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The autodependogram: A graphical device to investigate serial dependences
Abstract
In this article the serial dependences between the observed time series and the lagged series, taken into account one-by-one, are graphically analysed by what we have chosen to call the ‘autodependogram’. This tool is a sort of natural nonlinear counterpart of the well-known autocorrelogram used in the linear context. The autodependogram is based on the simple idea of using, instead of autocorrelations at varying time lags, the χ2-test statistics applied to convenient contingency tables. The efficacy of this graphical device is confirmed by real and artificial time series and by simulations from certain classes of well-known models, characterized by randomness and by different kinds of linear and nonlinear dependences.
... The considerations above have motivated the development of serial dependence diagrams (see Anderson and Vahid, 2005, Bagnato et al., 2012, Zhou, 2012, and Bagnato et al., 2014a, as well as serial independence tests (see Diks, 2009, pp. 6256-6257 andBagnato et al., 2014b), which are powerful against general types of dependence (omnibus procedures); however, the majority of such tests and diagrams have a drawback: they are based on pairs of lagged variables and, consequently, they can fail in detecting kinds of dependence involving more than two lagged variables simultaneously. ...
... To completely specify the test statistic δ l , the value of k must be selected. According to Bagnato et al. (2012), it is convenient to select k by matching the rules of Mann and Wald (1942) and Cochran (1954); in formula ...
... Operationally, δ l is computed for different, subsequent, values of l yielding the sample autodependence function (ADF) which is usually depicted (along with the critical value χ 2 [(k−1) 2 ;1−α] ) in the autodependogram of Bagnato et al. (2012). ...
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 paper 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.
... We discuss some of these measures again in Sect. 3. Also see Bagnato et al. (2012) and the references within for examples where more general association measures can be used for categorical time series. ...
... The special case λ = 0 gives us MI considered in, for example, Biswas and Guha (2009a,b). 2. The special case λ = 1 refers to the Pearson (χ 2 ) statistic (PS) considered by, for example, Weiß and Göb (2008), Bagnato et al. (2012) and Weiß (2013). ...
For stationary time series of nominal categorical data or ordinal categorical data (with arbitrary ordered numberings of the categories), autocorrelation does not make much sense. Biswas and Guha (J Stat Plan Infer 139:3076–3087, 2009a) used mutual information as a measure of association and introduced the concept of auto-mutual information in this context. In this present paper, we introduce general auto-association measures for this purpose and study several special cases. Theoretical properties and simulation results are given along with two illustrative real data examples.
... ; 30. Ifq l is the sample autocorrelation at lag l, the p-value is calculated as 2UðÀ jq l j ffiffi ffi T p Þ, where UðÁÞ denotes the standard normal distribution function (see, e.g., Romano andThombs, 1996 andNicolis, 2012). Figure 3 compares, separately for each series, "empirical" and "estimated" p-values. ...
We introduce multivariate models for the analysis of stock market returns. Our models are developed under hidden Markov and semi-Markov settings to describe the temporal evolution of returns, whereas the marginal distribution of returns is described by a mixture of multivariate leptokurtic-normal (LN) distributions. Compared to the normal distribution, the LN has an additional parameter governing excess kurtosis and this allows us a better fit to both the distributional and dynamic properties of daily returns. We outline an expectation maximization algorithm for maximum likelihood estimation which exploits recursions developed within the hidden semi-Markov literature. As an illustration, we provide an example based on the analysis of a bivariate time series of stock market returns. © The Author(s) 2018. Published by Oxford University Press. All rights reserved.
... Therefore, an ARMA/GARCH approach is usually applied to generate the filtered conditional residuals which can be assumed to be independent and identically distributed (Nystrom and Skoglund 2002;Ghrobel and Trabelsi 2009). Besides the copula approach, there is also an extensive literature on tests for serial independence (Rosenblatt 1975;Hallin and Melard 1988;Tsallis 1988;Tjøstheim 1993a, 1993b;Hall and Wolff 1995;Genest and R emillard 2004;Granger, Maasoumi and Racine 2004;Anderson and Vahid 2005;Diks 2009;Bagnato, Punzo, and Nicolis 2012;Punzo 2012, 2013;Zhou 2012;Bagnato, Capitani and Punzo 2014a, 2014b, 2016, 2017, 2018Bagnato et al. 2015). ...
Dependence measures, from linear correlation coefficients to recent copula-based methods, have been widely used to find out associations between variables. Although the latter type of measure has overcome many drawbacks of traditional measures, copula has intrinsically some undesirable characteristics for particular applications. In this paper, we discuss dependence modeling from a pattern recognition perspective and then introduce a new non-parametric approach based on anomaly detection through cluster analysis. The proposed methodology uses a weighting procedure based on Voronoi cells densities, named Weighted Voronoi Distance (WVD), to identify potentially atypical associations between univariate time series. The advantages are two-fold. First, the time series structure is respected and neither independence nor homoscedasticity is presumed within data. Second, any distribution of the data and any dependence function is allowed. An inference procedure is presented and simulation studies help to visualize the behavior and benefits of the proposed measure. Finally, real financial data is used to analyze the detection capacity of the contagion effect in financial markets during the 2007 sub-prime crisis. Different asset classes were included, and the WVD was able to signalize anomalies more strongly than the Extreme Value Theory and copula approach.
... which can be assumed to be independent and identically distributed (Ghorbel and Trabelsi, 2009;Nystrom and Skoglund, 2002). Besides the copula approach, there is also an extensive literature on tests for serial independence Punzo, 2018, 2017;Bagnato, Capitani, and Punzo, 2016;Bagnato et al., 2015;Bagnato, Capitani and Punzo, 2014a,b;Punzo, 2013, 2012;Zhou, 2012;Bagnato, Punzo, and Nicolis, 2011;Diks, 2009;Anderson and Vahid, 2005;Genest, and Rémillard, 2004;Granger, Maasoumi and Racine, 2004;Hall and Wolff, 1995;Skaug and Tjøstheim, 1993a,b;Tsallis, 1998;Hallin and Melard, 1988;Rosenblatt, 1975). ...
Dependence measures, from linear correlation coefficients to recent copula-based methods, have been widely used to find out associations between variables. Although the latter type of measure has overcome many drawbacks of traditional measures, copula has intrinsically some undesirable characteristics for particular applications. In this paper, we discuss dependence modeling from a pattern recognition perspective and then introduce a new non-parametric approach based on anomaly detection through cluster analysis. The proposed methodology uses a weighting procedure based on Voronoi cells densities, named Weighted Voronoi Distance (WVD), to identify potentially atypical associations between univariate time series. The advantages are two-fold. First, the time series structure is respected and neither independence nor homoscedasticity is presumed within data. Second, any distribution of the data and any dependence function is allowed. An inference procedure is presented and simulation studies help to visualize the behavior and benefits of the proposed measure. Finally, real financial data is used to analyze the detection capacity of the contagion effect in financial markets during the 2007 sub-prime crisis. Different asset classes were included, and the WVD was able to signalize anomalies more strongly than the Extreme Value Theory and copula approach.
... The value of k for the definition of b d L needs to be preliminarily determined. In particular, when L j j ¼ 1, k is chosen according to the rule of thumb given in Bagnato et al. (2012); in the other cases, k is selected through the following iterative procedure: 1. put k ¼ 3; 2. compute all the expected cell counts under the null; 3. if the lowest cell count is lower than five, then k ¼ k À 1 and the procedure ends; otherwise, put k ¼ k þ 1 and go to step 2. ...
The Ljung–Box test is typically used to test serial independence even if, by construction, it is generally powerful only in presence of pairwise linear dependence between lagged variables. To overcome this problem, Bagnato et al. recently proposed a simple statistic defining a serial independence test which, differently from the Ljung–Box test, is powerful also under a linear/nonlinear dependent process characterized by pairwise independence. The authors also introduced a normalized bar diagram, based on p-values from the proposed test, to investigate serial dependence. This paper proposes a balanced normalization of such a diagram taking advantage of the concept of reproducibility probability. This permits to study the strength and the stability of the evidence about the presence of the dependence under investigation. An illustrative example based on an artificial time series, as well as an application to a transport time series, are considered to appreciate the usefulness of the proposal.
... , d. We use contingency tables with classes such that the samples are equally distributed among √ n c classes in the x i -and x jdirection (Bagnato et al. 2012, section 3.1). Moreover, a χ 2 distribution with ( √ n c − 1) 2 degrees of freedom is applied. ...
The estimation of probability densities based on available data is a central task in many statistical applications. Especially in the case of large ensembles with many samples or high-dimensional sample spaces, computationally efficient methods are needed. We propose a new method that is based on a decomposition of the distribution to be estimated in terms of so-called distribution elements (DEs). These elements enable an adaptive and hierarchical discretization of the sample space with small or large elements in regions with high and variable or low densities, respectively. The refinement strategy that we propose is based on statistical goodness-of-fit and independence tests that evaluate the local approximation of the distribution in terms of DEs. The capabilities of our new method are inspected based on several low and high-dimensional examples.
... Despite the fast advancement in nonlinear time series models, there are few tools which can explore the complex dependence structures in nonlinear time series, like the autocorrelogram does for the linear ones. Starting from these considerations, several diagrams have been recently proposed which are very similar, in aspect and intent, to the autocorrelogram but they are widely applicable to both linear and nonlinear time series (see Anderson and Vahid 2005, Bagnato, Punzo, and Nicolis 2012, Bagnato, De Capitani, and Punzo 2014, 2013a, and Zhou 2012. ...
Detecting and measuring lag-dependencies is very important in time-series analysis.
This study is commonly carried out by focusing on the linear lag-dependencies via the
well-known autocorrelogram. However, in practice, there are many situations in which
the autocorrelogram fails because of the nonlinear structure of the serial dependence.
To cope with this problem, in this paper the R package SDD is introduced. Among
the available approaches to analyze the lag-dependencies in an omnibus way, the SDD
package considers the autodependogram and some of its variants. The autodependogram,
defined by computing the classical Pearson chi-square statistic at various lags, is a graphical
device recently proposed in the literature to analyze lag-dependencies. The concept of
reproducibility probability, and several density-based measures of divergence, are considered
to define the variants of the autodependogram. An application to daily returns of
the Swiss Market Index is also presented to exemplify the use of the package.
... The DBKGrad package also allows to graphically investigate the residuals dependence structure through the autocorrelogram, as implemented by the acf() function of the TSA package, and the autodependogram of Bagnato, Punzo, and Nicolis (2012), as implemented by the ADF() function of the SDD package (see also Punzo 2012a, 2013). The autodependogram looks like the autocorrelogram with the difference that the aucorrelations for each lag are substituted by the χ 2 statistics of (linear/nonlinear) dependence; in order to show possible "problematic" lags, a critical line is superimposed (for further details and developments on this diagram see, e.g., Bagnato and Punzo 2010and Bagnato, De Capitani, and Punzo 2014. ...
We introduce the R package DBKGrad, conceived to facilitate the use of kernel smoothing in graduating mortality rates. The package implements univariate and bivariate adaptive discrete beta kernel estimators. Discrete kernels have been preferred because, in this context, variables such as age, calendar year and duration, are pragmatically considered as discrete and the use of beta kernels is motivated since it reduces boundary bias. Furthermore, when data on exposures to the risk of death are available, the use of adaptive bandwidth, that may be selected by cross-validation, can provide additional bene�ts. To exemplify the use of the package, an application to Italian mortality rates, for di�fferent ages and calendar years, is presented.
Using a novel methodology, we offer new evidence that a threshold relationship exists for Okun's law (the well‐known output–unemployment co‐movement). We use a logistic smooth transition regression (LSTR) model where threshold endogeneity is addressed using copula transformations of the threshold variable. We also suggest a test of the linearity hypothesis against the LSTR model. In line with Okun's insight (and that of the subsequent literature) that the trade‐off can be affected by different margins, we consider several potential threshold variables. We find mainly a combination of structural and policy‐related variables accounts for changes in the Okun's law trade‐off for the United States in recent decades. This conclusion is bolstered by combing these threshold candidates into a single factor. Accordingly, we find that the unemployment gap is increasingly associated with a smaller output gap. Notably, while the Great Recession accelerated that rise, the bulk of the change occurred beforehand.
The partial autocorrelation function (PACF) is often used in time series analysis to identify the extent of the lag in an autoregressive model. However, the PACF is only suitable for detecting linear correlations. This article proposes the conditional distance autocovariance function (CDACF), which is zero if and only if measured time series components are conditionally independent. Due to the lack of this property, traditional tools for measuring partial correlations such as the PACF cannot work well for nonlinear sequences. Based on the CDACF, we introduce a tool known as an integrated conditional distance autocovariance function (ICDACF), which can test conditional temporal dependence structures of a sequence and estimate the order of an autoregressive process. Simulation studies reveal that the ICDACF can detect the conditional dependence of nonlinear autoregressive models efficiently while controlling for type‐I error rates. Finally, an analysis of a Bitcoin price dataset using the ICDACF demonstrates that our method has considerable advantages over other state‐of‐the‐art methods.
In this chapter the autodependogram is contextualized in model diagnostic checking for nonlinear models by studying the lag-dependencies of the residuals. Simulations are considered to evaluate its effectiveness in this context. An application to the Swiss Market Index is also provided.
The autodependogram is a graphical device recently proposed in the literature to analyze autodependencies. It is defined computing the classical Pearson -statistics of independence at various lags in order to point out the presence lag-depedencies. This paper proposes an improvement of this diagram obtained by substituting the -statistics with an estimator of the Kullback–Leibler divergence between the bivariate density of two delayed variables and the product of their marginal distributions. A simulation study, on well-established time series models, shows that this new autodependogram is more powerful than the previous one. An application to a well-known financial time series is also shown.
The NDARMA model is a discrete counterpart to the usual ARMA model, which can be applied to purely categorical processes. NDARMA processes are shown to be @f-mixing, so it is possible to find asymptotic expressions for the distribution of several types of statistics. Such asymptotic properties are useful for hypothesis testing or other inferential procedures. This is exemplified by considering the Gini index and the entropy as measures of marginal dispersion, the Pearson statistic for checking the goodness-of-fit with regard to a hypothetical marginal distribution, and several measures of signed or unsigned serial dependence. For each of these cases, the obtained asymptotic approximations are also compared to the empirically observed behavior in time series of finite length. Practical applications are illustrated by a real-data example.
The autodependogram is a graphical device recently proposed in the literature
to analyze autodependencies. It is defined computing the classical Pearson
chi-square statistics of independence at various lags in order to point out the
presence lag-depedencies. This paper proposes an improvement of this diagram
obtained by substituting the chi-square statistics with an estimator of the
Kulback-Leibler divergence between the bivariate density of two delayed
variables and the product of their marginal distributions. A simulation study,
on well-established time series models, shows that this new autodependogram is
more powerful than the previous one. An application to financial data is also
shown.
We extend the concept of distance correlation of Szekely et al. (2007) and propose the auto distance correlation function (ADCF) to measure the temporal dependence structure of time‐series. Unlike the classic measures of correlations such as the autocorrelation function, the proposed measure is zero if and only if the measured time‐series components are independent. In this article, we propose and theoretically verify a subsampling methodology for the inference of sample ADCF for dependent data. Our methodology provides a useful tool for exploring nonlinear dependence structures in time‐series.
The autodependogram is a graphical device recently proposed in the literature to analyze autodependencies. This paper proposes a normalization of this diagram taking into consideration the concept of reproducibility probability (RP). The result is a novel tool, named RP-autodependogram, which permits to study the strength and the stability of the evidence about the presence of lag-dependence. A simulation study on well-established time-series models is carried out to investigate the behavior of the RP-autodependogram also in comparison with other diagrams studying autodependencies. An application to financial data is finally considered to appreciate its usefulness in the identification of parametric/nonparametric models.
This article reviews some nonparametric serial independence tests based on measures of divergence between densities. Among others, the well-known Kullback–Leibler, Hellinger, Tsallis, and Rosenblatt divergences are analyzed. Moreover, their copula-based version is taken into account. Via a wide simulation study, the performances of the considered serial independence tests are compared under different settings. Both single-lag and multiple-lag testing procedures are investigated to find out the best “omnibus” solution.
This paper presents a graphical method to be used, in conjunction with a scatterplot, to investigate possible association of two variates as manifested in a sample of bivariate measurements. The method is designed so that the plot is approximately horizontal under independence, and under various forms of association it produces corresponding characteristic patterns. Examples given include application to study of regression residuals, dependence of two spatial point processes, serial association of time series, and comparison of two time series.
Ever since the publication in 1565 of Girolamo Cardano's treatise on gambling, Liber de Ludo Aleae (The Book of Games of Chance), statistics and nancial markets have become inextricably linked. Over the past few decades many of these links have become part of the canon of modern nance, and it is now impossible to fully appreciate the workings of nancial markets without them. This selective survey covers three of the most important ideas of nance|eecient markets, the random walk hypothesis, and derivative pricing models|that illustrate the enormous research opportunities that lie at the intersection of nance and statistics. I am grateful to Samantha Arrington and Mark Becker for helpful comments. (email).
This paper presents a new test of independence (linear and non-linear) among distributions based on the entropy of Shannon.
The main advantages of the presented approach are the fact that this measure does not need to assume any type of theoretical
probability distribution and has the ability to capture the linear and non-linear dependencies, without requiring the specification
of any kind of dependence model.
Glossary
Definition of the Subject
Introduction
Invariant Tests
Tests Based on Divergence Measures
Tests Based on Other Measures of Dependence
Bootstrap and Permutation Tests
Future Directions
Bibliography
The classical problem of choice of number of classes in testing goodness of fit is considered for a class of alternatives, for the chi-square and likelihood ratio statistics. Pitman and Bahadur efficiencies are used to compare the two statistics and also to analyse the effect for each statistic of changing the number of classes for the case where the number of classes increases asymptotically with the number of observations. Overall, the results suggest that if the class of alternatives is suitably restricted the number of classes should not be very large.
This collection of vignettes, the second in a series of four collections to appear in the Journal of the American Statistical Association in the year 2000, explores the interdigitation of statistics with business and social science. The 10 vignettes in the collection cover a broad range of methodologies and applications. The collection opens with vignettes on finance and marketing, followed by a series of methodologically focused vignettes on times series and forecasting, contingency tables, and causal inference. Attention then moves to disciplines, with coverage of political science, psychology, sociology, demography, and the law. The reader of this collection should gain an appreciation for the history and role of statistical thinking and methodology in the evolution of studies in business and social science, and for the considerable promise that exists for continued innovations at the interstices of statistics and these fields of inquiry.
In this article we consider the large-sample behavior of estimates of autocorrelations and autoregressive moving average (ARMA) coefficients, as well as their distributions, under weak conditions. Specifically. the usual text book formulas for variances of these estimates are based on strong assumptions and should nor be routinely applied without careful consideration. Such is the case when the time series follows an ARMA process with uncorrelated innovations that may not be assumed to be independent and identically distributed. As a specific case, ii is well known that if the process is independence and identically distributed, then the sample autocorrelation estimates, scaled by the square root of the sample size, are asymptotically standard normal, This result is used extensively as a diagnostic check on the residuals of a fitted model, or as an initial test on the observed time series to determine whether further model fitting is warranted. In this article we show that this result can be quite misleading. Specifically, if the underlying process is assumed to br uncorrelated rather than independent, then rile: asymptotic distribution is not necessarily standard normal. Although this distinction may appear superficial, the implications for making valid inference in time series modeling are broad. Usual procedures in time series analysis model correlation structure by fitting models whose estimated errors mimic an uncorrelated sequence. Therefore, testing for the presence of zero autocorrelation using a result that assumes independence may lead to incorrect conclusions. Furthermore, there exist stationary time series that have zero autocorrelation at all lags but yet are not independent, and so it is important to have valid procedures under general dependence structures. Were we present general asymptotic theory for the estimated autocorrelation function and discuss testing for lack of correlation without the further assumption of independence. We propose appropriate resampling methods that can be used to approximate the sampling distribution of the autocorrelation estimates under weak assumptions.
Introduction.- Stationary Time Series.- Smoothing in Time Series.- ARMA Modeling and Forecasting.- Parametric Nonlinear Time Series Models.- Nonparametric Models.- Hypothesis Testing.- Continuous Time Models in Finance.- Nonlinear Prediction.
The problem of testing a simple null hypothesis on multinomial distribution is considered. Biasedness of customarily used tests of fit is proved for unequal cell probabilities case.
This article describes in non-mathematical fashion the technique suggested by H. B. Mann and A. Wald for selecting the number and width of class intervals for the chi-square test of goodness of fit when the null hypothesis distribution is continuous and completely specified. The number of classes is selected by means of a formula depending upon the sample size and the level of significance and the class limits are chosen such that each class contains the same number of items under the null hypothesis. Finally it is suggested that the number of classes as given by the formula may be halved for practical purposes.
When values of a random variable X 1 ,X 2 ,···,X N are fitted to a hypothesized statistical distribution, the χ 2 test is sometimes used to determine whether the fit is acceptable. When the postulated distribution is continuous and completely specified, H. B. Mann and A. Wald [On the choice of the number of class intervals in the application of the chi-square test. Ann. Math. Stat. 13, 306-317 (1942)] suggest a method for determining the optimum number of classes k into which the N observations should be divided. They prove that for large enough N the power p of the χ 2 test against a fixed alternative is always one-half or greater, provided that k is determined by a formula involving N, a distance Δ between the distribution and the data and a significance level α. In this report we give more general formulas relating N, k, Δ, and α when the power must exceed some arbitrary constant β.
A selective review of nonlinear time series is presented. All of the three phases in the specification - estimation - verification modelling cycle are covered, but much of the emphasis is on nonparametric and restricted nonparametric methods. In particular, recent results on nonparametric tests of linearity and independence are included.
In this paper two tests of serial independence are proposed. The building block of these
procedures is the definition of a component chi-square test for testing independence between
pairs of lagged variables. With reference to different component chi-square tests, it is shown
that the corresponding test statistics are independent. Taking advantage of this result,
the component tests are used from both a simultaneous and a direct viewpoint
to define two different test procedures denoted by SIS (Serial Independence Simultaneous)
and SICS (Serial Independence Chi-Square). Simulations are used to explore
the performance of these tests in terms of size and power. Our results underline that
both the proposals are powerful against various types of alternatives. It is also shown,
through what we call Lag Subsets Dependence Plot (LSDP), how to detect possible
lag(s)-dependences graphically. Some examples are finally provided in order to evaluate
the effectiveness of the LSDP.
Note:
Republished in: Am J Psychol. 100(3-4) 441-71 (1987).
Republished in: Int J Epidemiol. 39(5):1137-50 (2010).
Environmental time series usually vary systematically in response to meteorological conditions and thus often are not stationary. In this article a class of additive models are introduced for environmental time series, in which both mean levels and variances of the series are nonlinear functions of relevant meteorological variables. Backfitting algorithms in nonlinear regression are adopted to estimate the unknown functions in the model, and the maximum likelihood method is used to estimate the parameters in the noise component. Asymptotic properties of the parameter estimates, including consistency and limiting distribution, are derived under mild conditions. The model is applied to daily maxima of ground-level ozone concentrations in the Chicago area for possible long-term trend assessment. Compared to alternative models, the proposed models gave more accurate estimations for the 95th and 99th percentiles of the ozone distribution.
When applied to frequency tables with small expected cell counts, Pearson chi-squared test statistics may be asymptotically inconsistent even in cases in which a satisfactory chi-squared approximation exists for the distribution under the null hypothesis. This problem is particularly important in cases in which the number of cells is large and the expected cell counts are quite variable. To illustrate this bias of the chi-squared test, this article considers the Pearson chi-squared test of the hypothesis that the cell probabilities for a multinomial frequency table have specified values. In this case, the expected value and variance of the Pearson chi-square may be evaluated under both the null and alternative hypotheses. When the number of cells is large, normal approximations and discrete Edgeworth expansions may also be used to assess the size and power of the Pearson chi-squared test. These analyses show that unless all cell probabilities are equal, it is possible to select a significance level and cell probabilities under the alternative hypothesis such that the power is less than the size of the test. As shown by exact calculations, the difference may be substantial even in cases in which all expected cell sizes are at least 5 under the null hypothesis. The use of moments shows that given any minimum expected cell size under the null hypothesis and given any significance level, it is possible to make the power arbitrarily close to 0 by the selection of a large enough number of cells in the table and suitable cell probabilities for the null and alternative hypotheses. The normal approximations for the distribution of the Pearson chi-squared statistic permit the size of this bias to be assessed in less-extreme cases involving tables with many cells. These results imply that caution must be exercised in the application of Pearson chi-squared statistics to sparse contingency tables with many cells. An alternative to the Pearson chi-square, proposed by Zelterman (1986), avoids some of the problems. Exact calculation, however, shows that the alternative statistic does not eliminate all problems of bias. The problems described in this article clearly extend to more general applications of the Pearson chi-squared statistic.
The power of Pearson chi-squared and likelihood ratio goodness-of-fit tests based on different partitions is studied by considering families of densities “between” the null density and fixed alternative densities. For sample sizes n → ∞, local asymptotic theory with respect to the number of classes k is developed for such families. Simple sufficient and almost necessary conditions are derived under which it is asymptotically optimal to let k tend to infinity with n. A numerical study shows that the results of the asymptotic local theory for contamination families agree well with the actual power performance of the tests. For heavy-tailed alternatives, the tests have the best power when k is relatively large. Unbalanced partitions with some small classes in the tails perform surprisingly well, in particular when the alternatives have fairly heavy tails.
The asymptotic non-null distribution is obtained for the modified form of the Pearson chi-square statistic studied by Dahiya and Gurland [3]. By utilizing this result the power is obtained for specific alternative distributions in testing for normality. This enables recommendations to be made as to the number of class intervals to be employed in applying the aforementioned modification of the Pearson chi-square test of normality.
The Helmert transformation of a set of observations {xj}, j = 1, 2, …, N is applied to give a direct approach to the problem of boundary determination in the case of the χ-test for location of the normal distribution, i.e. the choice of that set of boundaries which maximizes the power of the test. It is found that no significant increase in power can be achieved by taking a number of classes greater than 20; and even a number as small as 12 is sufficient. Moreover, a power of one-half is achieved for a number of classes much smaller than that required by the Mann-Wald approach. For a preassigned number of classes, symmetrically situated about the origin, it is found that the optimum partition corresponds to equal class-width of about 0.4 standard deviation with pooling of the terminal classes, and this is slightly more powerful than the equal class-probability partition.
This article describes in non-mathematical fashion the technique suggested by H. B. Mann and A. Wald for selecting the number and width of class intervals for the chi-square test of goodness of fit when the null hypothesis distribution is continuous and completely specified. The number of classes is selected by means of a formula depending upon the sample size and the level of significance and the class limits are chosen such that each class contains the same number of items under the null hypothesis. Finally it is suggested that the number of classes as given by the formula may be halved for practical purposes.* This article is based on a Master's Essay written at Columbia University under Professor T. W. Anderson, Jr.
Trapping records of the Canadian lynx show a strongly marked 10-year
cycle. The logarithms of the numbers trapped are analysed as if they were a
random process of autoregressive type. Such a process appears to fit the data
reasonably well. The significance of this for the explanation and prediction
of the cycle is discussed. The results will be used in a later paper to consider
how far meteorological phenomena influence the lynx population and may be
responsible for the observed synchronization of the cycle over the whole of
Canada.
In this article we consider the large-sample behavior of estimates of autocorrelations and autoregressive moving average (ARMA) coefficients, as well as their distributions, under weak conditions. Specifically, the usual text book formulas for variances of these estimates are based on strong assumptions and should not be routinely applied without careful consideration. Such is the case when the time series follows an ARMA process with uncorrelated innovations that may not be assumed to be independent and identically distributed. As a specific case, it is well known that if the process is independent and identically distributed, then the sample autocorrelation estimates, scaled by the square root of the sample size, are asymptotically standard normal. This result is used extensively as a diagnostic check on the residuals of a fitted model, or as an initial test on the observed time series to determine whether further model fitting is warranted. In this article we show that this result can be quite misleading. Specifically, if the underlying process is assumed to be uncorrelated rather than independent, then the asymptotic distribution is not necessarily standard normal. Although this distinction may appear superficial, the implications for making valid inference in time series modeling are broad. Usual procedures in time series analysis model correlation structure by fitting models whose estimated errors mimic an uncorrelated sequence. Therefore, testing for the presence of zero autocorrelation using a result that assumes independence may lead to incorrect conclusions. Furthermore, there exist stationary time series that have zero autocorrelation at all lags but yet are not independent, and so it is important to have valid procedures under general dependence structures. Here we present general asymptotic theory for the estimated autocorrelation function and discuss testing for lack of correlation without the further assumption of independence. We propose appropriate resampling methods that can be used to approximate the sampling distribution of the autocorrelation estimates under weak assumptions.
To use the Pearson chi-squared statistic to test the fit of a continuous distribution, it is necessary to partition the support of the distribution into k cells. A common practice is to partition the support into cells with equal probabilities. In that case, the power of the chi-squared test may vary substantially with the value of k. The effects of different values of k are investigated with a Monte Carlo power study of goodness-of-fit tests for distributions where location and scale parameters are estimated from the observed data. Allowing for the best choices of k, the Pearson and log-likelihood ratio chi-squared tests are shown to have similar maximum power for wide ranges of alternatives, but this can be substantially less than the power of other well-known goodness-of-fit tests.
Squared-residual autocorrelations have been found useful in detecting non-linear types of statistical dependence in the residuals of fitted autoregressive-moving average (ARMA) models [cf. C. W. J. Granger and A. P. Andersen, An introduction to bilinear time series models. (1978; Zbl 0379.62074)]. In this note it is shown that the normalized squared-residual autocorrelations are asymptotically unit multivariate normal. The results of a simulation experiment confirming the small- sample validity of the proposed tests is reported.
This empirical study of the chi-square approximations as commonly encountered in behavioral research involved: (1) both tests of goodness-of-fit and of independence, (2) uniform distributions and two levels of departure from uniform, (3) sample sizes ranging from 10 to 100. Excellent approximations were obtained with average expected frequencies of one or two in tests of goodness-of-fit to uniform; slightly higher expected frequencies were required with the nonuniform cases. Tests of independence were strikingly robust with respect to Type I errors; in almost all cases the errors were in the conservative direction.
The distribution of the chi-square goodness-of-fit statistic is studied in the equiprobable case. Tables of exact critical values are given for a = .1, .05, .01, .005; k = 2(1)4, N = 26(1)50; k = 5, N = 26(1)40; k = 6(1)10, N = 26(1)30, where a is the desired significance level, k is the number of cells and N is the sample size. Methods of fitting the true distribution are compared. If k> 3, it is found that a simple additive adjustment to the asymptotic chi-square fit leads to high accuracy even for N between 10 and 20. For k = 2, the Yates corrected chi-square statistic is very accurately fitted by the usual chi-square distribution.
The chi-squared goodness of fit test (also known as the PearsonX2test) is often used to test whether data are consistent with a specified continuous distribution. The value of X2(and hence its associated probability level) can be altered by the choice of (i) the number of classes and (ii) the class probabilities. The effect on the power of the X2test of varying the number of classes when class probabilities are chosen to be equal is investigated.
In the case of two signals with independent pairs of observations (x(n),y(n)) a statistic to estimate the variance of the histogram based mutual information estimator has been derived earlier. We present such a statistic for dependent pairs. To derive this statistic it is necessary to avail of a reliable statistic to estimate the variance of the sample mean in case of dependent observations. We derive and discuss this statistic and a statistic to estimate the variance of the mutual information estimator. These statistics are validated by simulations.
For over a decade, nonparametric modelling has been successfully applied to studying nonlinear structures in financial time series. It is well known that the usual nonparametric models often have less than satisfactory performance when dealing with more than one lag. When the mean has an additive structure, however, better estimation methods are available which fully exploit such a structure. Although in the past such nonparametric applications had been focused more on the estimation of the conditional mean, it is equally if not more important to measure the future risk of the series along with the mean. For the volatility function, i.e. the conditional variance given the past, a multiplicative structure is more appropriate than an additive structure, as the volatility is a positive scale function and a multiplicative model provides a better interpretation of each lagged value's influence on such a function. In this paper we consider the joint estimation of both the additive mean and the multiplicative volatility. The technique used is marginally integrated local polynomial estimation. The procedure is applied to the deutschmark/US dollar daily exchange returns.
The problem of describing hourly data of ground ozone is considered. The complexity of high frequency environmental data dynamics often requires models covering covariates, multiple frequency periodicities, long memory, non-linearity and heteroscedasticity. For these reasons we introduce a parametric model which includes seasonal fractionally integrated components, self-exciting threshold autoregressive components, covariates and autoregressive conditionally heteroscedastic errors with high tails. For the general model, we present estimation and identification techniques.
To show the model descriptive capability and its use, we analyse a five year hourly ozone data set from an air traffic pollution station located in Bergamo, Italy. The role of meteo and precursor covariates, periodic components, long memory and non-linearity is assessed. Copyright
This bibliography brings together and classifies a wide variety of nonparametric methods which are pocentially useful for the analysis of time series data, in particular for testing randomness. Inference on Markov chain models is also extensively surveyed.
The book is divided into four chapters. The first one introduces the structure of deregulated, competitive electricity markets with the power pools and power exchanges as the basic marketplaces for price discovery. Chapter 2 reviews the so-called stylized facts of selected power markets. In particular, the spiky nature of electricity prices, the different levels of seasonality inherent in load and price time series, the anti-persistent behavior of prices and the heavy-tailed distributions of returns. Chapter 3 reviews load forecasting techniques, with particular emphasis on statistical methods. Various models with and without exogenous variables are illustrated and compared in two comprehensive case studies. Finally, Chapter 4 discusses price modeling and forecasting. Six different approaches are surveyed and two – statistical and quantitative – are further studied. As in the previous chapter, the theoretical considerations and techniques are illustrated and evaluated using real-world data.
Past research into the evolution of Finnish stock returns focuses on modeling linear and nonlinear dependence using various ARIMA and GARCH formulations, respectively. This paper extends the extant work by using Grassberger-Procaccia correlations dimensions to explore the nature of the nonlinear dynamics in daily Finnish stock returns during the 1970s and 1980s. Nonlinear behavior in both periods is evident. A simple GARCH model removes the nonlinearity in the first decade and dramatically reduces the nonlinearity in the second period. This supports the notion that Finnish stock returns exhibit nonlinear dependence but that the form of dependence is not chaotic.
Due to inherent spatial and temporal variability of emission concentrations, influence of meteorological conditions, and uncertainties associated with initial and boundary conditions, it is very difficult to model, calibrate, and validate ozone variations from first principles. This paper describes a new procedure, based on dynamical systems theory, to model and predict ozone levels. A model is constructed by creating a multidimensional phase space map from observed ozone concentrations. Predictions are made by examining trajectories on a reconstructed phase space that applies to hourly ozone time series from the Cincinnati area. The proposed phase space model is used to make one-hour to one-day ahead predictions of ozone levels. Prediction accuracy is used as a diagnostic tool to characterize the nature, random vs deterministic, of ozone variations. To demonstrate the utility of this diagnostic tool, the proposed method is first applied to time series with known characteristics. Results for the ozone time series suggest that it can be characterized as a low-dimensional chaotic system. Then, the performance of the proposed phase space and optimum autoregressive model are compared for one-hour and one-day ahead predictions. Three performance measures, namely, root-mean-square error, prediction accuracy, and coefficient of determination yield similar results for both phase space and autoregressive models. However, the phase space model is clearly superior in terms of bias and normalized absolute error of predictions, specially for one-day ahead predictions. In addition, the ability of the proposed model to identify the underlying characteristics from a time series makes it a powerful tool to characterize and predict ozone concentrations.
We propose a wavelet multiscale decomposition-based autoregressive approach for the prediction of 1-h ahead load based on historical electricity load data. This approach is based on a multiple resolution decomposition of the signal using the non-decimated or redundant Haar à trous wavelet transform whose advantage is taking into account the asymmetric nature of the time-varying data. There is an additional computational advantage in that there is no need to recompute the wavelet transform (wavelet coefficients) of the full signal if the electricity data (time series) is regularly updated. We assess results produced by this multiscale autoregressive (MAR) method, in both linear and non-linear variants, with single resolution autoregression (AR), multilayer perceptron (MLP), Elman recurrent neural network (ERN) and the general regression neural network (GRNN) models. Results are based on the New South Wales (Australia) electricity load data that is provided by the National Electricity Market Management Company (NEMMCO).
Certain tests of independence based on the sample distribution function (d.f.) possess power properties superior to those of other tests of independence previously discussed in the literature. The characteristic functions of the limiting d.f.'s of a class of such test criteria are obtained, and the corresponding d.f. is tabled in the bivariate case, where the test is equivalent to one originally proposed by Hoeffding [4]. A discussion is included of the computational problems which arise in the inversion of characteristic functions of this type. Techniques for computing the statistics and for approximating the tail probabilities are considered.