Nonparametric predictive inference for stock returns

In finance, inferences about future asset returns are typically quantified with the use of parametric distributions and single-valued probabilities. It is attractive to use less restrictive inferential methods, including nonparametric methods which do not require distributional assumptions about variables, and imprecise probability methods which generalize the classical concept of probability to set-valued quantities. Main attractions include the flexibility of the inferences to adapt to the available data and that the level of imprecision in inferences can reflect the amount of data on which these are based. This paper introduces nonparametric predictive inference (NPI) for stock returns. NPI is a statistical approach based on few assumptions, with inferences strongly based on data and with uncertainty quantified via lower and upper probabilities. NPI is presented for inference about future stock returns, as a measure for risk and uncertainty, and for pairwise comparison of two stocks based on their future aggregate returns. The proposed NPI methods are illustrated using historical stock market data.     

Testing autocorrelation in a system perspective testing autocorrelation

The Breusch-Godfrey test for autocorrelated errors is generalised to cover systems of equations, and the properties of 18 versions of the test are studied using Monte Carlo methods. We show that only one group of tests regularly has actual size close to the nominal size; namely the likelihood ratio tests of the auxiliary regression system that are corrected in some manner for degrees-of-freedom. The Rao Ftest exhibits the best performance, whilst the commonly used TR2 test behaves badly even in single equations. However, the size and power properties of all tests deteriorate sharply as the number of equations increases, the system becomes more dynamic, the exogenous variables become more autocorrelated and the sample size decreases. This performance has, in general, an unknown degree since the interaction amongst these factors does not permit a predictive summary, as might be hoped for by response surface-type approaches.     

Testing autocorrelation in a system perspective testing autocorrelation

The Breusch-Godfrey test for autocorrelated errors is generalised to cover systems of equations, and the properties of 18 versions of the test are studied using Monte Carlo methods. We show that only one group of tests regularly has actual size close to the nominal size; namely the likelihood ratio tests of the auxiliary regression system that are corrected in some manner for degrees-of-freedom. The Rao Ftest exhibits the best performance, whilst the commonly used TR2 test behaves badly even in single equations. However, the size and power properties of all tests deteriorate sharply as the number of equations increases, the system becomes more dynamic, the exogenous variables become more autocorrelated and the sample size decreases. This performance has, in general, an unknown degree since the interaction amongst these factors does not permit a predictive summary, as might be hoped for by response surface-type approaches.     

Non-traditional analysis of stock returns

SUMMARY An investigation of the prices of eight individual stocks showed that pricechange returns are significantly less complex than are time-dependent returns. Timedependent returns computed every 15, 30 and 45 minutes were found to be more complex, using a complexity measure. Complexity is quantified by measuring the number of times that the estimated correlation dimension of an observed series is multiplied by when its original sequence is randomly shuffled.     

Jump-diffusion models of German stock returns

This paper discusses the statistical properties of jump-diffusion processes and reports on parameter estimates for the DAX stock index and 48 German stocks with traded options. It is found that a Poisson-type jump-diffusion process can explain the high levels of kurtosis and skewness of observed return distributions of German stocks. Furthermore, we demonstrate that the return dynamics of the DAX include a statistically significant jump component except for a few sample subperiods. This finding is seen to be inconsistent with asset pricing models assuming that the jump component of the stock's return is unsystematic and diversifiable in the market portfolio.     

Testing for autocorrelation in non-stationary dynamic systems of equations

Using Monte Carlo methods, the properties of systemwise generalizations of the Breusch-Godfrey test for autocorrelated errors are studied in integrated cointegrated systems of equations. Our analysis, regarding the size of the test, reveals that the corrected LR tests have been shown to perform satisfactorily even in cases when the exogenous variables follow a unit root process, whilst the commonly used TR2 test behaves badly even in single equations. All tests perform badly, however, when the number of equations increases and the exogenous variables are highly autocorrelated.     

Normal mixture method for stock daily returns over different sub-periods

In this paper, the normal mixture model, as an alternative distribution, is utilized to represent the characteristics of stock daily returns over different bull and bear markets. Firstly, we conduct the normality test for the return data and compare the Kolmogorov-Smirnov statistics of normal mixture models with different components. Secondly, we analyze the likely reasons why parameters change over different sub-periods. Our empirical examination proves that majority of the data series reject the normality assumption and mixture models with three components can model the behavior of daily returns more appropriately and steadily. This result has both statistical and economic significance.     

Dynamic component detection in a multifactor model for stock returns

Summary In this paper the factorial structure of some asset returns quoted at the Milan stock exchange is analyzed in order to detect the presence of a dynamic component. The maximum likelihood estimates of the dynamic model, for which a space-state representation and the Kalman filter were used, are compared with the estimates of the static model via the information criteria. There is no evidence of dynamic factors underlying the analyzed samples of asset returns, while one static factor seems to be relevant. Invited paper at the Conference held in Bologna, Italy, 27–28 May 1993, on ?Statistical Tests: Methodology and Econometric Applications?.     

The predictability of stock returns - a nonparametric approach

This paper reexamines the predictability of stock returns with a nonparametric model. We first identify, through a set of diagnostic tests, five lagged predictive factors from a linear model. Using these factors, we predict one-month-ahead stock index returns with a nonparametric approach. We find that our nonparametricmodel. We first identify, through a set of diagnostic tests, five lagged predictive factors from a linear model. Using these factors, we predict on -month-ahead stock index returns with a nonparametric approach. We find that our nonparametric model can correctly predict about 74% of stock index return signs. With various ex ante trading rules based on nonparametric predictions and transaction cost schedules, we then compare the performance of "managed" portfolios with that of the buy and hold portfolios. We fmd that the managed portfolios are mean-variance dominant over the buy-and-hold strategies when no or low transaction costs are assumed. When high transaction costs are assumed instead, the mean-variance dominance diminishes However,the Sharpe index of risk-adjusted portfolio performanceindicates that the managed portfolios significantly outperform the buy-and-hold strategies even for the high-transaction cost scenario. We show that the difference in performance between the managed portfolios and the buy-and-hold strategies can be partially explained by the January effect or the small firm effect. In sum, this paper demonstrates the merits of using a nonparametric approach for predicting stock returns and testing market efficiency.     

Testing for autocorrelation and random-effects in nonlinear mixed effects models based on M-estimation

M-estimation (robust estimation) for the parameters in nonlinear mixed effects models using Fisher scoring method is investigated in the article, which shares some of the features of the existing maximum likelihood estimation: consistency and asymptotic normality. Score tests for autocorrelation and random effects based on M-estimation, together with their asymptotic distribution are also studied. The performance of the test statistics are evaluated via simulations and a real data analysis of plasma concentrations data.     

Likelihood procedure for testing changes in skew normal model with applications to stock returns

The skew normal distribution family is an attractive distribution family due to its mathematical tractability and inclusion of the normal distribution as the special case. It has wide applications in many applied fields such as finance, economics, and medical research. Such a distribution family has been studied extensively since it was introduced by Azzalini in 1985 Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian Journal of Statistics 12:171–178. [Google Scholar] for the first time. Yet, few work has been done on the study of change point problem related to this distribution family. In this article, we propose the likelihood ratio test (LRT) to detect changes in the parameters of the skew normal distribution associated with some asymptotic results of the test statistic. Simulations have been conducted under different scenarios to investigate the performance of the proposed method. Comparisons to some other existing method indicate the comparable power of the method in detecting changes in parameters of the skew normal distribution model. Applications on two real data: Brazilian and Tanzanian stock returns illustrate the detection procedure.     

Estimates of the likelihood of extreme returns in international stock markets

This study applies extreme-value theory to daily international stock-market returns to determine (1) whether or not returns follow a heavy-tailed stable distribution, (2) the likelihood of an extreme return, such as a 20% drop in a single day, and (3) whether or not the likelihood of an extreme event has changed since October 1987. Empirical results reject a heavy-tailed stable distribution for returns. Instead, a Student-t distribution or an autoregressive conditional heteroscedastic process is better able to capture the salient features of returns. We find that the likelihood of a large single-day return diff ers widely across markets and, for the G-7 countries, the 1987 stock-market drop appears to be largely an isolated event. A drop of this magnitude, however, is not rare in the case of Hong Kong. Finally, there is only limited evidence that the chance of a large single-day decline is more likely since the October 1987 market drop; however, exceptions include stock markets in Germany, The Netherlands and the UK.     

Chi-square tests for multivariate normality with application to common stock prices

The theory of chi-square tests with data-dependent cells is applied to provide tests of fit to the family of p-variate normal distributions. The cells are bounded by hyperellipses (x-[Xbar])'S^-1 (x-[Xbar]) = c_i centered at the sample mean [Xbar] and having shape deter-mined by the sample covariance matrix S. The Pearson statistic with these cells is affine-invariant, has a null distribution not depending on the true mean and covariance, and has asymptotic critical points between those of x² (M-1) and x² (M-2) when M cells are employed. The test is insensitive to lack of symmetry, but peakedness, broad shoulders and heavy tails are easily discerned in the cell counts. Multivariate normality of logarithms of relative prices of common stocks, a common assumption in finan-cial markets theory, is studied using the statistic described here and a large data base.     

Maximum likelihood estimation of skew-t copulas with its applications to stock returns

The multivariate Student-t copula family is used in statistical finance and other areas when there is tail dependence in the data. It often is a good-fitting copula but can be improved on when there is tail asymmetry. Multivariate skew-t copula families can be considered when there is tail dependence and tail asymmetry, and we show how a fast numerical implementation for maximum likelihood estimation is possible. For the copula implicit in a multivariate skew-t distribution, the fast implementation makes use of (i) monotone interpolation of the univariate marginal quantile function and (ii) a re-parametrization of the correlation matrix. Our numerical approach is tested with simulated data with data-driven parameters. A real data example involves the daily returns of three stock indices: the Nikkei225, S&P500 and DAX. With both unfiltered returns and GARCH/EGARCH filtered returns, we compare the fits of the Azzalini–Capitanio skew-t, generalized hyperbolic skew-t, Student-t, skew-Normal and Normal copulas.     

Bayes factors for choosing among six common survival models

Lifetime Data Analysis - A super model that includes proportional hazards, proportional odds, accelerated failure time, accelerated hazards, and extended hazards models, as well as the model...     

A Cauchy estimator test for autocorrelation

This article presents a new test for serial correlation in an observed stationary time series. Rather than using the traditional portmanteau tests based on the sample autocorrelation function, we propose a test based on the Cauchy estimator of correlation. A goodness-of-fit statistic for fitted autoregressive moving average models is also derived and the asymptotic distribution of this statistic is quantified. The test can be employed using either this asymptotic distribution or by using Monte-Carlo quantiles. The small sample behaviour is studied via simulation and the Monte-Carlo-based test seems to be more precise. The method is demonstrated on monthly asset returns for Facebook, Incorporated.     

Testing for order among K populations: theory and examples

Testing for stochastic order among K populations is a common and important problem in statistical practice. It arises in the analysis of both planned experiments and observational studies. The authors develop a new nonparametric test for order among K populations that can accommodate any stochastic ordering. The test is based on a maximally selected chi‐bar‐square statistic. The authors find its limiting distribution and use simulations to derive critical values. Three important examples are used to illustrate the applicability of the general method. The authors find that the new tests outperform the existing methods in many practical cases. The Canadian Journal of Statistics 38: 97–115; 2010 © 2009 Statistical Society of Canada     

Testing hypotheses about the common mean of normal distributions

An overview of hypothesis testing for the common mean of independent normal distributions is given. The case of two populations is studied in detail. A number of different types of tests are studied. Among them are a test based on the maximum of the two available t-tests, Fisher's combined test, a test based on Graybill–Deal's estimator, an approximation to the likelihood ratio test, and some tests derived using some Bayesian considerations for improper priors along with intuitive considerations. Based on some theoretical findings and mostly based on a Monte Carlo study the conclusions are that for the most part the Bayes-intuitive type tests are superior and can be recommended. When the variances of the populations are close the approximate likelihood ratio test does best.     

Third-order inference for autocorrelation in nonlinear regression models

We propose third-order likelihood-based methods to derive highly accurate p-value approximations for testing autocorrelated disturbances in nonlinear regression models. The proposed methods are particularly accurate for small- and medium-sized samples whereas commonly used first-order methods like the signed log-likelihood ratio test, the Kobayashi (1991) test, and the standardized test can be seriously misleading in these cases. Two Monte Carlo simulations are provided to show how the proposed methods outperform the above first-order methods. An empirical example applied to US population census data is also provided to illustrate the implementation of the proposed method and its usefulness in practice.     

Testing hypotheses about the common mean of two normal populations

Two simple tests which allow for unequal sample sizes are considered for testing hypothesis for the common mean of two normal populations. The first test is an exact test of size a based on two available t-statistics based on single samples made exact through random allocation of α among the two available t-tests. The test statistic of the second test is a weighted average of two available t-statistics with random weights. It is shown that the first test is more efficient than the available two t-tests with respect to Bahadur asymptotic relative efficiency. It is also shown that the null distribution of the test statistic in the second test, which is similar to the one based on the normalized Graybill-Deal test statistic, converges to a standard normal distribution. Finally, we compare the small sample properties of these tests, those given in Zhou and Mat hew (1993), and some tests given in Cohen and Sackrowitz (1984) in a simulation study. In this study, we find that the second test performs better than the tests given in Zhou and Mathew (1993) and is comparable to the ones given in Cohen and Sackrowitz (1984) with respect to power..