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on Econometric Time Series |
| By: | Qiying Wang (University of Sydney); Peter C.B. Phillips (Cowles Foundation, Yale University); Ioannis Kasparis (Dept. of Economics, University of Cyprus) |
| Abstract: | This paper studies the asymptotic properties of empirical nonparametric regressions that partially misspecify the relationships between nonstationary variables. In particular, we analyze nonparametric kernel regressions in which a potential nonlinear cointegrating regression is misspecified through the use of a proxy regressor in place of the true regressor. Such regressions arise naturally in linear and nonlinear regressions where the regressor suffers from measurement error or where the true regressor is a latent variable. The model considered allows for endogenous regressors as the latent variable and proxy variables that cointegrate asymptotically with the true latent variable. Such a framework includes correctly specified systems as well as misspecified models in which the actual regressor serves as a proxy variable for the true regressor. The system is therefore intermediate between nonlinear nonparametric cointegrating regression (Wang and Phillips, 2009a, 2009b) and completely misspecified nonparametric regressions in which the relationship is entirely spurious (Phillips, 2009). The asymptotic results relate to recent work on dynamic misspecification in nonparametric nonstationary systems by Kasparis and Phillips (2012) and Duffy (2014). The limit theory accommodates regressor variables with autoregressive roots that are local to unity and whose errors are driven by long memory and short memory innovations, thereby encompassing applications with a wide range of economic and financial time series. |
| Keywords: | Cointegrating regression, Kernel regression, Latent variable, Local time, Misspecification, Nonlinear nonparametric nonstationary regression |
| JEL: | C23 |
| Date: | 2017–09 |
| URL: | http://d.repec.org/n?u=RePEc:cwl:cwldpp:3011&r=ets |
| By: | Degui Li (University of York); Peter C.B. Phillips (Cowles Foundation, Yale University); Jiti Gao (Dept. of Econometrics and Business Statistics, Monash University) |
| Abstract: | This paper studies nonlinear cointegrating models with time-varying coefficients and multiple nonstationary regressors using classic kernel smoothing methods to estimate the coefficient functions. Extending earlier work on nonstationary kernel regression to take account of practical features of the data, we allow the regressors to be cointegrated and to embody a mixture of stochastic and deterministic trends, complications which result in asymptotic degeneracy of the kernel-weighted signal matrix. To address these complications new \textsl{local} and \textsl{global rotation} techniques are introduced to transform the covariate space to accommodate multiple scenarios of induced degeneracy. Under certain regularity conditions we derive asymptotic results that differ substantially from existing kernel regression asymptotics, leading to new limit theory under multiple convergence rates. For the practically important case of endogenous nonstationary regressors we propose a fully-modified kernel estimator whose limit distribution theory corresponds to the prototypical pure (i.e., exogenous covariate) cointegration case, thereby facilitating inference using a generalized Wald-type test statistic. These results substantially generalize econometric estimation and testing techniques in the cointegration literature to accommodate time variation and complications of co-moving regressors. Finally an empirical illustration to aggregate US data on consumption, income, and interest rates is provided. |
| Keywords: | Cointegration, FM-kernel estimation, Generalized Wald test, Global rotation, Kernel degeneracy, Local rotation, Super-consistency, Time-varying coefficients |
| JEL: | C22 C65 |
| Date: | 2017–09 |
| URL: | http://d.repec.org/n?u=RePEc:cwl:cwldpp:3009&r=ets |
| By: | Todd Prono |
| Abstract: | Linear GARCH(1,1) and threshold GARCH(1,1) processes are established as regularly varying, meaning their heavy tails are Pareto like, under conditions that allow the innovations from the, respective, processes to be skewed. Skewness is considered a stylized fact for many financial returns assumed to follow GARCH-type processes. The result in this note aids in establishing the asymptotic properties of certain GARCH estimators proposed in the literature. |
| Keywords: | GARCH ; Pareto tail ; Heavy tail ; Regular variation ; Threshold GARCH |
| JEL: | C20 C22 C53 C58 |
| Date: | 2017–09–22 |
| URL: | http://d.repec.org/n?u=RePEc:fip:fedgfe:2017-95&r=ets |
| By: | McAleer, M.J. |
| Abstract: | One of the most widely-used multivariate conditional volatility models is the dynamic conditional correlation (or DCC) specification. However, the underlying stochastic process to derive DCC has not yet been established, which has made problematic the derivation of asymptotic properties of the Quasi-Maximum Likelihood Estimators (QMLE). To date, the statistical properties of the QMLE of the DCC parameters have purportedly been derived under highly restrictive and unverifiable regularity conditions. The paper shows that the DCC model can be obtained from a vector random coefficient moving average process, and derives the stationarity and invertibility conditions of the DCC model. The derivation of DCC from a vector random coefficient moving average process raises three important issues, as follows: (i) demonstrates that DCC is, in fact, a dynamic conditional covariance model of the returns shocks rather than a dynamic conditional correlation model; (ii) provides the motivation, which is presently missing, for standardization of the conditional covariance model to obtain the conditional correlation model; and (iii) shows that the appropriate ARCH or GARCH model for DCC is based on the standardized shocks rather than the returns shocks. The derivation of the regularity conditions, especially stationarity and invertibility, should subsequently lead to a solid statistical foundation for the estimates of the DCC parameters. Several new results are also derived for univariate models, including a novel conditional volatility model expressed in terms of standardized shocks rather than returns shocks, as well as the associated stationarity and invertibility conditions. |
| Keywords: | Dynamic conditional correlation, dynamic conditional covariance, vector random coefficient moving average, stationarity, invertibility, asymptotic properties. |
| JEL: | C22 C52 C58 G32 |
| Date: | 2017–09–01 |
| URL: | http://d.repec.org/n?u=RePEc:ems:eureir:101761&r=ets |
| By: | Tan, A.C.; McAleer, M.J. |
| Abstract: | The purpose of the paper is to explore the relative biases in the estimation of the Full BEKK model as compared with the Diagonal BEKK model, which is used as a theoretical and empirical benchmark. Chang and McAleer [4] show that univariate GARCH is not a special case of multivariate GARCH, specically, the Full BEKK model, and demonstrate that Full BEKK which, in practice, is estimated almost exclusively, has no underlying stochastic process, regularity conditions, or asymptotic properties. Diagonal BEKK (DBEKK) does not suf- fer from these limitations, and hence provides a suitable benchmark. We use simulated nancial returns series to contrast estimates of the conditional vari- ances and covariances from DBEKK and BEKK. The results of non-parametric tests suggest evidence of considerable bias in the Full BEKK estimates. The results of quantile regression analysis show there is a systematic relationship between the two sets of estimates as we move across the quantiles. Estimates of conditional variances from Full BEKK, relative to those from DBEKK, are lower in the left tail and higher in the right tail. |
| Keywords: | DBEKK, BEKK, Regularity Conditions, Asymptotic Properties, Non-Parametric, Bias, Qantile regression |
| JEL: | C13 C21 C58 |
| Date: | 2017–07–28 |
| URL: | http://d.repec.org/n?u=RePEc:ems:eureir:101765&r=ets |
| By: | Clark, Todd E. (Federal Reserve Bank of Cleveland); McCracken, Michael W. (Federal Reserve Bank of St. Louis); Mertens, Elmar (Bank for Inernational Settlements) |
| Abstract: | We develop uncertainty measures for point forecasts from surveys such as the Survey of Professional Forecasters, Blue Chip, or the Federal Open Market Committee’s Summary of Economic Projections. At a given point of time, these surveys provide forecasts for macroeconomic variables at multiple horizons. To track time-varying uncertainty in the associated forecast errors, we derive a multiple-horizon specification of stochastic volatility. Compared to constant-variance approaches, our stochastic-volatility model improves the accuracy of uncertainty measures for survey forecasts. |
| Keywords: | Stochastic volatility; survey forecasts; fan charts; |
| JEL: | C53 E37 |
| Date: | 2017–09–25 |
| URL: | http://d.repec.org/n?u=RePEc:fip:fedcwp:1715&r=ets |
| By: | Marcus Cordi; Damien Challet; Ioane Muni Toke |
| Abstract: | We show that univariate and symmetric multivariate Hawkes processes are only weakly causal: the true log-likelihoods of real and reversed event time vectors are almost equal, thus parameter estimation via maximum likelihood only weakly depends on the direction of the arrow of time. In ideal (synthetic) conditions, tests of goodness of parametric fit unambiguously reject backward event times, which implies that inferring kernels from time-symmetric quantities, such as the autocovariance of the event rate, only rarely produce statistically significant fits. Finally, we find that fitting financial data with many-parameter kernels may yield significant fits for both arrows of time for the same event time vector, sometimes favouring the backward time direction. This goes to show that a significant fit of Hawkes processes to real data with flexible kernels does not imply a definite arrow of time unless one tests it. |
| Date: | 2017–09 |
| URL: | http://d.repec.org/n?u=RePEc:arx:papers:1709.08516&r=ets |