On testing for causality in variance between two multivariate time series

Diagnostic checking of multivariate nonlinear time series models with martingale difference errors

This article investigates the causal relationships between gold and stock market performance or uncertainty by employing nonuniform weighting cross-correlations. In our sample period covering the last decade, we detect a unidirectional causality in mean from stock to gold, but find no causality in variance between the two. For subsample periods divided into pre- and post-current financial crisis, although we detect bidirectional causality in mean for the first sample period, there exists only a unilateral causality in mean and variance from stock to gold for the second sample period. These findings imply that flight-to-quality has occurred during the recent financial turmoil.

<abstract> <p>This paper examines the effects of volatility spillover between MIST stock markets. We used daily data from January 3, 2012, to November 16, 2021, and the test of causality in variance, including structural breaks. First, we observed no structural break for Turkey and Indonesia, but there is a structural break in each South Korean and Mexican stock markets. A structural break is found for Mexico the day, aka "El Betito" was arrested. Likewise, we detected a structural break on the first day of the Covid-19 case for South Korea. We surprisingly observed that Covid-19 did not cause a structural break in these markets during the period analyzed, except in South Korea. GARCH (1, 1) models, including structural breaks, show that all series are temporarily and permanently affected by their own shocks. The causality in variance test reveals that MIST countries have volatility spillover effects on each other. There is bidirectional causality in variance between all markets, except South Korea and Mexico. The Mexican stock market is a volatility transmitter for South Korea, but vice versa. The paper indicates a connection in terms of the financial markets of MIST countries, and they are affected by each other's shocks, according to the study results.</p> </abstract>

A new mathematical method was developed to analyze time series. Applications of this method to the delta component of all-night sleep electroencephalogram (EEG) revealed new variations with double-rapid eye movement (REM)-sleep interval. The proposed method entails repeated application of the least squares spectrum. First, the conventional least squares spectrum calculation is applied to the time series. Using the parameters of the peak components in the obtained spectrum, an intermediate time series is reconstructed. The residual time series is made by subtracting the intermediate time series from the previous time series. Next, the least squares calculation is again applied to the residual time series. These procedures are repeated until the component cannot be detected. We named this new method the Repeated Least Squares Spectrum for the Residual (RLSSR). The remarkably similar time series pattern to the original time series can be reconstructed using the obtained parameters of all components. The EEG was recorded during all-night sleep on five consecutive nights in five healthy adults, by a total of 25 recordings. The variations of the height of successive 1-min delta components in the EEG were analyzed. In addition, two autocorrelograms were made for two time series patterns. These were reconstructed from two specific sets of the components obtained by the proposed analysis method. These autocorrelograms demonstrated a long-span variation with near double-REM-sleep interval in not a few records. Based on these data, it appears that the delta component of sleep EEG does not always demonstrate a simple gradually decreasing trend.

SUMMARY The problem of testing for periodicities in multiple time series is considered. The test statistic proposed is based on the Euclidean norm of the matrix of periodograms. We first obtain the exact distribution of our test statistic under the hypothesis that observations are taken from an n-vector time series consisting of independent series of independent normal variables. We extend the theory to normal linear series with a specified spectral density matrix and then show that a suitable estimate can be used to replace this matrix in our test statistic. These results are asymptotic in nature. Extreme value theorems are stated for the case of normal variates and it is shown that the distributions for nonnormal series can be approximated adequately by those for normal series. An exact test for periodic components in an univariate time series was first proposed by Fisher (1929) who considered the null hypothesis of normal white noise. The test was based upon the maximum of a set of periodogram ordinates each divided by their sum, the division making it unnecessary to know in advance the variance of the components of the series. The use of the rth largest of these ordinates, r > 1, in testing for periodicities was also discussed by Fisher (1940). Whittle (1951, p. 101) showed how one could apply the Fisher tests to normal linear series provided the number of observations was large and one could specify the spectral density for the series. Hannan (1961) proposed a modification to this test that removed the necessity of knowing beforehand the spectral density function. Nicholls (1967) proposed a variation on Hannan's test statistic that improves its power under certain alternatives. Some extreme value results for periodogram ordinates were given by Walker (1965), who also demonstrated that dropping the normality condition has little effect on the large sample distributions, at least in the upper tails. In the sequel we consider the problem of testing for periodicities in multiple time series and pursue a course that parallels the historical development for univariate series as outlined in the paragraph above. Our tests are based on the Euclidean norm of the matrix of periodo- grams. In this connection Nicholls (1969) has discussed a test for discrete mass in a cross- spectrum. We consider real, discrete time parameter, n-vector, time series, Z(t) (t = 0, + 1, + 2, ...), where Z(t) = It+m(t) +X(t), with ,u+m(t) being a mean value function and X( ) being a weakly stationary, zero mean, time series possessing a spectral density matrix fxx(A) with tr{fxx(A)} <1 M < oo (JAI < 7r). We also assume the existence of a moving average representation 00 wo,t) , CO v - )

Test procedures for assessing whether two stationary and independent time series with unequal lengths have the same spectral density (or same auto‐covariance function) are investigated. A new test statistic is proposed based on the wavelet transform. It relies on empirical wavelet coefficients of the logarithm of two spectral densities' ratio. Under the null hypothesis that two spectral densities are the same, the asymptotic normal distribution of the empirical wavelet coeffcients is derived. Furthermore, these empirical wavelet coefficients are asymptotically uncorrelated. A test statistic is proposed based on these results. The performance of the new test statistic is compared to several recent test statistics, with respect to their exact levels and powers. Simulation studies show that our proposed test is very comparable to the current test statistics in most cases. The main advantage of our proposed test statistic is that it is constructed very simply and is easy to implement.

This article proposes a class of weighted differences of averages (WDA) statistics to test and estimate possible change-points in variance for time series with weakly dependent blocks and dependent panel data without specific distributional assumptions. We derive the asymptotic distributions of the test statistics for testing the existence of a single variance change-point under the null and local alternatives. We also study the consistency of the change-point estimator. Within the proposed class of the WDA test statistics, a standardized WDA test is shown to have the best consistency rate and is recommended for practical use. An iterative binary searching procedure is suggested for estimating the locations of possible multiple change-points in variance, whose consistency is also established. Simulation studies are conducted to compare detection power and number of wrong rejections of the proposed procedure to that of a cumulative sum (CUSUM) based test and a likelihood ratio-based test. Finally, we apply the proposed method to a stock index dataset and an unemployment rate dataset. Supplementary materials for this article are available online.

Semiparametric Innovation-Based Tests of Orthogonality and Causality between Two Infinite-Order Cointegrated Ceries with Application to Canada/US Monetary Interactions

Aims: Introduce two new test statistics in testing for the nondecreasing alternative in a mixed design consisting of a Completely Randomized Portion and a Randomized Complete Block Portion.&#x0D; Study Design: Simulation study comparing four test statistics for the nondecreasing alternative in a mixed design consisting of a CRD and an RCBD portion. The test statistics included two new test statistics and two existing test statistics. Random samples were taken from three different types of underlying distributions. Different percentages of the CRD portion will be considered as well as different sample sizes. Powers were estimated based on a variety location parameter shifts. Three, four, and five populations were considered.&#x0D; Place and Duration of Study: The simulation study took place on the campus of North Dakota State University during the calendar year 2019.&#x0D; Methodology: Levels of significance for each of the three types of underlying distributions, when the RCBD portion was larger than the CRD portion, when the CRD portion was larger than the RCBD portion, and when the CRD portion was equal to the RCBD portion, and when the number of populations were 3, 4, and 5.&#x0D; Results: Regardless of the underlying population types, the proposed test statistics did better than the existing test statistics when the difference between the last two parameters is large. This was true for 3, 4, and 5 populations.&#x0D; Conclusion: When the differences between the last two parameters is large, the two new test statistics performed better. Otherwise, the existing test statistics are better. In both cases, it is better to use the combined test statistic that first standardizes the individual test statistics for the CRD and RCBD portions before adding them together.

Conditional absolute mean calibration for partial linear multiplicative distortion measurement errors models

The likelihood ratio (LR) test for ranks is commonly used to test for the number of cointegrating relationships among multivariate time series. The distribution of the LR test in partially nonstationary models is nonstandard and contains nuisance parameters in the presence of non-iid errors and misspecified lags. In contrast, I show that under less-restrictive assumptions on the errors, the fully modified (FM) vector autoregressive (VAR) rank test has a chi-squared distribution for the null of cointegration but is degenerate for the null of no cointegration, unlike its LR counterpart, which is well defined for both nulls. It turns out that augmenting the VAR by an exogenous I(0) variable solves the degeneracy problem. The procedure can also be applied to testing for Granger causality and is in fact a generalization of Toda-Yamamoto's procedure of augmenting the VAR with additional lagged I(1) variables. Unlike the LR test for ranks or Toda-Yamamoto's test for causality, the FM tests do not require correct specification of the number of lagged variables in the VAR nor does it require that errors be iid.

A causal direction test for heterogeneous populations

This paper proposes a new test for the null hypothesis of panel unit roots for micropanels with short time dimensions (T) and large cross-sections (N). There are several distinctive features of this test. First, the test is based on a panel AR(1) model allowing for cross-sectional dependency, which is introduced by a factor structure of the initial condition. Second, the test employs the panel AR(1) model with AR(1) coefficients that are heterogeneous for finite N. Third, the test can be used both for the alternative hypothesis of stationarity and for that of explosive roots. Fourth, the test does not use the AR(1) coefficient estimator. The effectiveness of the test rests on the fact that the initial condition has permanent effects on the trajectory of a time series in the presence of a unit root. To measure the effects of the initial condition, the present paper employs cross-sectional regressions using the first time-series observations as a regressor and the last as a dependent variable. If there is a unit root in every individual time series, the coefficient of the regressor is equal to one. The t-ratios for the coefficient are this paper’s test statistics and have a standard normal distribution in the limit. The t-ratios are based on the OLS estimator and the instrumental variables estimator that uses reshuffled regressors as instruments. The test proposed in this paper makes it possible to test for a unit root even at T = 2 as long as N is large. Simulation results show that test statistics have reasonable empirical size and power. The test is applied to college graduates’ monthly real wage in South Korea. The number of time-series observations for this data is only two. The null hypothesis of a unit root is rejected against the alternative of stationarity.

Over the years several discrepancy functions have been introduced both in the literature and in the software of Structural Equation Modeling (SEM). The test statistics for the discrepancy functions associated with Maximum Likelihood (ML), Generalized Least Squares (GLS), and Normal Theory Weighted Least Squares (NWLS) are all asymptotically equivalent. These test statistics are all approximately distributed as central chi-square under correct model specification and if the observed variables are multivariate normally distributed. However, it is known that the distribution of these test statistics will not approximate a central Chi-square distribution for models containing specification error, but is more likely to follow a non-central Chi-square distribution (Browne 1984). This study investigates the empirical distributions of the ML and NWLS discrepancy functions. The study includes 13 different factor models with different types and degrees of specification error. It is found, except for small samples, that the empirical distribution of the ML-test statistic outperforms the empirical distribution of the NWLS-test statistic in terms of approximation to the theoretical non-central Chi-square distribution. Furthermore, in some cases, it turned out that the non-central Chi-square approximation was not appropriate even for models that contained minor and moderate degrees of specification error.

This paper is concerned with tests for seasonal roots in monthly univariate time series processes. The paper extends the procedures and tables of critical values due to Beaulieu and Miron (Seasonal unit roots in aggregate U.S. data. J. Economet. 55 (1993), 305–28) to obtain tests which are similar ( exactly and a symptotically) with respect to both the initial values of the process and the possibility of seasonal drifts under the seasonal unit root null hypothesis. We also develop test statistics which test simultaneously for a unit root at each frequency and for a unit root at each of the seasonal frequencies. Representations are derived for the limiting distributions of each of the test statistics proposed in this paper. We illustrate the practical usefulness of the proposed test statistics by a series of empirical applications