Asymptotic Critical Values Research Articles

The maximally selected likelihood ratio test in random coefficient models

Summary In a recent contribution, we developed a family of cumulative sum-based change-point tests in the context of a random coefficient autoregressive model of order 1. In the current paper, we complement the results in that contribution by studying the (maximally selected) likelihood ratio statistic. We show that this has power versus breaks occurring even as close as periods from the beginning/end of sample; moreover, the use of quasi-maximum likelihood-based estimates yields better power properties, with the added bonus of being nuisance-free. Our test statistic has the same distribution—of the Darling–Erdős type—irrespective of whether the data are stationary or not, and can therefore be applied with no prior knowledge of this. Our simulations show that our test has very good power and, when applying a suitable correction to the asymptotic critical values, the correct size. We illustrate the usefulness and generality of our approach through applications to economic and epidemiological time series.

Bootstrap Inference in Cointegrating Regressions: Traditional and Self-Normalized Test Statistics

Traditional tests of hypotheses on the cointegrating vector are well known to suffer from severe size distortions in finite samples, especially when the data are characterized by large levels of endogeneity or error serial correlation. To address this issue, we combine a vector autoregressive (VAR) sieve bootstrap to construct critical values with a self-normalization approach that avoids direct estimation of long-run variance parameters when computing test statistics. To asymptotically justify this method, we prove bootstrap consistency for the self-normalized test statistics under mild conditions. In addition, the underlying bootstrap invariance principle allows us to prove bootstrap consistency also for traditional test statistics based on popular modified OLS estimators. Simulation results show that using bootstrap critical values instead of asymptotic critical values reduces size distortions associated with traditional test statistics considerably, but combining the VAR sieve bootstrap with self-normalization can lead to even less size distorted tests at the cost of only small power losses. We illustrate the usefulness of the VAR sieve bootstrap in empirical applications by analyzing the validity of the Fisher effect in 19 OECD countries.

Improved tests for stock return predictability

– Predictive regression methods are widely used to examine the predictability of (excess) stock returns by lagged financial variables characterized by unknown degrees of persistence and endogeneity. We develop a new hybrid test for predictability in these circumstances based on simple regression t-statistics. Where the predictor is endogenous, the optimal, but infeasible, test for predictability is based on the t-statistic on the lagged predictor in the basic predictive regression augmented with the current period innovation driving the predictor. We propose a feasible version of this augmented test, designed for the case where the predictor is an endogenous near-unit root process, using a GLS-based estimate of the innovation used in the infeasible test regression. The limiting null distribution of this statistic depends on both the endogeneity correlation parameter and the local-to-unity parameter characterizing the predictor. A method for obtaining asymptotic critical values is discussed and response surfaces are provided. We compare the asymptotic power properties of the feasible augmented test with those of a (non augmented) t-test recently considered in Harvey et al. and show that the augmented test is more powerful in the strongly persistent predictor case. We then propose using a weighted combination of the augmented statistic and the t-statistic of Harvey et al., where the weights are obtained using the p-values from a unit root test on the predictor. We find this can further improve asymptotic power in cases where the predictor has persistence at or close to that of a unit root process. Our final hybrid testing procedure then embeds the weighted statistic within a switching-based procedure which makes use of a standard predictive regression t-test, compared with standard normal critical values, when there is evidence for the predictor being weakly persistent. Monte Carlo simulations suggest that overall our new hybrid test displays superior finite sample performance to comparable extant tests.

On bootstrapping tests of equal forecast accuracy for nested models

AbstractThe asymptotic distributions of the recursive out‐of‐sample forecast accuracy test statistics depend on stochastic integrals of Brownian motion when the models under comparison are nested. This often complicates their implementation in practice because the computation of their asymptotic critical values is burdensome. Hansen and Timmermann (2015, Econometrica) propose a Wald approximation of the commonly used recursive ‐statistic and provide a simple characterization of the exact density of its asymptotic distribution. However, this characterization holds only when the larger model has one extra predictor or the forecast errors are homoscedastic. No such closed‐form characterization is readily available when the nesting involves more than one predictor and heteroscedasticity or serial correlation is present. We first show through Monte Carlo experiments that both the recursive ‐test and its Wald approximation have poor finite‐sample properties, especially when the forecast horizon is greater than one and forecast errors exhibit serial correlation. We then propose a hybrid bootstrap method consisting of a moving block bootstrap and a residual‐based bootstrap for both statistics and establish its validity. Simulations show that the hybrid bootstrap has good finite‐sample performance, even in multi‐step ahead forecasts with more than one predictor, and with heteroscedastic or autocorrelated forecast errors. The bootstrap method is illustrated on forecasting core inflation and GDP growth.

A portmanteau-type test for detecting serial correlation in locally stationary functional time series

The portmanteau test provides the vanilla method for detecting serial correlations in classical univariate time series analysis. The method is extended to the case of observations from a locally stationary functional time series. Asymptotic critical values are obtained by a suitable block multiplier bootstrap procedure. The test is shown to asymptotically hold its level and to be consistent against general alternatives.

Lipschitz continuity of tangent directions at infinity

We relate the set of asymptotic critical values of a polynomial f:Rn→R with the set of discontinuity of two functions: the first one is a set-valued function which associates to each t∈R the set of tangent directions at infinity of the fiber f−1(t) and the second one is the (n−2)-dimensional volume of the first one.

Permutation Tests at Nonparametric Rates

ABSTRACT Classical two-sample permutation tests for equality of distributions have exact size in finite samples, but they fail to control size for testing equality of parameters that summarize each distribution. This article proposes permutation tests for equality of parameters that are estimated at root-n or slower rates. Our general framework applies to both parametric and nonparametric models, with two samples or one sample split into two subsamples. Our tests have correct size asymptotically while preserving exact size in finite samples when distributions are equal. They have no loss in local asymptotic power compared to tests that use asymptotic critical values. We propose confidence sets with correct coverage in large samples that also have exact coverage in finite samples if distributions are equal up to a transformation. We apply our theory to four commonly-used hypothesis tests of nonparametric functions evaluated at a point. Lastly, simulations show good finite sample properties, and two empirical examples illustrate our tests in practice. Supplementary materials for this article are available online.

Kpsstest: A command that implements the Kwiatkowski, Phillips, Schmidt, and Shin test with sample-specific critical values and reports p-values

Commonly used unit-root tests in time-series analysis—such as the Dickey–Fuller and Phillips–Perron tests—use a null hypothesis that the series contains a unit root. Such tests have low power against the alternative—when a time series is near integrated or highly autoregressive—implying that they do poorly in distinguishing such a series from having a unit root. Kwiatkowski et al. (1992, Journal of Econometrics 54: 159–178) introduced the Kwiatkowski, Phillips, Schmidt, and Shin test, in which the null hypothesis is that the series is stationary, to deal with this problem. One shortcoming of the presently available Kwiatkowski, Phillips, Schmidt, and Shin test in Stata is that it uses asymptotic critical values regardless of the sample size. This poses a problem in that researchers—especially social scientists—are often presented with short time series. I introduce kpsstest, a command that extends the previous implementation by including an option for a zero-mean-stationary null hypothesis, generating sample and test-specific critical values, and reporting appropriate p-values.

The probabilities of type I and II error of null of cointegration tests: A Monte Carlo comparison.

This paper evaluates the performance of eight tests with null hypothesis of cointegration on basis of probabilities of type I and II errors using Monte Carlo simulations. This study uses a variety of 132 different data generations covering three cases of deterministic part and four sample sizes. The three cases of deterministic part considered are: absence of both intercept and linear time trend, presence of only the intercept and presence of both the intercept and linear time trend. It is found that all of tests have either larger or smaller probabilities of type I error and concluded that tests face either problems of over rejection or under rejection, when asymptotic critical values are used. It is also concluded that use of simulated critical values leads to controlled probability of type I error. So, the use of asymptotic critical values may be avoided, and the use of simulated critical values is highly recommended. It is found and concluded that the simple LM test based on KPSS statistic performs better than rest for all specifications of deterministic part and sample sizes.

An Empirical Study on Relationship between Residents’ Disposable Income and CPI Based on TVECM: Application of Bootstrap SupLM Testing for Threshold Cointegration

Based on the fluctuation characteristics of residents’ disposable income and CPI (Consumer Price Index) in different periods, this article introduces a nonlinear threshold cointegration theory, establishes a TVECM (Threshold Vector Error Correction Model) of the residents’ disposable income and CPI. We propose an algorithm to obtain maximum likelihood estimation under the condition of cointegration vector and threshold value which are unknown and then propose a SupLM test for the presence of a threshold. The asymptotic distribution of the SupLM statistic is analyzed, and it appears to depend on the moment functionals, so tabulated critical values are unavailable. We discuss how the residual bootstrap can be used to calculate asymptotic critical values and p values, investigate the size and power of the SupLM test using Monte Carlo simulation, and find that the test works quite well. In the empirical section, we apply our methods to test and estimate the TVECM of residents’ disposable income and CPI. According to the experimental results, the causal relationship between residents’ disposable income and CPI under different mechanisms is tested and compared with the test results under the linear cointegration hypothesis. The empirical results show that the disposable income of residents and CPI belong to a two‐mechanism nonlinear threshold cointegration system. When the deviation from the equilibrium state exceeds the threshold value, the system may adjust to the equilibrium state, and the adjustment speed of CPI shall be faster than that of the residents’ disposable income.

An Asymptotic Test for A Single Outlier in Linear Regression Models

It is not uncommon to find an outlier in the response variable in linear regression. Such a deviant value needs to be detected and scrutinized to find out why it is not in agreement with its fitted value. Srikantan [1] has developed a test statistic for detecting the presence of an outlier in the response variable in a multiple linear regression model. Approximate critical values of this test statistic are available and are obtained based on the first-order Bonferroni upper bound. The exact critical values are not available and a result of that, tests carried out on the basis of this approximate critical values may not be very accurate. In this paper, we obtained more accurate and precise critical values of this test statistic for large sample sizes (herein called asymptotic critical values) to improve on the tests that use these critical values. The procedure involved using the exact probability density function of this test statistic to obtain its asymptotic critical values. We then compared these asymptotic critical values with the approximate critical values obtained. An application to simulation results for linear regression models was used to examine the power of this test statistic. The asymptotic critical values obtained were found to be more accurate and precise. Also, the test performed better under these asymptotic values (the power performance of this test statistic was found to better when the asymptotic critical values were used).

Bootstrap inference and diagnostics in state space models: With applications to dynamic macro models

AbstractThis paper investigates the potentials of the bootstrap as a tool for inference on the parameters of macroeconometric models which admit a state space representation. We consider a bootstrap estimator of the parameters of state space models and show that the bootstrap realizations of this estimator, usually employed to approximate asymptotic confidence intervals, p‐values, and critical values of tests, can be also constructively used to build a test for forms of misspecifications which invalidate asymptotic normality. The test evaluates how “close or distant” the estimated state space model is from the case where asymptotic inference based on the Gaussian distribution applies. We derive sufficient conditions on the number of bootstrap repetitions, B, relative to the number of sample observations, T, for the test statistic to have a well‐defined asymptotic distribution under the null. Throughout the paper, we focus on the state space form of small‐scale monetary dynamic stochastic general equilibrium (DSGE) models and investigate the usefulness of our approach through Monte Carlo experiments and empirical illustrations based on US quarterly data. Results show that (i) bootstrapping the state space form provides highly reliable inference and (ii) the suggested test detects weakly identified parameters reasonably well in finite samples.

Application of Change Point Analysis of Response Time Data to Detect Test Speededness.

Computer-based and web-based testing have become increasingly popular in recent years. Their popularity has dramatically expanded the availability of response time data. Compared to the conventional item response data that are often dichotomous or polytomous, response time has the advantage of being continuous and can be collected in an unobstrusive manner. It therefore has great potential to improve many measurement activities. In this paper, we propose a change point analysis (CPA) procedure to detect test speededness using response time data. Specifically, two test statistics based on CPA, the likelihood ratio test and Wald test, are proposed to detect test speededness. A simulation study has been conducted to evaluate the performance of the proposed CPA procedure, as well as the use of asymptotic and empirical critical values. Results indicate that the proposed procedure leads to high power in detecting test speededness, while keeping the false positive rate under control, even when simplistic and liberal critical values are used. Accuracy of the estimation of the actual change point, however, is highly dependent on the true change point. A real data example is also provided to illustrate the utility of the proposed procedure and its contrast to the response-only procedure. Implications of the findings are discussed at the end.

A New Class of Life Distributions Based on Moment Inequalities

In this paper, a new class of life distribution is named new better than renewal used in moment generating function (NBRUmgf). A new test for exponentiality versus NBRUmgf based on moment inequalities is established. Pitman's asymptotic efficiencies, powers and critical values of the new test are calculated to assess the performance of the test. The right censored data is handled also. Finally some applications are applied to the new test.

Specification Tests for Families of Discrete Distributions with Applications to Insurance Claims Data

Families of distributions are commonly used to model insurance claims data that require flexible distributional forms in a satisfactory manner, but the specification problem to assess the goodness-of-fit of the hypothesized model can sometimes be a challenge due to the complexity of the likelihood function of the family of distributions involved. The previous work shows that these specification problems can be attacked by means of semi-parametric tests based on generalized method of moment (GMM) estimators. While the approach can be directly applied to both discrete and continuous families of distributions, the paper focuses on developing a testing strategy within a framework of discrete families of distributions. Both the local power analysis and the approximate slope method demonstrate the excellent performance of these tests. The finite-sample performance of the tests, based on both asymptotic and bootstrap critical values, are also discussed and are compared with established methods that require the complete specification of likelihood functions.

Detecting departures from meta-ellipticity for multivariate stationary time series

Abstract A test for detecting departures from meta-ellipticity for multivariate stationary time series is proposed. The large sample behavior of the test statistic is shown to depend in a complicated way on the underlying copula as well as on the serial dependence. Valid asymptotic critical values are obtained by a bootstrap device based on subsampling. The finite-sample performance of the test is investigated in a large-scale simulation study, and the theoretical results are illustrated by a case study involving financial log returns.

А был ли сдвиг: эмпирический анализ тестов на структурные сдвиги в волатильности доходностей

In this article, two popular tests for structural breaks are considered for return volatilities: the ICSS algorithm employing the AIT test, and the least-squares (LS) estimator. We show that the AIT test is sensitive to many features of the time series, and the use of asymptotic critical values is not always justified. The LS method was found to detect breaks more accurately, especially if there are many, in comparative simulations. Real data analysis revealed that LS estimation yields results in better accordance with general economic intuition, although its results are somewhat sensitive to the sample length. In general, we recommend the LS estimator for practical purposes.

Statistical tests of a simple energy balance equation in a synthetic model of cotrending and cointegration

We develop new tests for the linear relationship between temperature and forcing, which is one of the most studied implications from a simple energy balance model. We consider a bivariate system of temperature and forcing where the time path of well-mixed-greenhouse-gases forcing is included as a potential common trend function in addition to a stochastic trend and a broken linear trend. Our test statistics are first devised as the likelihood ratio and then are modified to remove nuisance parameters in the asymptotic null distribution. The asymptotic null distribution and the required modification differ as to the existence of a stochastic trend. Thus, the test statistics are modified in two different ways and then are combined using the super-efficient estimator of the sum of autoregressive coefficients. The asymptotic critical values from the two cases remain close and we use the bigger one to control size for both cases. The proposed tests are applied to four temperature series and a forcing series. The null hypothesis of the linear relationship is not rejected with conventional sizes.

Testing serial independence with functional data

We consider tests of serial independence for a sequence of functional observations. The new methods are formulated as L2-type criteria based on empirical characteristic functions and are convenient from the computational point of view. We derive asymptotic normality of the proposed test statistics for both discretely and continuously observed functions. In a Monte Carlo study, we show that the new test is sensitive with respect to functional GARCH alternatives, investigate the choice of necessary tuning parameters, and demonstrate that critical values obtained by resampling lead to a test with good performance in any setup, whereas the asymptotic critical values may be recommended only for a sufficiently fine discretization grid. Finite-sample comparison with a distance (auto)covariance test criterion is also included, and the article concludes with application on a real data set.

Response Surface Regressions for Critical Value Bounds and Approximate p‐values in Equilibrium Correction Models1

AbstractWe consider the popular ‘bounds test’ for the existence of a level relationship in conditional equilibrium correction models. By estimating response surface models based on about 95 billion simulatedF‐statistics and 57 billiont‐statistics, we improve upon and substantially extend the set of available critical values, covering the full range of possible sample sizes and lag orders, and allowing for any number of long‐run forcing variables. By computing approximateP‐values, we find that the bounds test can be easily oversized by more than 5 percentage points in small samples when using asymptotic critical values.