"Frontmatter". In: Analysis of Financial Time Series

170 NONLINEAR TIME SERIESsummary(ibm.nn)# compute \& print the residual sum of squares.sse_sum((y-predict(ibm.nn,ibm.x))ˆ2)print(sse)#eigen(nnet.Hess(ibm.nn,ibm.x,y),T)$values# setup the input variables in the forecasting subsampleibm.p_cbind(x[864:887],x[863:886],x[862:885])# compute the forecastsyh_predict(ibm.nn,ibm.p)# The observed returns in the forecasting subsampleyo_x[865:888]# compute \& print the sum of squares of forecast errorsssfe_sum((yo-yh)ˆ2)print(ssfe)# quit S-Plusq()EXERCISES1. Consider the monthly log returns of General Electric (GE) stock from January1926 to December 1999. You may download the data from CRSP or use the file“m-ge2699.dat” on the Web. The log returns in the file are in percentages. Build athreshold GARCH model for the series using a t−1 as the threshold variable withzero threshold, where a t−1 is the shock at time t − 1. Check the fitted model.2. Suppose that the monthly log returns of GE stock, measured in percentages, followsa smooth threshold GARCH(1, 1) model. For the sampling period from January1926 to December 1999, the fitted model isr t = 1.06 + a t ,a t = σ t ɛ tσt 2 = 0.103at−1 2 + 0.952σ t−1 2 + 12(4.490 − 0.193σt−1 1 + exp(−10a t−1 ) ),where all of the estimates are highly significant, the coefficient 10 in the exponentis fixed a priori to simplify the estimation, and {ɛ t } are iid N(0, 1). Assume thata 888 = 16.0andσ888 2 = 50.2, what is the 1-step ahead volatility forecast ̂σ 888 2 (1)?Suppose instead that a 888 =−16.0, what is the 1-step ahead volatility forecast̂σ888 2 (1)?3. Suppose that the monthly log returns, in percentages, of a stock follow the followingMarkov switching modelr t = 1.25 + a t , a t = σ t ɛ t{σt 2 0.10a2= t−1+ 0.93σt−1 2 if s t = 14.24 + 0.10at−1 2 + 0.78σ t−1 2 if s t = 2,where the transition probabilities are