INTRODUCING MODEL UNCERTAINTY IN TIME SERIES BOOTSTRAP

It is common in parametric bootstrap to select the model from the data, and then treat it as it were the true model. Kilian (1998) have shown that ignoring the model uncertainty may seriously undermine the coverage accuracy of bootstrap confidence intervals for impulse response estimates which are closely related with multi-step-ahead prediction intervals. In this paper, we propose different ways of introducing the model selection step in the resampling algorithm. We present a Monte Carlo study comparing the finite sample properties of the proposed method with those of alternative methods in the case of prediction intervals.

Forecasting time series with sieve bootstrap

In this paper we consider bootstrap methods for constructing nonparametric prediction intervals for a general class of linear processes. Our approach uses the sieve bootstrap procedure of Biihlmann (1997) based on residual resampling from an autoregressive approximation to the given process. We show that the sieve bootstrap provides consistent estimators of the conditional distribution of future values given the observed data, assuming that the order of the autoregressive approximation increases with the sample size at a suitable rate and some restrictions about polynomial decay of the coefficients ~ j t:o of the process MA(oo) representation. We present a Monte Carlo study comparing the finite sample properties of the sieve bootstrap with those of alternative methods. Finally, we illustrate the performance of the proposed method with real data examples

Forecasting time series with sieve bootstrap.

In this paper we propose bootstrap methods for constructing nonparametric prediction intervals for a general class of linear processes. Our approach uses the AR(∞)-sieve bootstrap procedure based on residual resampling from an autoregressive approximation to the given process. We present a Monte Carlo study comparing the finite sample properties of the sieve bootstrap with those of alternative methods. Finally, we illustrate the performance of the proposed method with a real data example.We would like to thank Mike Wiper, two referees and the coordinating editor for carefully reading that greatly improved the paper. This research was partially supported by the Dirección General de Educación Superior project DGES PB96-0111 and Cátedra de Calidad BBVA.Publicad

Linearization Methods in Time Series Analysis

In this dissertation, we propose a set of computationally efficient methods based on approximating/representing nonlinear processes by linear ones, so-called linearization. Firstly, a linearization method is introduced for estimating the multiple frequencies in sinusoidal processes. It utilizes a regularized autoregressive (AR) approximation, which can be regarded as a "large p - small n" approach in a time series context. An appealing property of regularized AR is that it avoids a model selection step and allows for an efficient updating of the frequency estimates whenever new observations are obtained. The theoretical analysis shows that the regularized AR frequency estimates are consistent and asymptotically normally distributed. Secondly, a sieve bootstrap scheme is proposed using the linear representation of generalized autoregressive conditional heteroscedastic (GARCH) models to construct prediction intervals (PIs) for the returns and volatilities. Our method is simple, fast and distribution-free, while providing sharp and well-calibrated PIs. A similar linear bootstrap scheme can also be used for diagnostic testing. Thirdly, we introduce a robust lagrange multiplier (LM) test, which utilizes either the bootstrap or permutation procedure to obtain critical values, for detecting GARCH effects. We justify that both bootstrap and permutation LM tests are consistent. Intensive numerical studies indicate that the proposed resampling algorithms significantly improve the size and power of the LM test in both skewed and heavy-tailed processes. Moreover, fourthly, we introduce a nonparametric trend test in the presence of GARCH effects (NT-GARCH) based on heteroscedastic ANOVA. Our empirical evidence show that NT-GARCH can effectively detect non-monotonic trends under GARCH, especially in the presence of irregular seasonal components. We suggest to apply the bootstrap procedure for both selecting the window length and finding critical values. The newly proposed methods are illustrated by applications to astronomical data, to foreign currency exchange rates as well as to water and air pollution data. Finally, the dissertation is concluded by an outlook on further extensions of linearization methods, e.g., in model order selection and change point detection

Instrumental Variable Identification of Dynamic Variance Decompositions

Macroeconomists increasingly use external sources of exogenous variation for causal inference. However, unless such external instruments (proxies) capture the underlying shock without measurement error, existing methods are silent on the importance of that shock for macroeconomic fluctuations. We show that, in a general moving average model with external instruments, variance decompositions for the instrumented shock are interval-identified, with informative bounds. Various additional restrictions guarantee point identification of both variance and historical decompositions. Unlike SVAR analysis, our methods do not require invertibility. Applied to U.S. data, they give a tight upper bound on the importance of monetary shocks for inflation dynamics

Honest confidence sets in nonparametric IV regression and other ill-posed models

This paper provides novel methods for inference in a very general class of ill-posed models in econometrics, encompassing the nonparametric instrumental regression, different functional regressions, and the deconvolution. I focus on uniform confidence sets for the parameter of interest estimated with Tikhonov regularization, as in Darolles, Fan, Florens, and Renault (2011). I first show that it is not possible to develop inferential methods directly based on the uniform central limit theorem. To circumvent this difficulty I develop two approaches that lead to valid confidence sets. I characterize expected diameters and coverage properties uniformly over a large class of models (i.e. constructed confidence sets are honest). Finally, I illustrate that introduced confidence sets have reasonable width and coverage properties in samples commonly used in applications with Monte Carlo simulations and considering application to Engel curves

Honest confidence sets in nonparametric IV regression and other ill-posed models

This paper provides novel methods for inference in a very general class of ill-posed models in econometrics, encompassing the nonparametric instrumental regression, different functional regressions, and the deconvolution. I focus on uniform confidence sets for the parameter of interest estimated with Tikhonov regularization, as in Darolles, Fan, Florens, and Renault (2011). I first show that it is not possible to develop inferential methods directly based on the uniform central limit theorem. To circumvent this difficulty I develop two approaches that lead to valid confidence sets. I characterize expected diameters and coverage properties uniformly over a large class of models (i.e. constructed confidence sets are honest). Finally, I illustrate that introduced confidence sets have reasonable width and coverage properties in samples commonly used in applications with Monte Carlo simulations and considering application to Engel curves

Various Statistical Inferences for High-dimensional Time Series: Bootstrap, Homogeneity Pursuit and Autocovariance Test

This thesis aims to study various statistical inferences for high-dimensional data, especially high-dimensional time series, including sieve bootstrap, homogeneity pursuit, and an equivalence test for spiked eigenvalues of autocovariance matrix. The primary techniques used in this thesis are novel dimension-reduction methods developed from factor models and principal component analysis (PCA). Chapter 2 proposes a novel sieve bootstrap method for high-dimensional time series and applies it to sparse functional time series where the actual observations are not dense, and pre-smoothing is misleading. Chapter 3 introduces an iterative complement-clustering principal component analysis (CPCA) to study high-dimensional data with group structures, where both homogeneity and sub-homogeneity (group-specific information) can be identified and estimated. Lastly, Chapter 4 proposes a novel test statistic named the autocovariance test to compare the spiked eigenvalues of the autocovariance matrices for two high-dimensional time series. In all chapters, dimension-reduction methods are applied for novel statistical inferences. In particular, Chapters 2 and 4 focus on the spiked eigenstructure of autocovariance matrix and use factors to capture the temporal dependence of the high-dimensional time series. Meanwhile, Chapter 3 aims to simultaneously estimate homogeneity and sub-homogeneity, which form a more complicated spiked eigenstructure of the covariance matrix, despite that the group-specific information is relatively weak compared with the homogeneity and traditional PCA fails to capture it. The theoretical and asymptotic results of all three statistical inferences are provided in each chapter, respectively, where the numerical evidence on the finite-sample performance for each method is also discussed. Finally, these three statistical inferences are applied on particulate matter concentration data, stock return data, and age-specific mortality data for multiple countries, respectively, to provide valid statistical inferences

Rapid identification of oil contaminated soils using visible near infrared diffuse reflectance spectroscopy

Initially, 46 petroleum contaminated and non-contaminated soil samples were collected and scanned using visible near-infrared diffuse reflectance spectroscopy (VisNIR DRS) at three combinations of moisture content and pretreatment. The VisNIR spectra of soil samples were used to predict total petroleum hydrocarbon (TPH) content using partial least squares (PLS) regression and boosted regression tree (BRT) models. The field-moist intact scan proved best for predicting TPH content with a validation r2 of 0.64 and relative percent difference (RPD) of 1.70. Those 46 samples were used to calibrate a penalized spline (PS) model. Subsequently, the PS model was used to predict soil TPH content for 128 soil samples collected over an 80 ha study site. An exponential semivariogram using PS predictions revealed strong spatial dependence among soil TPH [r2 = 0.76, range = 52 m, nugget = 0.001 (log10 mg kg-1)2, and sill 1.044 (log10 mg kg-1)2]. An ordinary block kriging map produced from the data showed that TPH distribution matched the expected TPH variability of the study site. Another study used DRS to measure reflectance patterns of 68 artificially constructed samples with different clay content, organic carbon levels, petroleum types, and different levels of contamination per type. Both first derivative of reflectance and discrete wavelet transformations were used to preprocess the spectra. Principal component analysis (PCA) was applied for qualitative VisNIR discrimination of variable soil types, organic carbon levels, petroleum types, and concentration levels. Soil types were separated with 100% accuracy, and organic carbon levels were separated with 96% accuracy by linear discriminant analysis. The support vector machine produced 82% classification accuracy for organic carbon levels by repeated random splitting of the whole dataset. However, spectral absorptions for each petroleum hydrocarbon overlapped with each other and could not be separated with any classification scheme when contaminations were mixed. Wavelet-based multiple linear regression performed best for predicting petroleum amount with the highest residual prediction deviation (RPD) of 3.97. While using the first derivative of reflectance spectra, PS regression performed better (RPD = 3.3) than the PLS (RPD= 2.5) model. Specific calibrations considering additional soil physicochemical variability are recommended to produce improved predictions