Model selection for stock prices data

The geometric Brownian motion (GBM) is very popular in modeling the dynamics of stock prices. However, the constant volatility assumption is questionable and many models with nonconstant volatility have been developed. In the papers [7 M.L. Esquível and P.P. Mota, On some auto-induced regime switching double-threshold glued diffusions, J. Stat. Theory Pract. 8 (2014), pp. 760–771. doi: 10.1080/15598608.2013.854184.[Taylor &; Francis Online] , [Google Scholar],12 P. P. Mota and M.L. Esquível, On a continuous time stock price model with regime switching, delay, and threshold, Quant. Financ. 14 (2014), pp. 1479–1488. doi: 10.1080/14697688.2013.879990.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] the authors introduce a regime switching process where in each regime the process is driven by GBM and the change in regime is defined by the crossing of a threshold. In this paper we used Akaike's and Bayesian information criteria to show that the GBM with regimes provides a better fit than the GBM. We also perform a forecasting comparison of the models for two selected companies.     

Volatility estimation from short time series of stock prices

We consider estimation of the historical volatility of stock prices. It is assumed that the stock prices are represented as time series formed as samples of the solution of a stochastic differential equation with random and time-varying parameters; these parameters are not observable directly and have unknown evolution law. The price samples are available with limited frequency only. In this setting, the estimation has to be based on short time series, and the estimation error can be significant. We suggest some supplements to the existing nonparametric methods of volatility estimation. Two modifications of the standard summation formula for the volatility are derived. In addition, a linear transformation eliminating the appreciation rate and preserving the volatility is suggested.     

Time varying long memory parameter estimation for locally stationary long memory processes

The semiparametric estimators of time varying long memory parameter are investigated for locally stationary long memory processes. The GPH estimator and the local Whittle estimator are considered. Under some mild regularity assumptions, the weak consistency and the asymptotic normality of the estimators are obtained. The finite sample performance of the estimators is discussed through a small simulation study.     

Identification of long memory in GARCH models

Abstract: This work extends the analysis of Baillie, Bollerslev and Mikkelsen (1996) and Bollerslev and Mikkelsen (1996) on the estimation and identification problems of the Fractionally Integrated Generalized Autoregressive Conditional Heteroskedastik (FIGARCH) model. We assess the power of different information criteria and tests in identifying the presence of long memory in the conditional variances. The analysis is performed with a Montecarlo simulation study. In detail, the focus on the Akaike, Hannan-Quinn, Shibata and Schwarz information criteria and on the Jarque-Bera test for normality, Box-Pierce test for residual correlation and Engle test for ARCH effects. This study verifies that information criteria clearly distinguish the presence of long memory while tests do not evidence any difference between the fitted long and short memory models. An empirical application is provided; it analyses, on a high frequency dataset, the returns of the FIB30, the future on the MIB30, the Italian stock market index of highly capitalized firms.Massimiliano Caporin: mcaporin@unive.itThis paper was presented at the SIS 2002 Conference (Italian Statistical society annual meeting) held in Milan, University Bicocca, 5-7 June 2002. A short version of this work can be found in the proceedings of the conference     

Goodness-of-fit testing under long memory

This paper discusses the problem of fitting a distribution function to the marginal distribution of a long memory moving average process. Because of the uniform reduction principle, unlike in the i.i.d. set up, classical tests based on empirical process are relatively easy to implement. More importantly, we discuss fitting the marginal distribution of the error process in location, scale, location–scale and linear regression models. An interesting observation is that in the location model, location–scale model, or more generally in the linear regression models with non-zero intercept parameter, the null weak limit of the first order difference between the residual empirical process and the null model is degenerate at zero, and hence it cannot be used to fit an error distribution in these models for the large samples. This finding is in sharp contrast to a recent claim of Chan and Ling (2008) that the null weak limit of such a process is a continuous Gaussian process. This note also proposes some tests based on the second order difference for the location case. Another finding is that residual empirical process tests in the scale problem are robust against not knowing the scale parameter.     

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.     

Estimation in long memory time series models

A regression type estimator of the parameter d in fractionally differenced ARMA (p,q) processes is presented. The proposed estimator is shown to be mean square consistent. Its performance is compared with some of the existing estimators via a simulation study.     

Prediction of long memory processes on same-realisation

For the class of stationary Gaussian long memory processes, we study some properties of the least-squares predictor of _Xn+1$X_{n+1}$ based on (_Xn,…,_X1)$(X_n,\dots ,X_1)$. The predictor is obtained by projecting _Xn+1$X_{n+1}$ onto the finite past and the coefficients of the predictor are estimated on the same realisation. First we prove moment bounds for the inverse of the empirical covariance matrix. Then we deduce an asymptotic expression of the mean-squared error. In particular we give a relation between the number of terms used to estimate the coefficients and the number of past terms used for prediction, which ensures the L²$L^2$- sense convergence of the predictor. Finally we prove a central limit theorem when our predictor converges to the best linear predictor based on all the past.     

Approximate wavelet-based simulation of long memory processes

In this article, we investigate an algorithm for the fast O(N) and approximate simulation of long memory (LM) processes of length N using the discrete wavelet transform. The algorithm generates stationary processes and is based on the notion that we can improve standard wavelet-based simulation schemes by noting that the decorrelation property of wavelet transforms is not perfect for certain LM process. The method involves the simulation of circular autoregressive process of order one. We demonstrate some of the statistical properties of the processes generated, with some focus on four commonly used LM processes. We compare this simulation method with the white noise wavelet simulation scheme of Percival and Walden [Percival, D. and Walden, A., 2000, Wavelet Methods for Time Series Analysis (Cambridge: Cambridge University Press).].     

Detecting changes from short to long memory

This paper studies well-known tests by Kim et?al. (J Econom 109:389?C392, 2002) and Busetti and Taylor (J Econom 123:33?C66, 2004) for the null hypothesis of short memory against a change to nonstationarity, I (1). The potential break point is not assumed to be known but estimated from the data. First, we show that the tests are also applicable for a change from I (0) to a fractional order of integration I (d) with d?>?0 (long memory) in that the tests are consistent. The rates of divergence of the test statistics are derived as functions of the sample size T and d. Second, we compare their finite sample power experimentally. Third, we consider break point estimation for a change from I (0) to I (d) for finite samples in computer simulations. It turns out that the estimators proposed for the integer case (d?=?1) are practically reliable only if d is close enough to 1.     

On rapid change points under long memory

Estimation of points of rapid change in the mean function m(t) is considered under long memory residuals, irregularily spaced time points and smoothly changing marginal distributions obtained by local Gaussian subordination. The approach is based on kernel estimation of derivatives of the trend function. An asymptotic expression for the mean squared error is obtained. Limit theorems are derived for derivatives of m and the time points where rapid change occurs. The results are illustrated by an application to measurements of oxygen isotopes trapped in the Greenland ice sheets during the last 20,000 years.     

Out-of-sample forecast errors in misspecific perturbed long memory processes

In this paper we examine the relative increase in mean square forecast error fro fitting a weakly stationary process to the series of interest when in fact the true model is a so-called perturbed long-memory process recently introduced by Granger and Marmol (1997). This model has the property of being unidentifiable from a white noise process on the basis of the correlogram and the usual rule-of-thumbs in the Box-Jenkins methodology. We prove that this kind of missspecification can lead to serious errors in terms of forecasting. We also show that corrections based on the AR(1) model can in some cases partially solve the problem. Received: March 15, 1999; revised version: February 14, 2000     

Comparison of non-parametric and semi-parametric tests in detecting long memory

The first two stages in modelling times series are hypothesis testing and estimation. For long memory time series, the second stage was studied in the paper published in [M. Boutahar et al., Estimation methods of the long memory parameter: monte Carlo analysis and application, J. Appl. Statist. 34(3), pp. 261–301.] in which we have presented some estimation methods of the long memory parameter. The present paper is intended for the first stage, and hence completes the former, by exploring some tests for detecting long memory in time series. We consider two kinds of tests: the non-parametric class and the semi-parametric one. We precise the limiting distribution of the non-parametric tests under the null of short memory and we show that they are consistent against the alternative of long memory. We perform also some Monte Carlo simulations to analyse the size distortion and the power of all proposed tests. We conclude that for large sample size, the two classes are equivalent but for small sample size the non-parametric class is better than the semi-parametric one.     

Kernel type smoothed quantile estimation under long memory

This paper studies nonparametric kernel type (smoothed) estimation of quantiles for long memory stationary sequences. The uniform strong consistency and asymptotic normality of the estimates with rates are established. Finite sample behaviors are investigated in a small Monte Carlo simulation study.     

Change-of-variance problem for linear processes with long memory

In this paper we investigate the asymptotic properties of the test statistics for detecting change-points in the variance of infinite moving average sequences with long memory. This research is partly supported by NSFC Grants and SRF for ROCS, SEM.     

The S-estimator in the change-point random model with long memory

This paper considers two-phase random design linear regression models. Errors and regressors are stationary long-range-dependent Gaussian processes. The regression parameters, the scale parameter and the change-point are estimated using a method introduced by Rousseeuw and Yohai [Robust regression by means of S-estimators, in Robust and Nonlinear Time Series Analysis, J. Franke, W. Hrdle, and R.D. Martin, eds., Lecture Notes in Statistics, Vol. 26, Springer, New York, 1984, pp. 256–272], which is called the S-estimator and has the property be more robust than the classical estimators in the sense that the outliers do not bias the estimation results. Some asymptotic results, including the strong consistency and the convergence rate of the S-estimator are proved. Simulations and an application to the Nile River data are also presented. It is shown via Monte Carlo simulations that the S-estimator is better than two other estimators that are proposed in the literature.     

Misspecification tests for periodic long memory GARCH models

Distributional theory for Quasi-Maximum Likelihood estimators in long memory conditional heteroskedastic models is not formally defined, even asymptotically. Because of that, this paper analyses the real size and power of the likelihood ratio and the Lagrange multiplier misspecification tests when periodic long memory GARCH models are involved. The performance of these tests is studied by means of Monte Carlo simulations with respect to the class of generalized long memory GARCH models. For this class of models, analytical derivatives are developed. An application to the USD/JPY exchange rate is also provided.     

Estimation of slowly time-varying trend function in long memory regression models

ABSTRACT

We study the asymptotic properties of the least-squares estimator for the trend function of a particular class of locally stationary models, which are defined by considering a smooth variation of the trend function. Additionally, errors are assumed to be realizations from a long-range dependent stationary Gaussian process. Our findings are then illustrated through Monte Carlo simulations.     

A harmonically weighted filter for cyclical long memory processes

AStA Advances in Statistical Analysis - The estimation of the long memory parameter d is a widely discussed issue in the literature. The harmonically weighted (HW) process was recently introduced...