cambridge university press the econometric modelling of financial time series apr 2008 kho tài liệu bách khoa

This page intentionally left blank The Econometric Modelling of Financial Time Series Terence Mills’ best-selling graduate textbook provides detailed coverage of the latest research techniques and findings relating to the empirical analysis of financial markets In its previous editions it has become required reading for many graduate courses on the econometrics of financial modelling This third edition, co-authored with Raphael Markellos, contains a wealth of new material reflecting the developments of the last decade Particular attention is paid to the wide range of non-linear models that are used to analyse financial data observed at high frequencies and to the long memory characteristics found in financial time series The central material on unit root processes and the modelling of trends and structural breaks has been substantially expanded into a chapter of its own There is also an extended discussion of the treatment of volatility, accompanied by a new chapter on non-linearity and its testing Terence C Mills is Professor of Applied Statistics and Econometrics at Loughborough University He is the co-editor of the Palgrave Handbook of Econometrics and has over 170 publications Raphael N Markellos is Senior Lecturer in Quantitative Finance at Athens University of Economics and Business, and Visiting Research Fellow at the Centre for International Financial and Economic Research (CIFER), Loughborough University The Econometric Modelling of Financial Time Series Third edition Terence C Mills Professor of Applied Statistics and Econometrics Department of Economics Loughborough University Raphael N Markellos Senior Lecturer in Quantitative Finance Department of Management Science and Technology Athens University of Economics and Business CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521883818 © Terence C Mills and Raphael N Markellos 2008 This publication is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2008 ISBN-13 978-0-511-38103-4 eBook (Adobe Reader) ISBN-13 978-0-521-88381-8 hardback ISBN-13 978-0-521-71009-1 paperback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate Contents List of figures List of tables Preface to the third edition Introduction 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Univariate linear stochastic models: basic concepts Stochastic processes, ergodicity and stationarity Stochastic difference equations ARMA processes Linear stochastic processes ARMA model building Non-stationary processes and ARIMA models ARIMA modelling Seasonal ARIMA modelling Forecasting using ARIMA models 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Univariate linear stochastic models: testing for unit roots and alternative trend specifications Determining the order of integration of a time series Testing for a unit root Trend stationarity versus difference stationarity Other approaches to testing for unit roots Testing for more than one unit root Segmented trends, structural breaks and smooth transitions Stochastic unit root processes v Univariate linear stochastic models: further topics 4.1 Decomposing time series: unobserved component models and signal extraction page viii xi xiii 9 12 14 28 28 37 48 53 57 65 67 69 85 89 96 98 105 111 111 vi Contents 4.2 Measures of persistence and trend reversion 4.3 Fractional integration and long memory processes 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Univariate non-linear stochastic models: martingales, random walks and modelling volatility Martingales, random walks and non-linearity Testing the random walk hypothesis Measures of volatility Stochastic volatility ARCH processes Some models related to ARCH The forecasting performance of alternative volatility models 124 134 151 151 153 157 166 174 199 204 6.3 6.4 6.5 Univariate non-linear stochastic models: further models and testing procedures Bilinear and related models Regime-switching models: Markov chains and smooth transition autoregressions Non-parametric and neural network models Non-linear dynamics and chaos Testing for non-linearity 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 Modelling return distributions Descriptive analysis of returns series Two models for returns distributions Determining the tail shape of a returns distribution Empirical evidence on tail indices Testing for covariance stationarity Modelling the central part of returns distributions Data-analytic modelling of skewness and kurtosis Distributional properties of absolute returns Summary and further extensions 247 248 249 254 257 261 264 266 268 271 8.1 8.2 8.3 8.4 Regression techniques for non-integrated financial time series Regression models ARCH-in-mean regression models Misspecification testing Robust estimation 274 274 287 293 304 6.1 6.2 206 207 216 223 232 235 vii Contents 8.5 The multivariate linear regression model 8.6 Vector autoregressions 8.7 Variance decompositions, innovation accounting and structural VARs 8.8 Vector ARMA models 8.9 Multivariate GARCH models 316 319 323 Regression techniques for integrated financial time series Spurious regression Cointegrated processes Testing for cointegration in regression Estimating cointegrating regressions VARs with integrated variables Causality testing in VECMs Impulse response asymptotics in non-stationary VARs Testing for a single long-run relationship Common trends and cycles 329 330 338 346 352 356 373 375 377 383 Further topics in the analysis of integrated financial time series 10.1 Present value models, excess volatility and cointegration 10.2 Generalisations and extensions of cointegration and error correction models 388 388 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10 307 309 Data appendix References Index 401 411 412 446 Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 viii ACFs and simulations of AR(1) processes page 15 Simulations of MA(1) processes 18 ACFs of various AR(2) processes 20 Simulations of various AR(2) processes 22 Simulations of MA(2) processes 25 Real S&P returns (annual 1872–2006) 31 UK interest rate spread (monthly March 1952–December 2005) 32 Linear and quadratic trends 41 Explosive AR(1) model 42 Random walks 43 ‘Second difference’ model 46 ‘Second difference with drift’ model 47 Dollar/sterling exchange rate (daily January 1993–December 2005) 50 FTA All Share index (monthly 1965–2005) 51 Autocorrelation function of the absolute returns of the GIASE (intradaily, June–10 September 1998) 54 Autocorrelation function of the seasonally differenced absolute returns of the GIASE (intradaily, June–10 September 1998) 55 Nord Pool spot electricity prices and returns (daily averages, 22 March 2002–3 December 2004) À 56 Á Simulated limiting distribution of T ^T À 75 Simulated limiting distribution of  76 Simulated limiting distribution of  „ 77 FTA All Share index dividend yield (monthly 1965–2005) 84 Simulated limiting distribution of   86 UK interest rates (monthly 1952–2005) 97 Logarithms of the nominal S&P 500 index (1871–2006) with a smooth transition trend superimposed 103 Nikkei 225 index prices and seven-year Japanese government bond yields (end of year 1914–2003) 108 430 References Jones, C S (2003), ‘The Dynamics of Stochastic Volatility: Evidence from Underlying and Options Markets’, Journal of Econometrics, 116, 181–224 Jones, R (1978), ‘Nonlinear Autoregressive Processes’, Proceedings of the Royal Society of London, A, 360, 71–95 Jorion, P (1988), ‘On Jump Processes in the Foreign Exchange and Stock Markets’, Review of Financial Studies, 1, 427–45 Judge, G G., Griffiths, W E., Carter Hill, R., Luătkepohl, H., and Lee, T C (1985), The Theory and Practice of Econometrics, 2nd edn., New York: Wiley Kapetanios, G., Shin, Y., and Snell, A (2006), ‘Testing for Cointegration in Nonlinear Smooth Transition Error Correction Models’, Econometric Theory, 22, 279–303 Karanasos, M., Psaradakis, Z., and Sola, M (2004), ‘On the Autocorrelation Properties of Long-memory GARCH Processes’, Journal of Time Series Analysis, 25, 265–81 Keenan, D M (1985), ‘A Tukey Nonadditivity-type Test for Time Series Nonlinearity’, Biometrika, 72, 39–44 Kendall, M J (1953), ‘The Analysis of Economic Time Series, Part I: Prices’, Journal of the Royal Statistical Society, Series A, 96, 11–25 Kim, C J., Nelson, C R., and Startz, R (1998), ‘Testing for Mean Reversion in Heteroskedastic Data Based on Gibbs-sampling-augmented Randomization’, Journal of Empirical Finance, 5, 131–54 Kim, K., and Schmidt, P (1990), ‘Some Evidence on the Accuracy of Phillips–Perron Tests Using Alternative Estimates of Nuisance Parameters’, Economics Letters, 34, 345–50 Kleidon, A W (1986a), ‘Variance Bounds Tests and Stock Price Valuation Models’, Journal of Political Economy, 94, 953–1001 (1986b), ‘Bias in Small Sample Tests of Stock Price Rationality’, Journal of Business, 59, 237–61 Koedijk, K G., and Kool, C J M (1992), ‘Tail Estimates of East European Exchange Rates’, Journal of Business and Economic Statistics, 10, 83–96 Koedijk, K G., Schafgans, M M A., and de Vries, C G (1990), ‘The Tail Index of Exchange Rate Returns’, Journal of International Economics, 29, 93–108 Koedijk, K G., Stork, P A., and de Vries, C G (1992), ‘Differences between Foreign Exchange Rate Regimes: The View from the Tails’, Journal of International Money and Finance, 11, 462–73 Koenker, R (1982), ‘Robust Methods in Econometrics’, Econometric Reviews, 1, 213–55 Kokoszka, P S., and Taqqu, M S (1994), ‘Infinite Variance Stable ARMA Processes’, Journal of Time Series Analysis, 15, 203–20 (1996), ‘Infinite Variance Stable Moving Averages with Long Memory’, Journal of Econometrics, 73, 79–99 Kon, S (1984), ‘Models of Stock Returns: A Comparison’, Journal of Finance, 39, 147–65 Koop, G (1992), ‘ ‘‘Objective’’ Bayesian Unit Root Tests’, Journal of Applied Econometrics, 7, 65–82 Koop, G., Pesaran, M H., and Potter, S M (1996), ‘Impulse Response Analysis in Nonlinear Multivariate Models’, Journal of Econometrics, 74, 119–47 Koop, G., and Potter, S M (2001), ‘Are Apparent Findings of Nonlinearity Due to Structural Instability in Economic Time Series?’, Econometrics Journal, 4, 37–55 431 References Koopman, S J., Harvey, A C., Doornik, J A., and Shephard, N (2006), Stamp 6: Structural Time Series Analyser, Modeller and Predictor, London: Timberlake Consultants Koopman, S J., Jungbacker, B., and Hol, E (2005), ‘Forecasting Daily Variability of the S&P 100 Stock Index Using Historical, Realised and Implied Volatility Measurements’, Journal of Empirical Finance, 12, 445–75 Koopman, S J., Shephard, N., and Doornik, J A (1999), ‘Statistical Algorithms for Models in State Space Using SsfPack 2.2, Econometrics Journal, 2, 11366 Kraămer, W., and Ploberger, W (1990), ‘The Local Power of CUSUM and CUSUM of Squares Tests’, Econometric Theory, 6, 335–47 Kristensen, D (2005), On Stationarity and Ergodicity of the Bilinear Model with Applications to GARCH Models, Working Paper, University of Wisconsin Kristensen, D., and Linton, O (2006), ‘A Closed-form Estimator for the GARCH(1,1)Model’, Econometric Theory, 22, 323–37 Kuan, C.-M., and White, H (1994), ‘Artificial Neural Networks: An Econometric Perspective (with Discussion)’, Econometric Reviews, 13, 1143 Kuănsch, H R (1987), Statistical Aspects of Self-similar Processes, in Yu Prohorov and V V Sazanov (eds.), Proceedings of the First World Congress of the Bernoulli Society, Utrecht: VNU Science Press, 67–74 Kwiatkowski, D., Phillips, P C B., Schmidt, P., and Shin, Y (1992), ‘Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root’, Journal of Econometrics, 54, 159–78 Kyrtsou, C., and Serletis, A (2006), ‘Univariate Tests for Nonlinear Structure’, Journal of Macroeconomics, 28, 154–68 Lam, P S (1990), ‘The Hamilton Model with a General Autoregressive Component: Estimation and Comparison with Other Models of Economic Time Series’, Journal of Monetary Economics, 20, 409–32 Lanne, M., and Saikkonen, P (2005), ‘Nonlinear GARCH Models for Highly Persistent Volatility’, Econometrics Journal, 8, 251–76 Lee, D., and Schmidt, P (1996), ‘On the Power of the KPSS Test of Stationarity against Fractionally Integrated Alternatives’, Journal of Econometrics, 73, 285–302 Lee, J H H., and King, M L (1993), ‘A Locally Most Mean Powerful Based Score Test for ARCH and GARCH Regression Disturbances’, Journal of Business and Economic Statistics, 11, 17–27 Lee, T.-H (1994), ‘Spread and Volatility in Spot and Forward Exchange Rates’, Journal of International Money and Finance, 13, 375–83 Lee, T.-H., White, H., and Granger, C W J (1993), ‘Testing for Neglected Nonlinearity in Time Series Models: A Comparison of Neural Network Methods and Alternative Tests’, Journal of Econometrics, 56, 269–90 Lequeux, P (ed.) (1999), The Financial Markets Tick by Tick: Insights in Financial Markets Microstructure, Chichester: Wiley LeRoy, S (1973), ‘Risk Aversion and the Martingale Property of Stock Returns’, International Economic Review, 14, 436–46 (1982), ‘Expectations Models of Asset Prices: A Survey of the Theory’, Journal of Finance, 37, 185–217 432 References (1989) ‘Efficient Capital Markets and Martingales’, Journal of Economic Literature, 27, 1583–621 Levy, H., and Markowitz, H M (1979), ‘Approximating Expected Utility by a Function of Mean and Variance’, American Economic Review, 69, 308–17 Leybourne, S J., and McCabe, B P M (1994), ‘A Consistent Test for a Unit Root’, Journal of Business and Economic Statistics, 12, 157–66 (1999) ‘Modified Stationarity Tests with Data Dependent Model Selection Rules’, Journal of Business and Economic Statistics, 17, 264–70 Leybourne, S J., McCabe, B P M., and Mills, T C (1996), ‘Randomized Unit Root Processes for Modelling and Forecasting Financial Time Series: Theory and Applications’, Journal of Forecasting, 15, 253–70 Leybourne, S J., McCabe, B P M., and Tremayne, A R (1996), ‘Can Economic Time Series be Differenced to Stationarity?’, Journal of Business and Economic Statistics, 14, 435–46 Leybourne, S J., Mills, T C., and Newbold, P (1998), ‘Spurious Rejections by Dickey–Fuller Tests in the Presence of a Break under the Null’, Journal of Econometrics, 87, 191–203 Leybourne, S J., Newbold, P., and Vougas, D (1998), ‘Unit Roots and Smooth Transitions’, Journal of Time Series Analysis, 19, 83–97 Li, C W., and Li, W K (1996), ‘On a Double Threshold Autoregressive Heteroskedasticity Time Series Model’, Journal of Applied Econometrics, 11, 253–74 Li, Q., Yang, J., Hsiao, C., and Chang, Y.-J (2005), ‘The Relationship between Stock Returns and Volatility in International Stock Markets’, Journal of Empirical Finance, 12, 650–65 Li, W K., Ling, S., and McAleer, M (2002), ‘Recent Theoretical Results for Time Series Models with GARCH Errors’, Journal of Economic Surveys, 16, 245–69 Li, W K., and Mak, T K (1994), ‘On the Squared Residual Autocorrelations in Nonlinear Time Series with Conditional Heteroskedasticity’, Journal of Time Series Analysis, 15, 627–36 Lieberman, O (2001), ‘The Exact Bias of the Log-periodogram Regression Estimator’, Econometric Reviews, 20, 369–83 Liesenfeld, R., and Richard, J F (2003), ‘Univariate and Multivariate Stochastic Volatility Models: Estimation and Diagnostics’, Journal of Empirical Finance, 11, 1–27 Ling, S (2007), ‘Self-weighted and Local Quasi-maximum Likelihood Estimators for ARMAGARCH/IGARCH Models’, Journal of Econometrics, 140, 849–73 Ling, S., and McAleer, M (2003), ‘Asymptotic Theory for a Vector ARMA-GARCH Model’, Econometric Theory, 19, 280–310 Linton, O (1993), ‘Adaptive Estimation of ARCH Models’, Econometric Theory, 9, 539–69 (2007), ‘Semi- and Nonparametric ARCH/GARCH Modelling’, in T G Andersen, R A Davis, J P Kreiss and T Mikosch (eds.), Handbook of Financial Time Series, New York: Springer Linton, O., and Perron, B (2003), ‘The Shape of the Risk Premium: Evidence from a Semiparametric Generalized Autoregressive Conditional Heteroscedasticity Model’, Journal of Business and Economic Statistics, 21, 354–67 Liu, C Y., and He, J (1991), ‘A Variance Ratio Test of Random Walks in Foreign Exchange Rates’, Journal of Finance, 46, 773–85 Liu, M (2000), ‘Modeling Long Memory in Stock Market Volatility’, Journal of Econometrics, 99, 139–71 433 References Ljung, G M., and Box, G E P (1978), ‘On a Measure of Lack of Fit in Time Series Models’, Biometrika, 65, 297–303 Lo, A W (1991), ‘Long-term Memory in Stock Market Prices’, Econometrica, 59, 1279–313 Lo, A W., and MacKinlay, A C (1988), ‘Stock Prices not Follow Random Walks: Evidence from a Simple Specification Test’, Review of Financial Studies, 1, 41–66 (1989), ‘The Size and Power of the Variance Ratio Test in Finite Samples: A Monte Carlo Investigation’, Journal of Econometrics, 40, 203–38 Lobato, I N., Nankervis, J C., and Savin, N E (2001), ‘Testing for Autocorrelation Using a Modified Box–Pierce Q Test’, International Economic Review, 42, 187–205 (2002), ‘Testing for Zero Autocorrelation in the Presence of Statistical Dependence’, Econometric Theory, 18, 730–43 Loretan, M., and Phillips, P C B (1994), ‘Testing the Covariance Stationarity of Heavytailed Time Series: An Overview of the Theory with Applications to Several Financial Datasets’, Journal of Empirical Finance, 1, 211–48 Lucas, R E (1978), ‘Asset Prices in an Exchange Economy’, Econometrica, 46, 1429–46 Luger, R (2003), ‘Exact Non-parametric Tests for a Random Walk with Unknown Drift under Conditional Heteroskedasticity’, Journal of Econometrics, 115, 259–76 Lumsdaine, R L., and Ng, S (1999), ‘Testing for ARCH in the Presence of a Possibly Mispecified Conditional Mean’, Journal of Econometrics, 93, 257–79 Lundbergh, S., and Teraăsvirta, T (2002), Evaluating GARCH Models, Journal of Econometrics, 110, 41735 Luătkepohl, H (1985), Comparison of Criteria for Estimating the Order of a Vector Autoregresive Process’, Journal of Time Series Analysis, 6, 35–52 (1991), Introduction to Multiple Time Series Analysis, Berlin: Springer-Verlag MacKinlay, A C (1987), ‘On Mulivariate Tests of the CAPM’, Journal of Financial Economics, 18, 431–71 MacKinnon, J G (1991), ‘Critical Values for Cointegration Tests’, in R F Engle and C W J Granger (eds.), Long-run Economic Relationships, Oxford: Oxford University Press, 267–76 (1996), ‘Numerical Distribution Functions for Unit Root and Cointegration Tests’, Journal of Applied Econometrics, 11, 601–18 McCabe, B P M., and Tremayne, A R (1995), ‘Testing a Time Series for Difference Stationarity’, Annals of Statistics, 23, 1015–28 McCulloch, J H (1996), ‘Financial Applications of Stable Distributions’, in G S Maddala and C R Rao (eds.), Handbook of Statistics, Vol XIV: Statistical Methods in Finance, Amsterdam: Elsevier Science, 393–425 (1997), ‘Measuring Tail Thickness to Estimate the Stable Index fi: A Critique’, Journal of Business and Economic Statistics, 15, 74–81 McDonald, J B (1996), ‘Probability Distributions for Financial Models’, in G S Maddala and C R Rao (eds.), Handbook of Statistics, Vol XIV: Statistical Methods in Finance, Amsterdam: Elsevier Science, 427–61 McKenzie, E (1988), ‘A Note on Using the Integrated Form of ARIMA Forecasts’, International Journal of Forecasting, 4, 117–24 McLeod, A J., and Li, W K (1983), ‘Diagnostic Checking ARMA Time Series Models using Squared-residual Correlations’, Journal of Time Series Analysis, 4, 269–73 434 References Maddala, G S., and Kim, I.-M (1998), Unit Roots, Cointegration, and Structural Change, Cambridge: Cambridge University Press Malkiel, B G (2003) A Random Walk down Wall Street, New York: Norton Mandelbrot, B B (1963a), ‘New Methods in Statistical Economics’, Journal of Political Economy, 71, 421–40 (1963b), ‘The Variation of Certain Speculative Prices’, Journal of Business, 36, 394–419 (1966), ‘Forecasts of Future Prices, Unbiased Markets and ‘‘Martingale’’ Models’, Journal of Business, 39, 242–55 (1969), ‘Long-run Linearity, Locally Gaussian Process, H-Spectra, and Infinite Variances’, International Economic Review, 10, 82–111 (1972), ‘Statistical Methodology for Nonperiodic Cycles: From the Covariance to R/S Analysis’, Annals of Economic and Social Measurement, 1/3, 259–90 (1989), ‘Louis Bachelier’, in J Eatwell, M Milgate and P Newman (eds.), The New Palgrave: Finance, London: Macmillan, 86–8 Mandelbrot, B B., and Wallis, J R (1969), ‘Some Long-run Properties of Geophysical Records’, Water Resources Research, 5, 321–40 Mantegna, R N., and Stanley, H E (1994), ‘Stochastic Process with Ultraslow Convergence to a Gaussian: The Truncated Le´vy Flight’, Physical Review Letters, 73, 2946–9 (1995), ‘Scaling Behaviour in the Dynamics of an Economic Index’, Nature, 376, 46–9 Maravall, A (1983), ‘An Application of Nonlinear Time Series Forecasting’, Journal of Business and Economic Statistics, 3, 350–5 Markellos, R N., and Mills, T C (1998), ‘Complexity Reduction for Co-trending Variables’, Journal of Computational Intelligence in Finance, 6, 6–13 (2001), ‘Unit Roots in the CAPM?’, Applied Economics Letters, 8, 499–502 (2003), ‘Asset Pricing Dynamics’, European Journal of Finance, 9, 533–56 Markellos, R N., Mills, T C., and Siriopoulos, C (2003), ‘Intradaily Behavior of Listed and Unlisted Security Basket Indices in the Emerging Greek Stock Market’, Managerial Finance, 29, 29–54 Marsh, T A., and Merton, R C (1987), ‘Dividend Behaviour for the Aggregate Stock Market’, Journal of Business, 60, 1–40 Matacz, A (2000), ‘Financial Modeling and Option Theory with the Truncated Le´vy Process’, International Journal of Theoretical and Applied Finance, 3, 143–60 Meddahi, N., and Renault, E (2004), ‘Temporal Aggregation of Volatility Models’, Journal of Econometrics, 119, 355–79 Meese, R A (1986), ‘Testing for Bubbles in Exchange Markets: A Case of Sparkling Rates?’, Journal of Political Economy, 94, 345–73 Meese, R A., and Singleton, K J (1982), ‘On Unit Roots and the Empirical Modelling of Exchange Rates’, Journal of Finance, 37, 1029–35 Merton, R C (1973), ‘An Intertemporal Capital Asset Pricing Model’, Econometrica, 41, 867–87 (1976), ‘Option Prices when the Underlying Stock Returns Are Discontinuous’, Journal of Financial Economics, 3, 125–44 (1980), ‘On Estimating the Expected Return on the Market: An Exploratory Investigation’, Journal of Financial Economics, 8, 323–61 435 References Meyer, R., and Yu, J (2000), ‘BUGS for a Bayesian Analysis of Stochastic Volatility Models’, Econometrics Journal, 3, 198–215 Michael, P., Nobay, A R., and Peel, D A (1997), ‘Transactions Costs and Nonlinear Adjustment in Real Exchange Rates: An Empirical Investigation’, Journal of Political Economy, 105, 862–79 Mikosch, T., and Starica, C (2004), ‘Changes of Structure in Financial Time Series and the GARCH Model’, Revstat Statistical Journal, 2, 41–73 Milhøj, A (1985), ‘The Moment Structure of ARCH Processes’, Scandinavian Journal of Statistics, 12, 281–92 Miller, S M (1988), ‘The Beveridge–Nelson Decomposition of Economic Time Series: Another Economical Computational Method’, Journal of Monetary Economics, 21, 141–2 Mills, T C (1990), Time Series Techniques for Economists, Cambridge: Cambridge University Press (1991a), ‘Equity Prices, Dividends and Gilt Yields in the UK: Cointegration, Error Correction and ‘‘Confidence’’ ’, Scottish Journal of Political Economy, 38, 242–55 (1991b), ‘The Term Structure of UK Interest Rates: Tests of the Expectations Hypothesis’, Applied Economics, 23, 599–606 (1993), ‘Testing the Present Value Model of Equity Prices for the UK Stock Market’, Journal of Business Finance and Accounting, 20, 803–13 (1995), ‘Modelling Skewness and Kurtosis in the London Stock Exchange FT-SE Index Return Distributions’, The Statistician, 44, 323–32 (1996a), ‘Non-linear Forecasting of Financial Time Series: An Overview and Some New Models’, Journal of Forecasting, 15, 127–35 (1996b), ‘The Econometrics of the ‘‘Market Model’’: Cointegration, Error Correction and Exogeneity’, International Journal of Finance and Economics, 1, 275–86 (1997a), ‘Stylized Facts on the Temporal and Distributional Properties of Daily FT-SE Returns’, Applied Financial Economics, 7, 599–604 (1997b), ‘Technical Analysis and the London Stock Exchange: Testing Trading Rules Using the FT30’, International Journal of Finance and Economics, 2, 319–31 (1998), ‘Recent Developments in Modelling Nonstationary Vector Auto-regressions’, Journal of Economic Surveys, 12, 279–312 (2003), Modelling Trends and Cycles in Economic Time Series, Basingstoke: Palgrave Macmillan Mills, T C., and Coutts, J A (1995), ‘Anomalies and Calendar Effects in the New FT-SE Indices’, European Journal of Finance, 1, 79–93 (1996), ‘Misspecification and Robust Estimation of the Market Model: Estimating Betas for the FT-SE Industry Baskets’, European Journal of Finance, 2, 319–31 Mills, T C., and Stephenson, M J (1986), ‘An Empirical Analysis of the UK Treasury Bill Market’, Applied Economics, 17, 689–703 Mina, J., and Xiao, J Y (2001), Return to RiskMetrics: The Evolution of a Standard, New York: RiskMetrics Group Mittnik, S., and Rachev, S T (1993a), ‘Modeling Asset Returns with Alternative Stable Distributions’, Econometric Reviews, 12, 261–330 (1993b), ‘Reply to Comments on ‘‘Modeling Asset Returns with Alternative Stable Distributions’’ and Some Extensions’, Econometric Reviews, 12, 347–89 436 References Modha, D S., and Fainman, Y (1994), ‘A Learning Law for Density Estimation’, IEEE Transactions on Neural Networks, 5, 519–23 Morgan, M S (1990), The History of Econometric Ideas, Cambridge: Cambridge University Press Muth, J F (1960), ‘Optimal Properties of Exponentially Weighted Forecasts’, Journal of the American Statistical Association, 55, 299–305 Nawrocki, D N (1999), ‘A Brief History of Downside Risk Measures’, Journal of Investing, 8, 9–25 Nelson, C R., and Plosser, C I (1982), ‘Trends and Random Walks in Macroeconomic Time Series’, Journal of Monetary Economics, 10, 139–62 Nelson, C R., and Schwert, G W (1977), ‘Short-term Interest Rates as Predictors of Inflation: On Testing the Hypothesis that the Real Rate of Interest is Constant’, American Economic Review, 67, 478–86 Nelson, D B (1990a), ‘Stationarity and Persistence in the GARCH(1,1) Model’, Econometric Theory, 6, 318–34 (1990b), ‘ARCH Models as Diffusion Approximations’, Journal of Econometrics, 45, 7–38 (1991), ‘Conditional Heteroskedasticity in Asset Returns’, Econometrica, 59, 347–70 Nelson, D B., and Cao, C Q (1992), ‘Inequality Constraints in Univariate GARCH Models’, Journal of Business and Economic Statistics, 10, 229–35 Nelson, D B., and Foster, D P (1994), ‘Asymptotic Filtering for Univariate GARCH Models’, Econometrica, 62, 1–41 Nelson, S A (1902), The ABC of Stock Speculation, New York: S A Nelson Newbold, P (1990), ‘Precise and Efficient Computation of the Beveridge–Nelson Decomposition of Economic Time Series’, Journal of Monetary Economics, 26, 453–7 Newbold, P., and Agiakloglou, C (1993), ‘Bias in the Sample Autocorrelations of Fractional White Noise’, Biometrika, 80, 698–702 Newey, W K (1985), ‘Generalized Method of Moment Specification Testing’, Journal of Econometrics, 29, 229–56 Newey, W K., and West, K D (1987), ‘A Simple Positive Semidefinite, Heteroskedasticity Consistent Covariance Matrix’, Econometrica, 55, 703–8 Ng, S., and Perron, P (1995), ‘Unit Root Tests in ARMA Models with Data Dependent Methods for Good Size and Power’, Journal of the American Statistical Association, 90, 268–81 (2001), ‘Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power’, Econometrica, 69, 1519–54 Nickell, S (1985), ‘Error Correction, Partial Adjustment and All That: An Expository Note’, Oxford Bulletin of Economics and Statistics, 47, 119–29 Nychka, D W., Stephen, E., Gallant, A R., and McCaffrey, D F (1992), ‘Finding Chaos in Noisy Systems’, Journal of the Royal Statistical Society, Series B, 54, 399–426 Osborne, M M (1959), ‘Brownian Motion in the Stock Market’, Operations Research, 7, 145–73 Pagan, A R (1996), ‘The Econometrics of Financial Markets’, Journal of Empirical Finance, 3, 15–102 Pagan, A R., and Schwert, G W (1990), ‘Testing for Covariance Stationarity in Stock Market Data’, Economics Letters, 33, 165–70 437 References Park, J Y (1990), ‘Testing for Unit Roots and Cointegration by Variable Addition’, in G F Rhodes and T B Fomby (eds.), Advances in Econometrics, Vol VIII, Greenwich, CT: JAI Press, 107–33 Park, J Y., and Phillips, P C B (1988), ‘Statistical Inference in Regressions with Cointegrated Processes: Part I’, Econometric Theory, 4, 468–97 (1999) ‘Asymptotics for Nonlinear Transformations of Integrated Time Series’, Econometric Theory, 15, 269–98 (2001), ‘Nonlinear Regressions with Integrated Time Series’, Econometrica, 69, 117–61 Pearson, K., and Rayleigh, Lord (1905), ‘The Problem of the Random Walk’, Nature, 72, 294, 318, 342 Pena, D., and Rodriguez, J (2005), ‘Detecting Nonlinearity in Time Series by Model Selection Criteria’, International Journal of Forecasting, 21, 731–48 Perron, P (1988), ‘Trends and Random Walks in Macroeconomic Time Series: Further Evidence from a New Approach’, Journal of Economic Dynamics and Control, 12, 297– 332 (1989), ‘The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis’, Econometrica, 57, 1361–401 (1990), ‘Testing for a Unit Root in a Time Series with a Changing Mean’, Journal of Business and Economic Statistics, 8, 153–62 (1991), ‘Test Consistency with Varying Sampling Frequency’, Econometric Theory, 7, 341–68 (1997), ‘Further Evidence on Breaking Trend Functions in Macroeconomic Variables’, Journal of Econometrics, 80, 355–85 (2006), ‘Dealing with Structural Breaks’, in T C Mills and K Patterson (eds.), Palgrave Handbook of Econometrics, Vol I: Econometric Theory, Basingstoke: Palgrave Macmillan, 278–352 Perron, P., and Ng, S (1996), ‘Useful Modifications to Some Unit Root Tests with Dependent Errors and their Local Asymptotic Properties’, Review of Economic Studies, 63, 435–63 Perron, P., and Vogelsang, T J (1993), ‘A Note on the Asymptotic Distributions in the Additive Outlier Model with Breaks’, Revista de Econometrica, 8, 181–202 Pesaran, M H (1997), ‘The Role of Economic Theory in Modelling the Long Run’, Economic Journal, 107, 178–91 Pesaran, M H., and Pesaran, B (1997), Working with Microfit 4.0: Interactive Econometric Analysis, Oxford: Oxford University Press Pesaran, M H., and Shin, Y (1997), ‘Generalized Impulse Response Analysis in Linear Multivariate Models’, Economics Letters, 58, 17–29 (2002), ‘Long Run Structural Modelling’, Econometric Reviews, 21, 49–87 Pesaran, M H., Shin, Y., and Smith, R J (2001), ‘Bounds Testing Approaches to the Analysis of Levels Relationships’, Journal of Applied Econometrics, 16, 289–326 Pfann, G A., Schotman, P C., and Tschernig, R (1996), ‘Nonlinear Interest Rate Dynamics and Implications for the Term Structure’, Journal of Econometrics, 74, 149–76 Pham, D T (1993) ‘Bilinear Times Series Models’, in H Tong, (ed.), Dimension Estimation and Models, Singapore: World Scientific Publishing, 191–223 Phillips, A W (1957), ‘Stabilisation Policy and the Time Form of Lagged Responses’, Economic Journal, 67, 265–77 438 References Phillips, P C B (1986), ‘Understanding Spurious Regressions in Econometrics’, Journal of Econometrics, 33, 311–40 (1987a), ‘Time Series Regression with a Unit Root’, Econometrica, 55, 227–301 (1987b), ‘Towards a Unified Asymptotic Theory for Autoregression’, Biometrika, 74, 535–47 (1987c), ‘Asymptotic Expansions in Nonstationary Vector Autoregressions’, Econometric Theory, 3, 45–68 (1990), ‘Time Series Regression with a Unit Root and Infinite-variance Errors’, Econometric Theory, 6, 44–62 (1991), ‘Optimal Inference in Co-integrated Systems’, Econometrica, 59, 282–306 (1998), ‘Impulse Response and Forecast Error Asymptotics in Nonstationary VARs’, Journal of Econometrics, 83, 21–56 (2001), ‘Trending Time Series and Macroeconomic Activity: Some Present and Future Challenges’, Journal of Econometrics, 100, 21–7 (2003), ‘Laws and Limits of Econometrics’, Economic Journal, 113, C26–C52 (2005a), ‘HAC Estimation by Automated Regression’, Econometric Theory, 21, 116–42 (2005b), ‘Challenges of Trending Time Series Econometrics’, Mathematics and Computers in Simulation, 68, 401–16 Phillips, P C B., and Durlauf, S N (1986), ‘Multiple Time Series Regression with Integrated Processes’, Review of Economic Studies, 53, 99–125 Phillips, P C B., and Hansen, B E (1990), ‘Statistical Inference in Instrumental Variables Regression with I(1) Processes’, Review of Economic Studies, 57, 99–125 Phillips, P C B., and Loretan, M (1991), ‘Estimating Long-run Economic Equilibria’, Review of Economic Studies, 58, 407–36 Phillips, P C B., McFarland, J W., and McMahon, P C (1996), ‘Robust Tests of Forward Exchange Market Efficiency with Empirical Evidence from the 1920s’, Journal of Applied Econometrics, 11, 1–22 Phillips, P C B., and Ouliaris, S (1988), ‘Testing for Cointegration Using Principal Components Methods’, Journal of Economic Dynamics and Control, 12, 205–30 (1990), ‘Asymptotic Properties of Residual Based Tests for Cointegration’, Econometrica, 58, 165–94 Phillips, P C B., and Perron, P (1988), ‘Testing for Unit Roots in Time Series Regression’, Biometrika, 75, 335–46 Phillips, P C B., and Xiao, Z (1998), ‘A Primer on Unit Root Testing’, Journal of Economic Surveys, 12, 423–69 Pierce, D A (1979), ‘Signal Extraction Error in Nonstationary Time Series’, Annals of Statistics, 7, 1303–20 Pierse, R G., and Snell, A J (1995), ‘Temporal Aggregation and the Power of Tests for a Unit Root’, Journal of Econometrics, 65, 333–45 Pinske, H (1998), ‘A Consistent Nonparametric Test for Serial Independence’, Journal of Econometrics, 84, 20531 Ploberger, W., Kraămer, W., and Kontrus, K (1989), ‘A New Test for Structural Stability in the Linear Regression Model’, Journal of Econometrics, 40, 307–18 Plosser, C I., and Schwert, G W (1978), ‘Money, Income, and Sunspots: Measuring Economic Relationships and the Effects of Differencing’, Journal of Monetary Economics, 4, 637–60 439 References Pollock, D S G (2000), ‘Trend Estimation and De-trending via Rational Square Wave Filters’, Journal of Econometrics, 99, 317–34 Pong, E., Shackleton, M B., Taylor, S J., and Xu, X (2004), ‘Forecasting Currency Volatility: A Comparison of Implied Volatilities and AR(F)IMA Models’, Journal of Banking and Finance, 28, 2541–63 Poon, S.-H., and Granger, C W J (2003) ‘Forecasting Financial Market Volatility: A Review’, Journal of Economic Literature, 41, 478–539 Poskitt, D S., and Tremayne, A R (1987), ‘Determining a Portfolio of Linear Time Series Models’, Biometrika, 74, 125–37 Poterba, J M., and Summers, L H (1988), ‘Mean Reversion in Stock Prices: Evidence and Implications’, Journal of Financial Economics, 22, 27–59 Priestley, M B (1980), ‘State-dependent Models: A General Approach to Nonlinear Time Series Analysis’, Journal of Time Series Analysis, 1, 47–71 (1988) Non-linear and Non-stationary Time Series Analysis, London: Academic Press Prigent, J.-L., Renault, O., and Scaillet, O (2001), ‘An Autoregressive Conditional Binomial Option Pricing Model’, in H Geman, D Madan, S Pliska and T Vorst (eds.), Selected Papers from the First World Congress of the Bachelier Finance Society, Heidelberg: Springer, 353–73 Psaradakis, Z., Sola, M., and Spagnolo, F (2004), ‘On Markov Error Correction Models with an Application to Stock Prices and Dividends’, Journal of Applied Econometrics, 19, 69–88 Psychoyios, D., Dotsis, G., and Markellos, R N (2006), Does the VIX Jump? Implications for Pricing and Hedging Volatility Risk, working paper, Athens University of Economics and Business Quintos, C E (1997), ‘Stability Tests in Error Correction Models’, Journal of Econometrics, 82, 289–315 Quintos, C E., and Phillips, P C B (1993), ‘Parameter Constancy in Cointegrating Regressions’, Empirical Economics, 18, 675–706 Rabemananjara, R., and Zakoian, J M (1993), ‘Threshold ARCH Models and Asymmetries in Volatility’, Journal of Applied Econometrics, 8, 31–49 Rachev, S T., Menn, C., and Fabozzi, F J (2005), Fat Tailed and Skewed Asset Distributions, New York: Wiley Ramsey, J B (1969), ‘Tests for Specification Errors in Classical Linear Least Squares Regression Analysis’, Journal of the Royal Statistical Society, Series B, 31, 350–71 Ramsey, J B., and Rothman, P (1996), ‘Time Irreversibility and Business Cycle Asymmetry’, Journal of Money, Credit, and Banking, 28, 1–21 Reimers, H.-E (1992), ‘Comparisons of Tests for Multivariate Cointegration’, Statistical Papers, 33, 335–59 Rhea, R (1932), The Dow Theory: An Explanation of its Development and Attempt to Define its Usefulness as an Aid in Speculation, New York: Barron’s Richardson, M (1993), ‘Temporary Components of Stock Prices: A Skeptic’s View’, Journal of Business and Economic Statistics, 11, 199–207 Richardson, M., and Smith, T (1991), ‘Tests of Financial Models in the Presence of Overlapping Observations’, Review of Financial Studies, 4, 27–54 Richardson, M., and Stock, J H (1989), ‘Drawing Inferences from Statistics Based on Multiyear Asset Returns’, Journal of Financial Economics, 25, 323–48 440 References Roberts, H V (1959), ‘Stock-Market ‘‘Patterns’’ and Financial Analysis: Methodo-logical Suggestions’, Journal of Finance, 14, 1–10 Robinson, P M (1977), ‘The Estimation of a Non-linear Moving Average Model’, Stochastic Processes and their Applications, 5, 81–90 (1991), ‘Testing for Strong Serial Correlation and Dynamic Conditional Heteroskedasticity in Multiple Regression’, Journal of Econometrics, 47, 67–84 (1992), ‘Semiparametric Analysis of Long Memory Time Series’, Annals of Statistics, 22, 515–39 (1994), ‘Time Series with Strong Dependence’, in C A Sims (ed.), Advances in Econometrics: Sixth World Congress, Vol I, Cambridge: Cambridge University Press, 47–95 (1995a), ‘Log-periodogram Regression for Time Series with Long Range Dependence’, Annals of Statistics, 23, 1048–72 (1995b), ‘Gaussian Semiparametric Estimation of Long Run Dependence’, Annals of Statistics, 23, 1630–61 (2003), ‘Long Memory Time Series’, in P M Robinson (ed.) Time Series with Long Memory, Oxford: Oxford University Press, 4–32 Rogers, L C G (1997), ‘Stochastic Calculus and Markov Methods’, in M A H Dempster and S R Pliska (eds.), Mathematics of Derivative Securities, Cambridge: Cambridge University Press, 15–40 Romano, J L., and Thombs, L A (1996), ‘Inference for Autocorrelations under Weak Assumptions’, Journal of the American Statistical Association, 89, 1303–131 Rothman, P (1992), ‘The Comparative Power of the TR Test against Simple Threshold Models’, Journal of Applied Econometrics, 7, S187–S195 Rousseeuw, P J., and LeRoy, A M (2003), Robust Regression and Outlier Detection, New York: Wiley Runde, R (1997), ‘The Asymptotic Null Distribution of the Box–Pierce Q-statistic for Random Variables with Infinite Variance: An Application to German Stock Returns’, Journal of Econometrics, 78, 205–16 Ruppert, D., and Carroll, R J (1980), ‘Trimmed Least Squares Estimation in the Linear Model’, Journal of the American Statistical Association, 75, 828–38 Ryden, T., Teraăsvirta, T., and Asbrink, S (1998), Stylized Facts of Daily Return Series and the Hidden Markov Model’, Journal of Applied Econometrics, 13, 217–44 Said, S E., and Dickey, D A (1984), ‘Testing for Unit Roots in Autoregressive Movingaverage Models with Unknown Order’, Biometrika, 71, 599–607 (1985), ‘Hypothesis Testing in ARIMA(p,1,q) Models’, Journal of the American Statistical Association, 80, 369–74 Saikkonen, P (1991), ‘Asymptotically Efficient Estimation of Cointegrating Regressions’, Econometric Theory, 7, 121 Saikkonen, P., and Luătkepohl, H (1996), ‘Infinite-order Cointegrated Vector Autoregressive Processes: Estimation and Inference’, Econometric Theory, 12, 814–44 Saikkonen, P., and Luukkonnen R (1993), ‘Testing for a Moving Average Unit Root in Autoregressive Integrated Moving Average Models’, Journal of the American Statistical Association, 88, 596–601 Samarov, A., and Taqqu, M S (1988), ‘On the Efficiency of the Sample Mean in Long Memory Noise’, Journal of Time Series Analysis, 9, 191–200 441 References Samorodnitsky, G., and Taqqu, M S (1994), Stable Non-Gaussian Random Processes, New York: Chapman and Hall Samuelson, P A (1965), ‘Proof that Properly Anticipated Prices Fluctuate Randomly’, Industrial Management Review, 6, 41–9 (1973), ‘Proof that Properly Discounted Present Values of Assets Vibrate Randomly’, Bell Journal of Economics and Management Science, 4, 369–74 (1987), ‘Paradise Lost and Refound: The Harvard ABC Barometers’, Journal of Portfolio Management, 4, 4–9 Sargan, J D (1964), ‘Wages and Prices in the United Kingdom: A Study in Econometric Methodology’, in P E Hart, G Mills and J K Whitaker (eds.), Econometric Analysis for National Economic Planning, London: Butterworths, 25–63 Sargan, J D., and Bhargava, A S (1983), ‘Testing Residuals from Least Squares Regression for being Generated by the Gaussian Random Walk’, Econometrica, 51, 153–74 Schmidt, P (1990), ‘Dickey–Fuller Tests with Drift’, in G F Rhodes and T B Fomby (eds.), Advances in Econometrics, Vol VIII, Greenwich, CT: JAI Press, 161–200 Schwarz, G (1978), ‘Estimating the Dimension of a Model’, Annals of Statistics, 6, 461–4 Schwert, G W (1987), ‘Effects of Model Specification on Tests for Unit Roots in Macroeconomic Data’, Journal of Monetary Economics, 20, 73–105 (1989), ‘Why does Stock Market Volatility Change over Time?’, Journal of Finance, 44, 1115–53 (2003), ‘Anomalies and Market Efficiency’, in G M Constantinides, M Harris and R Stulz (eds.), Handbook of the Economics of Finance, Amsterdam: Elsevier, 935–71 Sensier, M., and van Dijk, D (2004), ‘Testing for Volatility Changes in US Macroeconomic Time Series’, Review of Economics and Statistics, 86, 833–9 Sentana, E (1995), ‘Quadratic ARCH Models’, Review of Economic Studies, 62, 639–61 SHAZAM (1993), SHAZAM User’s Reference Manual Version 7.0, New York: McGraw-Hill Shephard, N (1996), ‘Statistical Aspects of ARCH and Stochastic Volatility’, in D R Cox, D V Hinkley and O E Barndorff-Nielsen (eds.), Time Series Models in Econometrics, Finance and Other Fields, London: Chapman and Hall, 1–67 Shiller, R J (1979), ‘The Volatility of Long Term Interest Rates and Expectations Models of the Term Structure’, Journal of Political Economy, 87, 1190–209 (1981a), ‘Do Stock Prices Move Too Much to be Justified by Subsequent Changes in Dividends?’, American Economic Review, 71, 421–36 (1981b), ‘The Use of Volatility Measures in Assessing Market Efficiency’, Journal of Finance, 36, 291–304 Shiller, R J., and Perron, P (1985), ‘Testing the Random Walk Hypothesis: Power versus Frequency of Observation’, Economics Letters, 18, 381–6 Shintani, M., and Linton, O (2006), ‘Nonparametric Neural Network Estimation of Lyapunov Exponents and a Direct Test for Chaos’, Journal of Econometrics, 120, 1–33 Siklos, P L., and Granger, C W J (1997), ‘Temporary Cointegration with an Application to Interest Rate Parity’, Macroeconomic Dynamics, 1, 64057 Silvennoinen, A., and Teraăsvirta, T (2007), Multivariate GARCH Models’, in T G Andersen, R A Davis, J.-P Kreiss and T Mikosch (eds.), Handbook of Financial Time Series, New York: Springer 442 References Silverman, B W (1986), Density Estimation for Statistics and Data Analysis, London: Chapman and Hall Sims, C A (1980), ‘Macroeconomics and Reality’, Econometrica, 48, 1–48 (1981), ‘An Autoregressive Index Model for the US 1948–1975’, in J Kmenta and J B Ramsey (eds.), Large-scale Macroeconometric Models, Amsterdam: North-Holland, 283–327 (1988), ‘Bayesian Skepticism of Unit Root Econometrics’, Journal of Economic Dynamics and Control, 12, 463–75 Sims, C A., Stock, J H., and Watson, M W (1990), ‘Inference in Linear Time Series with Some Unit Roots’, Econometrica, 58, 113–44 Sims, C A., and Uhlig, H (1991), ‘Understanding Unit Rooters: A Helicopter Tour’, Econometrica, 59, 1591–9 Singleton, K (2001), ‘Estimation of Affine Asset Pricing Models Using the Empirical Characteristic Function’, Journal of Econometrics, 102, 111–41 Slutsky, E E (1937), ‘The Summation of Random Causes as the Source of Cyclic Processes, Problems of Economic Conditions’, Econometrica, 5, 105–46 Sollis, R., Leybourne, S J., and Newbold, P (1999), ‘Unit Roots and Asymmetric Smooth Transitions’, Journal of Time Series Analysis, 20, 671–7 Sowell, F B (1990), ‘The Fractional Unit Root Distribution’, Econometrica, 58, 498–505 (1992a), ‘Maximum Likelihood Estimation of Stationary Univariate Fractionally Integrated Time Series Models’, Journal of Econometrics, 53, 165–88 (1992b), ‘Modelling Long-run Behaviour with the Fractional ARIMA Model’, Journal of Monetary Economics, 29, 277–302 Spanos, A (1986), Statistical Foundations of Econometric Modelling, Cambridge: Cambridge University Press Stock, J H (1987), ‘Asymptotic Properties of Least Squares Estimators of Cointegrating Vectors’, Econometrica, 55, 1035–56 (1991), ‘Confidence Intervals for the Largest Autoregressive Root in US Macroeconomic Time Series’, Journal of Monetary Economics, 28, 435–69 (1994), ‘Unit Roots, Structural Breaks and Trends’, in R F Engle and D L McFadden (eds.), Handbook of Econometrics, Vol IV, New York: North-Holland, 2739–841 (1996), ‘VAR, Error Correction and Pretest Forecasts at Long Horizons’, Oxford Bulletin of Economics and Statistics, 58, 685–701 (1997), ‘Cointegration, Long-run Movements, and Long-horizon Forecasting’, in D M Kreps and K F Wallis (eds.), Advances in Economics and Econometrics, Vol III: Theory and Applications, Cambridge: Cambridge University Press, 34–60 Stock, J H., and Watson, M W (1988), ‘Testing for Common Trends’, Journal of the American Statistical Association, 83, 1097–107 (1993), ‘A Simple Estimator of Cointegrating Vectors in Higher Order Integrated Systems’, Econometrica, 61, 783–820 Stoll H R., and Whaley, R E (1990), ‘Stock Market Structure and Volatility’, Review of Financial Studies, 3, 37–71 Straumann, D (2004), Estimation in Conditionally Heteroskedastic Time Series Models, New York: Springer 443 References Subba Rao, T (1981), ‘On the Theory of Bilinear Models’, Journal of the Royal Statistical Society, Series B, 43, 244–55 Subba Rao, T., and Gabr, M (1980), ‘A Test for Linearity of Stationary Time Series Analysis’, Journal of Time Series Analysis, 1, 145–58 (1984), An Introduction to Bispectral Analysis and Bilinear Time Series Models, Berlin: Springer-Verlag Sullivan, E J., and Weithers, T M (1991), ‘Louis Bachelier: The Father of Modern Option Pricing Theory’, Journal of Economic Education, 22, 165–71 Sun, Y., and Phillips, P C B (2003), ‘Nonlinear Log-periodogram Regression for Perturbed Fractional Processes’, Journal of Econometrics, 115, 355–89 Swanson, N R., and Granger, C W J (1997), ‘Impulse Response Functions Based on a Causal Approach to Residual Orthogonalization in Vector Autoregressions’, Journal of the American Statistical Association, 92, 357–67 Tanaka, K (1990), ‘Testing for a Moving Average Unit Root’, Econometric Theory, 6, 433–44 (1999), ‘The Nonstationary Fractional Unit Root’, Econometric Theory, 15, 549–82 Taniguchi, M., and Kakizawa, Y (2000), Asymptotic Theory of Statistical Inference for Time Series, New York: Springer-Verlag Tauchen, G E., and Pitts, M (1983), ‘The Price Variability–Volume Relationship on Speculative Markets’, Econometrica, 51, 485–505 Taylor, J W (2004), ‘Volatility Forecasting with Smooth Transition Exponential Smoothing’, International Journal of Forecasting, 20, 273–86 Taylor, S J (1986), Modelling Financial Time Series, New York: Wiley (1994), ‘Modelling Stochastic Volatility: A Review and Comparative Study, Mathematical Finance, 4, 183204 Teraăsvirta, T (1994), Specification, Estimation, and Evaluation of Smooth Transition Autoregressive Models’, Journal of the American Statistical Association, 89, 208–18 (2007), ‘An Introduction to Univariate GARCH Models’, in T G Andersen, R A Davis, J P Kreiss and T Mikosch (eds.), Handbook of Financial Time Series, New York: Springer Teraăsvirta, T., Lin, C.-F., and Granger, C W J (1993), ‘Power of the Neural Network Test’, Journal of Time Series Analysis, 14, 20920 Teraăsvirta, T., Tjostheim, D., and Granger, C W J (1994), ‘Aspects of Modelling Nonlinear Time Series’, in R F Engle and D L McFadden (eds.), Handbook of Econometrics, Vol IV, New York: North-Holland, 2919–57 Thaler, R (1987a), ‘The January Effect’, Journal of Economic Perspectives, 1, 197–201 (1987b), ‘Seasonal Movements in Security Prices II: Weekend, Holiday, Turn of the Month, and Intraday Effects’, Journal of Economic Perspectives, 1, 169–77 Tieslau, M A., Schmidt, P., and Baillie, R T (1996), ‘A Minimum Distance Estimator for Long-memory Processes’, Journal of Econometrics, 71, 249–64 Timmermann, A., and Granger, C W J (2004), ‘Efficient Market Hypothesis and Forecasting’, International Journal of Forecasting, 20, 15–27 Toda, H Y (1994), ‘Finite Sample Properties of Likelihood Ratio Tests for Cointegrating Ranks when Linear Trends Are Present’, Review of Economics and Statistics, 76, 66–79 (1995), ‘Finite Sample Performance of Likelihood Ratio Tests for Cointegrating Ranks in Vector Autorgressions’, Econometric Theory, 11, 1015–32 444 References Toda, H Y., and Phillips, P C B (1993), ‘Vector Autoregression and Causality’, Econometrica, 61, 1367–93 (1994), ‘Vector Autoregression and Causality: A Theoretical Overview and Simulation Study’, Econometric Reviews, 13, 259–85 Toda, H Y., and Yamamoto, T (1995), ‘Statistical Inference in Vector Autoregressions with Possibly Integrated Processes’, Journal of Econometrics, 66, 225–50 Tomczyck, S., and Chatterjee, S (1984), ‘Estimating the Market Model Robustly’, Journal of Business Finance and Accounting, 11, 563–73 Tong, H., and Lim, K S (1980), ‘Threshold Autoregression, Limit Cycles, and Cyclical Data’, Journal of the Royal Statistical Society, Series B, 42, 245–92 Tremayne, A R (2006), ‘Stationary Linear Univariate Time Series Models’, in T C Mills and K Patterson (eds.), Palgrave Handbook of Econometrics, Vol I: Econometric Theory, Basingstoke: Palgrave Macmillan, 215–51 Tsay, R S (1986), ‘Nonlinearity Tests for Time Series’, Biometrika, 73, 461–6 Tse, Y K (2000), ‘A Test for Constant Correlations in a Multivariate GARCH Model’, Journal of Econometrics, 98, 107–27 Tukey, J W (1977), Exploratory Data Analysis, Reading, MA: Addison-Wesley Vahid, F (2006), ‘Common Cycles’, in T C Mills and K Patterson (eds.), Palgrave Handbook of Econometrics, Vol I: Econometric Theory, Basingstoke: Palgrave Macmillan, 610–30 Vahid, F., and Engle, R F (1993), ‘Common Trends and Common Cycles’, Journal of Applied Econometrics, 8, 341–60 (1997), ‘Codependent Cycles’, Journal of Econometrics, 80, 199–221 Van Dijk, D J C., Franses, P H., and Lucas, A (1999), ‘Testing for Smooth Transition Nonlinearity in the Presence of Outliers’, Journal of Business and Economic Statistics, 17, 217–35 Van Dijk, D J C., Teraăsvirta, T., and Franses, P H (2002), ‘Smooth Transition Autoregressive Models: A Survey of Recent Developments’, Econometric Reviews, 21, 1–47 Velasco, C (1999), ‘Non-stationary Log-periodogram Regression’, Journal of Econometrics, 91, 325–71 (2006), ‘Semiparametric Estimation of Long-memory Models’, in T C Mills and K Patterson (eds.), Palgrave Handbook of Econometrics, Vol I: Econometric Theory, Basingstoke: Palgrave Macmillan, 353–95 Vogelsang, T J (1997), ‘Wald-type Tests for Detecting Breaks in the Trend Function of a Dynamic Time Series’, Econometric Theory, 13, 818–49 Vogelsang, T J., and Perron, P (1998), ‘Additional Tests for a Unit Root Allowing for a Break in the Trend Function at an Unknown Point in Time’, International Economic Review, 39, 1073–100 Wecker, W (1981), ‘Asymmetric Time Series’, Journal of the American Statistical Association, 76, 16–21 Weiss, A A (1984), ‘ARMA Models with ARCH Errors’, Journal of Time Series Analysis, 5, 129–43 (1986a), ‘Asymptotic Theory for ARCH Models: Estimation and Testing’, Econometric Theory, 2, 107–31 (1986b), ‘ARCH and Bilinear Time Series Models: Comparison and Combination’, Journal of Business and Economic Statistics, 4, 59–70 ... implications of equation (1.3), that the stock price at the beginning of time t The Econometric Modelling of Financial Time Series equals the discounted sum of the next period’s expected future... integral part of the study of financial markets The first attempts to study the behaviour of financial time series were undertaken by financial professionals and journalists rather than by academics... 372 375 387 Preface to the third edition In the nine years since the manuscript for the second edition of The Econometric Modelling of Financial Time Series was completed there have continued to