AMS4322 Financial Time Series for Business Intelligence 2021/22 Assignment One Please write down your name, student ID no., and class no. on the first page of your assignment submission. Show all necessary steps of your work. Please submit the assignment by 29th October, 5pm. Question 1: Consider the time series data of U.S. retail sales and disposable income in Table 1. a) Choose a suitable trend model for the retail sales data. Fit the model using the data from 1970 to 1990. Use the fitted model to forecast the retail sales from 1991 to 1995. b) Consider the causal model of the the retail sales and the disposable income, RETt = β0 + β1DISt−1 + error. Fit the model using the data from 1970 to 1990. Make one-year-ahead prediction of the retail sales for Year 1991 using the fitted model. Question 2: Suppose Zt = 5 + 2t+Xt, where {Xt} is a zero-mean stationary series with autoco- variance function γk . (a) Find the mean function of {Zt}. (b) Find the autocovariance function of {Zt}. (c) Is {Zt} stationary? Why or why not. Question 3: Suppose that the data follows AR(1) model Xt − µ = φ(Xt−1 − µ) + t , where t is sequence of independent N(0, σ2) random variables. The model is fitted and the result is Coefficients: ar1 intercept 0.9341 3.9552 s.e. 0.0123 0.0274 sigma^2 estimated as 0.002756 (a) Suppose that Xn = 3.902 . Forecast Xn+3 and give a confidence interval of approximately 95%. (b) Find the autocovariance function γk for k = 0, 1, 2, . . .. Question 4: Suppose that the data follows MA(1) model Xt = µ+ t−θt−1 , where t is sequence of independent N(0, σ2) random variables. The model is fitted and the result is 1 Coefficients: ma1 intercept 0.8125 3.9552 s.e. 0.0143 0.0057 sigma^2 estimated as 0.008207 (a) Suppose that Xn−2 = 3.956 , Xn−1 = 3.941 , Xn = 3.902 , and ˆn−2 = 0.0948. Forecast Xn+1 , Xn+2 , and Xn+3 and give confidence intervals of approximately 95%. (b) Find the autocovariance function γk for k = 0, 1, 2, . . .. Question 5: Suppose that the data follows ARMA(1,1) model Xt−µ = φ(Xt−1−µ) + t− θt−1 , where t is sequence of independent N(0, σ 2) random variables. The model is fitted and the result is Coefficients: ar1 ma1 intercept 0.9078 0.2311 3.9552 s.e. 0.0152 0.0317 0.0235 sigma^2 estimated as 0.002592 (a) Suppose that Xn−2 = 3.956 , Xn−1 = 3.941 , Xn = 3.902 , and ˆn−2 = 0.0948. Forecast Xn+1 , Xn+2 , and Xn+3 and give confidence intervals of approximately 95%. (b) Find the autocovariance function γk for k = 0, 1, 2, . . .. DUE DATE: 29th, October, 2021 (Fri) 2 Table 1: Historical data for U.S. retail sales and disposable personal income (billions of dollars). Year RET SALES DIS INCOME 1970 375.2 710.0 1971 414.2 771.8 1972 458.5 834.8 1973 511.9 943.8 1974 542.0 1,033.7 1975 588.1 1,138.4 1976 656.4 1,248.8 1977 722.5 1,375.3 1978 804.2 1,546.5 1979 896.8 1,724.6 1980 957.4 1,914.3 1981 1,038.7 2,121.8 1982 1,069.3 2,255.1 1983 1,167.9 2,424.9 1984 1,281.7 2,662.1 1985 1,365.8 2,832.1 1986 1,435.9 3,007.6 1987 1,521.4 3,184.2 1988 1,629.2 3,467.9 1989 1,733.7 3,710.0 1990 1,807.2 3,949.1 3