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on Econometric Time Series |
| By: | Qiying Wang (University of Sydney); Peter C. B. Phillips (Cowles Foundation, Yale University, University of Auckland) |
| Abstract: | New limit theory is provided for a wide class of sample variance and covariance functionals involving both nonstationary and stationary time series. Sample functionals of this type commonly appear in regression applications and the asymptotics are particularly relevant to estimation and inference in nonlinear nonstationary regressions that involve unit root, local unit root or fractional processes. The limit theory is unusually general in that it covers both parametric and nonparametric regressions. Self normalized versions of these statistics are considered that are useful in inference. Numerical evidence reveals interesting strong bimodality in the finite sample distributions of conventional self normalized statistics similar to the bimodality that can arise in t-ratio statistics based on heavy tailed data. Bimodal behavior in these statistics is due to the presence of long memory innovations and is shown to persist for very large sample sizes even though the limit theory is Gaussian when the long memory innovations are stationary. Bimodality is shown to occur even in the limit theory when the long memory innovations are nonstationary. To address these complications new self normalized versions of the test statistics are introduced that deliver improved approximations that can be used for inference. |
| Keywords: | Bimodality, Endogeneity, Limit theory, Local time, Nonlinear functional, Nonstationarity, Sample covariance, Zero energy |
| JEL: | C13 C23 |
| Date: | 2024–06 |
| URL: | https://d.repec.org/n?u=RePEc:cwl:cwldpp:2337r1 |
| By: | Zhishui Hu (University of Science and Technology of China); Nan Liu (Xiamen University); Peter C. B. Phillips (Cowles Foundation, Yale University, University of Auckland); Qiying Wang (University of Sydney) |
| Abstract: | A new self-weighted least squares (LS) estimation theory is developed for local unit root (LUR) autoregression with heteroskedasticity. The proposed estimator has a mixed Gaussian limit distribution and the corresponding studentized statistic converges to a standard normal distribution free of the unknown localizing coefficient which is not consistently estimable. The estimator is super consistent with a convergence rate slightly below the OP (n) rate of LS estimation. The asymptotic theory relies on a new framework of convergence to the local time of a Gaussian process, allowing for the sample moments generated from martingales and many other integrated dependent sequences. A new unit root (UR) test in augmented autoregression is developed using self-weighted estimation and the methods are employed in predictive regression, providing an alternative approach to IVX regression. Simulation results showing good finite sample performance of these methods are reported together with a small empirical application. |
| Keywords: | Self-weighted least squares estimation, autoregression, super consistency, limit distribution, unit root test, predictive regression. |
| JEL: | C13 C22 |
| Date: | 2024–04 |
| URL: | https://d.repec.org/n?u=RePEc:cwl:cwldpp:2400 |
| By: | Emanuele Bacchiocchi (Department of Economics, University of Bologna); Andrea Bastianin (Department of Economics, Management and Quantitative Methods, University of Milan and Fondazione Eni Enrico Mattei); Toru Kitagawa (Department of Economics, Brown University); Elisabetta Mirto (Department of Economics, Management and Quantitative Methods, University of Milan) |
| Abstract: | This paper presents new results on the identification of heteroskedastic structural vector autoregressive (HSVAR) models. Point identification of HSVAR models fails when some shifts in the variances of the structural shocks are suspected to be statistically indistinguishable from each other. This paper presents a new strategy that allows researchers to continue using HSVAR models in this empirically relevant case. We show that a combination of heteroskedasticity and zero restrictions can recover point identification in HSVAR models even in the absence of heterogeneous variance shifts. We derive the identified sets for impulse responses and show how to compute them. We perform inference on the impulse response functions, building on the robust Bayesian approach developed for set-identified SVARs. To illustrate our proposal, we present an empirical example based on the literature on the global crude oil market, where standard identification is expected to fail under heteroskedasticity. |
| Keywords: | Heteroskedastic SVAR, point and set identification, robust Bayesian approach |
| JEL: | C11 C32 C51 Q41 |
| Date: | 2024–06 |
| URL: | https://d.repec.org/n?u=RePEc:fem:femwpa:2024.15 |
| By: | Ying Wang (Renmin University of China); Peter C. B. Phillips (Cowles Foundation, Yale University, University of Auckland); Yundong Tu (Peking University) |
| Abstract: | Functional coefficient (FC) cointegrating regressions offer empirical investigators flexibility in modeling economic relationships by introducing covariates that influence the direction and intensity of comovement among nonstationary time series. FC regression models are also useful when formal cointegration is absent, in the sense that the equation errors may themselves be nonstationary, but where the nonstationary series display well-defined FC linkages that can be meaningfully interpreted as correlation measures involving the covariates. The present paper proposes new nonparametric estimators for such FC regression models where the nonstationary series display linkages that enable consistent estimation of the correlation measures between them. Specifically, we develop Ãn-consistent estimators for the functional coefficient and establish their asymptotic distributions, which involve mixed normal limits that facilitate inference. Two novel features that appear in the limit theory are (i) the need for non- diagonal matrix normalization due to the presence of stationary and nonstationary components in the regression; and (ii) random bias elements that appear in the asymptotic distribution of the kernel estimators, again resulting from the nonstationary regression components. Numerical studies reveal that the proposed estimators achieve significant efficiency improvements compared to the estimators suggested in earlier work by Sun et al. (2011). Easily implementable specification tests with standard chi-square asymptotics are suggested to check for constancy of the functional coefficient. These tests are shown to have faster divergence rate under local alternatives and enjoy superior performance in simulations than tests proposed recently in Gan et al. (2014). An empirical application based on the quantity theory of money illustrates the practical use of correlated but non-cointegrated regression relations. |
| Keywords: | Cointegration; Correlation measure; Functional coefficient regression; Marginal integration; Nonstationary time series. |
| JEL: | C14 C22 |
| Date: | 2024–04 |
| URL: | https://d.repec.org/n?u=RePEc:cwl:cwldpp:2399 |
| By: | Paolo Maranzano (Department of Economics, Management and Statistics, University of Milano-Bicocca and Fondazione Eni Enrico Mattei); Matteo Pelagatti (Department of Economics, Management and Statistics, University of Milano-Bicocca) |
| Abstract: | The Hodrick-Prescott filter is a popular tool in macroeconomics for decomposing a time series into a smooth trend and a business cycle component. The last few years have witnessed global events, such as the Global Financial Crisis, the COVID-19 pandemic, and the war in Ukraine, that have had abrupt structural impacts on many economic time series. Moreover, new regulations and policy changes generally lead to similar behaviours. Thus, those events should be absorbed by the trend component of the trend-cycle decomposition, but the Hodrick-Prescott filter does not allow for jumps. We propose a modification of the Hodrick-Prescott filter that contemplates jumps and automatically selects the time points in which the jumps occur. We provide an efficient implementation of the new filter in an R package. We use our modified filter to assess what Italian labour market reforms impacted employment in different age groups. |
| Keywords: | Trend, State-space form, Unobserved component model, Structural change, LASSO, Business cycle, Employment |
| JEL: | C22 C63 E32 J21 |
| Date: | 2024–07 |
| URL: | https://d.repec.org/n?u=RePEc:fem:femwpa:2024.18 |
| By: | Ying Wang (Renmin University of China); Peter C. B. Phillips (Cowles Foundation, Yale University, University of Auckland) |
| Abstract: | Limit theory for functional coefficient cointegrating regression was recently found to be considerably more complex than earlier understood. The issues were explained and correct limit theory derived for the kernel weighted local constant estimator in Phillips and Wang (2023b). The present paper provides complete limit theory for the general kernel weighted local p-th order polynomial estimator of the functional coefficient and the coefficient deriva-tives. Both stationary and nonstationary regressors are allowed. Implications for bandwidth selection are discussed. An adaptive procedure to select the fit order p is proposed and found to work well. A robust t-ratio is constructed following the new correct limit theory, which corrects and improves the usual t-ratio in the literature. Furthermore, the robust t-ratio is valid and works well regardless of the properties of the regressors, thereby providing a unified procedure to compute the t-ratio and facilitating practical inference. Testing constancy of the functional coefficient is also considered. Supportive finite sample studies are provided that corroborate the new asymptotic theory. |
| Keywords: | bandwidth selection, functional-coefficient cointegration, local p-th order polyno-mial approximation, robust t-ratio |
| JEL: | C14 C22 |
| Date: | 2024–06 |
| URL: | https://d.repec.org/n?u=RePEc:cwl:cwldpp:2398 |
| By: | Martin Bruns (School of Economics, University of East Anglia); Helmut Lütkepohl (DIW Berlin and Freie Universität Berlin); James McNeil (Dalhousie University) |
| Abstract: | The shocks in structural vector autoregressive (VAR) analysis are typically assumed to be instantaneously uncorrelated. This condition may easily be violated in proxy VAR models if more than one shock is identified by a proxy variable. Correlated shocks may be obtained even if the proxies are uncorrelated and satisfy the usual relevance and exogeneity conditions individually. Examples from the recent proxy VAR literature are presented. It is shown that assuming uncorrelated proxies that satisfy the usual relevance and exogeneity conditions individually actually over-identifies the shocks of interest and a Generalized Method of Moments (GMM) algorithm is proposed that ensures orthogonal shocks and provides efficient estimators of the structural parameters. It generalizes an earlier GMM proposal that works only if at least K − 1 shocks are identified by proxies in a VAR with K variables. |
| Keywords: | Structural vector autoregression, proxy VAR, external instruments, correlated shocks, Generalized Method of Moments |
| JEL: | C32 C36 E52 |
| Date: | 2024–07 |
| URL: | https://d.repec.org/n?u=RePEc:uea:ueaeco:2024-05 |