"Frontmatter". In: Analysis of Financial Time Series

168 NONLINEAR TIME SERIESTable 4.4 shows the relative MSE of forecasts and mean forecast errors for thelinear model in Eq. (4.50), the TAR model in Eq. (4.51), and the MSA model inEq. (4.52), using the linear model as a benchmark. The comparisons are based onoverall performance as well as the status of the U.S. economy at the forecast origin.From the table, we make the following observations:1. For the overall comparison, TAR model and the linear model are very closein MSE, but the TAR model has smaller biases. Yet the MSA model has thehighest MSE, but smallest biases.2. For forecast origins in economic contractions, the TAR model shows improvementsover the linear model both in MSE and bias. The MSA model also showssome improvement over the linear model, but the improvement is not as largeas that of the TAR model.3. For forecast origins in economic expansions, the linear model outperformsboth nonlinear models.The results suggest that the contributions of nonlinear models over linear ones inforecasting the U.S. quarterly unemployment rate are mainly in the periods when theU.S. economy is in contractions. This is not surprising because, as mentioned before,it is during the economic contractions that government interventions and industrialrestructuring are most likely to occur. These external events could introduce nonlinearityin the U.S. unemployment rate. Intuitively, such improvements are importantbecause it is during the contractions that people pay more attention to economicforecasts.APPENDIX A. SOME RATS PROGRAMS FOR NONLINEARVOLATILITY MODELSA. This program was used to estimate an AR(2)-TAR-GARCH(1, 1) modelfor daily log returns of IBM stock. The data file is “d-ibmln99.dat.”all 0 9442:1open data d-ibmln99.datdata(org=obs) / rtset h = 0.0*nonlin mu p1 p2 a0 a1 a2 b0 b1 b2nonlin mu p2 a1 a2 b0 b1 b2*frml at = rt(t)-mu-p1*rt(t-1)-p2*rt(t-2)frml at = rt(t)-mu-p2*rt(t-2)frml u = (at(t-1)/abs(at(t-1))+1.0)/2.0frml gvar1 = a1*at(t-1)**2+a2*h(t-1)frml gvar = gvar1(t)+u(t)*(b0+b1*at(t-1)**2+b2*h(t-1))frml garchln = -0.5*log(h(t)=gvar(t))-0.5*at(t)**2/h(t)smpl 4 9442compute mu = 0.03, p1 = 0.1, p2 = -0.03