Marked Empirical Process Research Articles

Unified specification tests in partially linear time series models

Based on a residual marked empirical process, Cramér–von Mises and Kolmogorov–Smirnov tests are proposed for the correct specification of the nonparametric components in partially linear time series models. The tests are unified in the sense that the asymptotic distribution of residual marked empirical process is invariant across different nν-consistent estimators in calculating residuals (where ν>1/4) under the null. In addition, the residual marked empirical process has the same power property under the sequence of local alternatives regardless of the estimators used. Achieved through a projection method, these features also enable using a computationally convenient multiplier bootstrap to approximate the unified null distributions of the test statistics. Simulations show satisfactory finite-sample performance of the proposed method. The application to validate the parametric form of conditional variance in the ARCH-X model is also highlighted, along with an empirical analysis of the conditional variance of the FTSE 100 index return series.

Model Checking in Partially Linear Spatial Autoregressive Models

This article proposes two new classes of nonparametric tests for the correct specification of linear spatial autoregressive models based on the “integrated conditional moment” approach. Our test statistics are constructed as continuous functionals of a residual marked empirical process as well as its projected version. We derive asymptotic properties of the test statistics under the null hypothesis, the alternative hypothesis, and a sequence of local alternatives. The proposed tests do not involve the selection of tuning parameters such as bandwidths and are able to detect a broad class of local alternatives converging to the null at the parametric rate n − 1 / 2 , with n being the sample size. We also propose a multiplier bootstrap procedure that is computationally simple to approximate the critical values. Monte Carlo simulations illustrate that our tests have a reasonable size and satisfactory power for different types of data-generating processes. Finally, an empirical analysis of growth convergence is carried out to demonstrate the usefulness of the tests.

Jump‐robust testing of volatility functions in continuous time models

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Nonparametric volatility change detection

AbstractWe consider a nonparametric heteroscedastic time series regression model and suggest testing procedures to detect changes in the conditional variance function. The tests are based on a sequential marked empirical process and thus combine classical CUSUM tests from change point analysis with marked empirical process approaches known from goodness‐of‐fit testing. The tests are consistent against general alternatives of a change in the conditional variance function, a feature that classical CUSUM tests are lacking. We derive a simple limiting distribution and in the case of univariate covariates even obtain asymptotically distribution‐free tests. We demonstrate the good performance of the tests in a simulation study and consider exchange rates as a real data application.

Estimating change points in nonparametric time series regression models

In this paper we consider a regression model that allows for time series covariates as well as heteroscedasticity with a regression function that is modelled nonparametrically. We assume that the regression function changes at some unknown time lfloor ns_0rfloor , s_0in (0,1), and our aim is to estimate the (rescaled) change point s_0. The considered estimator is based on a Kolmogorov-Smirnov functional of the marked empirical process of residuals. We show consistency of the estimator and prove a rate of convergence of O_P(n^{-1}) which in this case is clearly optimal as there are only n points in the sequence. Additionally we investigate the case of lagged dependent covariates, that is, autoregression models with a change in the nonparametric (auto-) regression function and give a consistency result. The method of proof also allows for different kinds of functionals such that Cramér-von Mises type estimators can be considered similarly. The approach extends existing literature by allowing nonparametric models, time series data as well as heteroscedasticity. Finite sample simulations indicate the good performance of our estimator in regression as well as autoregression models and a real data example shows its applicability in practise.

Consistent nonparametric change point detection combining CUSUM and marked empirical processes

A weakly dependent time series regression model with multivariate covariates and univariate observations is considered, for which we develop a procedure to detect whether the nonparametric conditional mean function is stable in time against change point alternatives. Our proposal is based on a modified CUSUM type test procedure, which uses a sequential marked empirical process of residuals. We show weak convergence of the considered process to a centered Gaussian process under the null hypothesis of no change in the mean function and a stationarity assumption. This requires some sophisticated arguments for sequential empirical processes of weakly dependent variables. As a consequence we obtain convergence of Kolmogorov-Smirnov and Cramér-von Mises type test statistics. The proposed procedure acquires a very simple limiting distribution and nice consistency properties, features from which related tests are lacking. We moreover suggest a bootstrap version of the procedure and discuss its applicability in the case of unstable variances.

Weak convergence of marked empirical processes in a Hilbert space and its applications

We discuss goodness-of-fit tests for stationary ergodic processes based on limit theorems for marked empirical processes viewed as elements of an $L^{2}$ space. Our limit theorems cover a weighted process that enables an Anderson–Darling-type test statistic for the goodness-of-fit tests to be proposed. The procedures presented for these goodness-of-fit tests are novel.

Comparing two nonparametric regression curves in the presence of long memory in covariates and errors

This paper discusses the problem of testing the equality of two nonparametric regression functions against two-sided alternatives in the presence of long memory in the common covariate and errors. The proposed test is based on a marked empirical process of the differences between the response variables. We discuss asymptotic null distribution of this process and consistency of the test for a class of general alternatives. We also conduct a Monte Carlo simulation study to evaluate the finite sample level and power behavior of the test at some alternatives.

Goodness-of-fit tests for the functional linear model based on randomly projected empirical processes

We consider marked empirical processes indexed by a randomly projected functional covariate to construct goodness-of-fit tests for the functional linear model with scalar response. The test statistics are built from continuous functionals over the projected process, resulting in computationally efficient tests that exhibit root-n convergence rates and circumvent the curse of dimensionality. The weak convergence of the empirical process is obtained conditionally on a random direction, whilst the almost surely equivalence between the testing for significance expressed on the original and on the projected functional covariate is proved. The computation of the test in practice involves calibration by wild bootstrap resampling and the combination of several p-values, arising from different projections, by means of the false discovery rate method. The finite sample properties of the tests are illustrated in a simulation study for a variety of linear models, underlying processes, and alternatives. The software provided implements the tests and allows the replication of simulations and data applications.

Testing nonstationary and absolutely regular nonlinear time series models

We study some general methods for testing the goodness-of-fit of a general nonstationary and absolutely regular nonlinear time series model. These testing methods are based on some marked empirical processes that we show to converge in distribution to a zero-mean Gaussian process with respect to the Skorohod topology. We investigate the behavior of this process under fixed alternatives and under a sequence of local alternatives. Our results are applied to testing a general class of nonlinear semiparametric time series models. A simulation experiment shows that the Cramer–von Mises test studied behaves well on the examples considered.

Model checks for nonlinear cointegrating regression

Using the marked empirical processes, this paper develops a test of parametric specification in a nonlinear cointegrating regression model. Unlike the kernel-smoothed U-statistic considered in Gao et al. (2009) and Wang and Phillips (2012), our new test statistic avoids the use of bandwidth, which has some advantages for practitioners. Simulations and a real data example show that our new test has a good finite sample performance. Other contributions of this paper are to provide a rigorous proof on weak convergence for a class of martingales and construct a simulated estimator of the limiting null distribution, which are interesting in their own rights.

Lack of fit test for long memory regression models

This paper proposes a test for assessing the accuracy of an assumed nonlinear regression model with long memory design and heteroscedastic long memory errors. The test is based on the marked empirical process. The asymptotic distributions of the proposed test statistics are investigated. The leave-one-observation-out kernel type estimator of the conditional variance function is also constructed in order to implement the lack of fit test.

A nonparametric model check for time series when the random vectors are nonstationary and absolutely regular

In this Note, we study some general methods for testing the goodness-of-fit of a parametric model for a real-valued Markovian time series under nonstationarity and absolute regularity. For that, we define a marked empirical process based on residuals, which converges in distribution to a Gaussian process with respect to the Skorohod topology. This method was first introduced by Koul and Stute [1], and then widely developed by Ngatchou-Wandji [2,3] under more general conditions. Applications to general AR-ARCH models are given.

Testing Equality of Nonparametric Quantile Regression Functions

This article proposes a new approach for testing the equality of nonparametric quantile regression functions based on marked empirical processes. We develop test statistics that posses better Type I and power properties in comparison to all available procedures in the literature. Simulation results also indicate that our tests have superior local power properties over existing tests. A data analysis is given which highlights the usefulness of the proposed methodology.

Marked empirical processes for non-stationary time series

Consider a first-order autoregressive process $X_{i}=\beta X_{i-1}+\varepsilon_{i}$, where $\varepsilon_{i}=G(\eta_{i},\eta_{i-1},\ldots)$ and $\eta_{i}$, $i\in\mathbb{Z}$ are i.i.d. random variables. Motivated by two important issues for the inference of this model, namely, the quantile inference for $H_{0}\colon\ \beta=1$, and the goodness-of-fit for the unit root model, the notion of the marked empirical process $\alpha_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}g(X_{i}/a_{n})I(\varepsilon_{i}\leq x)$, $x\in\mathbb{R}$ is investigated in this paper. Herein, $g(\cdot)$ is a continuous function on $\mathbb{R}$ and $\{a_{n}\}$ is a sequence of self-normalizing constants. As the innovation $\{\varepsilon_{i}\}$ is usually not observable, the residual marked empirical process $\hat{\alpha}_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}g(X_{i}/a_{n})I(\hat{\varepsilon}_{i}\leq x)$, $x\in\mathbb{R}$, is considered instead, where $\hat{\varepsilon}_{i}=X_{i}-\hat{\beta}X_{i-1}$ and $\hat{\beta}$ is a consistent estimate of $\beta$. In particular, via the martingale decomposition of stationary process and the stochastic integral result of Jakubowski (Ann. Probab. 24 (1996) 2141–2153), the limit distributions of $\alpha_{n}(x)$ and $\hat{\alpha}_{n}(x)$ are established when $\{\varepsilon_{i}\}$ is a short-memory process. Furthermore, by virtue of the results of Wu (Bernoulli 95 (2003) 809–831) and Ho and Hsing (Ann. Statist. 24 (1996) 992–1024) of empirical process and the integral result of Mikosch and Norvaiša (Bernoulli 6 (2000) 401–434) and Young (Acta Math. 67 (1936) 251–282), the limit distributions of $\alpha_{n}(x)$ and $\hat{\alpha}_{n}(x)$ are also derived when $\{\varepsilon_{i}\}$ is a long-memory process.

Bootstrap based model checks with missing binary response data

Dikta, Kvesic, and Schmidt proposed a model-based resampling scheme to approximate critical values of tests for model checking involving binary response data. Their approach is inapplicable when the binary response variable is not always observed, however. We propose a missingness adjusted marked empirical process under the framework that the missing binary responses are missing at random. We introduce a resampling scheme for the bootstrap and prove its asymptotic validity. We present some numerical comparisons and illustrate our methodology using a real data set.

LACK-OF-FIT TESTING OF THE CONDITIONAL MEAN FUNCTION IN A CLASS OF MARKOV MULTIPLICATIVE ERROR MODELS

AbstractThe family of multiplicative error models, introduced by Engle (2002, Journal of Applied Econometrics 17, 425–446), has attracted considerable attention in recent literature for modeling positive random variables, such as the duration between trades at a stock exchange, volume transactions, and squared log returns. Such models are also applicable to other positive variables such as waiting time in a queue, daily/hourly rainfall, and demand for electricity. This paper develops a new method for testing the lack-of-fit of a given parametric multiplicative error model having a Markov structure. The test statistic is of Kolmogorov–Smirnov type based on a particular martingale transformation of a marked empirical process. The test is asymptotically distribution free, is consistent against a large class of fixed alternatives, and has nontrivial asymptotic power against a class of nonparametric local alternatives converging to the null hypothesis at the rate of O (n–1/2). In a simulation study, the test performed better overall than the general purpose Ljung–Box Q-test, a Lagrange multiplier type test, and a generalized moment test. We illustrate the testing procedure by considering two data examples.

The marked empirical process to test nonlinear time series against a large class of alternatives when the random vectors are nonstationary and absolutely regular

In this paper, we propose a method for testing absolutely regular and possibly nonstationary nonlinear time-series, with application to general AR-ARCH models. Our test statistic is based on a marked empirical process of residuals which is shown to converge to a Gaussian process with respect to the Skohorod topology. This testing procedure was first introduced by Stute [Nonparametric model checks for regression, Ann. Statist. 25 (1997), pp. 613–641] and then widely developed by Ngatchou-Wandji [Weak convergence of some marked empirical processes: Application to testing heteroscedasticity, J. Nonparametr. Stat. 14 (2002), pp. 325–339; Checking nonlinear heteroscedastic time series models, J. Statist. Plann. Inference 133 (2005), pp. 33–68; Local power of a Cramer-von Mises type test for parametric autoregressive models of order one, Compt. Math. Appl. 56(4) (2008), pp. 918–929] under more general conditions. Applications to general AR-ARCH models are given.

Estimation and testing of crossing‐points in fixed design regression

Many phenomena in the life sciences can be analyzed by using a fixed design regression model with a regression function m that exhibits a crossing‐point in the following sense: the regression function runs below or above its mean level, respectively, according as the input variable lies to the left or to the right of that crossing‐point, or vice versa. We propose a non‐parametric estimator and show weak and strong consistency as long as the crossing‐point is unique. It is defined as maximizing point arg max of a certain marked empirical process. For testing the hypothesis H0 that the regression function m actually is constant (no crossing‐point), a decision rule is designed for the specific alternative H1 that m possesses a crossing‐point. The pertaining test‐statistic is the ratio max/argmax of the maximum value and the maximizing point of the marked empirical process. Under the hypothesis the ratio converges in distribution to the corresponding ratio of a reflected Brownian bridge, for which we derive the distribution function. The test is consistent on the whole alternative and superior to the corresponding Kolmogorov–Smirnov test, which is based only on the maximal value max. Some practical examples of possible applications are given where a certain study about dental phobia is discussed in more detail.

Weak Convergence of Marked Empirical Processes for Focused Inference on AR(p) vs AR(p + 1) Stationary Time Series

The technique applied by the authors to construct consistent and focused tests of fit for i.i.d. samples and regression models is extended to AR models for stationary time series. This approach leads to construct a consistent goodness-of-fit test for the null hypothesis that a stationary series is governed by an autoregressive model of a given order p. In addition of the consistency, the test is focused to detect efficiently the alternative of an AR(p + 1) model. The basic functional statistic conveying the information provided by the series is the process of accumulated sums of the residuals computed under the model of the null hypothesis of fit, reordered as concomitants of the conveniently delayed process. This process is transformed in order to obtain a new process with the same limiting Gaussian law encountered in earlier applications of the technique. Therefore, a Watson type quadratic statistic computed from this process has the same asymptotic laws under the null hypothesis of fit, and also under the alternatives of focusing, than the test statistics used in those applications. As a consequence, the resulting test has the same desirable performance as the tests previously developed by applying the same kind of transformations of processes.