
on Econometric Time Series 
By:  Lui, Yiu Lim (Dongbei University of Finance and Economics); Phillips, Peter C.B. (Yale University); Yu, Jun (Singapore Management University) 
Abstract:  A heteroskedasticityautocorrelation robust (HAR) test statistic is proposed to test for the presence of explosive roots in financial or real asset prices when the equation errors are strongly dependent. Limit theory for the test statistic is developed and extended to heteroskedastic models. The new test has stable size properties unlike conventional test statistics that typically lead to size distortion and inconsistency in the presence of strongly dependent equation errors. The new procedure can be used to consistently timestamp the origination and termination of an explosive episode under similar conditions of long memory errors. Simulations are conducted to assess the finite sample performance of the proposed test and estimators. An empirical application to the S&P 500 index highlights the usefulness of the proposed procedures in practical work. 
Keywords:  HAR test; Long memory; Explosiveness; Unit root test; S&P 500 
JEL:  C12 C22 G01 
Date:  2022–10–28 
URL:  http://d.repec.org/n?u=RePEc:ris:smuesw:2022_011&r=ets 
By:  Jianwei Jin (Yokohama National University); Keiji Nagai (Yokohama National University) 
Abstract:  This paper examines the effect of initial values and smallsample properties in sequential unit root tests of the firstorder autoregressive (AR(1)) process with a coefficient expressed by a local parameter. Adopting a stopping rule based on observed Fisher information defined by Lai and Siegmund (1983), we use the sequential least squares estimator (LSE) of the local parameter as the test statistic. The sequential LSE is represented as a timechanged Brownian motion with drift. The stopping time is written as the integral of the reciprocal of twice of a Bessel process with drift generated by the timechanged Brownian motion. The time change is applied to the joint density and joint Laplace transform derived from the Bessel bridge of the squared Bessel process by Pitman and Yor (1982), by which we derive the limiting joint density and joint Laplace transform for the sequential LSE and stopping time. The joint Laplace transform is needed to calculate joint moments because the joint density oscillates wildly as the value of the stopping time approaches zero. Moreover, this paper also earns the exact distribution of stopping time by Imhof's formula for both normally distributed and fixed initial values. When the autoregressive coefficient is less than 1, the question arises as to whether the localtounity or the strong stationary model should be used. We make the decision by comparing joint moments for respective models with those calculated from the exact distribution or simulations. 
Keywords:  Stopping time, observed Fisher information, DDS Brownian motion, local asymptotic normality, Bessel process, initial values, exact distributions 
JEL:  C12 C22 C46 
Date:  2022–11 
URL:  http://d.repec.org/n?u=RePEc:kyo:wpaper:1085&r=ets 
By:  Greta Goracci; Davide Ferrari; Simone Giannerini; Francesco ravazzolo 
Abstract:  Threshold autoregressive movingaverage (TARMA) models are popular in time series analysis due to their ability to parsimoniously describe several complex dynamical features. However, neither theory nor estimation methods are currently available when the data present heavy tails or anomalous observations, which is often the case in applications. In this paper, we provide the first theoretical framework for robust Mestimation for TARMA models and also study its practical relevance. Under mild conditions, we show that the robust estimator for the threshold parameter is superconsistent, while the estimators for autoregressive and movingaverage parameters are strongly consistent and asymptotically normal. The Monte Carlo study shows that the Mestimator is superior, in terms of both bias and variance, to the least squares estimator, which can be heavily affected by outliers. The findings suggest that robust Mestimation should be generally preferred to the least squares method. Finally, we apply our methodology to a set of commodity price time series; the robust TARMA fit presents smaller standard errors and leads to superior forecasting accuracy compared to the least squares fit. The results support the hypothesis of a tworegime, asymmetric nonlinearity around zero, characterised by slow expansions and fast contractions. 
Date:  2022–11 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2211.08205&r=ets 
By:  Kohtaro Hitomi (Kyoto Institute of Technology); Jianwei Jin (Yokohama National University); Keiji Nagai (Yokohama National University); Yoshihiko Nishiyama (Institute of Economic Research, Kyoto University); Junfan Tao (Institute of Economic Research, Kyoto University) 
Abstract:  The DickeyFuller (DF) unit root tests are widely used in empirical studies on economics. In the localtounity asymptotic theory, the effects of initial values vanish as the sample size grows. However, for a small sample size, the initial value will affect the distribution of the test statistics. When ignoring the effect of the initial value, the leftsided unit root test sets the critical value smaller than it should be. Therefore, the size and power of the test become smaller. This paper investigates the effect of the initial value for the DF test (including the t test). Limiting approximations of the DF test statistics are the ratios of two integrals which are represented via a onedimensional squared Bessel process. We derive the joint density of the squared Bessel process and its integral, enabling us to compute this ratio's distribution. For independent normal errors, the exact distribution of the DickeyFuller coefficient test statistic is obtained using the Imhof (1961) method for noncentral chisquared distribution. Numerical results show that when the sample size is small, the limiting distributions of the DF test statistics with initial values fit well with the exact or simulated distributions. We transform the DF test with respect to a local parameter into the test for a shift in the location parameter of normal distributions. As a result, a concise method for computing the powers of DF tests is derived. 
Keywords:  DickeyFuller tests, Squared Bessel process, joint density, powers approximated by normal distribution, exact distribution 
JEL:  C12 C22 C46 
Date:  2022–11 
URL:  http://d.repec.org/n?u=RePEc:kyo:wpaper:1084&r=ets 
By:  Lutz Kilian; Michael D. Plante; Alexander W. Richter 
Abstract:  A common practice in empirical macroeconomics is to examine alternative recursive orderings of the variables in structural vector autoregressive (VAR) models. When the implied impulse responses look similar, the estimates are considered trustworthy. When they do not, the estimates are used to bound the true response without directly addressing the identification challenge. A leading example of this practice is the literature on the effects of uncertainty shocks on economic activity. We prove by counterexample that this practice is invalid in general, whether the data generating process is a structural VAR model or a dynamic stochastic general equilibrium model. 
Keywords:  Cholesky Decomposition; endogeneity; uncertainty; business cycle 
JEL:  C32 C51 E32 
Date:  2022–11–23 
URL:  http://d.repec.org/n?u=RePEc:fip:feddwp:95180&r=ets 
By:  James A. Duffy; Sophocles Mavroeidis; Sam Wycherley 
Abstract:  In the literature on nonlinear cointegration, a longstanding open problem relates to how a (nonlinear) vector autoregression, which provides a unified description of the short and longrun dynamics of a collection of time series, can generate 'nonlinear cointegration' in the profound sense of those series sharing common nonlinear stochastic trends. We consider this problem in the setting of the censored and kinked structural VAR (CKSVAR), which provides a flexible yet tractable framework within which to model time series that are subject to thresholdtype nonlinearities, such as those arising due to occasionally binding constraints, of which the zero lower bound (ZLB) on shortterm nominal interest rates provides a leading example. We provide a complete characterisation of how common linear and nonlinear stochastic trends may be generated in this model, via unit roots and appropriate generalisations of the usual rank conditions, providing the first extension to date of the GrangerJohansen representation theorem from a linear to a nonlinear setting, and thereby giving the first successful treatment of the open problem. The limiting common trend processes include regulated, censored and kinked Brownian motions, none of which have previously appeared in the literature on cointegrated VARs. Our results and running examples illustrate that the CKSVAR is capable of supporting a far richer variety of longrun behaviour than is a linear VAR, in ways that may be particularly useful for the identification of structural parameters. En route to establishing our main results, we also develop a set of sufficient conditions for the processes generated by a CKSVAR to be stationary, ergodic, and weakly dependent. 
Date:  2022–11 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2211.09604&r=ets 
By:  Marc S. Paolella (University of Zurich  Department of Banking and Finance; Swiss Finance Institute); Pawel Polak (Stony Brook UniversityDepartment of Applied Mathematics and Statistics) 
Abstract:  The CCCGARCH model, and its dynamic correlation extensions, form the most important model class for multivariate asset returns. For multivariate density and portfolio risk forecasting, a drawback of these models is the underlying assumption of Gaussianity. This paper considers the socalled COMFORT model class, which is the CCCGARCH model but endowed with multivariate generalized hyperbolic innovations. The novelty of the model is that parameter estimation is conducted by joint maximum likelihood, of all model parameters, using an EM algorithm, and so is feasible for hundreds of assets. This paper demonstrates that (i) the new model is blatantly superior to its Gaussian counterpart in terms of forecasting ability, and (ii) also outperforms adhoc three step procedures common in the literature to augment the CCC and DCC models with a fattailed distribution. An extensive empirical study confirms the COMFORT model’s superiority in terms of multivariate density and ValueatRisk forecasting. 
Keywords:  GJRGARCH, Multivariate Generalized Hyperbolic Distribution, NonEllipticity, ValueatRisk. 
JEL:  C51 C53 G11 G17 
Date:  2022–11 
URL:  http://d.repec.org/n?u=RePEc:chf:rpseri:rp2288&r=ets 
By:  Andrzej Kocięcki (University of Warsaw, Faculty of Economic Sciences); Marcin Kolasa (SGH Warsaw School of Economics; International Monetary Fund) 
Abstract:  We develop an analytical framework to study global identification in structural models with forwardlooking expectations. Our identification condition combines the similarity transformation linking the observationally equivalent state space systems with the constraints imposed on them by the model parameters. The key step of solving the identification problem then reduces to finding all roots of a system of polynomial equations. We show how it can be done using the concept of a Gröbner basis and recently developed algorithms to compute it analytically. In contrast to papers relying on numerical search, our approach can effectively prove whether a model is identified or not at the given parameter point, explicitly delivering the complete set of observationally equivalent parameter vectors. We present the solution to the global identification problem for several popular DSGE models. Our findings indicate that observational equivalence in mediumsized models of this class might be actually not as widespread as suggested by earlier, small modelbased evidence. 
Keywords:  global identification, state space systems, DSGE models, Gröbner basis 
JEL:  C10 C51 C65 E32 
Date:  2022 
URL:  http://d.repec.org/n?u=RePEc:war:wpaper:202201&r=ets 