Functional Coefficient Regression Models with Dependent Data

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Functional Coefficient Regression Models with
Dependent Data

• Yanrong Cao
• Joseph L. Rotman School of Management
• University of Toronto
• Haiqun Lin (Yale University)
• Zhou Wu (University of Cincinnati)
• Yan Yu (University of Cincinnati)

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Outline

• FC Regression Models for Nonlinear Time Series
• Estimation and Forecasting with Penalized Spline
  Approach
• Properties and Inferences of the Estimates
• Model Applications
• Simulation Study
• US Real GNP
• Conclusion

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FC Regression Models for Nonlinear TS

• Model
• u and x lagged values of y
• ?t
• iid,
• mean zero,
• finite variance ,
• Independent of ut , xtj,t?t , and ?t, for tltt

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Literature Review

• TAR Model (Tong 1990)
• FAR Model (Chen and Tsay 1993)
• EXPAR Model (Haggan and Ozaki 1981, and Ozaki
  1982)

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Literature Review Contd

• Local Linear Regression Estimation (Cai, Fan, and
  Yao 2000)
• Approximate aj(u) locally at u0 by a linear
  function
• Local Linear estimators
• Polynomial Spline Estimation (Huang and Shen
  2004)
• Polynomial basis functions Bjss, and constants
  djs,
• LS Estimation

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FC Regression Models PS Estimation Forecasting

• Assume
• Estimation of aj()
• Denote
• and column spline coefficient vector d.
• Mean Function Representation
• Let Dj be an appropriate semi-definite symmetric
  matrix, and

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PS Estimation Forecasting (Contd)

• PLS Estimation
• by minimizing
• Model Mean
• Forecasting

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PS Estimation Forecasting (Contd)

• Selection of Smoothing Parameters
• Smoothing Parameters
• Knots
• Penalty Weight ?j
• Selection Criterion
• Minimizing GCV
• where
• Algorithm
• Two-step Algorithm for Additive Penalized Spline
  (Ruppert and Carroll, 2000)

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Properties and Inferences of PS Estimates

• Finite Sample Properties Inference with ?
  Fixed
• Properties Inference with ?n?0
• Theorem l Under mild conditions, if ?no(1),
  then there exists a sequence of PLS estimators,
  and it is a strong consistent estimator of true
  spline parameter.
• Theorem 2 Under mild conditions, if ?no(n-1/2),
  then a sequence of PLS estimators exist and is
  asymptotically normally distributed
• where O is the a.s. limit of ZTZ/n.

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Model Applications

• Simulation Study -- EXPAR Model
• Model Settings
• et i.i.d. N(0,0.22)
• True Functional Coefficients
• Initialization of y0 and y1
• Basis
• Truncated Power Basis
• B-spline Basis
• Monte Carlo Variance Study
• Estimated covariance matrix by ridge regression
  with fixed ?,
• Asymptotic covariance matrix with ?0 with the
  true covariance matrix estimated by Monte Carlo
  simulation.
• Performance Measure
• RASE
• RMSE

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Simulation Study (Contd)Figure 1 Plots for
true function (red curve) and average PS
estimates (blue curve) and penalized spline
estimates (green curve)
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Simulation Study (Contd)

• Table 1 EXPAR Simulation Example. Mean (S.E.) of
  RASEs and RMSEs for Estimations of a1() and
  a2() by polynomial spline and penalized spline.

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Real Data Application US Real GNP

• Data Description
• 1947Q1 to 2002Q2 (222 observations)
• Data Transformation Growth Rate yt.
• where zt is the US real GNP series.
• Training subseries
• Forecasting subseries
• Model
• f(yt-1, yt-2)et
• Multi-step-ahead Forecast
• E(yt1?t) E(f(yt, yt-1)?t)

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US Real GNP (Contd)

• Penalized Spline Model
• Number of knots (40, 40)
• Estimation RASE, RMSE 0.9575, 0.9789
• 12-step-ahead forecast
• AAPE 0.6575
• Polynomial Spline Method
• Number of knots (4, 5)
• Estimation RASE, RMSE 0.9455, 0.9817
• 12-step-ahead forecast AAPE 0.7050

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Conclusion

• Different smoothness allowed for different aj()
  with smoothness controlled by ?j
• Computational expediency with fixed number of
  knots
• Strong consistency and asymptotic normality
• Explicit model expression
• Feasible multi-step-ahead forecasting

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• Thank You !