Tests for Parameter Instability in Regressions with 1(1) Processes

This article derives the large-sample distributions of Lagrange multiplier (LM) tests for parameter instability against several alternatives of interest in the context of cointegrated regression models. The fully modified estimator of Phillips and Hansen is extended to cover general models with stochastic and deterministic trends. The test statistics considered include the SupF test of Quandt, as well as the LM tests of Nyblom and of Nabeya and Tanaka. It is found that the asymptotic distributions depend on the nature of the regressor processes—that is, if the regressors are stochastic or deterministic trends. The distributions are noticeably different from the distributions when the data are weakly dependent. It is also found that the lack of cointegration is a special case of the alternative hypothesis considered (an unstable intercept), so the tests proposed here may also be viewed as a test of the null of cointegration against the alternative of no cointegration. The tests are applied to three data sets—an aggregate consumption function, a present value model of stock prices and dividends, and the term structure of interest rates.     

Tests on asymmetry for ordered categorical variables

Skewness is a well-established statistical concept for continuous and, to a lesser extent, for discrete quantitative statistical variables. However, for ordered categorical variables, limited literature concerning skewness exists, although this type of variables is common for behavioral, educational, and social sciences. Suitable measures of skewness for ordered categorical variables have to be invariant with respect to the group of strictly increasing, continuous transformations. Therefore, they have to depend on the corresponding maximal-invariants. Based on these maximal-invariants, we propose a new class of skewness functionals, show that members of this class preserve a suitable ordering of skewness and derive the asymptotic distribution of the corresponding skewness statistic. Finally, we show the good power behavior of the corresponding skewness tests and illustrate these tests by applying real data examples.     

Equal‐tailed confidence intervals for comparison of rates

Several methods are available for generating confidence intervals for rate difference, rate ratio, or odds ratio, when comparing two independent binomial proportions or Poisson (exposure‐adjusted) incidence rates. Most methods have some degree of systematic bias in one‐sided coverage, so that a nominal 95% two‐sided interval cannot be assumed to have tail probabilities of 2.5% at each end, and any associated hypothesis test is at risk of inflated type I error rate. Skewness‐corrected asymptotic score methods have been shown to have superior equal‐tailed coverage properties for the binomial case. This paper completes this class of methods by introducing novel skewness corrections for the Poisson case and for odds ratio, with and without stratification. Graphical methods are used to compare the performance of these intervals against selected alternatives. The skewness‐corrected methods perform favourably in all situations—including those with small sample sizes or rare events—and the skewness correction should be considered essential for analysis of rate ratios. The stratified method is found to have excellent coverage properties for a fixed effects analysis. In addition, another new stratified score method is proposed, based on the t‐distribution, which is suitable for use in either a fixed effects or random effects analysis. By using a novel weighting scheme, this approach improves on conventional and modern meta‐analysis methods with weights that rely on crude estimation of stratum variances. In summary, this paper describes methods that are found to be robust for a wide range of applications in the analysis of rates.     

Testing normalization and overidentification of cointegrating vectors in vector autoregressive processes

This paper develops test procedures for testing the validity of general linear identifying restrictions imposed on cointegrating vectors in the context of a vector autoregressive model. In addition to overidentifying restrictions the considered restrictions may also involve normalizing restrictions. Tests for both types of restrictions are developed and their asymptotic properties are obtained. Under the null hypothesis tests for normalizing restrictions have an asymptotic "multivariate unit root distribution", similar to that obtained for the likelihood ratio test for cointegration, while tests for overidentifying restrictions have a standard chi-square limiting distribution. Since these two types of tests are asymptotically independent they are easy to cotnbine to an overall test for the spccifed identifying restrictions. An overall test of this kind can consistently reveal the failure of the identifying restrictions in a wider class of cases than previous tests which only test for overidentifying restrictions.     

Measures of multivariate skewness and kurtosis for tests of nonnormality

Measures of multivariate skewness and kurtosis are proposed that are based on the skewness and kurtosis of individual components of standardized sample vectors. Asymptotic properties and small sample critical values of tests for nonnormality based on these measures are provided. It is demonstrated that the tests have favorable power properties. Extensions to time series data are pointed out.     

Testing for normality in linear regression models

The importance of the normal distribution for fitting continuous data is well known. However, in many practical situations data distribution departs from normality. For example, the sample skewness and the sample kurtosis are far away from 0 and 3, respectively, which are nice properties of normal distributions. So, it is important to have formal tests of normality against any alternative. D'Agostino et al. [A suggestion for using powerful and informative tests of normality, Am. Statist. 44 (1990), pp. 316–321] review four procedures Z ²(g ₁), Z ²(g ₂), D and K ² for testing departure from normality. The first two of these procedures are tests of normality against departure due to skewness and kurtosis, respectively. The other two tests are omnibus tests. An alternative to the normal distribution is a class of skew-normal distributions (see [A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Statist. 12 (1985), pp. 171–178]). In this paper, we obtain a score test (W) and a likelihood ratio test (LR) of goodness of fit of the normal regression model against the skew-normal family of regression models. It turns out that the score test is based on the sample skewness and is of very simple form. The performance of these six procedures, in terms of size and power, are compared using simulations. The level properties of the three statistics LR, W and Z ²(g ₁) are similar and close to the nominal level for moderate to large sample sizes. Also, their power properties are similar for small departure from normality due to skewness (γ₁≤0.4). Of these, the score test statistic has a very simple form and computationally much simpler than the other two statistics. The LR statistic, in general, has highest power, although it is computationally much complex as it requires estimates of the parameters under the normal model as well as those under the skew-normal model. So, the score test may be used to test for normality against small departure from normality due to skewness. Otherwise, the likelihood ratio statistic LR should be used as it detects general departure from normality (due to both skewness and kurtosis) with, in general, largest power.     

On estimating the box-cox transformation to normality

This paper studies four methods for estimating the Box-Cox parameter used to transform data to normality. Three of these are based on optimizing test statistics for standard normality tests (the Shapiro-Wilk. skewness, and kurtosis tests); the fourth uses the maximum likelihood estimator of the Box-Cox parameter. The four methods are compared and evaluated with a simulation study, where their performances under different skewness and kurtosis conditions are analyzed. The estimator based on optimizing the Shapiro-Wilk statistic generally gives rise to the best transformations, while the maximum likelihood estimator performs almost as well. Estimators based on optimizing skewness and kurtosis do not perform well in general.     

Omnibus tests of multinormality based on skewness and kurtosis

Measures of univariate skewness and kurtosis have long been used as a test of univariate normality, several omnibus test procedures based on a combination of the measures having been proposed, see Pearson, D’Agestino and Bowman (1977) and Mardia (1979). Mardia (1970) proposed measures of multivariate skewness and kurtosis, and constructed a test of multinormality based on these measures. we obtain the correlation between these measures and propose several omnibus tests using the two measures. The performances of these tests are compared by means of a Monte Carlo study.     

Measures of tail asymmetry for bivariate copulas

Three tail asymmetry measures for bivariate copulas are introduced using two different approaches—univariate skewness of a projection and distance between a copula and its survival/reflected copula. We compare the asymmetry measures based on certain desirable properties. Bounds for each measure are obtained and also copulas which attain these extreme values are identified. Two data examples show the amount of asymmetry that might be expected in practice.     

A continuously adaptive nonparametric two–sample test

A two–sample test statistic for detecting shifts in location is developed for a broad range of underlying distributions using adaptive techniques. The test statistic is a linear rank statistics which uses a simple modification of the Wilcoxon test; the scores are Winsorized ranks where the upper and lower Winsorinzing proportions are estimated in the first stage of the adaptive procedure using sample the first stage of the adaptive procedure using sample measures of the distribution's skewness and tailweight. An empirical relationship between the Winsorizing proportions and the sample skewness and tailweight allows for a ‘continuous’ adaptation of the test statistic to the data. The test has good asymptotic properties, and the small sample results are compared with other populatr parametric, nonparametric, and two–stage tests using Monte Carlo methods. Based on these results, this proposed test procedure is recommended for moderate and larger sample sizes.     

Markov-Switching and Stochastic Volatility Diffusion Models of Short-Term Interest Rates

This article empirically compares the Markov-switching and stochastic volatility diffusion models of the short rate. The evidence supports the Markov-switching diffusion model. Estimates of the elasticity of volatility parameter for single-regime models unanimously indicate an explosive volatility process, whereas the Markov-switching models estimates are reasonable. Itis found that either Markov switching or stochastic volatility, but not both, is needed to adequately fit the data. A robust conclusion is that volatility depends on the level of the short rate. Finally, the Markov-switching model is the best for forecasting. A technical contribution of this article is a presentation of quasi-maximum likelihood estimation techniques for the Markov-switching stochastic-volatility model.     

Conditional Asymmetries in Real GNP: A Seminonparametric Approach

Two critical assumptions are often made in empirical research regarding the relationship between economic variables and economic disturbances—linearity and Gaussianity. Together, these two assumptions place strong restrictions on the time series behavior of a model. Most important, these restrictions imply conditional symmetry. Using seminonparametric (SNP) techniques, this article presents evidence that real gross national product growth displays conditional asymmetry. Although these results confirm related results of Brock and Sayers, Sichel, and Hamilton, the SNP approach is novel in that it emphasizes the relationship between common modeling assumptions and the restrictions that these assumptions place on data.     

The limiting power of point optimal autocorrelation tests

This paper considers the point optimal tests for AR(1) errors in the linear regression model. It is shown that these tests have the same limiting power characteristics as the Durbin-Watson test. . The limiting power is zero or one when the regression has no intercept, but lies strictly between these values when an intercept is included.     

A powerful and interpretable alternative to the Jarque–Bera test of normality based on 2nd-power skewness and kurtosis,using the Rao's score test on the APD family

We introduce the 2nd-power skewness and kurtosis, which are interesting alternatives to the classical Pearson's skewness and kurtosis, called 3rd-power skewness and 4th-power kurtosis in our terminology. We use the sample 2nd-power skewness and kurtosis to build a powerful test of normality. This test can also be derived as Rao's score test on the asymmetric power distribution, which combines the large range of exponential tail behavior provided by the exponential power distribution family with various levels of asymmetry. We find that our test statistic is asymptotically chi-squared distributed. We also propose a modified test statistic, for which we show numerically that the distribution can be approximated for finite sample sizes with very high precision by a chi-square. Similarly, we propose a directional test based on sample 2nd-power kurtosis only, for the situations where the true distribution is known to be symmetric. Our tests are very similar in spirit to the famous Jarque–Bera test, and as such are also locally optimal. They offer the same nice interpretation, with in addition the gold standard power of the regression and correlation tests. An extensive empirical power analysis is performed, which shows that our tests are among the most powerful normality tests. Our test is implemented in an R package called PoweR.     

Control charts for fraction nonconforming in a bivariate binomial process

Many multivariate quality control techniques are used for multivariate variable processes, but few work for multivariate attribute processes. To monitor multivariate attributes, controlling the false alarms (type I errors) and considering the correlation between attributes are two important issues. By taking into account these two issues, a new control chart is presented to monitor a bivariate binomial process. An example is illustrated for the proposed method. To evaluate the performance of the proposed method, a simulation study is conducted to compare the results with those using both the multivariate np chart and skewness reduction approaches. The results show that the correlation is taken into account in the designed chart and the overall false alarm is controlled at the nominal value. Moreover, the process shift can be quickly detected and the variable that is responsible for a signal can be determined.     

Business Cycle Asymmetries

Tests for business cycle asymmetries are developed for Markov-switching autoregressive models. The tests of deepness, steepness, and sharpness are Wald statistics, which have standard asymptotics. For the standard two-regime model of expansions and contractions, deepness is shown to imply sharpness (and vice versa), whereas the process is always nonsteep. Two and three-state models of U.S. GNP growth are used to illustrate the approach, along with models of U.S. investment and consumption growth. The robustness of the tests to model misspecification, and the effects of regime-dependent heteroscedasticity, are investigated.     

Skewness and mixtures of normal distributions

The effect of skewness on hypothesis tests for the existence of a mixture of univariate and bivariate normal distributions is examined through a Monte Carlo study. A likelihood ratio test based on results of the simultaneous estimation of skewness parameters, derived from power transformations, with mixture parameters is proposed. This procedure detects the difference between inherent distributional skewness and the apparent skewness which is a manifestation of the mixture of several distributions. The properties of this test are explored through a simulation study.     

Behaviour of skewness, kurtosis and normality tests in long memory data

We establish the limiting distributions for empirical estimators of the coefficient of skewness, kurtosis, and the Jarque–Bera normality test statistic for long memory linear processes. We show that these estimators, contrary to the case of short memory, are neither ${\sqrt{n}}We establish the limiting distributions for empirical estimators of the coefficient of skewness, kurtosis, and the Jarque–Bera normality test statistic for long memory linear processes. We show that these estimators, contrary to the case of short memory, are neither ?n{\sqrt{n}}-consistent nor asymptotically normal. The normalizations needed to obtain the limiting distributions depend on the long memory parameter d. A direct consequence is that if data are long memory then testing normality with the Jarque–Bera test by using the chi-squared critical values is not valid. Therefore, statistical inference based on skewness, kurtosis, and the Jarque–Bera normality test, needs a rescaling of the corresponding statistics and computing new critical values of their nonstandard limiting distributions.     

The effect of non-normal disturbances and conditional heteroskedasticity on multiple cointegration tests

!t is well-known that Johansen's multiple cointegration tests' results and those of Johansen and Juselius' tests for restricrions on cointegrating vectors and their weights have far-reaching implications for economic modelling and analysis. Therefore, it is important to ensure that the tests have desirable finite sample properties. Although the statistics are derived under Gaussian distribution,the asympotic results are derived under a much wider class of distributions. Using simulation, this paper investigates the effect of non-normal disturbances on these tests in finite samples. Further, ARCH/GARCH type conditional heteroskedasticity is present in many economic and financial time series. This paper examines the finite properties of the tests when the error term follows ARCH/GARCH type processes. From the evidence, it appears that researchers should not be overly concerned by the possibility of small departures from non-normality when using Johansen's suggested techniques even in finite samples. ARCH and GARCH effects may be more problematic, however. In particular it becomes more important ro test whether the restriction implicit in the integrated (or near-integrated) ARCH-type Drocess actually holds in time series for the application of the cointegraiion rank tests and the test for restrictions on cointegrating weights. The tests for restrictions on cointegrating vectors apper to be robust for non-normal errors and for all ARCH and GARCH type processes considered.     

Tests for multivariate normality based on canonical correlations

We propose new affine invariant tests for multivariate normality, based on independence characterizations of the sample moments of the normal distribution. The test statistics are obtained using canonical correlations between sets of sample moments in a way that resembles the construction of Mardia’s skewness measure and generalizes the Lin–Mudholkar test for univariate normality. The tests are compared to some popular tests based on Mardia’s skewness and kurtosis measures in an extensive simulation power study and are found to offer higher power against many of the alternatives.