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on Econometric Time Series |
| By: | Bent Jesper Christensen (Aarhus University and CREATES); Paolo Santucci de Magistris (University of Pavia and CREATES) |
| Abstract: | We propose a simple model in which realized stock market return volatility and implied volatility backed out of option prices are subject to common level shifts corresponding to movements between bull and bear markets. The model is estimated using the Kalman filter in a generalization to the multivariate case of the univariate level shift technique by Lu and Perron (2008). An application to the S&P500 index and a simulation experiment show that the recently documented empirical properties of strong persistence in volatility and forecastability of future realized volatility from current implied volatility, which have been interpreted as long memory (or fractional integration) in volatility and fractional cointegration between implied and realized volatility, are accounted for by occasional common level shifts. |
| Keywords: | Common level shifts, fractional cointegration, fractional VECM, implied volatility, long memory, options, realized volatility. |
| JEL: | C32 G13 G14 |
| Date: | 2010–09–09 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2010-60&r=ets |
| By: | James G. MacKinnon (Queen's University); Morten Ørregaard Nielsen (Queen?s University and CREATES) |
| Abstract: | We calculate numerically the asymptotic distribution functions of likelihood ratio tests for fractional unit roots and cointegration rank. Because these distributions depend on a real-valued parameter, b, which must be estimated, simple tabulation is not feasible. Partly due to the presence of this parameter, the choice of model specification for the response surface regressions used to obtain the numerical distribution functions is more involved than is usually the case. We deal with model uncertainty by model averaging rather than by model selection. We make available a computer program which, given the dimension of the problem, q, and a value of b, provides either a set of critical values or the asymptotic P value for any value of the likelihood ratio statistic. The use of this program is illustrated by means of an empirical example involving opinion poll data. |
| Keywords: | Cofractional process, fractional unit root, fractional cointegration, response surface regression, cointegration rank, numerical distribution function, model averaging. |
| JEL: | C12 C16 C22 C32 |
| Date: | 2010–08–03 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2010-59&r=ets |
| By: | Francesco Ravazzolo (Norges Bank (Central Bank of Norway)); Philip Rothman (East Carolina University) |
| Abstract: | We study the real-time Granger-causal relationship between crude oil prices and US GDP growth through a simulated out-of-sample (OOS) forecasting exercise; we also provide strong evidence of in-sample predictability from oil prices to GDP. Comparing our benchmark model "without oil" against alternatives "with oil," we strongly reject the null hypothesis of no OOS predictability from oil prices to GDP via our point forecast comparisons from the mid-1980s through the Great Recession. Further analysis shows that these results may be due to our oil price measures serving as proxies for a recently developed measure of global real economic activity omitted from the alternatives to the benchmark forecasting models in which we only use lags of GDP growth. By way of density forecast OOS comparisons, we find evidence of such oil price predictability for GDP for our full 1970-2009 OOS period. Examination of the density forecasts reveals a massive increase in forecast uncertainty following the 1973 post-Yom Kippur War crude oil price increases. |
| Date: | 2010–09–15 |
| URL: | http://d.repec.org/n?u=RePEc:bno:worpap:2010_18&r=ets |
| By: | Trenkler, Carsten; Weber, Enzo |
| Abstract: | We analyze non-stationary time series that do not only trend together in the long run, but restore the equilibrium immediately in the period following a deviation. While this represents a common serial correlation feature, the framework is extended to codependence, allowing for delayed adjustment. We show which restrictions are implied for VECMs and lay out a likelihood ratio test. In addition, due to identification problems in codependent VECMs a GMM test approach is proposed. We apply the concept to US and European interest rate data, examining the capability of the Fed and ECB to control overnight money market rates. |
| Keywords: | Serial correlation common features; codependence; cointegration; overnight interest rates; central banks |
| JEL: | C32 E52 |
| Date: | 2010–09–15 |
| URL: | http://d.repec.org/n?u=RePEc:bay:rdwiwi:16478&r=ets |
| By: | Trenkler, Carsten; Weber, Enzo |
| Abstract: | In this paper we discuss identification of codependent VAR and VEC models. Codependence of order q is given if a linear combination of autocorrelated variables eliminates the serial correlation after q lags. Importantly, maximum likelihood estimation and corresponding likelihood ratio testing are only possible if the codependence restrictions can be uniquely imposed. However, our study reveals that codependent VAR and VEC models are not generally identified. Nevertheless, we show that one can guarantee identification in case of serial correlation common features, i.e. when q=0, and for a single vector generating codependence of order q=1. |
| Keywords: | Codependence; identification; VAR; cointegration; serial correlation common features |
| JEL: | C32 |
| Date: | 2010–09–15 |
| URL: | http://d.repec.org/n?u=RePEc:bay:rdwiwi:16477&r=ets |
| By: | Massimiliano Caporin; Michael McAleer (University of Canterbury) |
| Abstract: | This paper focuses on the selection and comparison of alternative non-nested volatility models. We review the traditional in-sample methods commonly applied in the volatility framework, namely diagnostic checking procedures, information criteria, and conditions for the existence of moments and asymptotic theory, as well as the out-of-sample model selection approaches, such as mean squared error and Model Confidence Set approaches. The paper develops some innovative loss functions which are based on Value-at-Risk forecasts. Finally, we present an empirical application based on simple univariate volatility models, namely GARCH, GJR, EGARCH, and Stochastic Volatility that are widely used to capture asymmetry and leverage. |
| Keywords: | Volatility model selection; volatility model comparison; non-nested models; model confidence set; Value-at-Risk forecasts; asymmetry, leverage |
| JEL: | C11 C22 C52 |
| Date: | 2010–09–01 |
| URL: | http://d.repec.org/n?u=RePEc:cbt:econwp:10/58&r=ets |
| By: | Peter C. B. Phillips (Cowles Foundation, Yale University); Jun Yu (Singapore Management University) |
| Abstract: | A new recursive regression methodology is introduced to analyze the bubble characteristics of various financial time series during the subprime crisis. The methods modify a technique proposed in Phillips, Wu and Yu (2010) and provide a technology for identifying bubble behavior and consistent dating of their origination and collapse. The tests also serve as an early warning diagnostic of bubble activity. Seven relevant financial series are investigated, including three financial assets (the Nasdaq index, home price index and asset-backed commercial paper), two commodities (the crude oil price and platinum price), one bond rate (Baa), and one exchange rate (Pound/USD). Statistically significant bubble characteristics are found in all of these series. The empirical estimates of the origination and collapse dates suggest an interesting migration mechanism among the financial variables: a bubble first emerged in the equity market during mid-1995 lasting to the end of 2000, followed by a bubble in the real estate market between January 2001 and July 2007 and in the mortgage market between November 2005 and August 2007. After the subprime crisis erupted, the phenomenon migrated selectively into the commodity market and the foreign exchange market, creating bubbles which subsequently burst at the end of 2008, just as the effects on the real economy and economic growth became manifest. Our empirical estimates of the origination and collapse dates match well with the general datetimes of this crisis put forward in a recent study by Caballero, Farhi and Gourinchas (2008). |
| Keywords: | Financial bubbles, Crashes, Date stamping, Explosive behavior, Mildly explosive process, Subprime crisis, Timeline |
| JEL: | C15 C12 |
| Date: | 2010–09 |
| URL: | http://d.repec.org/n?u=RePEc:cwl:cwldpp:1770&r=ets |
| By: | Jiti Gao (School of Economics, University of Adelaide); Peter C. B. Phillips (Cowles Foundation, Yale University) |
| Abstract: | A system of vector 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 strictly exogenous and represent trends. The parametric regressors may be stationary or nonstationary and the nonparametric regressors are nonstationary time series. This framework allows for the nonparametric treatment of stochastic trends and subsumes many practical cases. Semiparametric least squares (SLS) estimation is considered and its asymptotic properties are derived. Due to endogeneity in the parametric regressors, SLS is generally inconsistent for the parametric component and a semiparametric instrumental variable least squares (SIVLS) method is proposed instead. Under certain regularity conditions, the SIVLS estimator of the parametric component is shown to be consistent with a limiting normal distribution that is amenable to inference. The rate of convergence in the parametric component is the usual /n rate and is explained by the fact that the common (nonlinear) trend in the system is eliminated nonparametrically by stochastic detrending. |
| Keywords: | Simultaneous equation, Stochastic detrending, Vector semiparametric regression |
| JEL: | C23 C25 |
| Date: | 2010–09 |
| URL: | http://d.repec.org/n?u=RePEc:cwl:cwldpp:1769&r=ets |
| By: | Peter C. B. Phillips (Cowles Foundation, Yale University) |
| Abstract: | Trends are ubiquitous in economic discourse, play a role in much economic theory, and have been intensively studied in econometrics over the last three decades. Yet the empirical economist, forecaster, and policy maker have little guidance from theory about the source and nature of trend behavior, even less guidance about practical formulations, and are heavily reliant on a limited class of stochastic trend, deterministic drift, and structural break models to use in applications. A vast econometric literature has emerged but the nature of trend remains elusive. In spite of being the dominant characteristic in much economic data, having a role in policy assessment that is often vital, and attracting intense academic and popular interest that extends well beyond the subject of economics, trends are little understood. This essay discusses some implications of these limitations, mentions some research opportunities, and briefly illustrates the extent of the difficulties in learning about trend phenomena even when the time series are far longer than those that are available in economics. |
| Keywords: | Climate change, Etymology of trend, Paleoclimatology, Policy, Stochastic trend |
| JEL: | C22 |
| Date: | 2010–09 |
| URL: | http://d.repec.org/n?u=RePEc:cwl:cwldpp:1771&r=ets |
| By: | Xiaoxia Shi (Dept. of Economics, Yale University); Peter C. B. Phillips (Cowles Foundation, Yale University) |
| Abstract: | An asymptotic theory is developed for a weakly identified cointegrating regression model in which the regressor is a nonlinear transformation of an integrated process. Weak identification arises from the presence of a loading coefficient for the nonlinear function that may be close to zero. In that case, standard nonlinear cointegrating limit theory does not provide good approximations to the finite sample distributions of nonlinear least squares estimators, resulting in potentially misleading inference. A new local limit theory is developed that approximates the finite sample distributions of the estimators uniformly well irrespective of the strength of the identification. An important technical component of this theory involves new results showing the uniform weak convergence of sample covariances involving nonlinear functions to mixed normal and stochastic integral limits. Based on these asymptotics, we construct confidence intervals for the loading coefficient and the nonlinear transformation parameter and show that these confidence intervals have correct asymptotic size. As in other cases of nonlinear estimation with integrated processes and unlike stationary process asymptotics, the properties of the nonlinear transformations affect the asymptotics and, in particular, give rise to parameter dependent rates of convergence and differences between the limit results for integrable and asymptotically homogeneous functions. |
| Keywords: | Integrated process, Local time, Nonlinear regression, Uniform weak convergence, Weak identification |
| JEL: | C13 C22 |
| Date: | 2010–09 |
| URL: | http://d.repec.org/n?u=RePEc:cwl:cwldpp:1768&r=ets |
| By: | Feldkircher, Martin (Oesterreichische Nationalbank) |
| Abstract: | In this study the forecast performance of model averaged forecasts is compared to that of alternative single models. Following Eklund and Karlsson (2007) we form posterior model probabilities - the weights for the combined forecast - based on the predictive likelihood. Extending the work of Fernández et al. (2001a) we carry out a prior sensitivity analysis for a key parameter in Bayesian model averaging (BMA): Zellner's g. The main results based on a simulation study are fourfold: First the predictive likelihood does always better than the traditionally employed 'marginal' likelihood in settings where the true model is not part of the model space. Secondly, and more striking, forecast accuracy as measured by the root mean square error (rmse) is maximized for the median probability model put forward by Barbieri and Berger (2003). On the other hand, model averaging excels in predicting direction of changes, a finding that is in line with Crespo Cuaresma (2007). Lastly, our recommendation concerning the prior on g is to choose the prior proposed by Laud and Ibrahim (1995) with a hold-out sample size of 25% to minimize the rmse (median model) and 75% to optimize direction of change forecasts (model averaging). We finally forecast the monthly industrial production output of six Central Eastern and South Eastern European (CESEE) economies for a one step ahead forecasting horizon. Following the aforementioned forecasting recommendations improves the out-of-sample statistics over a 30-period horizon beating for almost all countries the first order autoregressive benchmark model. |
| Keywords: | Forecast Combination; Bayesian Model Averaging; Median Probability Model; Predictive Likelihood; Industrial Production; Model Uncertainty |
| JEL: | C11 C15 C53 |
| Date: | 2010–09–15 |
| URL: | http://d.repec.org/n?u=RePEc:ris:sbgwpe:2010_014&r=ets |
| By: | Athanasopoulos, George; Guillén, Osmani Teixeira de Carvalho; Issler, João Victor; Vahid, Farshid |
| Abstract: | We study the joint determination of the lag length, the dimension of the cointegrating space andthe rank of the matrix of short-run parameters of a vector autoregressive (VAR) model using modelselection criteria. We consider model selection criteria which have data-dependent penalties aswell as the traditional ones. We suggest a new two-step model selection procedure which is ahybrid of traditional criteria and criteria with data-dependant penalties and we prove its consistency.Our Monte Carlo simulations measure the improvements in forecasting accuracy that can arisefrom the joint determination of lag-length and rank using our proposed procedure, relative to anunrestricted VAR or a cointegrated VAR estimated by the commonly used procedure of selecting thelag-length only and then testing for cointegration. Two empirical applications forecasting Brazilianinflation and U.S. macroeconomic aggregates growth rates respectively show the usefulness of themodel-selection strategy proposed here. The gains in different measures of forecasting accuracy aresubstantial, especially for short horizons. |
| Date: | 2010–09–13 |
| URL: | http://d.repec.org/n?u=RePEc:fgv:epgewp:707&r=ets |
| By: | Jin Feng; Jean-Pierre Fouque; Rohini Kumar |
| Abstract: | In this paper, we study stochastic volatility models in regimes where the maturity is small but large compared to the mean-reversion time of the stochastic volatility factor. The problem falls in the class of averaging/homogenization problems for nonlinear HJB type equations where the "fast variable" lives in a non-compact space. We develop a general argument based on viscosity solutions which we apply to the two regimes studied in the paper. We derive a large deviation principle and we deduce asymptotic prices for Out-of-The-Money call and put options, and their corresponding implied volatilities. The results of this paper generalize the ones obtained in \cite{FFF} (J. Feng, M. Forde and J.-P. Fouque, {\it Short maturity asymptotic for a fast mean reverting Heston stochastic volatility model}, SIAM Journal on Financial Mathematics, Vol. 1, 2010) by a moment generating function computation in the particular case of the Heston model. |
| Date: | 2010–09 |
| URL: | http://d.repec.org/n?u=RePEc:arx:papers:1009.2782&r=ets |
| By: | Frantisek Slanina |
| Abstract: | We compare systematically several classes of stochastic volatility models of stock market fluctuations. We show that the long-time return distribution is either Gaussian or develops a power-law tail, while the short-time return distribution has generically a stretched-exponential form, but can assume also an algebraic decay, in the family of models which we call ``GARCH''-type. The intermediate regime is found in the exponential Ornstein-Uhlenbeck process. We calculate also the decay of the autocorrelation function of volatility. |
| Date: | 2010–09 |
| URL: | http://d.repec.org/n?u=RePEc:arx:papers:1009.2696&r=ets |
| By: | Shelley, Gary; Wallace, Frederick |
| Abstract: | his paper examines whether the CPI and real GDP for the U.S. exhibit nonlinear reversion to trend as recently concluded by Beechey and Österholm [Beechey, M. and Österholm, P., 2008. Revisiting the uncertain unit root in GDP and CPI: testing for non-linear trend reversion. Economics Letters 100, 221-223]. The wild bootstrap is used to correct for non-normality and heteroscedasticity in a nonlinear unit root test. Test results are found to be sensitive to the sample period examined. |
| Keywords: | nonlinear unit root test; wild bootstrap; non-normality |
| JEL: | E32 E31 C22 |
| Date: | 2010–01 |
| URL: | http://d.repec.org/n?u=RePEc:pra:mprapa:24962&r=ets |