An approximate entropy test for randomness

We develop an entropy-based test for randomness of binary time series of finite length. The test uses the frequencies of contiguous blocks of different lengths. A simple condition ib the block lengths and the length of the time series enables one to estimate the entropy rate for the data, and this information is used to develop a statistic to test the hypothesis of randomness. This static measures the deviation of the estimated entropy of the observed data from the theoretical maximum under the randomness hypothesis. This test offers a real alternative to the conventional runs test. Critical percentage points, based on simulations, are provided for testing the hypothesis of randomness. Power calculations using dependent data show that the proposed test has higher power against the runs test for short series, and it is similar to the runs test for long series. The test is applied to two published data sets that wree investigated by others with respect to their randomness.     

Testing for Serial Independence: Beyond the Portmanteau Approach

Portmanteau tests are typically used to test serial independence even if, by construction, they are generally powerful only in presence of pairwise dependence between lagged variables. In this article, we present a simple statistic defining a new serial independence test, which is able to detect more general forms of dependence. In particular, differently from the Portmanteau tests, the resulting test is powerful also under a dependent process characterized by pairwise independence. A diagram, based on p-values from the proposed test, is introduced to investigate serial dependence. Finally, the effectiveness of the proposal is evaluated in a simulation study and with an application on financial data. Both show that the new test, used in synergy with the existing ones, helps in the identification of the true data-generating process. Supplementary materials for this article are available online.     

Tests for serial correlation in mean and variance of a sequence of time series objects

In this era of Big Data, large-scale data storage provides the motivation for statisticians to analyse new types of data. The proposed work concerns testing serial correlation in a sequence of sets of time series, here referred to as time series objects. An example is serial correlation of monthly stock returns when daily stock returns are observed. One could consider a representative or summarized value of each object to measure the serial correlation, but this approach would ignore information about the variation in the observed data. We develop Kolmogorov–Smirnov-type tests with the standard bootstrap and wild bootstrap Ljung–Box test statistics for serial correlation in mean and variance of time series objects, which take the variation within a time series object into account. We study the asymptotic property of the proposed tests and present their finite sample performance using simulated and real examples.     

Nonparametric Entropy-Based Tests of Independence Between Stochastic Processes

This article develops nonparametric tests of independence between two stochastic processes satisfying β-mixing conditions. The testing strategy boils down to gauging the closeness between the joint and the product of the marginal stationary densities. For that purpose, we take advantage of a generalized entropic measure so as to build a whole family of nonparametric tests of independence. We derive asymptotic normality and local power using the functional delta method for kernels. As a corollary, we also develop a class of entropy-based tests for serial independence. The latter are nuisance parameter free, and hence also qualify for dynamic misspecification analyses. We then investigate the finite-sample properties of our serial independence tests through Monte Carlo simulations. They perform quite well, entailing more power against some nonlinear AR alternatives than two popular nonparametric serial-independence tests.     

Tests for Serial Dependence and Other Specification Analysis in Models of Markets in Disequilibrium

The assumption of serial independence of disturbances is the starting point of most of the work done on analyzing market disequilibrium models. We derive tests for serial dependence given normality and homoscedasticity using the Lagrange multiplier (LM) test principle. Although the likelihood function under serial dependence is very complicated and involves multiple integrals of dimensions equal to the sample size, the test statistic we obtain through the LM principle is very simple. We apply the test to the housing-start data of Fair and Jaffee (1972) and study its finite sample properties through simulation. The test seems to perform quite well in finite samples in terms of size and power. We present an analysis of disequilibrium models that assumes that the disturbances are logistic rather than normal. The relative performances of these distributions are investigated by simulation.     

Efficient run orders for a two-factor response surface experiment on a correlated process

The standard assumption in the design and analysis of response surface experiments is that the errors are uncorrelated with common variance. In practice however there may be serial dependence between the errors, in which case the efficiency of the design depends on the order in which the runs are carried out. We investigate the effect of run order on a two-factor response surface experiment when there is positive serial correlation, and recommend some particular run orders which are efficient under a wide range of conditions.     

Kendall's tau for serial dependence

The authors show how Kendall's tau can be adapted to test against serial dependence in a univariate time series context. They provide formulas for the mean and variance of circular and noncircular versions of this statistic, and they prove its asymptotic normality under the hypothesis of independence. They present also a Monte Carlo study comparing the power and size of a test based on Kendall's tau with the power and size of competing procedures based on alternative parametric and nonparametric measures of serial dependence. In particular, their simulations indicate that Kendall's tau outperforms Spearman's rho in detecting first‐order autoregressive dependence, despite the fact that these two statistics are asymptotically equivalent under the null hypothesis, as well as under local alternatives.     

Generalized run tests for statistical process control

ABSTRACT

In a sequence of elements, a run is defined as a maximal subsequence of like elements. The number of runs or the length of the longest run has been widely used to test the randomness of an ordered sequence. Based on two different sampling methods and two types of test statistics used, run tests can be classified into one of four cases. Numerous researchers have derived the probability distributions in many different ways, treating each case separately. In the paper, we propose a unified approach which is based on recurrence arguments of two mutually exclusive sub-sequences. We also consider the sequence of nominal data that has more than two classes. Thus, the traditional run tests for a binary sequence are special cases of our generalized run tests. We finally show that the generalized run tests can be applied to many quality management areas, such as testing changes in process variation, developing non-parametric multivariate control charts, and comparing the shapes and locations of more than two process distributions.     

Testing for pairwise serial independence via the empirical distribution function

Built on Skaug and Tjøstheim's approach, this paper proposes a new test for serial independence by comparing the pairwise empirical distribution functions of a time series with the products of its marginals for various lags, where the number of lags increases with the sample size and different lags are assigned different weights. Typically, the more recent information receives a larger weight. The test has some appealing attributes. It is consistent against all pairwise dependences and is powerful against alternatives whose dependence decays to zero as the lag increases. Although the test statistic is a weighted sum of degenerate Cramér–von Mises statistics, it has a null asymptotic N (0, 1) distribution. The test statistic and its limit distribution are invariant to any order preserving transformation. The test applies to time series whose distributions can be discrete or continuous, with possibly infinite moments. Finally, the test statistic only involves ranking the observations and is computationally simple. It has the advantage of avoiding smoothed nonparametric estimation. A simulation experiment is conducted to study the finite sample performance of the proposed test in comparison with some related tests.     

Adaptive positive and negative runs test

As a nonparametric randomness test, the positive and negative runs test is widely used in practice due to the simplicity of its procedures. The test can lose efficiency if the alternative distribution is symmetrical at 0.5. In addition, the test can only be applied to test the randomness of a sequence from the uniform distribution. In this paper, we introduce an adaptive positive and negative runs test method to maximize the power function by choosing the optimal cut point. Also, the test is extended to check the randomness of a sequence generated from any other given distributions. Furthermore, we derive the exact distribution and obtain the asymptotical critical values of the proposed test statistics. Compared with the existed test, the efficiency of the proposed adaptive positive and negative runs test is competitive through simulation study.     

On the Monitoring of Autocorrelated Linear Profiles

Control charts are commonly used to monitor quality of a process or product characterized by a quality characteristic or a vector of quality characteristics. However, in many practical situations the quality of a process or product can be characterized by a function or profile. Here we consider a linear function and investigate the violation of common independence assumption implicitly considered in most control charting applications. We specifically consider the case when profiles are not independent from each other over time. In this article, the effect of autocorrelation between profiles is investigated using average run length (ARL) criterion. Simulation results indicate significant impact on the ARL values when autocorrelation is overlooked. In addition, three methods based on time series approach are used to eliminate the effect of autocorrelation. Their performances are compared using ARL criterion.     

Sampled autocovariance and autocorrelation results for linear time processes

We derive an exact formula for the covariance between the sampled autocovariances at any two lags for a finite time series realisation from a general stationary autoregressive moving average process. We indicate, through one particular example, how this result can be used to deduce analogous formulae for any nonstationary model of the ARUMA class, a generalisation of the ARIMA models. Such formulae then allow us to obtain approximate expressions for the convariances between all pairs of serial correlations for finite realisations from the ARUMA model. We also note that, in the limit as the series length n → ∞, our results for the ARMA class retrieve those of Bartlett (1946). Finally, we investigate an improvement to the approximation that is obtained by applying Bartlett's general asymptotic formula to finite series realisations. That such an improvement should exist can immediately be seen by consideration of out results for the simplest case of a white noise process. However, we deduce the final improved approapproximation, for general models, in two ways - from (corrected) results due to Davies and Newbold (1980), and by an alternative approach to theirs.     

Tests for an end-of-sample bubble in financial time series

ABSTRACT

In this paper, we examine the issue of detecting explosive behavior in economic and financial time series when an explosive episode is both ongoing at the end of the sample and of finite length. We propose a testing strategy based on a subsampling method in which a suitable test statistic is calculated on a finite number of end-of-sample observations, with a critical value obtained using subsample test statistics calculated on the remaining observations. This approach also has the practical advantage that, by virtue of how the critical values are obtained, it can deliver tests which are robust to, among other things, conditional heteroskedasticity and serial correlation in the driving shocks. We also explore modifications of the raw statistics to account for unconditional heteroskedasticity using studentization and a White-type correction. We evaluate the finite sample size and power properties of our proposed procedures and find that they offer promising levels of power, suggesting the possibility for earlier detection of end-of-sample bubble episodes compared to existing procedures.     

Universal codes as a basis for time series testing

We suggest a new approach to hypothesis testing for ergodic and stationary processes. In contrast to standard methods, the suggested approach gives a possibility to make tests, based on any lossless data compression method even if the distribution law of the codeword lengths is not known. We apply this approach to the following four problems: goodness-of-fit testing (or identity testing), testing for independence, testing of serial independence and homogeneity testing and suggest nonparametric statistical tests for these problems. It is important to note that practically used so-called archivers can be used for suggested testing.     

Powerful Trend Function Tests That Are Robust to Strong Serial Correlation,With an Application to the Prebisch–Singer Hypothesis

We propose tests for hypotheses on the parameters of the deterministic trend function of a univariate time series. The tests do not require knowledge of the form of serial correlation in the data, and they are robust to strong serial correlation. The data can contain a unit root and still have the correct size asymptotically. The tests that we analyze are standard heteroscedasticity autocorrelation robust tests based on nonparametric kernel variance estimators. We analyze these tests using the fixed-b asymptotic framework recently proposed by Kiefer and Vogelsang. This analysis allows us to analyze the power properties of the tests with regard to bandwidth and kernel choices. Our analysis shows that among popular kernels, specific kernel and bandwidth choices deliver tests with maximal power within a specific class of tests. Based on the theoretical results, we propose a data-dependent bandwidth rule that maximizes integrated power. Our recommended test is shown to have power that dominates a related test proposed by Vogelsang. We apply the recommended test to the logarithm of a net barter terms of trade series and we find that this series has a statistically significant negative slope. This finding is consistent with the well-known Prebisch–Singer hypothesis.     

Strong laws of large numbers for sub-linear expectation without independence

In this paper, we investigate some strong laws of large numbers for sub-linear expectation without independence which generalize the classical ones. We give some strong laws of large numbers for sub-linear expectation on some moment conditions with respect to the partial sum and some conditions similar to Petrov’s. We can reduce the conclusion to a simple form when the the sequence of random variables is i.i.d. We also show a strong law of large numbers for sub-linear expectation with assumptions of quasi-surely.     

A Comparison of the Runs Test for Volatility Forecastability and the LM Test for GARCH Using Aggregated Returns

Christoffersen and Diebold (2000) have introduced a runs test for forecastable volatility in aggregated returns. In this note, we compare the size and power of their runs test and the more conventional LM test for GARCH by Monte Carlo simulation. When the true daily process is GARCH, EGARCH, or stochastic volatility, the LM test has better power than the runs test for the moderate-horizon returns considered by Christoffersen and Diebold. For long-horizon returns, however, the tests have very similar power. We also consider a qualitative threshold GARCH model. For this process, we find that the runs test has greater power than the LM test. Theresults support the use of the runs test with aggregated returns.     

Generalized spectral tests for serial dependence

Two tests for serial dependence are proposed using a generalized spectral theory in combination with the empirical distribution function. The tests are generalizations of the Cramér-von Mises and Kolmogorov-Smirnov tests based on the standardized spectral distribution function. They do not involve the choice of a lag order, and they are consistent against all types of pairwise serial dependence, including those with zero autocorrelation. They also require no moment condition and are distribution free under serial independence. A simulation study compares the finite sample performances of the new tests and some closely related tests. The asymptotic distribution theory works well in finite samples. The generalized Cramér-von Mises test has good power against a variety of dependent alternatives and dominates the generalized Kolmogorov-Smirnov test. A local power analysis explains some important stylized facts on the power of the tests based on the empirical distribution function.     

Measures of Dependence and Tests of Independence

Measures of dependence and resulting tests of independence are surveyed. Measures arising both from linear and nonlinear modeling are examined. Tests based on chaos theory are briefly discussed. The main emphasis, however, is on some recently developed nonparametric tests using estimated distribution and density functions. Most of the paper is phrased in terms of serial dependence for a univariate stationary time series, but it is indicated how more general situations can be analysed. The bootstrap is an essential tool for determining the critical value of the new tests.     

The Finite-Sample Performance of White's Test for Heteroskedasticity Under Stochastic Regressors

We study the finite-sample properties of White's test for heteroskedasticity in stochastic regression models where explanatory variables are random and not given. We investigate by simulation the effect of non independence of explanatory variables and error term and heteroskedasticity on White's test. A standard bootstrap method in the computationally convenient form is found to work well with respect to the size and power.