Run Variance Research Articles

Testing for Equal Average Forecast Accuracy in Possibly Unstable Environments *

We consider the issue of testing the null of equal average forecast accuracy in a model where the forecast error loss differential series has a potentially non-constant mean function over time. We show that when time variation is present in the loss differential mean, the standard Diebold and Mariano (1995) test, which was proposed for evaluating forecasts in a stable environment, has an asymptotic size of zero, and, whilst consistent, can have reduced local power. This arises due to inconsistent estimation of the implicit long run variance estimator, which diverges under a time varying mean. We suggest a modified statistic that replaces the standard long run variance estimator based on full-sample demeaning of the loss differential series with one based on nonparametric local demeaning. The new long run variance estimator is consistent under both the null and alternative when the mean function is time varying or constant, and in both cases, the modified test recovers the asymptotic size and power properties associated with the original test in the constant mean case. The modified test therefore provides a robust method for testing the equal average forecast accuracy null, allowing for instability in the loss differential mean. The benefits of our test are demonstrated via Monte Carlo simulation and two empirical applications.

High-dimensional IV cointegration estimation and inference

A semiparametric triangular systems approach shows how multicointegrating linkages occur naturally in an I(1) cointegrated regression model when the long run error variance matrix in the system is singular. Under such singularity, cointegrated I(1) systems embody a multicointegrated structure that makes them useful in many empirical settings. Earlier work shows that such systems may be analyzed and estimated without appealing to the associated I(2) system but with suboptimal convergence rates and potential asymptotic bias. The present paper develops a robust approach to estimation and inference of such systems using high dimensional IV methods that have appealing asymptotic properties like those known to apply in the optimal estimation of cointegrated systems (Phillips, 1991). The approach uses an extended version of high-dimensional trend IV (Phillips, 2006, 2014) estimation with deterministic orthonormal instruments. The methods and derivations involve new results on high-dimensional IV techniques and matrix normalization in the limit theory that are of independent interest. Wald tests of general linear restrictions are constructed using a fixed-b long run variance estimator that leads to robust pivotal HAR inference in both cointegrated and multicointegrated cases. Simulations show good properties of the estimation and inferential procedures in finite samples. An empirical illustration to housing stocks, starts and completions is provided.

Some impossibility results for inference with cluster dependence with large clusters

This paper focuses on a setting with observations having a cluster dependence structure and presents two main impossibility results. First, we show that when there is only one large cluster, i.e., the researcher does not have any knowledge on the dependence structure of the observations, it is not possible to consistently discriminate the mean. When within-cluster observations satisfy the uniform central limit theorem, we also show that a sufficient condition for consistent n-discrimination of the mean is that we have at least two large clusters. This result shows some limitations for inference when we lack information on the dependence structure of observations. Our second result provides a necessary and sufficient condition for the cluster structure that the long run variance is consistently estimable. Our result implies that when there is at least one large cluster, the long run variance is not consistently estimable.

Asymptotic F test in regressions with observations collected at high frequency over long span

This paper proposes tests of linear hypotheses when the variables may be continuous-time processes with observations collected at a high sampling frequency over a long span. Utilizing series long run variance (LRV) estimation in place of the traditional kernel LRV estimation, we develop easy-to-implement and more accurate F tests in both stationary and nonstationary environments. The nonstationary environment accommodates exogenous regressors that are general semimartingales. Endogenous regressors are allowed in a nonstationary environment similar to cointegration models in the usual discrete-time setting. The F tests can be implemented in exactly the same way as in the discrete-time setting. The F tests are, therefore, robust to the continuous-time or discrete-time nature of the data. Simulations demonstrate the improved size accuracy and competitive power of the F tests relative to existing continuous-time testing procedures and their improved versions. The F tests are of practical interest as recent work by Chang et al. (2021) demonstrates that traditional inference methods can become invalid and produce spurious results when continuous-time processes are observed on finer grids over a long span.

Inference in time series models using smoothed-clustered standard errors

This paper proposes a long run variance estimator for conducting inference in time series regression models that combines the nonparametric approach with a cluster approach. The basic idea is to divide the time periods into non-overlapping clusters. The long run variance estimator is constructed by first aggregating within clusters and then kernel smoothing across clusters or applying the nonparametric series method to the clusters with Type II discrete cosine transform. We develop an asymptotic theory for test statistics based on these “smoothed-clustered” long run variance estimators. We derive asymptotic results holding the number of clusters fixed and also treating the number of clusters as increasing with the sample size. For the kernel smoothing approach, these two asymptotic limits are different whereas for the cosine series approach, the two limits are the same. When clustering before kernel smoothing, we find that the “fixed-number-of-clusters” asymptotic approximation works well whether the number of clusters is small or large. Finite sample simulations suggest that the naive i.i.d. bootstrap mimics the fixed-number-of-clusters critical values. The simulations also suggest that clustering before kernel smoothing can reduce over-rejections caused by strong serial correlation at a cost of power. When there is a natural way of clustering, clustering can reduce over-rejection problems and achieve small gains in power for the kernel approach. In contrast, the cosine series approach does not benefit from clustering.

Testing-optimal kernel choice in HAR inference

The paper investigates the optimal kernel choice in heteroskedasticity and autocorrelation robust tests based on the fixed-b asymptotics. In parallel with the optimality of the quadratic spectral kernel under the asymptotic mean squared error criterion of the point estimator of the long run variance as considered in Andrews (1991) , we show that the optimality of the quadratic spectral kernel continues to hold under the testing-oriented criterion of Sun, Philips and Jin (2008) which takes a weighted average of the probabilities of type I and type II errors of the fixed-b asymptotic test.

Multiscale change point detection for dependent data

AbstractIn this article we study the theoretical properties of the simultaneous multiscale change point estimator (SMUCE) in piecewise‐constant signal models with dependent error processes. Empirical studies suggest that in this case the change point estimate is inconsistent, but it is not known if alternatives suggested in the literature for correlated data are consistent. We propose a modification of SMUCE scaling the basic statistic by the long run variance of the error process, which is estimated by a difference‐type variance estimator calculated from local means from different blocks. For this modification we prove model consistency for physical‐dependent error processes and illustrate the finite sample performance by means of a simulation study.

Appraisal of Shipping Trade Influence on Economic Growth in Nigeria

The study examined appraisal of shipping trade influence on the economic growth in Nigeria from 1981-2016 by the use of secondary data collected from the Central Bank of Nigeria. The main objectives are to examine the trend of shipping trade in Nigeria as well as determine the influence of economic growth on gross domestic product (GDP). The study employed both descriptive and influential tools. The study adopted cointegration regression method for the analysis of each of the variables (shipping trade, external reserves and external debts). The results show that there is a statistically significant relationship between GDP and external reserves with p-value 0.0190. Also the result revealed that there is a statistically significant relationship between GDP and shipping trade with p-value 0.000. However, shipping trade and external reserves contributed positively at 1% and 5% level of significance respectively while external debts impacted negatively to GDP at 5% level of significance with a long run variance of cointegration regression. Therefore, the study suggests that government should do all it can to curtail foreign dominance and repatriation of profit for a more sustainable economy as well as upgrade port facilities, prevent unnecessary delays and encourage exportable goods for maximum deployment of ships. Keywords : External debts, External reserve, GDP and Shipping trade. DOI : 10.7176/CER/12-1-04 Publication date: January 31 st 2020

Identifying Global and National Output and Fiscal Policy Shocks Using a GVAR

The paper contributes to the growing global VAR (GVAR) literature by showing how global and national shocks can be identified within a GVAR framework. The usefulness of the proposed approach is illustrated in an application to the analysis of the interactions between public debt and real output growth in a multicountry setting, and the results are compared to those obtained from standard single country VAR analysis. We find that on average (across countries) global shocks explain about one third of the long-horizon forecast error variance of output growth, and about one fifth of the long run variance of the rate of change of debt-to-GDP. Evidence on the degree of cross-sectional dependence in these variables and their innovations are exploited to identify the global shocks, and priors are used to identify the national shocks within a Bayesian framework. It is found that posterior median debt elasticity with respect to output is much larger when the rise in output is due to a fiscal policy shock, as compared to when the rise in output is due to a positive technology shock. The cross country average of the median debt elasticity is 1.58 when the rise in output is due to a fiscal expansion as compared to 0.75 when the rise in output follows from a favorable output shock.

Identifying Global and National Output and Fiscal Policy Shocks Using a Gvar

The paper contributes to the growing global VAR (GVAR) literature by showing how global and national shocks can be identified within a GVAR framework. The usefulness of the proposed approach is illustrated in an application to the analysis of the interactions between public debt and real output growth in a multicountry setting, and the results are compared to those obtained from standard single country VAR analysis. We find that on average (across countries) global shocks explain about one third of the long-horizon forecast error variance of output growth, and about one fifth of the long run variance of the rate of change of debt-to-GDP. Evidence on the degree of cross-sectional dependence in these variables and their innovations are exploited to identify the global shocks, and priors are used to identify the national shocks within a Bayesian framework. It is found that posterior median debt elasticity with respect to output is much larger when the rise in output is due to a fiscal policy shock, as compared to when the rise in output is due to a positive technology shock. The cross country average of the median debt elasticity is 1.58 when the rise in output is due to a fiscal expansion as compared to 0.75 when the rise in output follows from a favorable output shock.

Are shocks to ecological footprint in OECD countries permanent or temporary?

The main goal of this study is to check the unit root properties of the ecological footprint and its six subcomponents—cropland footprint, grazing land footprint, fishing grounds footprint, forest land footprint, built-up land footprint, and carbon footprint—in 25 OECD (The Organisation for Economic Co-operation and Development) countries during the period 1961–2013. This study uses a new stationarity test with sharp and smooth breaks in the panel setting, developed by Bahmani-Oskooee et al. (2014). Empirical findings demonstrate that the fishing grounds footprint does not have mean-reverting behavior while the remaining ecological indicators display stationary properties under the assumption of both homogenous and heterogeneous long run variance. Therefore, if policymakers decide to decrease the size of the fishing grounds footprint, they will not face stiff resistance. In addition to the panel results, the univariate stationarity test results can also help policymakers in each country design effective emission reduction policies.

HAR Inference: Recommendations for Practice

ABSTRACTThe classic papers by Newey and West (1987) and Andrews (1991) spurred a large body of work on how to improve heteroscedasticity- and autocorrelation-robust (HAR) inference in time series regression. This literature finds that using a larger-than-usual truncation parameter to estimate the long-run variance, combined with Kiefer-Vogelsang (2002, 2005) fixed-b critical values, can substantially reduce size distortions, at only a modest cost in (size-adjusted) power. Empirical practice, however, has not kept up. This article therefore draws on the post-Newey West/Andrews literature to make concrete recommendations for HAR inference. We derive truncation parameter rules that choose a point on the size-power tradeoff to minimize a loss function. If Newey-West tests are used, we recommend the truncation parameter rule S = 1.3T1/2 and (nonstandard) fixed-b critical values. For tests of a single restriction, we find advantages to using the equal-weighted cosine (EWC) test, where the long run variance is estimated by projections onto Type II cosines, using ν = 0.4T2/3 cosine terms; for this test, fixed-b critical values are, conveniently, tν or F. We assess these rules using first an ARMA/GARCH Monte Carlo design, then a dynamic factor model design estimated using a 207 quarterly U.S. macroeconomic time series.

Finite sample performance of a long run variance estimator based on exactly (almost) unbiased autocovariance estimators

This paper proposes a bias reduced long run variance (LRV) estimator of a univariate time series with unknown mean that addresses well known finite sample bias problems. The LRV estimator is based on the (almost) exactly unbiased autocovariance estimator proposed by Vogelsang and Yang (2016). Whereas using fixed-b critical values is known to correct downward bias in LRV estimates generated by demeaning the data, the approach we take also corrects the classic Parzen bias that is not captured by the fixed-b approach. When applied to the tests of the null hypothesis of the mean in a simple location model, a simulation study shows that the proposed LRV estimator leads to tests with less over-rejections while maintaining power at least as high and often higher as the standard robust t test based on fixed-b critical values. These simulations suggest further theoretical analysis of the bias reduction approach is warranted.

Asymptotic [formula omitted] and [formula omitted] tests in an efficient GMM setting

This paper considers two-step efficient GMM estimation and inference where the weighting matrix and asymptotic variance matrix are based on the series long run variance estimator. We propose a simple and easy-to-implement modification to the trinity of test statistics in the two-step efficient GMM setting and show that the modified test statistics are all asymptotically F distributed under the so-called fixed-smoothing asymptotics. The modification is multiplicative and involves the J statistic for testing over-identifying restrictions. This leads to convenient asymptotic F tests whose critical values, i.e., the standard F critical values, are readily available from standard statistical tables and programming environments. For testing a single restriction with a one-sided alternative, an asymptotic t test theory using the standard t distribution as the reference distribution is also developed.

Fixed bandwidth asymptotics for the studentized mean of fractionally integrated processes

We consider inference for the mean of a general stationary process based on standardizing the sample mean by a frequency domain estimator of the long run variance. Here, the main novelty is that we consider alternative asymptotics in which the bandwidth is kept fixed. This does not yield a consistent estimator of the long run variance, but, for the weakly dependent case, the studentized sample mean has a Student-t limit distribution, which, for any given bandwidth, appears to be more precise than the traditional Gaussian limit. When data are fractionally integrated, the fixed bandwidth limit distribution of the studentized mean is not standard, and we derive critical values for various bandwidths. By a Monte Carlo experiment of finite sample performance we find that this asymptotic result provides a better approximation than other proposals like the test statistic based on the Memory Autocorrelation Consistent (MAC) estimator of the variance of the sample mean.

Exchange Rate, Economic Growth and Foreign Direct Investment in Emerging Asian Economies: Fresh Evidence from Long Run Estimation and Variance Decomposition Approach

This paper investigates the potential relationship between exchange rates, economic growth and foreign direct investment (FDI) for emerging Asian economies. We tested a long run relationship through the ARDL bounds testing approach and the robust regression function which incorporated financial development, capital, openness and trade balance as the additional variables of our model. The findings confirm that exchange rate depreciation affects the volume of FDI and promotes growth in the long run. Economic development and inflows are also associated with exchange rates. The causality test also suggested a long run interdependent link between these three macroeconomic factors. The results of the variance decomposition method suggest that the findings are robust and reliable.

6 1/2 Decades of Global Trade and Income: 'New Normal' or 'Back to Normal' after GTC and GFC?

Global merchandise trade expanded rapidly over the last 6 1/2 decades and its relationship with global income has seen ebbs and flows. This paper examines the shifts in this relationship using time series data over 1950-2014 and situates it in the current and longer term context. The conjunctural context comes from, among other things, the 'great trade collapse' (GTC) and the global financial crisis (GFC) in 2009, and developments since then. The longer term context comes from the relative role of 'globalization' and 'technology' shocks in accounting for the short and long run variance of global exports and income. The paper estimates trade and income elasticities using ADL models taking account of structural breaks, and impulse response functions from structural VARs. The estimated SVAR model provides a lens to ask whether global trade and income are in a 'new normal' or only 'back to (an old) normal' after the GTC and GFC.

61/2 Decades of Global Trade and Income: 'New Normal' or 'Back to Normal' after GTC and GFC?

Global merchandise trade expanded rapidly over the last 6½ decades and its relationship with global income has seen ebbs and flows. This paper examines the shifts in this relationship using time series data over 1950-2014 and situates it in the current and longer term context. The conjunctural context comes from, among other things, the “great trade collapse” (GTC) and the global financial crisis (GFC) in 2009, and developments since then. The longer term context comes from the relative role of “globalization” and “technology” shocks in accounting for the short and long run variance of global exports and income. The paper estimates trade and income elasticities using ADL models taking account of structural breaks, and impulse response functions from structural VARs. The estimated SVAR model provides a lens to ask whether global trade and income are in a “new normal’ or only “back to (an old) normal” after the GTC and GFC.

Integrated modified OLS estimation and fixed-[formula omitted] inference for cointegrating regressions

This paper is concerned with parameter estimation and inference in a cointegrating regression, where as usual endogenous regressors as well as serially correlated errors are considered. We propose a simple, new estimation method based on an augmented partial sum (integration) transformation of the regression model. The new estimator is labeled integrated modified ordinary least squares (IM-OLS). IM-OLS is similar in spirit to the fully modified OLS approach of Phillips and Hansen (1990) and also bears similarities to the dynamic OLS approach of Phillips and Loretan (1991), Saikkonen (1991) and Stock and Watson (1993), with the key difference that IM-OLS does not require estimation of long run variance matrices and avoids the need to choose tuning parameters (kernels, bandwidths, lags). Inference does require that a long run variance be scaled out, and we propose traditional and fixed-b methods for obtaining critical values for test statistics. The properties of IM-OLS are analyzed using asymptotic theory and finite sample simulations. IM-OLS performs well relative to other approaches in the literature.

Sieve inference on possibly misspecified semi-nonparametric time series models

This paper establishes the asymptotic normality of plug-in sieve M estimators of possibly irregular functionals of semi-nonparametric time series models. We show that, even when the sieve score process is not a martingale difference sequence, the asymptotic variance in the case of irregular functionals is the same as those for independent data. Using an orthonormal series long run variance estimator, we construct a “pre-asymptotic” Wald statistic and show that it is asymptotically F distributed. Simulations indicate that our “pre-asymptotic” Wald test with F critical values has more accurate size in finite samples than the conventional Wald test with chi-square critical values.