Lag Truncation Research Articles

Self‐Normalized KPSS Tests With Power Enhancement

ABSTRACTMuch research has focused on testing the null hypothesis of stationarity against the unit root alternative. In this paper, we propose a novel class of self‐normalized KPSS tests without needing a consistent estimation of the long‐run variance. Under persistent autocorrelation, the widely used heteroskedasticity and autocorrelation consistent long‐run variance estimator proposed by Newey and West [Econometrica (1987) Vol. 55, pp. 703–708], Newey and West [The Review of Economic Studies (1994) Vol. 61, pp. 631–653], and Andrews [Econometrica (1991) Vol. 59, pp. 817–858] may not be reliable and often lead to tests with size distortions and power losses in finite samples. In addition, the practitioner has to choose the truncation lag that is ultimately arbitrary. To improve the finite sample performance of the stationarity tests and make them robust to realistic amounts of dependence, we propose the use of self‐normalizing methods, for example, the range‐based self‐normalization by Hong et al. [Journal of Econometrics (2024) Vol. 238, 105603] and the fixed‐ asymptotics by Kiefer and Vogelsang [Econometric Theory (2005) Vol. 21, pp. 1130–1164] and Amsler, Schmidt, and Vogelsang [Journal of Time Series Econometrics (2009) Vol. 1, pp. 1–44], to control the effect of the long‐run variance. The self‐normalized tests are inconsistent in that, under the unit root alternative, they do not diverge as the sample size approaches infinity. To recover the consistency of the tests, we devise a mechanism similar to the power enhancement mechanism proposed by Fan, Liao, and Yao [Econometrica (2015) Vol. 83, pp. 1497–1541]. Under the null hypothesis, this mechanism is asymptotically negligible. However, under the alternative hypothesis, the mechanism diverges as the sample size increases. We show that this mechanism also enhances the power of the self‐normalized stationarity tests. A simulation study and an empirical application demonstrate the merits of the approach advocated.

Under the same (Chole)sky: DNK models, timing restrictions and recursive identification of monetary policy shocks

Recent structural VAR studies of the monetary transmission mechanism have voiced concerns about the use of recursive identification schemes based on short-run exclusion restrictions. This paper characterizes the effects on impulse propagation of informational constraints embodying classical Cholesky-type timing restrictions in otherwise standard DSGE models. We formally show that timing restrictions can produce non-trivial moving average components of rational expectations solutions, or even serve as an independent source of model-based nonfundamentalness, thereby hampering impulse response analysis via VAR procedures. We then derive population conditions for existence of VAR representations of DSGE economies exhibiting timing restrictions, and numerically explore their bearing on shock identification in a range of monetary models of the business cycle. Our analysis reveals that dynamic New Keynesian models admit invertible equilibrium representations as well as fast-converging VAR coefficient matrices under empirically tenable parameterizations. This alleviates concerns about identification and lag truncation bias: low-order Cholesky-VARs do well at retrieving the true aggregate effects of monetary policy shocks in a Cholesky world.

Under the same (Chole)sky: DNK models, timing restrictions and recursive identification of monetary policy shocks

Recent structural VAR studies of the monetary transmission mechanism have voiced concerns about the use of recursive identification schemes based on short-run exclusion restrictions. We trace out the effects on impulse propagation of informational constraints embodying classical Cholesky-timing restrictions in otherwise standard Dynamic New Keynesian (DNK) models. By reinforcing internal propagation mechanisms and enlarging a model's equilibrium state space, timing restrictions may produce a non-trivial moving average component of the equilibrium representation, making finite order VARs a poor approximation of true adjustment paths to monetary impulses, albeit correctly identified. They can even serve as an independent source of model-based nonfundamentalness, thereby hampering shock identification via VAR methods. This notwithstanding, restricted DNK models are shown to feature (i) invertible equilibrium representations for the observables and (ii) fast-converging VAR coefficient matrices under empirically tenable parameterizations. This alleviates concerns about identification and lag truncation bias: low-order Cholesky-VARs do well at uncovering the transmission of monetary impulses in a truly Cholesky world.

GMM Long-Run Variance Estimation via VAR Averaging

This paper examines the heteroskedasticity and autocorrelation consistent (HAC) estimation of the long-run variance (LRV) matrix of a random vector process in a GMM estimation framework via vector autoregression (VAR) model averaging. By combining a VAR representation of GMM moments and VAR model averaging techniques, we present two new averaging-VAR-based LRV estimators: One constructs the LRV estimator based on the data-dependent weighted VAR residuals, and the other additionally combines with prewhitened kernel estimators in the spirit of Andrews and Monahan (1992). Our former proposal does not suffer from problems involving the choices of a truncation lag (bandwidth) or VAR lag length that are inherent in existing methods; the latter can be viewed as employing the VAR averaging strategy for prewhitening purposes. Theoretical justifications of the proposed methods rely on new consistency results for the averaging least-squares estimation of VAR coefficients and for the proposed LRV estimators. In extensive simulation experiments, our new averaging-VAR-based LRV estimators perform well in finite samples by offering more inference accuracy relative to existing competing estimators, with the improvement particularly prominent when combining VAR averaging with prewhitened kernel estimators. We empirically test the forward rate unbiasedness hypothesis to illustrate the utility of our methodology.

Lag truncation and the local asymptotic distribution of the ADF test for a unit root

The issue of lag selection in ADF unit root testing is important, even asymptotically, for if the number of lags is not allowed to increase at a certain rate the test might not be correctly sized. However, size control is not the only concern. Indeed, simulations have repeatedly shown how increasing lag lengths tend to be associated with reductions in power, thus adding to the well-known low power problem when the alternative is local to the unit root. But while the simulation evidence is plentiful, there is as of yet almost no asymptotic results that can be used to ascertain whether lag length has any effect on the local asymptotic power of the ADF test. The purpose of the present paper is to fill this gap in the literature.

A fixed-bandwidth view of the pre-asymptotic inference for kernel smoothing with time series data

This paper develops robust testing procedures for nonparametric kernel methods in the presence of temporal dependence of unknown forms. Based on the …xed-bandwidth asymptotic variance and the pre-asymptotic variance, we propose a heteroskedasticity and autocorrelation robust (HAR) variance estimator that achieves double robustness — it is asymptotically valid regardless of whether the temporal dependence is present or not, and whether the kernel smoothing bandwidth is held constant or allowed to decay with the sample size. Using the HAR variance estimator, we construct the studentized test statistic and examine its asymptotic properties under both the …xed-smoothing and increasing-smoothing asymptotics. The …xed-smoothing approximation and the associated convenient t-approximation achieve extra robustness — it is asymptotically valid regardless of whether the truncation lag parameter governing the covariance weighting grows at the same rate as or a slower rate than the sample size. Finally, we suggest a simulation-based calibration approach to choose smoothing parameters that optimize testing oriented criteria. Simulation shows that the proposed procedures work very well in …nite samples.

Measuring nonfundamentalness for structural VARs

As nonfundamental vector moving averages do not have causal VAR representations, standard structural VAR methods are deemed inappropriate for recovering the economic shocks of general equilibrium models with nonfundamental reduced forms. In the previous literature it has been pointed out that, despite nonfundamentalness, structural VARs may still be good approximating models. I characterize nonfundamentalness as bias depending on the zeros of moving average filters. However, measuring the nonfundamental bias is not trivial because of the simultaneous occurrence of lag truncation bias. I propose a method to disentangle the bias based on population spectral density and derive a measure for the nonfundamental bias in population. In the application, I find that the SVAR exercises of Sims (2012) are accurate because the nonfundamental bias is mild.

White noise testing and model diagnostic checking for functional time series

Abstract This paper is concerned with white noise testing and model diagnostic checking for stationary functional time series. To test for the functional white noise null hypothesis, we propose a Cramer–von Mises type test based on the functional periodogram introduced by Panaretos and Tavakoli (2013a). Using the Hilbert space approach, we derive the asymptotic distribution of the test statistic under suitable assumptions. A new block bootstrap procedure is introduced to obtain the critical values from the non-pivotal limiting distribution. Compared to existing methods, our approach is robust to the dependence within white noise and it does not involve the choices of functional principal components and lag truncation number. We employ the proposed method to check the adequacy of functional linear models and functional autoregressive models of order one by testing the uncorrelatedness of the residuals. Monte Carlo simulations are provided to demonstrate the empirical advantages of the proposed method over existing alternatives. Our method is illustrated via an application to cumulative intradaily returns.

On the Application of Fast Fractional Differencing in Modeling Long Memory of Conditional Variance

In econometrics, long memory models for variance modeling like FIGARCH or FIAPARCH are characterized by a Fractional Differencing term. In order to estimate and apply these models, the infinite MacLaurin expansion of the differencing term has to be truncated at a certain level. We transfer the recently introduced fast fractional differencing that utilizes fast Fourier transforms (FFT) to long memory conditional variance models and show that this FFT approach offers immense speed-ups. This allows to further increase the truncation lag while ensuring a feasible computation time. We demonstrate how calculation times of parameter estimations of these models benefit from this new approach, relative to sample length and truncation lag. In this simulation study, allowing for higher truncation lags implies better and more precise results in parameter estimations. In order to emphasize the importance for practitioners and research in risk management, we carry out different time consuming rolling-window analyses for WTI and Brent crude oil returns and show that total computation times can be reduced by a factor 20 to 30 for FIGARCH. The speed-ups for FIAPARCH are found to be significantly higher. The FFT approach offers a computational advantage to all ARCH(infinity)-representations of widely-used long memory models like FIGARCH, FIAPARCH, HYGARCH, and FIEGARCH, especially for large data sets which are common in high frequency analyses.

Lag length specification in Engle-Granger cointegration test: a modified Koyck mean lag approach based on partial correlation

The Engle-Granger cointegration test is highly sensitive to the choice of lag length and the poor performance of conventional lag selection criteria such as standard information criteria in selecting appropriate optimal lag length for the implementation of the Engle-Granger cointegration test is well-established in the statistical literature. Testing for cointegration within the framework of the residual-based Engle-Granger cointegration methodology is the same as testing for the stationarity of the residual series via the augmented Dickey-Fuller test which is well known to be sensitive to the choice of lag length. Given an array of candidate optimal lag lengths that may be suggested by different standard information criteria, the applied researchers are faced with the problem of deciding the best optimal lag among the candidate optimal lag lengths suggested by different standard information criteria, which are often either underestimated or overestimated. In an attempt to address this well-known major pitfall of standard information criteria, this paper introduces a new lag selection criterion called a modified Koyck mean lag approach based on partial correlation criterion for the selection of optimal lag length for the residual-based Engle-Granger cointegration test. Based on empirical findings, it was observed that in some instances over-specification of lag length can bias the Engle-Granger cointegration test towards the rejection of a true cointegration relationship and non-rejection of a spurious cointegration relationship. Using real-life data, we present an empirical illustration which demonstrates that our proposed criterion outperformed the standard information criteria in selecting appropriate optimal truncation lag for the implementation of the Engle-Granger cointegration test using both augmented Dickey-Fuller and generalized least squares Dickey-Fuller tests.

Does income distribution matter for development? Trends in labour share of income and their economic impacts in developing countries

Does income distribution matter for development? Trends in labour share of income and their economic impacts in developing countries

Let’s fix it: Fixed-[formula omitted] asymptotics versus small-[formula omitted] asymptotics in heteroskedasticity and autocorrelation robust inference

In the presence of heteroscedasticity and autocorrelation of unknown forms, the covariance matrix of the parameter estimator is often estimated using a nonparametric kernel method that involves a lag truncation parameter. Depending on whether this lag truncation parameter is speci…ed to grow at a slower rate than or the same rate as the sample size, we obtain two types of asymptotic approximations: the small-b asymptotics and the …xed-b asymptotics. Using techniques for probability distribution approximation and high order expansions, this paper shows that the …xed-b asymptotic approximation provides a higher order re…nement to the …rst order small-b asymptotics. This result provides a theoretical justi…cation on the use of the …xed-b asymptotics in empirical applications. On the basis of the …xed-b asymptotics and higher order small-b asymptotics, the paper introduces a new and easy-to-use asymptotic F test that employs a …nite sample corrected Wald statistic and uses an F-distribution as the reference distribution. Finally, the paper develops a bandwidth selection rule that is testing-optimal in that the bandwidth minimizes the type II error of the asymptotic F test while controlling for its type I error. Monte Carlo simulations show that the asymptoticF test with the testing-optimal bandwidth works very well in …nite samples.

Un-Truncating VARs

Macroeconomic research often relies on structural vector autoregressions to uncover empirical regularities. Critics argue the method goes awry due to lag truncation: short lag-lengths imply a poor approximation to DSGE-models. Empirically, short lag-length is deemed necessary as increased parametrization induces excessive uncertainty. The paper shows that this argument is incomplete. Longer lag-length simultaneously reduces misspecification, which in turn reduces variance. For data generated by frontier DSGE-models long-lag VARs are feasible, reduce bias and variance, and have better coverage. Thus, contrary to conventional wisdom, the trivial solution to the critique actually works.

A New Criterion for Lag-Length Selection in Unit Root Tests

This paper examines lag selection problem in unit root tests which has become a major specification problem in empirical analysis of non-stationary time series data. It is known that the implementation of unit root tests requires the choice of optimal truncation lag for good power proper ties and it is equally unrealistic to assume that the true optimal truncation lag is known a prior to the practitioners and other applied researchers. Consequently, these users rely largely on the use of standard information criteria for selection of truncation lag in unit root tests. A number of previous studies have shown that these criteria have problem of over-specification of truncation lag-length leading to the well-known low power problem that is commonly associated with most unit root tests in the literature. This paper focuses on the problem of over-specification of truncation lag-length within the context of augmented Dickey-Fuller (ADF) and generalized least squares Dickey-Fuller (DF-GLS)unit root tests. In an attempt to address this lag selection problem, we propose a new criterion for the selection of truncation lag in unit root tests based on Koyck distributed lag model and we show that this new criterion avoids the problem of over-specification of truncationlag-length that is commonly associated with standard information criteria.

A Strategy to Reduce the Count of Moment Conditions in Panel Data GMM

The problem of instrument proliferation and its consequences (overfitting of endogenous variables, bias of estimates, weakening of Sargan/Hansen test) are well known. The literature provides little guidance on how many instruments is too many. It is common practice to report the instrument count and to test the sensitivity of results to the use of more or fewer instruments. Strategies to alleviate the instrument proliferation problem are the lag-depth truncation and/or the collapse of the instrument set (the latter being an horizontal squeezing of the instrument matrix). However, such strategies involve either a certain degree of arbitrariness (based on the ability and the experience of the researcher) or of trust in the restrictions implicitly imposed (and hence untestable) on the instrument matrix. The aim of the paper is to introduce a new strategy to reduce the instrument count. The technique we propose is statistically founded and purely datadriven and, as such, it can be considered a sort of benchmark solution to the problem of instrument proliferation. We apply the principal component analysis (PCA) on the instrument matrix and exploit the PCA scores as the instrument set for the panel generalized method-of-moments (GMM)estimation. Through extensive Monte Carlo simulations, under alternative characteristics of persistence of the endogenous variables, we compare the performance of the Difference GMM, Level and System GMM estimators when lag truncation, collapsing and our principal component-based IV reduction (PCIVR henceforth) are applied to the instrument set. The same comparison has been carried out with two empirical applications on real data: the first replicates the estimates of Blundell and Bond [1998]; the second exploits a new and large panel data-set in order to assess the role of tangible and intangible capital on productivity. Results show that PCIVR is a promising strategy of instrument reduction.

A strategy to reduce the count of moment conditions in panel data GMM

The problem of instrument proliferation and its consequences (overfitting of endogenous variables, bias of estimates, weakening of Sargan/Hansen test) are well known. The literature provides little guidance on how many instruments is too many. It is common practice to report the instrument count and to test the sensitivity of results to the use of more or fewer instruments. Strategies to alleviate the instrument proliferation problem are the lag-depth truncation and/or the collapse of the instrument set (the latter being an horizontal squeezing of the instrument matrix). However, such strategies involve either a certain degree of arbitrariness (based on the ability and the experience of the researcher) or of trust in the restrictions implicitly imposed (and hence untestable) on the instrument matrix. The aim of the paper is to introduce a new strategy to reduce the instrument count. The technique we propose is statistically founded and purely datadriven and, as such, it can be considered a sort of benchmark solution to the problem of instrument proliferation. We apply the principal component analysis (PCA) on the instrument matrix and exploit the PCA scores as the instrument set for the panel generalized method-of-moments (GMM) estimation. Through extensive Monte Carlo simulations, under alternative characteristics of persistence of the endogenous variables, we compare the performance of the Difference GMM, Level and System GMM estimators when lag truncation, collapsing and our principal component-based IV reduction (PCIVR henceforth) are applied to the instrument set. The same comparison has been carried out with two empirical applications on real data: the first replicates the estimates of Blundell and Bond [1998]; the second exploits a new and large panel data-set in order to assess the role of tangible and intangible capital on productivity. Results show that PCIVR is a promising strategy of instrument reduction.

ON AUGMENTED HEGY TESTS FOR SEASONAL UNIT ROOTS

In this paper we extend the large-sample results provided for the augmented Dickey–Fuller test by Said and Dickey (1984, Biometrika 71, 599–607) and Chang and Park (2002, Econometric Reviews 21, 431–447) to the case of the augmented seasonal unit root tests of Hylleberg, Engle, Granger, and Yoo (1990, Journal of Econometrics 44, 215–238), inter alia. Our analysis is performed under the same conditions on the innovations as in Chang and Park (2002), thereby allowing for general linear processes driven by (possibly conditionally heteroskedastic) martingale difference innovations. We show that the limiting null distributions of the t-statistics for unit roots at the zero and Nyquist frequencies and joint F-type statistics are pivotal, whereas those of the t-statistics at the harmonic seasonal frequencies depend on nuisance parameters that derive from the lag parameters characterizing the linear process. Moreover, the rates on the lag truncation required for these results to hold are shown to coincide with the corresponding rates given in Chang and Park (2002); in particular, an o(T1/2) rate is shown to be sufficient.

A modified LLC panel unit root test of the PPP hypothesis

In a recent study, Westerlund (Empir Econ 37:517–531, 2009) shows that the performance of the popular LLC (Levin et al., J Econ 108:1–24, 2002) panel unit root test depends critically on the choice of lag truncation used when correcting for serial correlation, and that it is only when this parameter is set as a function of time that the power raises above size. The purpose of the current paper is to propose a modified test that does not suffer from this drawback. The new test is not only simpler to compute but also superior in terms of small-sample performance, which is illustrated using an example purchasing power parity for less developed countries.

A Strategy to Reduce the Count of Moment Conditions in Panel Data GMM

The problem of instrument proliferation and its consequences (overfitting of endogenous variables, bias of estimates, weakening of Sargan/Hansen test) are well known. The literature provides little guidance on how many instruments is too many. It is common practice to report the instrument count and to test the sensitivity of results to the use of more or fewer instruments. Strategies to alleviate the instrument proliferation problem are the lag-depth truncation and/or the collapse of the instrument set (the latter being an horizontal squeezing of the instrument matrix). However, such strategies involve either a certain degree of arbitrariness (based on the ability and the experience of the researcher) or of trust in the restrictions implicitly imposed (and hence untestable) on the instrument matrix. The aim of the paper is to introduce a new strategy to reduce the instrument count. The technique we propose is statistically founded and purely datadriven and, as such, it can be considered a sort of benchmark solution to the problem of instrument proliferation. We apply the principal component analysis (PCA) on the instrument matrix and exploit the PCA scores as the instrument set for the panel generalized method-of-moments (GMM) estimation. Through extensive Monte Carlo simulations, under alternative characteristics of persistence of the endogenous variables, we compare the performance of the Difference GMM, Level and System GMM estimators when lag truncation, collapsing and our principal component-based IV reduction (PCIVR henceforth) are applied to the instrument set. The same comparison has been carried out with two empirical applications on real data: the first replicates the estimates of Blundell and Bond [1998]; the second exploits a new and large panel data-set in order to assess the role of tangible and intangible capital on productivity. Results show that PCIVR is a promising strategy of instrument reduction.

Covariate unit root tests with good size and power

Abstract The selection of the truncation lag for covariate unit root tests is analyzed using Monte Carlo simulation. It is shown that standard information criteria such as the BIC or the AIC select lag orders that are too small and can result in tests with large size distortions. Modified information criteria select higher lag orders and can be used to construct covariate unit root tests with good size and power. A strategy to select the lead and lag orders that yields tests with better power is discussed. When unit root tests are constructed for the United States inflation rate using macroeconomic variables as covariates and the BIC to select the lag order, the unit root hypothesis can be rejected. On the other hand, when modified information criteria are used, the unit root hypothesis cannot be rejected.