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i Applied Financial Econometrics Slides Rolf Tschernig | Florian Brezina University of Regensburg Version: 18 July 20121 1 c Rolf Tschernig. I very much thank Joachim Schnurbus for his important corrections and suggestions. i ii Contents 1 Introduction6 1.1 Themes........................................ 6 1.2 Some basics...................................... 8 1.3 Stochastic processes................................. 12 1.4 R Code........................................ 14 2 The basics of time series modeling 15 2.1 Autoregressive processes............................... 15 2.2 Moving average processes............................... 29 2.3 ARMA processes................................... 35 2.4 Trajectory examples.................................. 38 2.5 Estimation...................................... 41 2.6 R Code........................................ 62 ii iii 3 Forecasting (financial) time series 65 3.1 Some general remarks and definitions......................... 65 3.2 Decomposition of prediction errors.......................... 70 3.3 AR Model Specification............................... 76 3.4 Prediction with AR models.............................. 79 3.5 Evaluation of forecasts................................ 84 3.6 R Code........................................ 88 4 More on modeling time series 89 4.1 Unit root tests.................................... 89 4.2 Model Checking.................................... 103 4.3 Estimating dynamic regression models........................ 110 4.4 R Code........................................ 114 5 Modeling volatility dynamics 117 5.1 Standard conditional volatility models........................ 119 5.2 Maximum Likelihood Estimation........................... 125 5.3 Estimation of GARCH(m; n) models......................... 131 5.4 Asymmetry and leverage effects........................... 140 5.5 Testing for the presence of conditional heteroskedasticity............... 142 5.6 Model selection.................................... 144 iii iv 5.7 Prediction of conditional volatility.......................... 145 5.8 R Code........................................ 147 6 Long-run forecasting 151 6.1 Estimating/predicting unconditional means...................... 151 6.2 Predicting long-term wealth: the role of arithmetic and geometric means...... 160 6.3 Are long-term returns predictable?.......................... 169 7 Explaining returns and estimating factor models 171 7.1 The basics of the theory of finance.......................... 171 7.2 Is the asset pricing equation empirically relevant?.................. 200 7.3 Factor-pricing models................................. 201 7.4 Regression based tests of linear factor models.................... 208 7.5 Supplement: Organisation of an empirical project.................. 212 iv Applied Financial Econometrics | General Information | U Regensburg | July 2012 1 General Information Schedule and locations see http://www-wiwi.uni-regensburg.de/Institute/VWL/Tschernig/Lehre/Applied.html.en 1 Applied Financial Econometrics | General Information | U Regensburg | July 2012 2 Contact Rolf Tschernig Florian Brezina Building RW(L), 5th floor, Room 515 RW(L), 5th floor, Room 517 Tel. (+49) 941/943 2737 (+49) 941/943 2739 Fax (+49) 941/943 4917 (+49) 941/943 4917 Email [email protected] fl[email protected] Homepage http://www-wiwi.uni-regensburg.de/Institute/VWL/Tschernig/Team/index.html.en 2 Applied Financial Econometrics | General Information | U Regensburg | July 2012 3 Grading The grade for the course will be based on the written exam, the graded presentations of the exercises (up to a maximum of 15 points), and a mid-term exam. Details are provided at: http://www-wiwi.uni-regensburg.de/images/institute/vwl/tschernig/lehre/Hinweise_Kurse_Noten_ab_ SS2010.pdf. Final Exam early in August: precise date and time will be announced. Prerequisites Okonometrie¨ I or even better Methoden der Okonometrie¨ or comparable courses. Literature • Basic textbook(s): { Cochrane, J.H. (2005). Asset Pricing, rev. ed., Princeton University Press. 3 Applied Financial Econometrics | General Information | U Regensburg | July 2012 4 { Kirchg¨assner,G. and Wolters, J. (2008, 2007). Introduction to modern time series anal- ysis, Springer, Berlin. (In the campus network full text available) { L¨utkepohl, Helmut und Kr¨atzig,Markus (2004, 2008). Applied Time Series Econometrics, Cambridge University Press. • Introduction to software R: { Kleiber, C. and Zeileis, A. (2008). Applied econometrics with R, Springer, New York. (In the campus network full text available) { Ligges, U. (2008). Programmieren mit R, Springer, Berlin. (In the campus network full text available) Additional Reading: • Introductory level: { Brooks, C. (2008). Introductory econometrics for finance, 2nd ed., Cambridge University Press. { Diebold, F.X. (2007). Elements of forecasting, 4. ed., Thomson/South-Western. { Wooldridge, J.M. (2009). Introductory Econometrics. A Modern Approach, Thomson South-Western. 4 Applied Financial Econometrics | General Information | U Regensburg | July 2012 5 • Graduate level: { Campbell, J.Y., A.W. Lo, and A.C. MacKinlay (1997). The Econometrics of Financial Markets, Princeton University Press. { Enders, W. (2010). Applied econometric time series, Wiley. { Franke, J., H¨ardle,W., and Hafner, C. (2011). Statistics of financial markets. An intro- duction, Springer, (Advanced, old edition in German available) { Tsay, R.S. (2010). Analysis of financial time series, Wiley. • German Reading: { Franke, J., H¨ardle, W., and Hafner, C. (2004). Einf¨uhrungin die Statistik der Fi- nanzm¨arkte, 2. ed., Springer. (Advanced, newer English edition available) { Kreiß, J.-P. and Neuhaus, G. (2006). Einf¨uhrungin die Zeitreihenanalyse, Springer. (Ad- vanced, In the campus network full text available) { Neusser, K. (2009). Zeitreihenanalyse in den Wirtschaftswissenschaften, 2. Auflage, Teubner. (In the campus network full text available, many copies of the first edition (2006) available for lending) 5 Applied Financial Econometrics | 1 Introduction | U Regensburg | July 2012 6 1 Introduction 1.1 Themes • How to measure returns and risks of financial assets? • Are asset returns predictable? In the short run - in the long run? −! requires command of time series econometrics • Does the risk of an asset vary with time? Is this important? How can one model time-varying risk? • Is the equity premium (excess returns of stocks over bonds) really that high? • How can one explain variations in stock returns across various stocks? For outline of the course see Contents 6 Applied Financial Econometrics | 1.1 Themes | U Regensburg | July 2012 7 This course provides an introduction to the basics of financial econometrics, mainly to analyzing financial time series. There are many more topics in financial econometrics that cannot be covered by this course but are treated in advanced textbooks such as Franke et al.(2011) or Tsay(2010). A selection of advanced topics not treated here is: • Statistics of extreme risks • Credit risk management and probability of default • Interest rate models and term structure models • Analyzing high-frequency data and modeling market microstructure • Analyzing and estimating models for options • Multivariate time series models • Technical methods such as state-space models and the Kalman filter, principal components and factor models, copulae, nonparametric methods, .... 7 Applied Financial Econometrics | 1.2 Some basics | U Regensburg | July 2012 8 1.2 Some basics • Return Rt (or gross return) Pt + Dt Rt = Pt−1 • Net return (Pt − Pt−1) + Dt = Rt − 1 Pt−1 • Log returns rt or continuously compounded returns @ ln(x) 1 { Recall: ln(1) = 0, @x = x. Taking a Taylor expansion of degree 1 at x0 delivers @ ln(x) 1 ln x ≈ ln x0 + (x − x0) = ln x0 + (x − x0) @x jx0 x0 Thus, expanding at x0 = 1, one has for x close to 1 ln x ≈ x − 1 { Replacing x by Rt gives rt = log(Rt) ≈ Rt − 1 8 Applied Financial Econometrics | 1.2 Some basics | U Regensburg | July 2012 9 • Real prices Pt real pricet(t) = CPIt Note that if real prices should be given in prices of year s, then one has to compute CPIs real pricet(s) = Pt CPIt • Real return Pt=CP It + Dt=CP It Pt + Dt CPIt−1 CPIt−1 real returnt = = = Rt Pt−1=CP It−1 Pt−1 CPIt CPIt • Log real returns CPIt−1 logged real returnt = log Rt CPIt = log(Rt) + log CPIt−1 − log CPIt = rt + log CPIt−1 − log CPIt • Excess log returns of asset A over asset B A B excess log returnt = log(Rt ) − log(Rt ) A B = rt − rt A B = rt + log CPIt−1 − log CPIt − rt + log CPIt−1 − log CPIt = excess log real returnt 9 Applied Financial Econometrics | 1.2 Some basics | U Regensburg | July 2012 10 A first look at data: S&P 500 Composite Index Real prices of the S&P 500 Composite and real earnings, January 1871 { March 2012, Source: homepage of Robert Shiller: www.econ.yale.edu/~shiller/ 2500 450 400 2000 350 300 1500 250 Price 200 1000 150 Real S&P Composite Earnings Real S&P Real S&P 500Stock Price Index Real S&P 100 500 50 Earnings 0 0 1870 1890 1910 1930 1950 1970 1990 2010 Year Are real prices RPt predictable? 10 Applied Financial Econometrics | 1.2 Some basics | U Regensburg | July 2012 11 • Estimating a simple model RPt = α0 + α1RPt−1 + ut; t = 1;:::; 1695 (1.1) Call: lm(formula = RP.t ~ 1 + RP.tm1) Residuals: Min 1Q Median 3Q Max -251.036 -5.128 0.356 6.094 155.825 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.707306 0.742315 0.953 0.341 RP.tm1 1.000171 0.001299 769.665 <2e-16 *** --- Signif. codes: 0 *** 0.001 ** 0.01 *
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