance matrix for W, and γ is a column vector of length 2 containing the covariances of Y with each of the 2 components of W. The coefficients φ, for the two variables X₃ and X₁, are then obtained as φ = Γ^-1γ