Throughout, let {et} be a white noise process with constant variance o² 1. Consider the Yule-Walker equations for an AR(2) process, and estimation of the coefficients of Xt-1 and Xt-2. Assume that the necessary autocorrelation coefficient(s) are known. a. Write down the two equations in the unknown coefficients and solve for those parameters. b. Suppose a time series {X+} is observed, and its partial autocorrelation function (say for lags 1 to 20) estimated. Will either result in (a) be used in this function? If so, explain.

Suppose we have the following trend-plus-noise model xt = 1+t+wt) where {wi} is white noise with variance 1. (a) Show that {xt}is NOT weakly stationary. (b) Show that the differenced time series y; Væt is weakly stationary.

If a random variable X has the moment generating function Mx (t)= 2 - ť Determine the variance of X.

A time series {yt} follows an MA(2) model: Yt = 2 + Ut +0.54t-1 + 0.4ut-2. Assume that ut is a white noise series with a mean of O and a variance of 2. Please calculate Var(yt) (i.e. the variance of) 2.96 1.41 2.82 O 1.98

If the alleles A and B of the cystic fibrosis gene occur in a population with frequencies p and 1 - p (where pis between 0 and 1), then the frequency of heterozygous carriers (carriers with both alleles) is 2p(l - p). Which value of p gives the largest frequency of heterozygous carriers?

Suppose that a drug company wishes to test the effects of five new compounds on the growth rate of white rats. It is possible that rats within the same litter may have similar response. Hence blocks are defined as litters. Twenty-five rats within each of 4 litters are chosen at random from a large group of rats, and 5 rats of each litter are placed in one pen to be given one of the five treatments. Body weight gains (g/day/animal) on the pen basis are given in the following table after 3 months of feeding the compound.   a. Does any difference exist among litters? b. Do all the compounds have the same effect on body weight gains?

If the moment-generating function of X is 2 M(t) = e' + find the mean, variance, and pmf of X. 2t

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Consider the time series data for the real consumption of households and government from year 1950 to year 2019. Five autoregressive models, AR(1) to AR(5), are estimated as follows, where the t statistics for the corresponding estimated coefficients are in the parentheses, and the size of the samples, n, used for each estimation is also shown in the table. According to the significant lag approach, which model is the optimal under the 5% significance level? AR(1) AR(2) AR(3) AR(4) AR(5) 5365.55 7681.51 intercept t stat 5854.66 5041.07 5689.50 (2.44) (2.05) (2.13) 0.92 (2.13) (2.82) rcont-1 0.82 0.92 0.92 0.92 t stat (315.53) (9.49) (9.41) (9.33) (9.57) -0.26 rcont-2 -0.18 -0.28 -0.27 t stat (-1.38) (-1.38) (-1.31) (-1.32) rcon-s 0.08 0.02 0.07 t stat (0.64) (0.09) (0.32) rcont-4 0.06 -0.32 t stat (0.38) (-1.47) rcont-s 0.34 t stat (2.32) 65 65 65 65 65 AIC 18.30 18.30 18.33 18.36 18.29 BIC 18.37 18.40 18.46 18.52 18.50 O A. AR(2). O B. AR(1). O C. none. D. AR(5).