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on Econometric Time Series |
| By: | Adrian Pagan (School of Economics, University of Sydney); Don Harding (School of Economics and Finance, La Trobe University) |
| Abstract: | Economic events such as expansions and recessions in economic activity, bull and bear markets in stock prices and financial crises have long attracted substantial interest. In recent times there has been a focus upon predicting the events and constructing Early Warning Systems of them. Econometric analysis of such recurrent events is however in its infancy. One can represent the events as a set of binary indicators. However they are different to the binary random variables studied in micro-econometrics, being constructed from some (possibly) continuous data. The lecture discusses what difference this makes to their econometric analysis. It sets out a framework which deals with how the binary variables are constructed, what an appropriate estimation procedure would be, and the implications for the prediction of them. An example based on Turkish business cycles is used throughout the lecture. |
| Keywords: | Business and Financial Cycles, Binary Time Series, BBQ Algorithm |
| JEL: | C22 E32 E37 |
| Date: | 2011–09–19 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2011-33&r=ets |
| By: | Qi Gao (The school of Public Finance and Taxation, Southwestern University of Finance and Economics); Jingping Gu (Department of Economics, University of Arkansas); Paula Hernandez-Verme (Department of Economics & Finance, University of Guanajuato, UCEA-Campus Marfil) |
| Abstract: | In this paper, we propose a new semiparametric varying coefficient model which extends the existing semi-parametric varying coefficient models to allow for a time trend regressor with smooth coefficient function. We propose to use the local linear method to estimate the coefficient functions and we provide the asymptotic theory to describe the asymptotic distribution of the local linear estimator. We present an application to evaluate credit rationing in the U.S. credit market. Using U.S. monthly data (1952.1-2008.1) and using inflation as the underlying state variable, we find that credit is not rationed for levels of inflation that are either very low or very high; and for the remaining values of inflation, we find that credit is rationed and the Mundell-Tobin effect holds. |
| Keywords: | non-stationarity, semi-parametric smooth coefficients, nonlinearity, credit rationing |
| JEL: | C14 C22 E44 |
| Date: | 2011 |
| URL: | http://d.repec.org/n?u=RePEc:cuf:wpaper:515&r=ets |
| By: | Jushan Bai (Department of Economics, Columbia University; CEMA, Central University of Finance and Economics); Shuzhong Shi (Department of Finance, Guanghua School of Management) |
| Abstract: | Estimating covariance matrices is an important part of portfolio selection, risk management, and asset pricing. This paper reviews the recent development in estimating high dimensional covariance matrices, where the number of variables can be greater than the number of observations. The limitations of the sample covariance matrix are discussed. Several new approaches are presented, including the shrinkage method, the observable and latent factor method, the Bayesian approach, and the random matrix theory approach. For each method, the construction of covariance matrices is given. The relationships among these methods are discussed. |
| Keywords: | Factor analysis, Principal components, Singular value decomposition, Random matrix theory, Empirical Bayes, Shrinkage method, Optimal portfolios, CAPM, APT, GMM |
| JEL: | C33 |
| Date: | 2011–11 |
| URL: | http://d.repec.org/n?u=RePEc:cuf:wpaper:516&r=ets |
| By: | Elena-Ivona Dumitrescu (LEO - Laboratoire d'économie d'Orleans - CNRS : UMR6221 - Université d'Orléans); Christophe Hurlin (LEO - Laboratoire d'économie d'Orleans - CNRS : UMR6221 - Université d'Orléans); Jaouad Madkour (LEO - Laboratoire d'économie d'Orleans - CNRS : UMR6221 - Université d'Orléans) |
| Abstract: | This paper proposes a new evaluation framework for interval forecasts. Our model free test can be used to evaluate intervals forecasts and High Density Regions, potentially discontinuous and/or asymmetric. Using a simple J-statistic, based on the moments de ned by the orthonormal polynomials associated with the Binomial distribution, this new approach presents many advantages. First, its implementation is extremely easy. Second, it allows for a separate test for unconditional coverage, independence and conditional coverage hypotheses. Third, Monte-Carlo simulations show that for realistic sample sizes, our GMM test has good small-sample properties. These results are corroborated by an empirical application on SP500 and Nikkei stock market indexes. It con rms that using this GMM test leads to major consequences for the ex-post evaluation of interval forecasts produced by linear versus nonlinear models. |
| Keywords: | Interval forecasts, High Density Region, GMM. |
| Date: | 2011–08 |
| URL: | http://d.repec.org/n?u=RePEc:hal:wpaper:halshs-00618467&r=ets |
| By: | Degui Li; Zudi Lu; Oliver Linton |
| Abstract: | Local linear fitting is a popular nonparametric method in statistical and econometric modelling. Lu and Linton (2007) established the pointwise asymptotic distribution for the local linear estimator of a nonparametric regression function under the condition of near epoch dependence. In this paper, we further investigate the uniform consistency of this estimator. The uniform strong and weak consistencies with convergence rates for the local linear fitting are established under mild conditions. Furthermore, general results regarding uniform convergence rates for nonparametric kernel-based estimators are provided. The results of this paper will be of wide potential interest in time series semiparametric modelling. |
| Keywords: | α-mixing, local linear fitting, near epoch dependence, convergence rates, uniform consistency |
| JEL: | C13 C14 C22 |
| Date: | 2011–09 |
| URL: | http://d.repec.org/n?u=RePEc:msh:ebswps:2011-16&r=ets |
| By: | Jiti Gao; Dag Tjøstheim; Jiying Yin |
| Abstract: | This paper treats estimation in a class of new nonlinear threshold autoregressive models with both a stationary and a unit root regime. Existing literature on nonstationary threshold models have basically focused on models where the nonstationarity can be removed by differencing and/or where the threshold variable is stationary. This is not the case for the process we consider, and nonstandard estimation problems are the result. This paper proposes a parameter estimation method for such nonlinear threshold autoregressive models using the theory of null recurrent Markov chains. Under certain assumptions, we show that the ordinary least squares (OLS) estimators of the parameters involved are asymptotically consistent. Furthermore, it can be shown that the OLS estimator of the coefficient parameter involved in the stationary regime can still be asymptotically normal while the OLS estimator of the coefficient parameter involved in the nonstationary regime has a nonstandard asymptotic distribution. In the limit, the rate of convergence in the stationary regime is asymptotically proportional to n-1/4, whereas it is n-1 in the nonstationary regime. The proposed theory and estimation method are illustrated by both simulated data and a real data example. |
| Keywords: | Autoregressive process; null-recurrent process; semiparametric model; threshold time series; unit root structure. |
| JEL: | C14 C22 |
| Date: | 2011–09 |
| URL: | http://d.repec.org/n?u=RePEc:msh:ebswps:2011-21&r=ets |
| By: | Jiti Gao; Maxwell King |
| Abstract: | This paper considers a class of parametric models with nonparametric autoregressive errors. A new test is proposed and studied to deal with the parametric specification of the nonparametric autoregressive errors with either stationarity or nonstationarity. Such a test procedure can initially avoid misspecification through the need to parametrically specify the form of the errors. In other words, we propose estimating the form of the errors and testing for stationarity or nonstationarity simultaneously. We establish asymptotic distributions of the proposed test. Both the setting and the results differ from earlier work on testing for unit roots in parametric time series regression. We provide both simulated and real-data examples to show that the proposed nonparametric unit-root test works in practice. |
| Keywords: | Autoregressive process; nonlinear time series; nonparametric method; random walk; semiparametric model; unit root test. |
| JEL: | C12 C14 C22 |
| URL: | http://d.repec.org/n?u=RePEc:msh:ebswps:2011-20&r=ets |
| By: | Pipat Wongsaart; Jiti Gao |
| Abstract: | A crucially important advantage of the semiparametric regression approach to the nonlinear autoregressive conditional duration (ACD) model developed in Wongsaart et al. (2011), i.e. the so-called Semiparametric ACD (SEMI-ACD) model, is the fact that its estimation method does not require a parametric assumption on the conditional distribution of the standardized duration process and, therefore, the shape of the baseline hazard function. The research in this paper complements that of Wongsaart et al. (2011) by introducing a nonparametric procedure to test the parametric density function of ACD error through the use of the SEMI-ACD based residual. The hypothetical structure of the test is useful, not only to the establishment of a better parametric ACD model, but also to the specification testing of a number of financial market microstructure hypotheses, especially those related to the information asymmetry in finance. The testing procedure introduced in this paper differs in many ways from those discussed in existing literatures, for example Aït-Sahalia (1996), Gao and King (2004) and Fernandes and Grammig (2005). We show theoretically and experimentally the statistical validity of our testing procedure, while demonstrating its usefulness and practicality using datasets from New York and Australia Stock Exchange. |
| Keywords: | Duration model, hazard rates and random measures, nonparametric kernel testing. |
| JEL: | C14 C41 F31 |
| Date: | 2011–09 |
| URL: | http://d.repec.org/n?u=RePEc:msh:ebswps:2011-18&r=ets |
| By: | Jiti Gao; Degui Li; Dag Tjøstheim |
| Abstract: | This paper establishes a suite of uniform consistency results for nonparametric kernel density and regression estimators when the time series regressors concerned are nonstationary null-recurrent Markov chains. Under suitable conditions, certain rates of convergence are also obtained for the proposed estimators. Our results can be viewed as an extension of some well-known uniform consistency results for the stationary time series case to the nonstationary time series case. |
| Keywords: | β-null recurrent Markov chain, nonparametric estimation, rate of convergence, uniform consistency |
| JEL: | C13 C14 C22 |
| Date: | 2011–09 |
| URL: | http://d.repec.org/n?u=RePEc:msh:ebswps:2011-13&r=ets |
| By: | Jia Chen; Jiti Gao; Degui Li |
| Abstract: | A semiparametric fixed effects model is introduced to describe the nonlinear trending phenomenon in panel data analysis and it allows for the cross-sectional dependence in both the regressors and the residuals. A pooled semiparametric profile likelihood dummy variable approach based on the first-stage local linear fitting is developed to estimate both the parameter vector and the nonparametric time trend function. As both the time series length T and the cross-sectional size N tend to infinity simultaneously, the resulting estimator of the parameter vector is asymptotically normal with a rate of convergence of Op(NT)^{-1/2}. Meanwhile, the asymptotic distribution for the estimator of the nonparametric trend function is also established with a rate of convergence of Op(NTh)^{-1/2}. Two simulated examples are provided to illustrate the finite sample performance of the proposed method. In addition, the proposed model and estimation method is applied to analyze a CPI data set as well as an input-output data set. |
| Keywords: | Cross-sectional dependence, nonlinear time trend, panel data, profile likelihood, semiparametric regression. |
| JEL: | C13 C14 C23 |
| Date: | 2011–09 |
| URL: | http://d.repec.org/n?u=RePEc:msh:ebswps:2011-15&r=ets |
| By: | Chaohua Dong; Jiti Gao |
| Abstract: | Two types of Brownian motion functionals, both time-homogeneous and time-inhomogeneous, are expanded in terms of orthonormal bases in respective Hilbert spaces. Meanwhile, different time horizons are treated from the applicability point of view. Moreover, the degrees of approximation of truncation series to the corresponding series are established. An asymptotic theory is established. Both the proposed expansions and asymptotic theory are applied to establish consistent estimators in a class of time series econometric models. |
| Keywords: | Asymptotic theory; Brownian motion; econometric estimation, series expansion. |
| JEL: | C14 C32 |
| Date: | 2011–09 |
| URL: | http://d.repec.org/n?u=RePEc:msh:ebswps:2011-19&r=ets |
| By: | Jiti Gao; Peter C.B. Phillips |
| Abstract: | A system of multivariate semiparametric nonlinear time series models is studied with possible dependence structures and nonstationarities in the parametric and nonparametric components. The parametric regressors may be endogenous while the nonparametric regressors are assumed to be strictly exogenous. The parametric regressors may be stationary or nonstationary and the nonparametric regressors are nonstationary integrated time series. Semiparametric least squares (SLS) estimation is considered and its asymptotic properties are derived. Due to endogeneity in the parametric regressors, SLS is not consistent for the parametric component and a semiparametric instrumental variable (SIV) method is proposed instead. Under certain regularity conditions, the SIV estimator of the parametric component is shown to have a limiting normal distribution. The rate of convergence in the parametric component depends on the properties of the regressors. The conventional √n rate may apply even when nonstationarity is involved in both sets of regressors. |
| Keywords: | Endogeneity; integrated process, nonstationarity; partial linear model; simultaneity; vector semiparametric regression. |
| JEL: | C23 C25 |
| Date: | 2011–09–05 |
| URL: | http://d.repec.org/n?u=RePEc:msh:ebswps:2011-17&r=ets |
| By: | Calhoun, Gray |
| Abstract: | This paper weakens the size and moment conditions needed for typical block bootstrap methods (i.e. the moving blocks, circular blocks, and stationary bootstraps) to be valid for the sample mean of Near-Epoch-Dependent functions of mixing processes; they are consistent under the weakest conditions that ensure the original process obeys a Central Limit Theorem (those of de Jong, 1997, Econometric Theory). In doing so, this paper extends de Jong's method of proof, a blocking argument, to hold with random and unequal block lengths. This paper also proves that bootstrapped partial sums satisfy a Functional CLT under the same conditions. |
| Keywords: | Resampling; Time Series; Near Epoch Dependence; Functional Central Limit Theorem |
| JEL: | C12 C15 |
| Date: | 2011–09–23 |
| URL: | http://d.repec.org/n?u=RePEc:isu:genres:34313&r=ets |
| By: | Carles Bretó; Helena Veiga |
| Abstract: | In this paper we compare the forecast performance of continuous and discrete-time volatility models. In discrete time, we consider more than ten GARCH-type models and an asymmetric autoregressive stochastic volatility model. In continuous-time, a stochastic volatility model with mean reversion, volatility feedback and leverage. We estimate each model by maximum likelihood and evaluate their ability to forecast the two scales realized volatility, a nonparametric estimate of volatility based on highfrequency data that minimizes the biases present in realized volatility caused by microstructure errors. We find that volatility forecasts based on continuous-time models may outperform those of GARCH-type discrete-time models so that, besides other merits of continuous-time models, they may be used as a tool for generating reasonable volatility forecasts. However, within the stochastic volatility family, we do not find such evidence. We show that volatility feedback may have serious drawbacks in terms of forecasting and that an asymmetric disturbance distribution (possibly with heavy tails) might improve forecasting. |
| Keywords: | Asymmetry, Continuous and discrete-time stochastic volatility models, GARCH-type models, Maximum likelihood via iterated filtering, Particle filter, Volatility forecasting |
| Date: | 2011–07 |
| URL: | http://d.repec.org/n?u=RePEc:cte:wsrepe:ws112518&r=ets |
| By: | Sergey Ivashchenko |
| Abstract: | This article compares properties of different non-linear Kalman filters: well-known Unscented Kalman filter (UKF), Central Difference Kalman Filter (CDKF) and unknown Quadratic Kalman filter (QKF). Small financial DSGE model is repeatedly estimated by maximum quasi-likelihood methods with different filters for data generated by the model. Errors of parameters estimation are measure of filters quality. The result is that QKF has reasonable advantage in quality over CDKF and UKF with some loose in speed. |
| Keywords: | DSGE, QKF, CDKF, UKF, quadratic approximation, Kalman filtering |
| JEL: | C13 C32 E32 |
| Date: | 2011–09–20 |
| URL: | http://d.repec.org/n?u=RePEc:eus:wpaper:ec0711&r=ets |
| By: | Mattéo Luciani; David Veredas |
| Abstract: | Realized volatilities, when observed through time, share the following stylized facts: co–movements, clustering, long–memory, dynamic volatility, skewness and heavy–tails. We propose a simple dynamic factor model that captures these stylized facts and that can be applied to vast panels of volatilities as it does not suffer from the curse of dimensionality. It is an enhanced version of Bai and Ng (2004) in the following respects: i) we allow for long–memory in both the idiosyncratic and the common components, ii) the common shocks are conditionally heteroskedastic, and iii) the idiosyncratic and common shocks are skewed and heavy–tailed. Estimation of the factors, the idiosyncratic components and the parameters is straightforward: principal components and low dimension maximum likelihood estimations. A throughout Monte Carlo study shows the usefulness of the approach and an application to 90 daily realized volatilities, pertaining to S&P100, from January 2001 to December 2008, evinces, among others, the following findings: i) All the volatilities have long–memory, more than half in the nonstationary range, that increases during financial turmoil. ii) Tests and criteria point towards one dynamic common factor driving the co–movements. iii) The factor has larger long–memory than the assets volatilities, suggesting that long–memory is a market characteristic. iv) The volatility of the realized volatility is not constant and common to all. v) A forecasting horse race against univariate short– and long–memory models and short–memory dynamic factor models shows that our model outperforms short–, medium–, and long–run predictions, in particular in periods of stress. |
| Keywords: | realized volatilities; vast dimensions; factor models; long-memory; forecasting |
| JEL: | C32 C51 |
| Date: | 2011–09 |
| URL: | http://d.repec.org/n?u=RePEc:eca:wpaper:2013/97304&r=ets |
| By: | Mattéo Luciani |
| Abstract: | In this paper we investigate whether accounting for non-pervasive shocks improves the forecast of a factor model. We compare four models on a large panel of US quarterly data: factor models, factor models estimated on selected variables, Bayesian shrinkage, and factor models together with Bayesian shrinkage for the idiosyncratic component. The results of the forecasting exercise show that the four approaches considered perform equally well and produce highly correlated forecasts, meaning that non-pervasive shocks are of no helps in forecasting. We conclude that comovements captured by factor models are informative enough to make accurate forecasts. |
| JEL: | C13 C32 C33 C52 C53 |
| Date: | 2011–07 |
| URL: | http://d.repec.org/n?u=RePEc:eca:wpaper:2013/97308&r=ets |
| By: | Heinen, Florian; Willert, Juliane |
| Abstract: | We consider the detection of a change in persistence of a long range dependent time series. The usual approach is to use one-shot tests to detect a change in persistence a posteriori in a historical data set. However, as breaks can occur at any given time and data arrives steadily it is desirable to detect a change in persistence as soon as possible. We propose the use of a MOSUM type test which allows sequential application whenever new data arrives. We derive the asymptotic distribution of the test statistic and prove consistency. We further study the finite sample behavior of the test and provide an empirical application. |
| Keywords: | Change in persistence, long range dependency, MOSUM test |
| JEL: | C12 C22 |
| Date: | 2011–09 |
| URL: | http://d.repec.org/n?u=RePEc:han:dpaper:dp-479&r=ets |