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Abstract: We develop a theory of evolutionary spectra for heteroskedasticity and autocorrelation robust (HAR) inference when the data may not satisfy second-order stationarity. Nonstationarity is a common feature of economic time series which may arise either from parameter variation or model misspecification. In such a context, the theories that support HAR inference are either not applicable or do not provide accurate approximations. HAR tests standardized by existing long-run variance estimators then may display size distortions and little or no power. This issue can be more severe for methods that use long bandwidths (i.e., fixed-b HAR tests). We introduce a class of nonstationary processes that have a time-varying spectral representation which evolves continuously except at a finite number of time points. We present an extension of the classical heteroskedasticity and autocorrelation consistent (HAC) estimators that applies two smoothing procedures. One is over the lagged autocovariances, akin to classical HAC estimators, and the other is over time. The latter element is important to flexibly account for nonstationarity. We name them double kernel HAC (DK-HAC) estimators. We show the consistency of the estimators and obtain an optimal DK-HAC estimator under the mean squared error (MSE) criterion. Overall, HAR tests standardized by the proposed DK-HAC estimators are competitive with fixed-b HAR tests, when the latter work well, with regards to size control even when there is strong dependence. Notably, in those empirically relevant situations in which previous HAR tests are undersized and have little or no power, the DK-HAC estimator leads to tests that have good size and power.

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Topics: Estimator (58%), Heteroscedasticity (52%), Mean squared error (52%) ... read more

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6 results found

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Abstract: An important and widely used class of semiparametric models is formed by the varying-coefficient models. Although the varying coefficients are traditionally assumed to be smooth functions, the varying-coefficient model is considered here with the coefficient functions containing a finite set of discontinuities. Contrary to the existing nonparametric and varying-coefficient estimation of piecewise smooth functions, the varying-coefficient models are considered here under dependence and are applicable in time series with heteroskedastic and serially correlated errors. Additionally, the conditional error variance is allowed to exhibit discontinuities at a finite set of points too. The (uniform) consistency and asymptotic normality of the proposed estimators are established and the finite-sample performance is tested via a simulation study and in a real-data example.

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Topics: Classification of discontinuities (56%), Asymptotic distribution (55%), Estimator (53%) ... read more

3 Citations

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Abstract: It is well known that the conventional CUSUM test suffers from low power and large detection delay. We therefore propose two alternative detector statistics. The backward CUSUM detector sequentially cumulates the recursive residuals in reverse chronological order, whereas the stacked backward CUSUM detector considers a triangular array of backward cumulated residuals. While both the backward CUSUM detector and the stacked backward CUSUM detector are suitable for retrospective testing, only the stacked backward CUSUM detector can be monitored on-line. The limiting distributions of the maximum statistics under suitable sequences of alternatives are derived for retrospective testing and fixed endpoint monitoring. In the retrospective testing context, the local power of the tests is shown to be substantially higher than that for the conventional CUSUM test if a single break occurs after one third of the sample size. When applied to monitoring schemes, the detection delay of the stacked backward CUSUM is shown to be much shorter than that of the conventional monitoring CUSUM procedure. Moreover, an infinite horizon monitoring procedure and critical values are presented.

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Topics: CUSUM (68%)

1 Citations

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Abstract: This paper develops change-point methods for the spectrum of a locally stationary time series. We focus on series with a bounded spectral density that change smoothly under the null hypothesis but exhibits change-points or becomes less smooth under the alternative. We address two local problems. The first is the detection of discontinuities (or breaks) in the spectrum at unknown dates and frequencies. The second involves abrupt yet continuous changes in the spectrum over a short time period at an unknown frequency without signifying a break. Both problems can be cast into changes in the degree of smoothness of the spectral density over time. We consider estimation and minimax-optimal testing. We determine the optimal rate for the minimax distinguishable boundary, i.e., the minimum break magnitude such that we are able to uniformly control type I and type II errors. We propose a novel procedure for the estimation of the change-points based on a wild sequential top-down algorithm and show its consistency under shrinking shifts and possibly growing number of change-points. Our method can be used across many fields and a companion program is made available in popular software packages.

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Topics: Spectral density (54%), Smoothness (probability theory) (51%), Series (mathematics) (51%)

1 Citations

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Abstract: We show that the nonstandard limiting distribution of HAR test statistics under fixed-b asymptotics is not pivotal (even after studentization) when the data are nonstationarity. It takes the form of a complicated function of Gaussian processes and depends on the integrated local long-run variance and on on the second moments of the relevant series (e.g., of the regressors and errors for the case of the linear regression model). Hence, existing fixed-b inference methods based on stationarity are not theoretically valid in general. The nuisance parameters entering the fixed-b limiting distribution can be consistently estimated under small-b asymptotics but only with nonparametric rate of convergence. Hence, We show that the error in rejection probability (ERP) is an order of magnitude larger than that under stationarity and is also larger than that of HAR tests based on HAC estimators under conventional asymptotics. These theoretical results reconcile with recent finite-sample evidence in Casini (2021) and Casini, Deng and Perron (2021) who showing that fixed-b HAR tests can perform poorly when the data are nonstationary. They can be conservative under the null hypothesis and have non-monotonic power under the alternative hypothesis irrespective of how large the sample size is.

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Topics: Asymptotic distribution (55%), Studentization (54%), Statistical hypothesis testing (53%) ... read more

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Abstract: Autoregressive (AR) models such as the heterogeneous autoregressive (HAR) model capture the linear footprint inherent in realized volatility. We draw upon the fact that the HAR model is a constrain...

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Topics: Realized variance (53%), Autoregressive model (53%), Model selection (52%)

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53 results found

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Abstract: This paper describes a simple method of calculating a heteroskedasticity and autocorrelation consistent covariance matrix that is positive semi-definite by construction. It also establishes consistency of the estimated covariance matrix under fairly general conditions.

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Topics: Covariance matrix (60%), Positive-definite matrix (52%), Consistent estimator (50%)

17,401 Citations

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Topics: Method of moments (statistics) (67%), L-moment (61%), Generalized method of moments (61%) ... read more

12,211 Citations

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Abstract: This paper proposes a very tractable approach to modeling changes in regime. The parameters of an autoregression are viewed as the outcome of a discrete-state Markov process. For example, the mean growth rate of a nonstationary series may be subject to occasional, discrete shifts. The econometrician is presumed not to observe these shifts directly, but instead must draw probabilistic inference about whether and when they may have occurred based on the observed behavior of the series. The paper presents an algorithm for drawing such probabilistic inference in the form of a nonlinear iterative filter

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Topics: Markov process (53%)

8,684 Citations

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Abstract: We consider the null hypothesis that a time series has a unit root with possibly nonzero drift against the alternative that the process is «trend-stationary». The interest is that we allow under both the null and alternative hypotheses for the presence for a one-time change in the level or in the slope of the trend function. We show how standard tests of the unit root hypothesis against trend stationary alternatives cannot reject the unit root hypothesis if the true data generating mechanism is that of stationary fluctuations around a trend function which contains a one-time break

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Topics: KPSS test (71%), Unit root (69%), Unit root test (68%) ... read more

7,132 Citations

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Abstract: We propose and evaluate explicit tests of the null hypothesis of no difference in the accuracy of two competing forecasts. In contrast to previously developed tests, a wide variety of accuracy measures can be used (in particular, the loss function need not be quadratic and need not even be symmetric), and forecast errors can be non-Gaussian, nonzero mean, serially correlated, and contemporaneously correlated. Asymptotic and exact finite-sample tests are proposed, evaluated, and illustrated.

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Topics: Sign test (50%)

5,619 Citations