The Annals of Statistics

Asymptotic distribution-free tests for semiparametric regressions with dependent data

Juan Carlos Escanciano, Juan Carlos Pardo-Fernández, and Ingrid Van Keilegom

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This article proposes a new general methodology for constructing nonparametric and semiparametric Asymptotically Distribution-Free (ADF) tests for semiparametric hypotheses in regression models for possibly dependent data coming from a strictly stationary process. Classical tests based on the difference between the estimated distributions of the restricted and unrestricted regression errors are not ADF. In this article, we introduce a novel transformation of this difference that leads to ADF tests with well-known critical values. The general methodology is illustrated with applications to testing for parametric models against nonparametric or semiparametric alternatives, and semiparametric constrained mean–variance models. Several Monte Carlo studies and an empirical application show that the finite sample performance of the proposed tests is satisfactory in moderate sample sizes.

Article information

Ann. Statist., Volume 46, Number 3 (2018), 1167-1196.

Received: July 2016
Revised: February 2017
First available in Project Euclid: 3 May 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E20: Asymptotic distribution theory 62G08: Nonparametric regression 62G20: Asymptotic properties 62H15: Hypothesis testing

Beta-mixing error distribution goodness-of-fit tests local polynomial estimation nonparametric regression


Escanciano, Juan Carlos; Pardo-Fernández, Juan Carlos; Van Keilegom, Ingrid. Asymptotic distribution-free tests for semiparametric regressions with dependent data. Ann. Statist. 46 (2018), no. 3, 1167--1196. doi:10.1214/17-AOS1581.

Export citation


  • [1] Bekaert, G. and Harvey, C. R. (1995). Time-varying world market integration. J. Finance 50 403–444.
  • [2] Chen, X. (2007). Large sample sieve estimation of semi-nonparametric models. In Handbook of Econometrics (J. J. Heckman and E. E. Leamer, eds.) 6 5549–5632.
  • [3] Dedecker, J. and Louhichi, S. (2002). Maximal inequalities and empirical central limit theorems. In Empirical Process Techniques for Dependent Data (H. Dehling, T. Mikosch and M. Sørensen, eds.) 137–159. Birkhäuser, Boston, MA.
  • [4] Delgado, M. A. and González Manteiga, W. (2001). Significance testing in nonparametric regression based on the bootstrap. Ann. Statist. 29 1469–1507.
  • [5] Dette, H., Marchlewski, M. and Wagener, J. (2012). Testing for a constant coefficient of variation in nonparametric regression by empirical processes. Ann. Inst. Statist. Math. 64 1045–1070.
  • [6] Dette, H., Neumeyer, N. and Van Keilegom, I. (2007). A new test for the parametric form of the variance function in non-parametric regression. J. R. Stat. Soc. Ser. B. Stat. Methodol. 69 903–917.
  • [7] Dette, H., Pardo-Fernández, J. C. and Van Keilegom, I. (2009). Goodness-of-fit tests for multiplicative models with dependent data. Scand. J. Stat. 36 782–799.
  • [8] De Santis, G. and Gerard, B. (1997). International asset pricing and portfolio diversification with time-varying risk. J. Finance 52 1881–1912.
  • [9] Doukhan, P. (1994). Mixing. Lecture Notes in Statistics: Properties and Examples 85. Springer, New York.
  • [10] Einmahl, U. and Mason, D. M. (2005). Uniform in bandwidth consistency of kernel-type function estimators. Ann. Statist. 33 1380–1403.
  • [11] Escanciano, J. C. (2009). On the lack of power of omnibus specification tests. Econometric Theory 25 162–194.
  • [12] Escanciano, J. C., Jacho-Chávez, D. T. and Lewbel, A. (2014). Uniform convergence of weighted sums of non and semiparametric residuals for estimation and testing. J. Econometrics 178 426–443.
  • [13] Escanciano, J. C., Pardo-Fernández, J. C. and Van Keilegom, I. (2017). Semiparametric estimation of risk-return relationships. J. Bus. Econom. Statist. 35 40–52.
  • [14] Escanciano, J. C., Pardo-Fernández, J. C. and Van Keilegom, I. (2018). Supplement to “Asymptotic distribution-free tests for semiparametric regressions with dependent data.” DOI:10.1214/17-AOS1581SUPP.
  • [15] Fan, J. and Yao, Q. (1998). Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85 645–660.
  • [16] Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods. Springer, New York.
  • [17] Ferson, W. (1989). Changes in expected security returns, risk and level of interest rates. J. Finance 44 1191–1217.
  • [18] Ferson, W., Foerster, S. R. and Keim, D. B. (1993). General tests of latent variable models and mean–variance spanning. J. Finance 48 131–156.
  • [19] González-Manteiga, W. and Crujeiras, R. M. (2013). An updated review of goodness-of-fit tests for regression models. TEST 22 361–411.
  • [20] Hansen, B. E. (2008). Uniform convergence rates for kernel estimation with dependent data. Econometric Theory 24 726–748.
  • [21] Härdle, W., Hall, P. and Ichimura, H. (1993). Optimal smoothing in single-index models. Ann. Statist. 21 157–178.
  • [22] Härdle, W. and Mammen, E. (1993). Comparing nonparametric versus parametric regression fits. Ann. Statist. 21 1926–1947.
  • [23] Harvey, C. R. (1989). Time-varying conditional covariances in tests of asset pricing models. J. Financ. Econom. 24 289–317.
  • [24] Janssen, A. (2000). Global power functions of goodness of fit tests. Ann. Statist. 28 239–253.
  • [25] Khmaladze, E. V. and Koul, H. L. (2004). Martingale transforms goodness-of-fit tests in regression models. Ann. Statist. 32 995–1034.
  • [26] Koul, H. L. and Stute, W. (1999). Nonparametric model checks for time series. Ann. Statist. 27 204–236.
  • [27] Lee, T.-H., Tu, Y. and Ullah, A. (2015). Forecasting equity premium: Global historical average versus local historical average and constraints. J. Bus. Econom. Statist. 33 393–402.
  • [28] Masry, E. (1996). Multivariate local polynomial regression for time series: Uniform strong consistency and rates. J. Time Series Anal. 17 571–599.
  • [29] Merton, R. C. (1973). An intertemporal capital asset pricing model. Econometrica 41 867–887.
  • [30] Merton, R. C. (1980). On estimating the expected return on the market. An explanatory investigation. J. Financ. Econom. 8 323–361.
  • [31] Neumeyer, N. and Van Keilegom, I. (2010). Estimating the error distribution in nonparametric multiple regression with applications to model testing. J. Multivariate Anal. 101 1067–1078.
  • [32] Newey, W. K. (1994). The asymptotic variance of semiparametric estimators. Econometrica 62 1349–1382.
  • [33] Robinson, P. M. (1988). Root-$N$-consistent semiparametric regression. Econometrica 56 931–954.
  • [34] Stute, W. (1997). Nonparametric model checks for regression. Ann. Statist. 25 613–641.
  • [35] Stute, W., Xu, W. L. and Zhu, L. X. (2008). Model diagnosis for parametric regression in high-dimensional spaces. Biometrika 95 451–467.
  • [36] Tang, Y. and Whitelaw, R. F. (2011). Time-varying sharpe ratios and market timing. Quarterly J. Finance 1 465–493.
  • [37] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.
  • [38] Van Keilegom, I., González Manteiga, W. and Sánchez Sellero, C. (2008). Goodness-of-fit tests in parametric regression based on the estimation of the error distribution. TEST 17 401–415.

Supplemental materials

  • Supplement to “Asymptotic distribution-free tests for semiparametric regressions with dependent data”. The supplement contains further Monte Carlo simulations and the proofs of some asymptotic results.