A parametric bootstrap for heavytailed distributions

It is known that Efron's resampling bootstrap of the mean of random variables with common distribution in the domain of attraction of the stable laws with infinite variance is not consistent, in the sense that the limiting distribution of the bootstrap mean is not the same as the limiting distribution of the mean from the real sample. Moreover, the limiting distribution of the bootstrap mean is random and unknown. The conventional remedy for this problem, at least asymptotically, is either the m out of n bootstrap or subsampling. However, we show that both these procedures can be quite unreliable in other than very large samples. A parametric bootstrap is derived by considering the distribution of the bootstrap P value instead of that of the bootstrap statistic. The quality of inference based on the parametric bootstrap is examined in a simulation study, and is found to be satisfactory with heavy-tailed distributions unless the tail index is close to 1 and the distribution is heavily skewed.bootstrap inconsistency, stable distribution, domain of attraction, infinite variance

The Explicit Formula for the Hodrick-Prescott Filter in Finite Sample

We derive the exact expression for the weights of the Hodrick-Prescott (HP) filter in finite sample without making any assumptions about the statistical properties of the time series. We use the results to give insights about the properties of the HP filter and to build a fast algorithm with computational improvements by a factor of up to three times in samples typical in economics

A parametric bootstrap for heavytailed distributions

It is known that Efron's resampling bootstrap of the mean of random variables with common distribution in the domain of attraction of the stable laws with infinite variance is not consistent, in the sense that the limiting distribution of the bootstrap mean is not the same as the limiting distribution of the mean from the real sample. Moreover, the limiting distribution of the bootstrap mean is random and unknown. The conventional remedy for this problem, at least asymptotically, is either the m out of n bootstrap or subsampling. However, we show that both these procedures can be quite unreliable in other than very large samples. A parametric bootstrap is derived by considering the distribution of the bootstrap P value instead of that of the bootstrap statistic. The quality of inference based on the parametric bootstrap is examined in a simulation study, and is found to be satisfactory with heavy-tailed distributions unless the tail index is close to 1 and the distribution is heavily skewed

Link of moments before and after transformations, with an application to resampling from fat-tailed distributions

Let x be a transformation of y, whose distribution is unknown. We derive an expansion formulating the expectations of x in terms of the expectations of y. Apart from the intrinsic interest in such a fundamental relation, our results can be applied to calculating E(x) by the low-order moments of a transformation which can be chosen to give a good approximation for E(x). To do so, we generalize the approach of bounding the terms in expansions of characteristic functions, and use our result to derive an explicit and accurate bound for the remainder when a finite number of terms is taken. We illustrate one of the implications of our method by providing accurate naive bootstrap confidence intervals for the mean of any fat-tailed distribution with an infinite variance, in which case currently available bootstrap methods are asymptotically invalid or unreliable in finite samples

Bootstrapping Structural Change Tests

This paper demonstrates the asymptotic validity of methods based on the wild recursive and wild fixed bootstraps for testing hypotheses about discrete parameter change in linear models estimated via Two Stage Least Squares. The framework allows for the errors to exhibit conditional and/or unconditional heteroscedasticity, and for the reduced form to be unstable. Simulation evidence indicates the bootstrap tests yield reliable inferences in the sample sizes often encountered in macroeconomics. If the errors exhibit unconditional heteroscedasticity and/or the reduced form is unstable then the bootstrap methods are particularly attractive because the limiting distributions of the test statistics are not pivotal

Replication data for: "The Explicit Formula for the Hodrick-Prescott Filter in Finite Sample"

Cornea-Madeira, Adriana, (2017) "The Explicit Formula for the Hodrick-Prescott Filter in Finite Sample." Review of Economics and Statistics 99:2, 314-318

A Parametric Bootstrap for Heavy Tailed Distributions

ACL-1International audienceIt is known that Efron’s bootstrap of the mean of a distribution in the domain of attraction of the stable laws with infinite variance is not consistent, in the sense that the limiting distribution of the bootstrap mean is not the same as the limiting distribution of the mean from the real sample. Moreover, the limiting bootstrap distribution is random and unknown. The conventional remedy for this problem, at least asymptotically, is either the m out of n bootstrap or subsampling. However, we show that both these procedures can be unreliable in other than very large samples. We introduce a parametric bootstrap that overcomes the failure of Efron’s bootstrap and performs better than the m out of n bootstrap and subsampling. The quality of inference based on the parametric bootstrap is examined in a simulation study, and is found to be satisfactory with heavy-tailed distributions unless the tail index is close to 1 and the distribution is heavily skewed

Econometric analysis of switching expectations in UK inflation

We estimate with UK data a Phillips curve model with backward-looking and forward-looking methods of determining inflation expectations and with agents switching between these based on their recent performance. We find that, while on average backward-looking and forward-looking methods have about equal weight, there are considerable movements in the weight given to each method. We show this model has better in-sample fit than other Phillips curve models and this is robust to the methodology chosen. The model out-of-sample forecasts on certain dates do better than other Phillips curve models and the Atkeson and Ohanian model

Disentangling the effect of measures, variants, and vaccines on SARS-CoV-2 infections in England:A dynamic intensity model

In this paper, we estimate the path of daily SARS-CoV-2 infections in England from the beginning of the pandemic until the end of 2021. We employ a dynamic intensity model, where the mean intensity conditional on the past depends both on past intensity of infections and past realized infections. The model parameters are time-varying, and we employ a multiplicative specification along with logistic transition functions to disentangle the time-varying effects of nonpharmaceutical policy interventions, of different variants, and of protection (waning) of vaccines/boosters. Our model results indicate that earlier interventions and vaccinations are key to containing an infection wave. We consider several scenarios that account for more infectious variants and different protection levels of vaccines/boosters. These scenarios suggest that, as vaccine protection wanes, containing a new wave in infections and an associated increase in hospitalizations in the near future may require further booster campaigns and/or nonpharmaceutical interventions

Bootstrapping structural change tests

This paper analyses the use of bootstrap methods to test for parameter change in linear models estimated via Two Stage Least Squares (2SLS). Two types of test are considered: one where the null hypothesis is of no change and the alternative hypothesis involves discrete change at k unknown break-points in the sample; and a second test where the null hypothesis is that there is discrete parameter change at l break-points in the sample against an alternative in which the parameters change at l + 1 break-points. In both cases, we consider inferences based on a sup-Wald-type statistic using either the wild recursive bootstrap or the wild fixed bootstrap. We establish the asymptotic validity of these bootstrap tests under a set of general conditions that allow the errors to exhibit conditional and/or unconditional heteroskedasticity, and report results from a simulation study that indicate the tests yield reliable inferences in the sample sizes often encountered in macroeconomics. The analysis covers the cases where the first-stage estimation of 2SLS involves a model whose parameters are either constant or themselves subject to discrete parameter change. If the errors exhibit unconditional heteroskedasticity and/or the reduced form is unstable then the bootstrap methods are particularly attractive because the limiting distributions of the test statistics are not pivotal