PSTAT 174f274 Fall 2020 Homework 7 Note: {33} ~ WN(0,0'22) denotes white noise. 1. {Updating AR(2} forecast in presence of more information.) You are given the following AR(2} model: X, = 33 + 0.533-; + 2;, where Z, is a white noise process. After observing 375 = 54 and 1:77 2 68, your forecast for X33 is a. After observing, in addition, 3:73 = , your forecast for X30 beoomes b. Calculate b a. all Hint: Check Erampte 13.2 in lecture notes or an slide 11 of Week \"I. 2. (Standard error of forecast error.) You are given the following tted AR(1) model: X3 = 5 + 0.852124 + Zg. The estimated mean squared error {that is, variance of Z!) is 13.545. Calculate the twostep ahead forecast standard error. Hint: Check: Emample 13.1 in lecture notes or on slide 10 of Week '3'. 3. {Box-Pierce Statistics.) A modeler tted 100 observations, using assume} model. For the 100 residuals, she have determined the rst seven autocorrelation coefficients: k 1 2 3 4 5 7 awn) .20 .15 _.1s .16 .ns .07 .ns Calculate the value of the BoxPierce x2 statistics and determine the number of degrees of {reedom for its distribution. 4. (Prediction intervals.) A Gaussian ARH) model was tted to a time series based on a sample of size n. You are given $1 = 5.8, ,. = 2, 6% = 9 x 10\"}, 1:... = 2.05. Write the 95% prediction interval for the observation three periods ahead. Hint: renters Erample 13.1 of Week \"i",- stide M. Do not forget that the mean is not 0! The following problem is for students enrolled in PSTAT 274 ONLY G1. A Gaussian AiR} model was tted to a time series based on a sample of size n = 51. You are given: {1. = 16.75, d1 = 0.?5. The last observation was :51 = 20.25, and the sum of the squares of the 51 residuals is 75.72 Determine the upper bound of the shortest 95% probability limit for the forecast of the observation two time periods ahead