Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Cover of the book

2015 | OriginalPaper | Chapter

1. Prediction: Some Heuristic Notions

Author : Dimitris N. Politis

Published in: Model-Free Prediction and Regression

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

To explain or to predict? The problem of prediction is revisited with a view towards going beyond the typical nonparametric setting and reaching a fully model-free environment for predictive inference, i.e., point predictors and predictive intervals.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
next chapter The Model-Free Prediction Principle
Footnotes
1
Interestingly, for the dataset of Fig. 1.1, straight-line regression of logY on X gives a fitted curve that is almost identical to the straight-line regression of Y on X so long as X is in the interval [0,2]; the difference between the two models becomes pronounced only for large X.
 
Literature
Alonso AM, Peña D, Romo J (2002) Forecasting time series with sieve bootstrap. J Stat Plan Infer 100(1):1–11CrossRefMATH
Altman NS (1992) An introduction to kernel and nearest-neighbor nonparametric regression. Am Stat 46(3):175–185MathSciNet
Andersen TG, Bollerslev T (1998) Answering the sceptics: yes, standard volatility models do provide accurate forecasts. Int Econ Rev 39(4):885–905MathSciNetCrossRef
Andersen TG, Bollerslev T, Christoffersen PF, Diebold FX (2006) Volatility and correlation forecasting. In: Elliott G, Granger CWJ, Timmermann A (eds) Handbook of economic forecasting. North-Holland, Amsterdam, pp 778–878
Andersen TG, Bollerslev T, Meddahi N (2004) Analytic evaluation of volatility forecasts. Int Econ Rev 45:1079–1110MathSciNetCrossRef
Atkinson AC (1985) Plots, transformations and regression. Clarendon Press, OxfordMATH
Antoniadis A, Paparoditis E, Sapatinas T (2006) A functional wavelet-kernel approach for time series prediction. J R Stat Soc Ser B 68(5):837–857MathSciNetCrossRefMATH
Bachelier L (1900) Theory of speculation. Reprinted in Cootner PH (ed) The random character of stock market prices. MIT Press, Cambridge, MA, pp 17–78, 1964
Barndorff-Nielsen OE, Nielsen B, Shephard N, Ysusi C (1996) Measuring and forecasting financial variability using realized variance with and without a model. In: Harvey AC, Koopman SJ, Shephard N (eds) State space and unobserved components models: theory and applications. Cambridge University Press, Cambridge, pp 205–235
Bartlett MS (1946) On the theoretical specification of sampling properties of autocorrelated time series. J R Stat Soc Suppl 8:27–41MathSciNetCrossRefMATH
Berkes I, Gombay E, Horvath L, Kokoszka P (2004) Sequential change-point detection in GARCH (p, q) models. Econ Theory 20(6):1140–1167MathSciNetCrossRefMATH
Bickel P, Gel YR (2011) Banded regularization of autocovariance matrices in application to parameter estimation and forecasting of time series. J R Stat Soc Ser B 73(5):711–728MathSciNetCrossRefMATH
Bollerslev T, Chou R, Kroner K (1992) ARCH modelling in finance: a review of theory and empirical evidence. J Econ 52:5–60CrossRefMATH
Bose A (1988) Edgeworth correction by bootstrap in autoregressions. Ann Stat 16:1345–1741CrossRef
Bose A, Chatterjee S (2002) Comparison of bootstrap and jackknife variance estimators in linear regression: second order results. Stat Sin 12:575–598MathSciNetMATH
Box GEP, Draper NR (1987) Empirical model-building and response surfaces. Wiley, New YorkMATH
Box GEP, Jenkins GM (1976) Time series analysis, control, and forecasting. Holden Day, San Francisco
Breidt FJ, Davis RA, Dunsmuir W (1995) Improved bootstrap prediction intervals for autoregressions. J Time Ser Anal 16(2):177–200MathSciNetCrossRefMATH
Breiman L, Friedman J (1985) Estimating optimal transformations for multiple regression and correlation. J Am Stat Assoc 80:580–597MathSciNetCrossRefMATH
Brockwell PJ, Davis RA (1991) Time series: theory and methods, 2nd edn. Springer, New YorkCrossRef
Brockwell PJ, Davis RA (1988) Simple consistent estimation of the coefficients of a linear filter. Stoch Process Appl 22:47–59MathSciNetCrossRef
Cai TT, Ren Z, Zhou HH (2013) Optimal rates of convergence for estimating Toeplitz covariance matrices. Probab Theory Relat Fields 156(1–2):101–143MathSciNetCrossRefMATH
Cao R, Febrero-Bande M, Gonzalez-Manteiga W, Prada-Sanchez JM, GarcIa-Jurado I (1997) Saving computer time in constructing consistent bootstrap prediction intervals for autoregressive processes. Commun Stat Simul Comput 26(3):961–978MathSciNetCrossRefMATH
Carmack PS, Schucany WR, Spence JS, Gunst RF, Lin Q, Haley RW (2009) Far casting cross-validation. J Comput Graph Stat 18(4):879–893MathSciNetCrossRef
Carroll RJ, Ruppert D (1988) Transformations and weighting in regression. Chapman and Hall, New YorkCrossRef
Caroll RJ, Ruppert D (1991) Prediction and tolerance intervals with transformation and/or weighting. Technometrics 33(2):197–210MathSciNetCrossRef
Chen X, Xu M, Wu W-B (2013) Covariance and precision matrix estimation for high-dimensional time series. Ann Stat 41(6):2994–3021MathSciNetCrossRefMATH
Cheng T-ZF, Ing C-K, Yu S-H (2015) Inverse moment bounds for sample autocovariance matrices based on detrended time series and their applications. Linear Algebra Appl (to appear)
Cox DR (1975) Prediction intervals and empirical Bayes confidence intervals. In: Gani J (eds) Perspectives in probability and statistics. Academic, London, pp 47–55
Dahlhaus R (2012) Locally stationary processes. In: Handbook of statistics, vol 30. Elsevier, Amsterdam, pp 351–412
Dai J, Sperlich S (2010) Simple and effective boundary correction for kernel densities and regression with an application to the world income and Engel curve estimation. Comput Stat Data Anal 54(11):2487–2497MathSciNetCrossRefMATH
DasGupta A (2008) Asymptotic theory of statistics and probability. Springer, New YorkMATH
Davison AC, Hinkley DV (1997) Bootstrap methods and their applications. Cambridge University Press, CambridgeCrossRef
Dowla A, Paparoditis E, Politis DN (2003) Locally stationary processes and the local block bootstrap. In: Akritas MG, Politis DN (eds) Recent advances and trends in nonparametric statistics. Elsevier, Amsterdam, pp 437–444CrossRef
Draper NR, Smith H (1998) Applied regression analysis, 3rd edn. Wiley, New YorkMATH
Engle R (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation. Econometrica 50:987–1008MathSciNetCrossRefMATH
Fan J (1993) Local linear regression smoothers and their minimax efficiencies. Ann Stat 21(1):196–216CrossRefMATH
Fan J, Gijbels I (1996) Local polynomial modelling and its applications. Chapman and Hall, New YorkMATH
Fan J, Yao Q (2003) Nonlinear time series: nonparametric and parametric methods. Springer, New YorkCrossRef
Ferraty F, Vieu P (2006) Nonparametric functional data analysis. Springer, New YorkMATH
Francq C, Zakoian JM (2011) GARCH models: structure, statistical inference and financial applications. Wiley, New York
Franke J, Härdle W (1992) On bootstrapping kernel spectral estimates. Ann Stat 20:121–145CrossRefMATH
Franke J, Kreiss J-P, Mammen E (2002) Bootstrap of kernel smoothing in nonlinear time series. Bernoulli 8(1):1–37MathSciNetMATH
Freedman D (1984) On bootstrapping two-stage least-squares estimates in stationary linear models. Ann Stat 12:827–842CrossRefMATH
Fryzlewicz P, Van Bellegem S, Von Sachs R (2003) Forecasting non-stationary time series by wavelet process modelling. Ann Inst Stat Math 55(4):737–764CrossRefMATH
Gangopadhyay AK, Sen PK (1990) Bootstrap confidence intervals for conditional quantile functions. Sankhya Ser A 52(3):346–363MathSciNetMATH
Ghysels E, Santa-Clara P, Valkanov R (2006) Predicting volatility: getting the most out of return data sampled at different frequencies. J Econ 131(1–2):59–95MathSciNetCrossRef
Gray RM (2005) Toeplitz and circulant matrices: a review. Commun Inf Theory 2(3):155–239
Grenander U, Szegö G (1958) Toeplitz forms and their applications, vol 321. University of California Press, BerkeleyMATH
Hahn J (1995) Bootstrapping quantile regression estimators. Econ Theory 11(1):105–121CrossRef
Hall P (1993) On Edgeworth expansion and bootstrap confidence bands in nonparametric curve estimation. J R Stat Soc Ser B 55:291–304MATH
Hall P, Wehrly TE (1991) A geometrical method for removing edge effects from kernel type nonparametric regression estimators. J Am Stat Assoc 86:665–672MathSciNetCrossRef
Hamilton JD (1994) Time series analysis. Princeton University Press, Princeton, NJMATH
Hansen BE (2004) Nonparametric estimation of smooth conditional distributions. Working paper, Department of Economics, University of Wisconsin
Hansen PR, Lunde A (2005) A forecast comparison of volatility models: does anything beat a GARCH (1,1)? J Appl Econ 20:873–889MathSciNetCrossRef
Hansen PR, Lunde A, Nason JM (2003) Choosing the best volatility models: the model confidence set approach. Oxf Bull Econ Stat 65:839–861CrossRef
Härdle W, Bowman AW (1988) Bootstrapping in nonparametric regression: local adaptive smoothing and confidence bands. J Am Stat Assoc 83:102–110MATH
Härdle W, Marron JS (1991) Bootstrap simultaneous error bars for nonparametric regression. Ann Stat 19:778–796CrossRefMATH
Hastie T, Tibshirani R, Friedman JH (2009) The elements of statistical learning: data mining, inference, and predictions, 2nd edn. Springer, New YorkCrossRef
Hong Y (1999) Hypothesis testing in time series via the empirical characteristic function: a generalized spectral density approach. J Am Stat Assoc 94:1201–1220CrossRefMATH
Hong Y, White H (2005) Asymptotic distribution theory for nonparametric entropy measures of serial dependence. Econometrica 73(3):837–901MathSciNetCrossRefMATH
Huber PJ (1973) Robust regression: asymptotics, conjectures and Monte Carlo. Ann Stat 1:799–821CrossRefMATH
Hurvich CM, Zeger S (1987) Frequency domain bootstrap methods for time series. Technical Report, New York University, Graduate School of Business Administration
Jentsch C, Politis DN (2015) Covariance matrix estimation and linear process bootstrap for multivariate time series of possibly increasing dimension. Ann Stat 43(3):1117–1140MathSciNetCrossRef
Kirch C, Politis DN (2011) TFT-Bootstrap: resampling time series in the frequency domain to obtain replicates in the time domain. Ann Stat 39(3):1427–1470MathSciNetCrossRefMATH
Kokoszka P, Politis DN (2011) Nonlinearity of ARCH and stochastic volatility models and Bartlett’s formula. Probab Math Stat 31:47–59MathSciNetMATH
Koopman SJ, Jungbacker B, Hol E (2005) Forecasting daily variability of the S&P 100 stock index using historical, realised and implied volatility measurements. J Empir Finance 12:445–475CrossRef
Kreiss J-P, Paparoditis E, Politis DN (2011) On the range of validity of the autoregressive sieve bootstrap. Ann Stat 39(4):2103–2130MathSciNetCrossRefMATH
Künsch H (1989) The jackknife and the bootstrap for general stationary observations. Ann Stat 17:1217–1241CrossRefMATH
Lahiri SN (2003) A necessary and sufficient condition for asymptotic independence of discrete Fourier transforms under short-and long-range dependence. Ann Stat 31(2):613–641CrossRefMATH
Li Q, Racine JS (2007) Nonparametric econometrics. Princeton University Press, PrincetonMATH
Linton OB, Chen R, Wang N, Härdle W (1997) An analysis of transformations for additive nonparametric regression. J Am Stat Assoc 92:1512–1521CrossRefMATH
Linton OB, Sperlich S, van Keilegom I (2008) Estimation of a semiparametric transformation model. Ann Stat 36(2):686–718CrossRefMATH
Loader C (1999) Local regression and likelihood. Springer, New YorkMATH
Mandelbrot B (1963) The variation of certain speculative prices. J Bus 36:394–419CrossRef
Maronna RA, Martin RD, Yohai VJ (2006) Robust statistics: theory and methods. Wiley, New YorkCrossRef
Masarotto G (1990) Bootstrap prediction intervals for autoregressions. Int J Forecast 6(2):229–239CrossRef
Masry E, Tjøstheim D (1995) Nonparametric estimation and identification of nonlinear ARCH time series. Econ Theory 11:258–289CrossRef
McMurry T, Politis DN (2008) Bootstrap confidence intervals in nonparametric regression with built-in bias correction. Stat Probab Lett 78:2463–2469MathSciNetCrossRefMATH
McMurry T, Politis DN (2010) Banded and tapered estimates of autocovariance matrices and the linear process bootstrap. J Time Ser Anal 31:471–482 [Corrigendum: J Time Ser Anal 33, 2012]
McMurry T, Politis DN (2015) High-dimensional autocovariance matrices and optimal linear prediction (with discussion). Electr J Stat 9:753–822MathSciNetCrossRefMATH
Meddahi N (2001) An eigenfunction approach for volatility modeling. Technical report, CIRANO Working paper 2001s–70, University of Montreal
Mikosch T, Starica C (2004) Changes of structure in financial time series and the GARCH model. Revstat Stat J 2(1):41–73MathSciNetMATH
Nadaraya EA (1964) On estimating regression. Theory Probab Appl 9:141–142CrossRef
Pagan A, Ullah A (1999) Nonparametric econometrics. Cambridge University Press, CambridgeCrossRef
Pan L, Politis DN (2014) Bootstrap prediction intervals for Markov processes. Discussion paper, Department of Economics, University of California, San Diego. Retrievable from http://​escholarship.​org/​uc/​item/​7555757g. Accepted for publication in CSDA Annals of Computational and Financial Econometrics
Pan L, Politis DN (2015) Bootstrap prediction intervals for linear, nonlinear and nonparametric autoregressions (with discussion). J Stat Plan Infer (to appear)
Paparoditis E, Politis DN (1998) The backward local bootstrap for conditional predictive inference in nonlinear time series. In: Lipitakis EA (ed) Proceedings of the 4th Hellenic-European conference on computer mathematics and its applications (HERCMA’98). Lea Publishing, Athens, pp 467–470
Paparoditis E, Politis DN (2001) A Markovian local resampling scheme for nonparametric estimators in time series analysis. Econ Theory 17(3):540–566MathSciNetCrossRefMATH
Patton AJ (2011) Volatility forecast evaluation and comparison using imperfect volatility proxies. J Econ 160(1):246–256MathSciNetCrossRef
Politis DN (1998) Computer-intensive methods in statistical analysis. IEEE Signal Process Mag 15(1):39–55CrossRef
Politis DN (2003a) A normalizing and variance-stabilizing transformation for financial time series. In: Akritas MG, Politis DN (eds) Recent advances and trends in nonparametric statistics. Elsevier, Amsterdam, pp 335–347CrossRef
Politis DN (2004) A heavy-tailed distribution for ARCH residuals with application to volatility prediction. Ann Econ Finance 5:283–298
Politis DN (2007a) Model-free vs. model-based volatility prediction. J Financ Econ 5(3):358–389MathSciNet
Politis DN (2007b) Model-free prediction, vol LXII. Bulletin of the International Statistical Institute, Lisbon, pp 1391–1397
Politis DN (2014) Bootstrap confidence intervals in nonparametric regression without an additive model. In: Akritas MG, Lahiri SN, Politis DN (eds) Proceedings of the first conference of the international society for nonParametric statistics. Springer, New York, pp 271–282
Politis DN, Romano JP (1992) A general resampling scheme for triangular arrays of alpha-mixing random variables with application to the problem of spectral density estimation. Ann Stat 20:1985–2007MathSciNetCrossRefMATH
Politis DN, Thomakos D (2008) Financial time series and volatility prediction using NoVaS transformations. In: Rapach DE, Wohar ME (eds) Forecasting in the presence of structural breaks and model uncertainty. Emerald Group Publishing, Bingley, pp 417–447CrossRef
Politis DN, Thomakos DD (2012) NoVaS transformations: flexible inference for volatility forecasting. In: Chen X, Swanson N (eds) Recent advances and future directions in causality, prediction, and specification analysis: essays in honor of Halbert L. White Jr. Springer, New York, pp 489–528
Pourahmadi M (1999) Joint mean-covariance models with applications to longitudinal data: unconstrained parameterisation. Biometrika 86(3):677–690MathSciNetCrossRefMATH
Poon S, Granger C (2003) Forecasting volatility in financial markets: a review. J Econ Lit 41:478–539CrossRef
Priestley MB (1965) Evolutionary spectra and non-stationary processes. J R Stat Soc Ser B 27:204–237MathSciNetMATH
Priestley MB (1988) Nonlinear and nonstationary time series analysis. Academic, London
Raïs N (1994) Méthodes de rééchantillonnage et de sous-échantillonnage pour des variables aléatoires dépendantes et spatiales. Ph.D. thesis, University of Montreal
Resnick S, Samorodnitsky G, Xue F (1999) How misleading can sample ACF’s of stable MA’s be? (Very!) Ann Appl Probab 9(3):797–817MathSciNetMATH
Rissanen J, Barbosa L (1969) Properties of infinite covariance matrices and stability of optimum predictors. Inf Sci 1:221–236MathSciNetCrossRef
Ruppert D, Cline DH (1994) Bias reduction in kernel density estimation by smoothed empirical transformations. Ann Stat 22:185–210MathSciNetCrossRefMATH
Schucany WR (2004) Kernel smoothers: an overview of curve estimators for the first graduate course in nonparametric statistics. Stat Sci 19:663–675MathSciNetCrossRefMATH
Shephard N (1996) Statistical aspects of ARCH and stochastic volatility. In: Cox DR, Hinkley DV, Barndorff-Nielsen OE (eds) Time series models in econometrics, finance and other fields. Chapman and Hall, London, pp 1–67
Starica C, Granger C (2005) Nonstationarities in stock returns. Rev Econ Stat 87(3):503–522CrossRef
Tibshirani R (1988) Estimating transformations for regression via additivity and variance stabilization. J Am Stat Assoc 83:394–405MathSciNetCrossRef
Tibshirani R (1996) Regression shrinkage and selection via the Lasso. J R Stat Soc Ser B 58(1):267–288MathSciNetMATH
Thomakos DD, Klepsch J, Politis DN (2015) Multivariate NoVaS and inference on conditional correlations. Working paper, Department of Economics, University of California, San Diego
Wang L, Brown LD, Cai TT, Levine M (2008) Effect of mean on variance function estimation in nonparametric regression. Ann Stat 36:646–664MathSciNetCrossRefMATH
Wang L, Politis DN (2015) Asymptotic validity of bootstrap confidence intervals in nonparametric regression without an additive model. Working paper, Department of Mathematics, University of California, San Diego
Wu S, Harris TJ, McAuley KB (2007) The use of simplified or misspecified models: linear case. Can J Chem Eng 75:386–398
Wu W-B, Pourahmadi M (2009) Banding sample autocovariance matrices of stationary processes. Stat Sin 19(4):1755–1768MathSciNetMATH
Zhou Z, Wu W-B (2009) Local linear quantile estimation for non-stationary time series. Ann Stat 37:2696–2729CrossRefMATH
Zhou Z, Wu W-B (2010) Simultaneous inference of linear models with time-varying coefficients. J R Stat Soc Ser B 72:513–531CrossRef
Metadata
Title
Prediction: Some Heuristic Notions
Author
Dimitris N. Politis
Copyright Year
2015
Book
Model-Free Prediction and Regression

Print ISBN: 978-3-319-21346-0

Electronic ISBN: 978-3-319-21347-7

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-21347-7_1

Premium Partner

    Image Credits