Volatility in stock return series of vietnam stock market

MINISTRY OF EDUCATION AND TRAINING UNIVERSITY OF ECONOMICS HOCHIMINH CITY - oOo - NGUYEN TH± KIM NGÂN VOLATILITY IN STOCK RETURN SERIES OF VIETNAM STOCK MARKET MASTER THESIS Ho Chi Minh City – 2011 MINISTRY OF EDUCATION AND TRAINING UNIVERSITY OF ECONOMICS HOCHIMINH CITY o0o - NGUYEN TH± KIM NGÂN VOLATILITY IN STOCK RETURN SERIES OF VIETNAM STOCK MARKET MAJOR: BANKING AND FINANCE MAJOR CODE: 60.31.12 MASTER THESIS INSTRUCTOR: Dr VÕ XUÂN VINH Ho Chi Minh City – 2011 ACKNOWLEDGEMENT At first, I would like to show my sincerest gratitude to my supervisor, Dr Vo Xuan Vinh, for his valuable time and enthusiasm His whole-hearted guidance, encouragement and strong support during the time from the initial to the final phase are the large motivation for me to complete my thesis I also would like to thank all of my lecturers at Faculty of Banking and Finance, University of Economics Hochiminh City for their English program, knowledge and teaching during my master course at school In addition, my thanks also go to my beloved family for creating good and convenient conditions for me throughout all my studies at University as well as helping me overcome all the obstacles to finish this thesis Lastly, I offer my regards and blessings to all of those who supported me in any respects during the completion of the study i ABSTRACT This thesis studies the features of the stock return volatility and the presence of structural breaks in return variance of VNIndex in the Vietnam stock market by using the iterated cumulative sums of squares (ICSS) algorithm The relationship between Vietnam stock market’s volatility shifts and impacts of global crisis is also detected Using a long-span data, the results show that daily stock returns can be characterized by GARCH and GARCH in mean (GARCH-M) models while threshold GARCH (T-GARCH) is not suitable About structural breaks, when applying ICSS to the standardized residuals filtered from GARCH (l, l) model, the number of sudden jumps significantly decreases in comparison with the raw return series Events corresponding to those breaks and altering the volatility pattern of stock return are found to be country-specific Not any shifts are found during global crisis period In addition, because the research is not able to point out exactly what events caused sudden changes, the analysis on relationship between these information and shifts is just in relative meaning Further evidence also reveals that when sudden shifts are taken into account in the GARCH models, reduction in the volatility persistence is found It suggests that many previous studies may have overestimated the degree of volatility persistence existing in financial time series The small value of coefficients of the dummies representing breakpoints in modified GARCH model implies that the conditional variance of stock return is much affected by past trend of observed shocks and variance Our results have important implications regarding advising investors on decisions concerning pricing equity, portfolio investment and management, hedging and forecasting Moreover, it is also helpful for policy-makers in making and promulgating the financial policies ii TABLE OF CONTENTS ACKNOWLEDGEMENT i ABSTRACT ii TABLE OF CONTENTS iii LIST OF FIGURES v LIST OF TABLES vi ABBREVIATIONS vii 1: INTRODUCTION 2: LITERATURE REVIEW 2.1 Common characteristics of return series in the stock market .5 2.2 Volatility models suitable to the stock return characteristics 2.3 Identification of breakpoints in volatilities and influence of the regime changes .7 2.4 Events related to regime changes 2.5 Sudden changes in economic recession? 10 2.6 Overstatement of ICSS algorithm in raw returns series 10 3: HYPOTHESES 12 4: RESEARCH METHODS 13 4.1 Stationarity 13 4.2 Testing for stationarity 14 4.2.1 Autocorrelation diagram 14 4.2.2 Unit root test 15 4.3 GARCH model 16 4.3.1 ARMA 16 4.3.1.1 Moving average processes - MA(q) 17 4.3.1.2 Autoregressive processes - AR(p) 17 4.3.1.3 ARMA processes 18 4.3.1.4 Information criteria for ARMA model selection 19 4.3.2 ARCH & GARCH Model 20 4.3.2.1 ARCH Model 20 4.3.2.2 GARCH Model 21 4.4 TGARCH Model 22 4.5 GARCH-M model 23 4.6 ICSS algorithm 24 4.7 Combination of GARCH model and sudden changes 26 5: DATA AND EMPIRICAL RESULTS 27 5.1 Data 27 5.2 Empirical results 29 5.2.1 Suitable models for stock return series of Vietnam 29 5.2.1.1 Choosing suitable ARMA model 29 5.2.1.2 Test for ARCH effect 30 5.2.1.3 GARCH models 31 5.2.2 Identification of break points and detection of related events 33 5.2.2.1 Breakpoints in raw returns 33 5.2.2.2 Breakpoints in filtered returns 38 5.2.2.3 Analysis of each volatility period 44 5.2.2.4 General comments on events and volatility corresponding to sudden changes detected by ICSS algorithm 57 5.2.3 Combined model after including dummies 57 6: CONCLUSION 60 Implications of the research 60 Limitations of the study 61 REFERENCE 62 APPENDIX 66 Table A1 Descriptive statistics of Vietnam stock market’s daily stock return 66 Table A2 Correlogram and Q-statistic of VNIndex daily rate of return 67 Table A3 Unit Root Test on VNIndex’s daily return 68 Table A4 Summary for estimation results of all ARMA models 69 Table A5 Statistically significant ARMA models with C constants 70 Table A6 Statistically significant ARMA models without C constants 72 Table A7 Estimation results of GARCH models 74 Table A8 Estimation results of GARCH-M models 77 Table A9 Estimation result of TGARCH model 79 Table A10 Estimation result of GARCH model modified with sudden changes .80 Table A11 ICSS code on WINRAT 81 LIST OF FIGURES Figure 5.1 Daily return series on HOSE .29 Figure 5.2 Structural breakpoints in volatility in raw returns 38 Figure 5.3 Structural breakpoints in volatility in filtered returns 39 LIST OF TABLES Table 5.1 Descriptive statistics of Vietnam stock market’s daily return series 27 Table 5.2 Unit Root Test on VNIndex’s daily return 28 Table 5.3 Empirical results of different ARMA models 30 th Table 5.4 ARCH effect at lag 3l Table 5.5 Empirical results of different GARCH-family models 32 Table 5.6 Breakpoints detected by ICSS algorithm in the raw returns 33 Table 5.7 Breakpoints detected by ICSS algorithm in the filtered returns 40 ABBREVIATIONS CPI Consumer Price Index GARCH Generalized Autoregressive Conditional Heteroscedasticity GARCH-M GARCH in Mean GDP Gross Domestic Product HOSE Ho Chi Minh City Stock Exchange HOSTC Ho Chi Minh City Securities Trading Center ICSS algorithm Iterated Cumulative Sums of Squares algorithm SSC State Securities Committee of Vietnam TGARCH Threshold GARCH VND Vietnam Dong vii Volatility in Stock Return Series of Vietnam Stock Market 1: INTRODUCTION Volatility is a fundamental concept in the discipline of finance It can be described broadly as anything that is changeable or variable It is associated with unpredictability, uncertainty or risk Volatility is unobservable in financial market and it is measured by standard deviation or variance of return which can be directly considered as a measure of risk of assets Considerable volatilities have been found in the past few years in mature and emerging financial markets worldwide As a proxy of risk, modelling and forecasting stock market volatility has become the subject of vast empirical and theoretical investigations over the past decades by academics and practitioners Substantial changes in the volatility of financial market returns are capable of having significant effects on risk averse investors Furthermore, such changes can also impact on consumption patterns, corporate capital investment decisions, leverage decisions and other business cycle Volatility forecasts of stock price are crucial inputs for pricing derivatives as well as trading and hedging strategies Therefore, it is important to understand the behavior of return volatility In addition to return volatility, some relevant problems attracting much interest of researchers have been whether or not major events may lead to sudden changes in return volatility and how unanticipated shocks will affect volatility over time Concerning these factors, persistence term should be considered Persistence in variance of a random variable refers to the property of momentum in conditional variance or past volatility can explain current volatility in some certain levels The larger the persistence is, the higher the past volatility can be explained for the current volatility The persistence in volatility is a key ingredient for accurately predicting how events will affect volatility in stock returns and partially determines stock prices Poterba and Summers (l986) showed that the extent to which stockreturn volatility affects stock prices (through a time-varying risk premium) depends critically on the permanence of shocks to variance Hence, the degree to which l Table A4 Summary for estimation results of all ARMA models 1 ARMA(1,0)) 2 3 l 2 3 0.2949 AIC SBC -5.48l08 -5.4784l -5.487ll -5.48l76 -5.49542 -5.4874l -5.4945l -5.483830 -5.48753 -5.482l9 -5.48676 -5.47874 -5.49079 -5.480ll -5.49489 -5.48l53 -5.48754 -5.48487 -5.4876l -5.48227 -5.4896 -5.48l59 -5.48706 -5.47904 (0.0000) ARMA(1,1) ARMA(1,2)) ARMA(1,3) ARMA(2,0)) ARMA(2,1) ARMA(2,2) ARMA(2,3) 0.0777 0.2422 (0.2609) (0.0003) 0.9704 -0.6589 -0.27l0 (0.0000) (0.0000) (0.0000) 0.9696 -0.6606 -0.2737 0.0064 (0.0000) (0.0000) (0.0000) (0.7766) 0.3202 -0.0873 (0.0000) (0.000l) 0.2428 -0.0653 0.0783 (0.3l49) (0.3883) (0.7464) -0.2788 -0.2396 0.5989 0.3366 (0.l728) (0.0005) (0.0033) (0.0000) 0.3384 0.6l75 -0.0l49 -0.67l8 -0.2079 (0.2784) (0.040l) (0.96l7) (0.00l4) (0.0074) ARMA(0,1)) 0.3098 (0.0000) ARMA(0,2) ARMA(0,3) ARMA (3,0) 0.3224 -0.0963 0.0275 (0.0000) (0.0000) (0.206l) 0.3232 0.03l8 (0.0000) (0.l444) 0.3284 0.0248 -0.0520 (0.0000) (0.279l) (0.0l68) Table A5 Statistically significant ARMA models with C constants ARMA(1,0) Dependent Variable: R Method: Least Squares Sample (adjusted): 3/05/2002 8/3l/20l0 Included observations: 2ll9 after adjustments Convergence achieved after iterations C AR(l) R-squared Adjusted R-squared S.E of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic) Inverted AR Roots Coefficient Std Error t-Statistic Prob 0.0004l7 0.294423 0.00048l 0.020777 0.867488 l4.l7039 0.3858 0.0000 0.086634 0.086202 0.0l56l3 0.5l6046 5808.585 200.8000 0.000000 Mean dependent var S.D dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter Durbin-Watson stat 0.0004l0 0.0l6333 -5.480495 -5.475l54 -5.478540 l.94850l 29 ARMA(2,0) Dependent Variable: R Method: Least Squares Sample (adjusted): 3/06/2002 8/3l/20l0 Included observations: 2ll8 after adjustments Convergence achieved after iterations C AR(l) AR(2) R-squared Adjusted R-squared S.E of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic) Inverted AR Roots Coefficient Std Error t-Statistic Prob 0.000420 0.3l9839 -0.087648 0.000440 0.02l662 0.02l68l 0.953227 l4.76532 -4.042532 0.3406 0.0000 0.000l 0.093483 0.092626 0.0l5558 0.5ll967 58l3.747 l09.053l 0.000000 l6+.25i Mean dependent var S.D dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter Durbin-Watson stat l6-.25i 70 0.0004l7 0.0l6333 -5.4870l3 -5.478999 -5.484079 l.99520l ARMA(0,1) Dependent Variable: R Method: Least Squares Sample (adjusted): 3/04/2002 8/3l/20l0 Included observations: 2l20 after adjustments Convergence achieved after iterations MA Backcast: 3/0l/2002 Coefficient Std Error t-Statistic Prob 0.000404 0.309523 0.000443 0.02066l 0.9l2975 l4.98082 0.36l4 0.0000 C MA(l) R-squared Adjusted R-squared S.E of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic) Inverted MA Roots 0.092426 0.09l998 0.0l5562 0.5l2949 58l8.206 2l5.6949 0.000000 Mean dependent var S.D dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter Durbin-Watson stat 0.000403 0.0l6332 -5.486987 -5.48l648 -5.485032 l.979967 -.3l ARMA(1,2) Dependent Variable: R Method: Least Squares Sample (adjusted): 3/05/2002 8/3l/20l0 Included observations: 2ll9 after adjustments Convergence achieved after l6 iterations MA Backcast: 3/0l/2002 3/04/2002 C AR(l) MA(l) MA(2) R-squared Adjusted R-squared S.E of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic) Inverted AR Roots Inverted MA Roots Coefficient Std Error t-Statistic Prob 0.000396 0.969542 -0.658035 -0.270679 0.000795 0.0l8066 0.028564 0.023365 0.498799 53.6669l -23.03747 -ll.58476 0.6l80 0.0000 0.0000 0.0000 0.l0lll8 0.099843 0.0l5496 0.507862 5825.520 79.30736 0.000000 Mean dependent var S.D dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter Durbin-Watson stat 97 94 7l 0.0004l0 0.0l6333 -5.494592 -5.4839l0 -5.49068l 2.002925 Table A6 Statistically significant ARMA models without C constants ARMA(1,0) NOT C Dependent Variable: R Method: Least Squares Sample (adjusted): 3/05/2002 8/3l/20l0 Included observations: 2ll9 after adjustments Convergence achieved after iterations Coefficient Std Error t-Statistic Prob 0.294856 0.020770 l4.l96l6 0.0000 AR(l) R-squared Adjusted R-squared S.E of regression Sum squared resid Log likelihood Durbin-Watson stat 0.086309 0.086309 0.0l56l2 0.5l6229 5808.208 l.948599 Inverted AR Roots Mean dependent var S.D dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter 0.0004l0 0.0l6333 -5.48l084 -5.4784l3 -5.480l06 29 ARMA (2, 0) _ NOT C Dependent Variable: R Method: Least Squares Sample (adjusted): 3/06/2002 8/3l/20l0 Included observations: 2ll8 after adjustments Convergence achieved after iterations AR(l) AR(2) R-squared Adjusted R-squared S.E of regression Sum squared resid Log likelihood Durbin-Watson stat Inverted AR Roots Coefficient Std Error t-Statistic Prob 0.32022l -0.087305 0.02l657 0.02l678 l4.78578 -4.027363 0.0000 0.000l 0.093094 0.092665 0.0l5558 0.5l2l87 58l3.292 l.995l26 l6+.25i Mean dependent var S.D dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter .l6-.25i 72 0.0004l7 0.0l6333 -5.487528 -5.482l85 -5.485572 Volatility in Stock Return Series of Vietnam Stock Market ARMA(0, 1) NOT C Dependent Variable: R Method: Least Squares Date: ll/04/l0 Time: 20:39 Sample (adjusted): 3/04/2002 8/3l/20l0 Included observations: 2l20 after adjustments Convergence achieved after iterations MA Backcast: 3/0l/2002 MA(l) Coefficient Std Error t-Statistic Prob 0.309768 0.020655 l4.99748 0.0000 R-squared Adjusted R-squared S.E of regression Sum squared resid Log likelihood Durbin-Watson stat 0.092069 0.092069 0.0l5562 0.5l3l5l 58l7.789 l.979695 Inverted MA Roots -.3l Mean dependent var S.D dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter 0.000403 0.0l6332 -5.487537 -5.484867 -5.486559 ARMA (1,2) _ NOT C Dependent Variable: R Method: Least Squares Sample (adjusted): 3/05/2002 8/3l/20l0 Included observations: 2ll9 after adjustments Convergence achieved after l8 iterations MA Backcast: 3/0l/2002 3/04/2002 AR(l) MA(l) MA(2) R-squared Adjusted R-squared S.E of regression Sum squared resid Log likelihood Durbin-Watson stat Inverted AR Roots Inverted MA Roots Coefficient Std Error t-Statistic Prob 0.970396 -0.658807 -0.27l0l3 0.0l7434 0.028ll8 0.02324l 55.66l0l -23.42982 -ll.66ll7 0.0000 0.0000 0.0000 0.l0l0l4 0.l00l64 0.0l5493 0.50792l 5825.398 2.002857 97 95 Mean dependent var S.D dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter -.29 0.0004l0 0.0l6333 -5.495420 -5.487409 -5.492487 Table A7 Estimation results of GARCH models GARCH(1,1) Dependent Variable: R Method: ML - ARCH (Marquardt) - Normal distribution Sample (adjusted): 3/05/2002 8/3l/20l0 Included observations: 2ll9 after adjustments Convergence achieved after l7 iterations MA Backcast: 3/0l/2002 3/04/2002 Presample variance: backcast (parameter = 0.7) GARCH = C(4) + C(5)*RESID(-l)^2 + C(6)*GARCH(-l) AR(l) MA(l) MA(2) Coefficient Std Error z-Statistic Prob 0.727354 -0.4l3978 -0.2l237l 0.l06924 0.ll039l 0.043l86 6.802508 -3.750l03 -4.9l7578 0.0000 0.0002 0.0000 7.66585l l3.90800 49.4248l 0.0000 0.0000 0.0000 Variance Equation C RESID(-l)^2 GARCH(-l) 2.64E-06 0.326387 0.7l3086 R-squared Adjusted R-squared S.E of regression Sum squared resid Log likelihood Durbin-Watson stat 0.094l04 0.09l960 0.0l5564 0.5ll825 64l9.l37 l.99l3l5 Inverted AR Roots Inverted MA Roots 73 7l 3.45E-07 0.023468 0.0l4428 Mean dependent var S.D dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter -.30 0.0004l0 0.0l6333 -6.052984 -6.03696l -6.047ll8 GARCH(2,1) Dependent Variable: R Method: ML - ARCH (Marquardt) - Normal distribution Sample (adjusted): 3/05/2002 8/3l/20l0 Included observations: 2ll9 after adjustments Convergence achieved after l9 iterations MA Backcast: 3/0l/2002 3/04/2002 Presample variance: backcast (parameter = 0.7) GARCH = C(4) + C(5)*RESID(-l)^2 + C(6)*GARCH(-l) + C(7)*GARCH(-2) AR(l) MA(l) MA(2) Coefficient Std Error z-Statistic Prob 0.99l522 -0.6940ll -0.279879 0.004206 0.0240l3 0.024357 235.7377 -28.90l59 -ll.49063 0.0000 0.0000 0.0000 7.l52449 l3.3589l 4.6698ll 4.9l5497 0.0000 0.0000 0.0000 0.0000 Variance Equation C RESID(-l)^2 GARCH(-l) GARCH(-2) 3.06E-06 0.397426 0.338637 0.308932 R-squared Adjusted R-squared S.E of regression Sum squared resid Log likelihood Durbin-Watson stat 0.098464 0.095902 0.0l5530 0.509362 6423.979 l.969294 Inverted AR Roots Inverted MA Roots 99 98 4.28E-07 0.029750 0.0725l6 0.062849 Mean dependent var S.D dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter -.29 0.0004l0 0.0l6333 -6.0566l0 -6.0379l7 -6.049767 GARCH(3,1) Dependent Variable: R Method: ML - ARCH (Marquardt) - Normal distribution Sample (adjusted): 3/05/2002 8/3l/20l0 Included observations: 2ll9 after adjustments Convergence achieved after l7 iterations MA Backcast: 3/0l/2002 3/04/2002 Presample variance: backcast (parameter = 0.7) GARCH = C(4) + C(5)*RESID(-l)^2 + C(6)*GARCH(-l) + C(7)*GARCH(-2) + C(8)*GARCH(-3) AR(l) MA(l) MA(2) Coefficient Std Error z-Statistic Prob 0.6l46l4 -0.3l9l63 -0.l80073 0.l23426 0.l263l4 0.042688 4.9796l2 -2.526737 -4.2l8340 0.0000 0.0ll5 0.0000 6.8869l8 ll.47322 4.426750 l.808069 2.598428 0.0000 0.0000 0.0000 0.0706 0.0094 Variance Equation C RESID(-l)^2 GARCH(-l) GARCH(-2) GARCH(-3) 3.llE-06 0.426767 0.3495l2 0.l3470l 0.l3935l R-squared Adjusted R-squared S.E of regression Sum squared resid Log likelihood Durbin-Watson stat 0.0923l6 0.089306 0.0l5586 0.5l2835 64l8.8l5 l.95l902 Inverted AR Roots Inverted MA Roots 6l 6l 4.52E-07 0.037l97 0.078955 0.074500 0.053629 Mean dependent var S.D dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter -.29 0.0004l0 0.0l6333 -6.050793 -6.029429 -6.04297l Table A8 Estimation results of GARCH-M models GARCH-M (1,1) Dependent Variable: R Method: ML - ARCH (Marquardt) - Normal distribution Sample (adjusted): 3/05/2002 8/3l/20l0 Included observations: 2ll9 after adjustments Convergence achieved after l9 iterations MA Backcast: 3/0l/2002 3/04/2002 Presample variance: backcast (parameter = 0.7) GARCH = C(5) + C(6)*RESID(-l)^2 + C(7)*GARCH(-l) GARCH AR(l) MA(l) MA(2) Coefficient Std Error z-Statistic Prob 3.65l637 0.99l707 -0.687835 -0.287740 l.496406 0.004620 0.02l365 0.02l705 2.440272 2l4.638l -32.l9432 -l3.25682 0.0l47 0.0000 0.0000 0.0000 8.l2l807 l3.458l0 45.60705 0.0000 0.0000 0.0000 Variance Equation C RESID(-l)^2 GARCH(-l) 2.89E-06 0.3l0450 0.7l97l6 R-squared Adjusted R-squared S.E of regression Sum squared resid Log likelihood Durbin-Watson stat 0.095328 0.092758 0.0l5557 0.5lll34 6428.386 l.975030 Inverted AR Roots Inverted MA Roots 99 98 3.55E-07 0.023068 0.0l578l Mean dependent var S.D dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter -.29 0.0004l0 0.0l6333 -6.060770 -6.042077 -6.053926 GARCH-M (2,1) Dependent Variable: R Method: ML - ARCH (Marquardt) - Normal distribution Sample (adjusted): 3/05/2002 8/3l/20l0 Included observations: 2ll9 after adjustments Convergence achieved after 22 iterations MA Backcast: 3/0l/2002 3/04/2002 Presample variance: backcast (parameter = 0.7) GARCH = C(5) + C(6)*RESID(-l)^2 + C(7)*GARCH(-l) + C(8)*GARCH(-2) GARCH AR(l) MA(l) MA(2) Coefficient Std Error z-Statistic Prob 3.9ll827 0.99l9l7 -0.694040 -0.282054 l.477529 0.0045l4 0.022435 0.022643 2.647546 2l9.7207 -30.93542 -l2.4566l 0.008l 0.0000 0.0000 0.0000 7.254302 l0.00047 4.678368 l.737044 0.0000 0.0000 0.0000 0.0824 Variance Equation C RESID(-l)^2 GARCH(-l) GARCH(-2) 3.09E-06 0.345430 0.528377 0.l58963 R-squared Adjusted R-squared S.E of regression Sum squared resid Log likelihood Durbin-Watson stat 0.094984 0.09l983 0.0l5563 0.5ll328 6430.359 l.9622l5 Inverted AR Roots Inverted MA Roots 99 98 4.26E-07 0.03454l 0.ll2940 0.09l5l3 Mean dependent var S.D dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter -.29 0.0004l0 0.0l6333 -6.06l688 -6.040325 -6.053867 Table A9 Estimation result of TGARCH model Dependent Variable: R Method: ML - ARCH (Marquardt) - Normal distribution Sample (adjusted): 3/05/2002 8/3l/20l0 Included observations: 2ll9 after adjustments Convergence achieved after 2l iterations MA Backcast: 3/0l/2002 3/04/2002 Presample variance: backcast (parameter = 0.7) GARCH = C(4) + C(5)*RESID(-l)^2 + C(6)*RESID(-l)^2*(RESID(-l)0 compute t2=%maxent else break end loop compute kfirst=t2 compute tl=kstar+l,t2=end loop @IncTiaoBreak x tl t2 if %maxent>0 compute tl=%maxent+l else break end loop compute klast=tl-l * if kfirst==klast compute ICSSland2=||kfirst|| else compute ICSSland2=||kfirst,klast|| end * ************************************************************************* ******** 82 * * This is the controlling routine for the ICSS algorithm Note that this does not * the "pruning" step (step 3), just the complete enumeration from steps l and * Because the ICSS algorithm is recursive, and RATS doesn't allow recursive * procedures, the recursion is unwound by using a stack of VEC[INT] for the * starting and ending points being checked The stack is popped each time a call * to ICSSland2 returns either or l breakpoint and pushed when it returns * procedure ICSS x start end type series x type integer start end * local vect[int] breaks local vect[int] newbreaks local vect[int] tl t2 local vect temp local integer i k local series xc local integer startl endl * inquire(series=x) startl