THE LOCAL PROJECTION RESIDUAL BOOTSTRAP FOR AR(1) MODELS

This paper considers inference in a broad class of nonregular models. The models considered are nonregular in the sense that standard test statistics have asymptotic distributions that are discontinuous in some parameters. It is shown in Andrews and Guggenberger (2009a) that standard fixed critical value, subsampling, and m out of n bootstrap methods often have incorrect asymptotic size in such models. This paper introduces general methods of constructing tests and confidence intervals that have correct asymptotic size. In particular, we consider a hybrid subsampling/fixed-critical-value method and size-correction methods. The paper discusses two examples in detail. They are (i) confidence intervals in an autoregressive model with a root that may be close to unity and conditional heteroskedasticity of unknown form and (ii) tests and confidence intervals based on a post-conservative model selection estimator. Copyright 2009 The Econometric Society.

This study offers a new technique for constructing percentile bootstrap intervals to predict the regression of univariate local polynomials. Bootstrap regression uses resampling derived from paired and residual bootstrap methods. The main objective of this study is to perform a comparative analysis between the two resampling methods by considering the nominal coverage probability. Resampling uses a nonparametric bootstrap technique with the return method, where each sample point has an equal chance of being selected. The principle of nonparametric bootstrapping uses the original sample data as a source of diversity in contrast to parametric bootstrapping, where the variety comes from generating a particular distribution. The simulation results show that the paired and residual bootstrap interval coverage probabilities are close to nominal coverage. The results showed no significant difference between paired bootstrap interval and percentile residual. Increasing the bootstrap sample size sufficiently large gives the scatterplot smoothness of the confidence interval. Applying the smoothing parameter by choice gives a second-order polynomial regression with a smoother distribution than the first-order polynomial regression. The scatterplot shows that the second-degree polynomial regression can capture the data curvature feature compared to the first-degree polynomial. The bands made from second-degree polynomials give a narrower width than first-degree polynomials. In contrast, applying optimal smoothing parameters to the model provides different conclusions by using smoothing parameters based on choice. In addition to the differences based on the scatterplot, the bootstrap estimates of the coverage probability are also other. Selecting smoothing parameters based on a particular value provides probability coverage with the paired bootstrap method for the first-degree local polynomial regression is 0.93, while the second-degree local polynomial is 0.96. The probability of coverage based on the residual bootstrap method for the first-degree local polynomial regression is 0.95, while the second-degree local polynomial is 0.96. The probability coverage based on the optimal parameters of the paired bootstrap method for the first-degree local polynomial regression is 0.945, while the second-degree local polynomial is 0.93. The residual bootstrap method gives the first-degree local polynomial regression of 0.95, while the second-degree local polynomial is 0.93. In general, both bootstrap methods work well for estimating prediction confidence intervals.

The growing interest in personalized medicine requires making inferences from descriptive indexes estimated from individual recordings of physiological signals, with statistical analyses focused on individual differences between/within subjects, rather than comparing supposedly homogeneous cohorts. To this end, methods to compute confidence limits of individual estimates of descriptive indexes are needed. This study introduces numerical methods to compute such confidence limits and perform statistical comparisons between indexes derived from autoregressive (AR) modeling of individual time series. Analytical approaches are generally not viable, because the indexes are usually nonlinear functions of the AR parameters. We exploit Monte Carlo (MC) and Bootstrap (BS) methods to reproduce the sampling distribution of the AR parameters and indexes computed from them. Here, these methods are implemented for spectral and information-theoretic indexes of heart-rate variability (HRV) estimated from AR models of heart-period time series. First, the MS and BC methods are tested in a wide range of synthetic HRV time series, showing good agreement with a gold-standard approach (i.e. multiple realizations of the "true" process driving the simulation). Then, real HRV time series measured from volunteers performing cognitive tasks are considered, documenting (i) the strong variability of confidence limits' width across recordings, (ii) the diversity of individual responses to the same task, and (iii) frequent disagreement between the cohort-average response and that of many individuals. We conclude that MC and BS methods are robust in estimating confidence limits of these AR-based indexes and thus recommended for short-term HRV analysis. Moreover, the strong inter-individual differences in the response to tasks shown by AR-based indexes evidence the need of individual-by-individual assessments of HRV features. Given their generality, MC and BS methods are promising for applications in biomedical signal processing and beyond, providing a powerful new tool for assessing the confidence limits of indexes estimated from individual recordings.

Economic and financial theories postulate that stocks should provide a hedge against expected inflation. Since the work of Fisher (1930), there has been on-going empirical investigation into testing the relationships between stock price indices and consumer price indices, in levels and first differences. The findings of these investigations are mixed. One concern is that the methodological issues associated with these studies are not adequately addressed in the literature. The main contribution of this thesis is to identify and improve upon the weaknesses of some of the methodologies employed for testing this relationship and apply the improved methods to Australian data. This thesis conducts investigation into the short run and long run relationships between Australian stock and consumer price indices, in levels and first differences, using bivariate and multivariate frameworks. In addition, this thesis examines, whether or not, the major monetary policy change introduced by the Reserve Bank of Australia in January 1990, has had any significant influence on these relationships. During the period leading up to this change, Australia experienced a high inflationary environment. Using the quarterly data for the period 1969 to 2008 and employing vector autoregression (VAR), autoregressive distributed lag (ARDL) models and bootstrap methods, this thesis presents robust statistical inference on the relationships between stock and consumer price indices. A review of the literature suggests that previous empirical studies investigating this relationship paid inadequate attention to improving the statistical inference on the long run parameters. This thesis makes two major improvements to the methodologies used by previous empirical studies: one is the construction of bootstrap confidence intervals for VAR impulse responses. The other is the estimation of the long run model parameters that are nonlinear functions of those of ARDL models by employing bootstrap methods. Traditionally, OLS and delta methods are used to estimate these long run parameters, although the latter method is known to work well only with large samples under normality. Such strong requirements do not appear to be satisfied for the empirical models studied in the thesis. Here, a bias-corrected bootstrap method for estimating long run model parameters and their confidence intervals is adopted when the normality assumption is violated, and the wild bootstrap method is adopted when both normality and homoscedasticity assumptions are violated. Based on the VAR impulse response functions and bootstrap confidence intervals, this thesis finds that there is a short run negative relationship between stock returns and inflation. The long run ARDL model estimates indicate that the real stock returns are independent of expected inflation, suggesting that Australian stocks constitute a good hedge against expected inflation. Furthermore, the empirical results indicate that the relationship between stock returns and inflation is not affected by the major monetary policy change introduced in January 1990. No evidence of a long run relation between real stock prices and consumer prices was found for the more recent low inflationary period. Based on the empirical evidence presented in the thesis, the overall conclusion is that Australian stocks provide a hedge against inflation, from which domestic and foreign investors can benefit.

Statistical analysis which aims to analyze a linear relationship between the independent variable and the dependent variable is known as regression analysis. To estimate parameters in a regression analysis method commonly used is the Ordinary Least Square (OLS). But the assumption is often violated in the OLS, the assumption of normality due to one outlier. As a result of the presence of outliers is parameter estimators produced by the OLS will be biased. Bootstrap Residual is a bootstrap method that is applied to the residual resampling process. The results showed that the residual bootstrap method is only able to overcome the bias on the number of outliers 5% with 99% confidence intervals. The resulting parameters estimators approach the residual bootstrap values ??OLS initial allegations were also able to show that the bootstrap is an accurate prediction tool.

This paper proposes a GMM-based method for asymptotic confidence interval construction in stationary autoregressive models, which is robust to the presence of conditional heteroskedasticity of unknown form. The confidence regions are obtained by inverting the asymptotic acceptance region of the distance metric test for the continuously updated GMM (CU-GMM) estimator. Unlike the predetermined symmetric shape of the Wald confidence intervals, the shape of the proposed confidence intervals is data-driven owing an estimated sequence of nonuniform weights. It appears that the flexibility of the CU-GMM estimator in downweighting certain observations proves advantageous for confidence interval construction. This stands in contrast to some other generalized empirical likelihood estimators with appealing optimality properties such as the empirical likelihood estimator whose objective function prevents such downweighting. A Monte Carlo simulation study illustrates the excellent small-sample properties of the method for AR models with ARCH errors. The procedure is applied to study the dynamics of the federal funds rate.

Generalizability theory is widely applied in psychological and educational measurement.The variability of estimated variance component,which is constrained by sampling,is the "Achilles heel" of generalizability theory. Therefore,estimating the variability of estimated variance components needs to be further explored. In previous literature,some problems remain to be settled:first,the previous studies failed to compare the variability of estimated variance components among different methods simultaneously:traditional,bootstrap,jackknife and Markov Chain Monte Carlo (MCMC); second,some studies only focused on such variability of estimated variance components as the standard error,while neglected other variability such as confidence interval; last but not least,MCMC method which can be used in generalizability theory hasn't gained sufficient exploration. There are different methods to estimate the variability of variance components:standard error and confidence interval,including traditional,bootstrap,jackknife and MCMC. Based on these four methods,the study adopts Monte Carlo data simulation technique to compare the variability of estimated variance components for normal distribution data. For traditional method,ANOVA is used to estimate the variance components and their standard errors. Satterthwaite and TBGJL are used to estimate the confidence intervals. For bootstrap method,twelve bootstrap strategies are adopted,but only three strategies are considered in jackknife method and only two strategies,i.e.,informative and non-informative priors,for MCMC method. Some criteria are set to compare the four methods. The bias is cared about when variance components and their standard errors are estimated. The smaller the absolute bias is,the more reliable the result is. The criterion of confidence intervals is "80% interval coverage". If the "80% interval coverage" is more closed to 0.80,the confidence interval is more reliable. The simulation is implemented in R statistical programming environment. To link R program with WinBUGS,R2WinBUGS and Coda package are adopted. And the simulation results are as follows. First,it is more accurate to use traditional,jackknife,adjusted bootstrap and MCMC method with informative priors to estimate variance components. But unadjusted bootstrap and MCMC method with non-informative priors are not adequate. Second,traditional and MCMC method with informative priors are accurate to estimate standard errors of three variance components,while jackknife method is not. Bootstrap method needs to adopt a "divide-and-conquer" strategy to obtain good estimated standard errors:the most accurate estimation of standard error for person is consistently provided by adjusted boot-p or boot-ir; adjusted boot-pi is the clear winner in estimating standard error for item; adjusted boot-ir,boot-pir or boot-pr give good estimate of standard error for person and item. Finally,using traditional and MCMC method to estimate confidence intervals is suitable because their respective interval coverages are close to 0.80. But jackknife method is not accurate in estimating confidence intervals. There is no great difference between Bootstrap-PC and Bootstrap-BCa method that are used to estimate confidence intervals of three variance components. Bootstrap method should apply the "divide-and-conquer" strategy to get desirable estimated confidence intervals as following:adjusted boot-p for person; adjusted boot-pi for item; adjusted boot-ir,boot-pr or boot-p for person and item. This study shows that jackknife method is not accurate to estimate the variability of estimated variance components. If the "divide-and-conquer" strategy is not used in bootstrap method,it is advisable to use traditional method or MCMC method with informative priors.

Inference in VARs with conditional heteroskedasticity of unknown form

Bootstrap resampling methods: something for nothing?

Bootstrapping autoregressions with conditional heteroskedasticity of unknown form

A version of the nonparametric bootstrap, which resamples the entire subjects from original data, called the case bootstrap, has been increasingly used for estimating uncertainty of parameters in mixed-effects models. It is usually applied to obtain more robust estimates of the parameters and more realistic confidence intervals (CIs). Alternative bootstrap methods, such as residual bootstrap and parametric bootstrap that resample both random effects and residuals, have been proposed to better take into account the hierarchical structure of multi-level and longitudinal data. However, few studies have been performed to compare these different approaches. In this study, we used simulation to evaluate bootstrap methods proposed for linear mixed-effect models. We also compared the results obtained by maximum likelihood (ML) and restricted maximum likelihood (REML). Our simulation studies evidenced the good performance of the case bootstrap as well as the bootstraps of both random effects and residuals. On the other hand, the bootstrap methods that resample only the residuals and the bootstraps combining case and residuals performed poorly. REML and ML provided similar bootstrap estimates of uncertainty, but there was slightly more bias and poorer coverage rate for variance parameters with ML in the sparse design. We applied the proposed methods to a real dataset from a study investigating the natural evolution of Parkinson's disease and were able to confirm that the methods provide plausible estimates of uncertainty. Given that most real-life datasets tend to exhibit heterogeneity in sampling schedules, the residual bootstraps would be expected to perform better than the case bootstrap.

Different methods arising from scientific investigation for removing noise from chaotic signals have been introduced and one of their best is Local Projection approach. The local projection approach projects the chaotic data in the neighborhood onto the hyper-plane. Selection of neighborhood radius has a direct impact on its performance. A noticeable feature associated with chaotic trajectories is their stretching and folding. An idea would be making use of the large neighborhood radius in those points of the trajectory where the behavior is in the form of stretching. On the other hand in folding points where the trajectory changes direction abruptly, we should use small neighborhood radius. We think using this method may improve local projection approach efficiency.

The main contribution of this paper is a proof of the asymptotic validity of the application of the bootstrap to AR(∞) processes with unmodelled conditional heteroskedasticity. We first derive the asymptotic properties of the least-squares estimator of the autoregressive sieve parameters when the data are generated by a stationary linear process with martingale difference errors that are possibly subject to conditional heteroskedasticity of unknown form. These results are then used in establishing that a suitably constructed bootstrap estimator will have the same limit distribution as the least-squares estimator. Our results provide theoretical justification for the use of either the conventional asymptotic approximation based on robust standard errors or the bootstrap approximation of the distribution of autoregressive parameters. A simulation study suggests that the bootstrap approach tends to be more accurate in small samples.

A confidence interval is an interval estimate of a parameter of a population calculated from a sample drawn from the population. Bootstrapping method, which involves producing several new data sets that are resampled from the original data in order to estimate parameter for each newly created data set, allowing an empirical distribution for the parameter to be estimated. Since certain statistics are harder to estimate, confidence intervals are rarely employed. Several statistics might necessitate multi-step formulas assuming that are impractical for calculating confidence intervals. This paper reviews research on the concept of bootstrapping and bootstrap confidence interval. The current narrative analysis was developed to answer the main research question: (1) What is the concept of the bootstrap method and bootstrap confidence interval? (2) What are the methods of bootstrapping to obtain confidence interval? This study has found general bootstrap method idea, various techniques of bootstrap methods, its advantages and disadvantages, and its limitations. There are normal interval method, percentile bootstrap method, basic method, first-order normal approximation method, bias-corrected bootstrap, accelerated bias-corrected bootstrap and bootstrap-t method. This study concludes that the advantages of using bootstrap CI is that it does not require any assumptions about the shape of distribution and universality of the approach. Bootstrapping is a computer-intensive statistical technique that relies significantly on modern high-speed digital computers to do massive computations.

Tests for error autocorrelation (AC) are derived under the assumption of independent and identically distributed errors. The tests are not asymptotically valid if the errors are conditionally heteroskedastic. In this article we propose wild bootstrap (WB) Lagrange multiplier tests for error AC in vector autoregressive (VAR) models. We show that the WB tests are asymptotically valid under conditional heteroskedasticity of unknown form. WB tests based on a version of the heteroskedasticity-consistent covariance matrix estimator are found to have the smallest error in rejection probability under the null and high power under the alternative. We apply the tests to VAR models for credit default swap prices and Euribor interest rates. An important result that we find is that the WB tests lead to parsimonious models while the asymptotic tests suggest that a long lag length is required to get white noise residuals.