Evaluating Direct Multistep Forecasts

ABSTRACT

This paper examines the asymptotic and finite-sample properties of tests of equal forecast accuracy and encompassing applied to direct, multistep predictions from nested regression models. We first derive asymptotic distributions; these nonstandard distributions depend on the parameters of the data-generating process. We then use Monte Carlo simulations to examine finite-sample size and power. Our asymptotic approximation yields good size and power properties for some, but not all, of the tests; a bootstrap works reasonably well for all tests. The paper concludes with a reexamination of the predictive content of capacity utilization for inflation.     

Bootstrapping Noncausal Autoregressions: With Applications to Explosive Bubble Modeling

In this article, we develop new bootstrap-based inference for noncausal autoregressions with heavy-tailed innovations. This class of models is widely used for modeling bubbles and explosive dynamics in economic and financial time series. In the noncausal, heavy-tail framework, a major drawback of asymptotic inference is that it is not feasible in practice as the relevant limiting distributions depend crucially on the (unknown) decay rate of the tails of the distribution of the innovations. In addition, even in the unrealistic case where the tail behavior is known, asymptotic inference may suffer from small-sample issues. To overcome these difficulties, we propose bootstrap inference procedures using parameter estimates obtained with the null hypothesis imposed (the so-called restricted bootstrap). We discuss three different choices of bootstrap innovations: wild bootstrap, based on Rademacher errors; permutation bootstrap; a combination of the two (“permutation wild bootstrap”). Crucially, implementation of these bootstraps do not require any a priori knowledge about the distribution of the innovations, such as the tail index or the convergence rates of the estimators. We establish sufficient conditions ensuring that, under the null hypothesis, the bootstrap statistics estimate consistently particular conditionaldistributions of the original statistics. In particular, we show that validity of the permutation bootstrap holds without any restrictions on the distribution of the innovations, while the permutation wild and the standard wild bootstraps require further assumptions such as symmetry of the innovation distribution. Extensive Monte Carlo simulations show that the finite sample performance of the proposed bootstrap tests is exceptionally good, both in terms of size and of empirical rejection probabilities under the alternative hypothesis. We conclude by applying the proposed bootstrap inference to Bitcoin/USD exchange rates and to crude oil price data. We find that indeed noncausal models with heavy-tailed innovations are able to fit the data, also in periods of bubble dynamics. Supplementary materials for this article are available online.     

Evaluating Direct Multistep Forecasts

This paper examines the asymptotic and finite-sample properties of tests of equal forecast accuracy and encompassing applied to direct, multistep predictions from nested regression models. We first derive asymptotic distributions; these nonstandard distributions depend on the parameters of the data-generating process. We then use Monte Carlo simulations to examine finite-sample size and power. Our asymptotic approximation yields good size and power properties for some, but not all, of the tests; a bootstrap works reasonably well for all tests. The paper concludes with a reexamination of the predictive content of capacity utilization for inflation.     

Consistent Nonparametric Tests for Lorenz Dominance

This article proposes consistent nonparametric methods for testing the null hypothesis of Lorenz dominance. The methods are based on a class of statistical functionals defined over the difference between the Lorenz curves for two samples of welfare-related variables. We present two specific test statistics belonging to the general class and derive their asymptotic properties. As the limiting distributions of the test statistics are nonstandard, we propose and justify bootstrap methods of inference. We provide methods appropriate for case where the two samples are independent as well as the case where the two samples represent different measures of welfare for one set of individuals. The small sample performance of the two tests is examined and compared in the context of a Monte Carlo study and an empirical analysis of income and consumption inequality.     

Bootstrap methods for heteroskedastic regression models: evidence on estimation and testing

This paper uses Monte Carlo simulation analysis to study the finite-sample behavior of bootstrap estimators and tests in the linear heteroskedastic model. We consider four different bootstrapping schemes, three of them specifically tailored to handle heteroskedasticity. Our results show that weighted bootstrap methods can be successfully used to estimate the variances of the least squares estimators of the linear parameters both under normality and under nonnormality. Simulation results are also given comparing the size and power of the bootstrapped Breusch-Pagan test with that of the original test and of Bartlett and Edgeworth-corrected tests. The bootstrap test was found to be robust against unfavorable regression designs.     

Linear bootstrap methods for vector autoregressive moving-average models

We provide the theoretical justification of bootstrapping stationary invertible echelon vector autoregressive moving-average (VARMA) models using linear methods. The asymptotic validity of the bootstrap is established with strong white noise under parametric and nonparametric assumptions. Our methods are practical and useful for building reliable simulation-based inference and forecasting without implementing nonlinear estimation techniques such as ML which is usually burdensome, time demanding or impractical, particularly in big or highly persistent systems. The relevance of our procedures is more pronounced in the context of dynamic simulation-based techniques such as maximized Monte Carlo (MMC) tests [see Dufour J-M. Monte Carlo tests with nuisance parameters: a general approach to finite-sample inference and nonstandard asymptotics in econometrics. J Econom. 2006;133(2):443–477 and Dufour J-M, Jouini T. Finite-sample simulation-based tests in VAR models with applications to Granger causality testing. J Econom. 2006;135(1–2):229–254 for the VAR case]. Simulation evidence shows that, compared with conventional asymptotics, our bootstrap methods have good finite-sample properties in approximating the actual distribution of the studentized echelon VARMA parameter estimates, and in providing echelon parameter confidence sets with satisfactory coverage.     

QuickMMCTest: quick multiple Monte Carlo testing

Multiple hypothesis testing is widely used to evaluate scientific studies involving statistical tests. However, for many of these tests, p values are not available and are thus often approximated using Monte Carlo tests such as permutation tests or bootstrap tests. This article presents a simple algorithm based on Thompson Sampling to test multiple hypotheses. It works with arbitrary multiple testing procedures, in particular with step-up and step-down procedures. Its main feature is to sequentially allocate Monte Carlo effort, generating more Monte Carlo samples for tests whose decisions are so far less certain. A simulation study demonstrates that for a low computational effort, the new approach yields a higher power and a higher degree of reproducibility of its results than previously suggested methods.     

Bootstrap Testing for Changes in Persistence with Heavy-Tailed Innovations

This article considers the problem of detection for changes in persistence with heavy-tailed innovations. We adopt a ratio type test and derive its null asymptotic distribution which is dependent on the stable index. Then a residual-based bootstrap is proposed when the stable index is unknown. Our procedure requires drawing bootstrap samples of size m < T, T being the size of original sample. We establish the convergence in probability of the bootstrap distribution function assuming that m → ∞ and m/T → 0. A Monte Carlo study has shown that the bootstrap improve the finite sample size and power compared to the asymptotic test, especially for small stable index.     

Permutation tests for homogeneity based on some characterizations

In this work, non parametric tests are proposed for testing the homogeneity of two or more populations. The tests are based on recently obtained characterizations. The test procedure is based on the permutation bootstrap technique. For the two-sample case the new tests are compared with permutation tests based on the empirical characteristic function and some other tests. The comparison is fulfilled via a Monte Carlo simulation.     

Bias evaluation in the proportional hazards model

We consider two approaches for bias evaluation and reduction in the proportional hazards model proposed by Cox. The first one is an analytical approach in which we derive the n^-1 bias term of the maximum partial likelihood estimator. The second approach consists of resampling methods, namely the jackknife and the bootstrap. We compare all methods through a comprehensive set of Monte Carlo simulations. The results suggest that bias-corrected estimators have better finite-sample performance than the standard maximum partial likelihood estimator. There is some evidence oithe bootstrap-correction superiority over the jackknife-correction but its performance is similar to the analytical estimator. Finaily an application iliustrates the proposed approaches.     

A robust I-sample analysis of means type randomization test for variances for unbalanced designs

Two analysis of means type randomization tests for testing the equality of I variances for unbalanced designs are presented. Randomization techniques for testing statistical hypotheses can be used when parametric tests are inappropriate. Suppose that I independent samples have been collected. Randomization tests are based on shuffles or rearrangements of the (combined) sample. Putting each of the I samples ‘in a bowl’ forms the combined sample. Drawing samples ‘from the bowl’ forms a shuffle. Shuffles can be made with replacement (bootstrap shuffling) or without replacement (permutation shuffling). The tests that are presented offer two advantages. They are robust to non-normality and they allow the user to graphically present the results via a decision chart similar to a Shewhart control chart. A Monte Carlo study is used to verify that the permutation version of the tests exhibit excellent power when compared to other robust tests. The Monte Carlo study also identifies circumstances under which the popular Levene's test fails.     

A Simulation Study of White's Test for Heteroskedasticity in Fixed and Stochastic Regression Models

We study the finite-sample properties of White's test for heteroskedasticity in fixed and stochastic regression models. We compare by simulation White and bootstrap methods when the underlying distribution is symmetric as well as asymmetric. The superior performance of the bootstrap method in small samples does not hold when the underlying distribution is asymmetric.     

Bootstrap power of the generalized correlation coefficient

We present a bootstrap Monte Carlo algorithm for computing the power function of the generalized correlation coefficient. The proposed method makes no assumptions about the form of the underlying probability distribution and may be used with observed data to approximate the power function and pilot data for sample size determination. In particular, the bootstrap power functions of the Pearson product moment correlation and the Spearman rank correlation are examined. Monte Carlo experiments indicate that the proposed algorithm is reliable and compares well with the asymptotic values. An example which demonstrates how this method can be used for sample size determination and power calculations is provided.     

Asymptotically refined score and GOF tests for inverse Gaussian models

ABSTRACT

The score test and the GOF test for the inverse Gaussian distribution, in particular the latter, are known to have large size distortion and hence unreliable power when referring to the asymptotic critical values. We show in this paper that with the appropriately bootstrapped critical values, these tests become second-order accurate, with size distortion being essentially eliminated and power more reliable. Two major generalizations of the score test are made: one is to allow the data to be right-censored, and the other is to allow the existence of covariate effects. A data mapping method is introduced for the bootstrap to be able to produce censored data that are conformable with the null model. Monte Carlo results clearly favour the proposed bootstrap tests. Real data illustrations are given.     

Bias correction for the regression-based LM fractional integration test

This paper examines the finite-sample behavior of the Lagrange Multiplier (LM) test for fractional integration proposed by Breitung and Hassler (J. Econom. 110:167–185, 2002). We find by extensive Monte Carlo simulations that size distortions can be quite large in small samples. These are caused by a finite-sample bias towards the alternative. Analytic expressions for this bias are derived, based on which the test can easily be corrected.     

On finite sample properties of the tests of robinson (1994) for fractional integration

This article proposes a bootstrap version of the tests of Robinson (1994) for testing unit and/or fractional roots. The finite-sample behaviour of the tests, based on these bootstrap critical values is compared with those based on asymptotic and on finite-sample results and with a number of leading unit-root tests. The Monte-Carlo simulations indicate that the bootstrap version of the tests of Robinson (1994) outperforms the other tests, including the one using finite-sample critical values. The improvement in the size and the power is particularly important under AR(1) alternatives. A small empirical application is also carried out with inflation for a panel of 16 European countries. The results show that the differences across countries depend on the critical values used: whereas the I (1) property of inflation is unclear with the asymptotic tests in some countries, the bootstrap version of Robinson's (1994) tests cannot reject the presence of a unit-root in inflation.     

A note on tests for interaction in quantal response data

There are few distribution-free methods for detecting interaction in fixed-dose trials involving quantal response data, despite the fact that such trials are common. We present three new tests to address this issue, including a simple bootstrap procedure. We examine the power of the likelihood ratio test and our new bootstrap test statistic using an innovative linear extrapolation power-estimation technique described in Boos, D. D. and Zhang, J. (2000) in Monte Carlo evaluation of resampling-based hypothesis tests. Journal of the American Statistical Association, 95, 486–492.     

Implementation of saddlepoint approximations in resampling problems

In many situations saddlepoint approximations can replace the Monte Carlo simulation typically used to find the bootstrap distribution of a statistic. We explain how bootstrap and permutation distributions can be expressed as conditional distributions and how methods for linear programming and for fitting generalized linear models can be used to find saddlepoint approximations to these distributions. The ideas are illustrated using an example from insurance.     

Two permutation tests of equality of variances

The F-ratio test for equality of dispersion in two samples is by no means robust, while non-parametric tests either assume a common median, or are not very powerful. Two new permutation tests are presented, which do not suffer from either of these problems. Algorithms for Monte Carlo calculation of P values and confidence intervals are given, and the performance of the tests are studied and compared using Monte Carlo simulations for a range of distributional types. The methods used to speed up Monte Carlo calculations, e.g. stratification, are of wider applicability.     

Estimating probabilities from invariant permutation distributions

Summary We introduce variance reduction techniques as general tools for estimating probabilities from invariant permutation distributions. The paper discusses importance sampling, antithetic sampling and control variates sampling as alternatives to uniform Monte Carlo sampling for estimating exact critical values orP-values in a broad class of permutation tests. Results may be extended to permutation confidence intervals and linear rank tests. An asymptotic theory is provided for each proposed variance reduction method. Invited paper at the Conference held in Bologna, Italy, 27–28 May 1993, on ?Statistical Tests: Methodology and Econometric Applications?.