EF4822 Financial Econometrics Problem Set 2 Due: Right before the class next week. Please submit hard copies in groups and write down the names and EID of all group members. If R programming is used, please show the R commands. 1. Suppose that the simple return of a stock follows the model rt = 0.1 + at − 0.5at−2, where {at}T t=1 follows a white noise process with mean zero and variance σ2 a = 0.04. Please compute the following: (a). What is the mean of rt? (b). What is the variance of rt?

1. Suppose that the simple return of a stock follows the modelrt = 0.1 + at − 0.5at−2,where {at}Tt=1 follows a white noise process with mean zero and variance σ2a = 0.04. Pleasecompute the following:(a). What is the mean of rt?(b). What is the variance of rt?

1. Suppose that the simple return of a stock follows the model rt = 0.1 + at − 0.5at−2, where {at}T t=1 follows a white noise process with mean zero and variance σ2 a = 0.04. Please compute the following: (a). What is the mean of rt? (b). What is the variance of rt? (c). Compute the first order autocorrelation of rt. (d). Compute the second order autocorrelation of rt. (e). Assume that you are standing at period 100 and a100 = 0.01. Compute the 2-step-ahead forecast of the return.

2. Suppose that we have an AR(2) model as follows:yt = 0.1 + ϕ1yt−1 + ϕ2yt−2 + ut,where ut follows a white noise process with mean zero and variance σ2u = 0.2. Please computethe following:(a). Assume that ϕ1 and ϕ2 take values that make the model stationary. What is the meanof yt?(b). Compute E[yt+1|Ωt].(c). Compute E[yt+2|Ωt]

(d). Fit the return data with the following GJR model:yt = µ + utut = vtσtσ2t = α0 + α1u2t−1 + β1σ2t−1 + γu2t−1It−1Write down the estimated model. Hint: Search the function ‘ugarchfit’ in the package‘rugarch’. You can first specify the GJR model as:ugarchspec(variance.model=list(model=’gjrGARCH’), mean.model=list(armaOrder=c(0,0)))Then use ugarchfit to estimate the specified model.(e). Is there leverage effect according to the evidence suggested by the GJR model? Specifyyour reason.

DTF Investments Ltd has a two-stock portfolio: 40% invested in Apple stock and 60% invested in Pear stock. The table below sets out the expected return on the stocks for various states of the economy (returns are shown in decimals). What is the expected return of DTF Investments Ltd’s portfolio? Record your answer to the nearest four decimal places (e.g., If your answer is 3.40%, then report 0.0340). State of economy Probability Apple Stock Pear Stock Boom 0.21 0.27 0.26 Normal ? 0.08 0.11 Bust 0.15 -0.17 -0.17