sup-Wald Test Research Articles

Forecasting crude oil volatility and stock volatility: New evidence from the quantile autoregressive model

This paper employs the quantile autoregressive (QAR) model to examine the forecasting relationship between stock volatility and crude oil volatility. We first utilize the sup-Wald test to evaluate Granger causality across various quantile levels, providing valuable information for forecasting. Our findings reveal that the causal effects between stock volatility and crude oil volatility differ considerably across different quantiles, with a V-shaped relationship evident at the quantile level. Results from out-of-sample forecasts indicate that the forecasting effect of oil volatility on stock volatility has both positive and negative impacts. In contrast, when using stock volatility to forecast crude oil volatility, predictability improves relative to the benchmark, particularly at more extreme quantiles. Further analysis highlights the necessity of forecast combinations to achieve an overall improvement in forecasting tasks.

Testing Granger non-causality in expectiles

This article aims to derive a consistent test of Granger causality at a given expectile. We also propose a sup-Wald test for jointly testing Granger causality at all expectiles that has the correct asymptotic size and power properties. Expectiles have the advantage of capturing similar information as quantiles, but they also have the merit of being much more straightforward to use than quantiles, since they are defined as least squares analog of quantiles. Studying Granger causality in expectiles is practically simpler and allows us to examine the causality at all levels of the conditional distribution. Moreover, testing Granger causality at all expectiles provides a sufficient condition for testing Granger causality in distribution. A Monte Carlo simulation study reveals that our tests have good finite-sample size and power properties for a variety of data-generating processes and different sample sizes. Finally, we provide two empirical applications to illustrate the usefulness of the proposed tests.

Factor-augmented forecasting regressions with threshold effects

Summary This paper introduces a factor-augmented forecasting regression model in the presence of threshold effects. We consider least squares estimation of the regression parameters and establish asymptotic theories for estimators of both slope coefficients and the threshold parameter. Prediction intervals are also constructed for factor-augmented forecasts. Moreover, we develop a likelihood ratio statistic for tests on the threshold parameter and a sup-Wald test statistic for tests on the presence of threshold effects, respectively. Simulation results show that the proposed estimation method and testing procedures work very well in finite samples. Finally, we demonstrate the usefulness of the proposed model through an application to forecasting stock market returns.

Buffered vector error-correction models: an application to the U.S. Treasury bond rates

Abstract This paper extends the buffered autoregressive model to the buffered vector error-correction model (VECM). Least squares estimation and a reduced-rank estimation are discussed, and the consistency of the estimators on the delay parameter and threshold parameters is derived. We also propose a supWald test for the presence of buffer-type threshold effect. Under the null hypothesis of no threshold, the supWald test statistic converges to a function of Gaussian process. A bootstrap method is proposed to obtain the p-value for the supWald test. We investigate the effectiveness of our methods by simulation studies. We apply our model to study the monthly Federal bond rates of United States. We find the evidences of buffering regimes and the asymmetric error-correction effect.

Testing for Changes in Forecasting Performance

We consider the issue of forecast failure (or breakdown) and propose methods to assess retrospectively whether a given forecasting model provides forecasts which show evidence of changes with respect to some loss function. We adapt the classical structural change tests to the forecast failure context. First, we recommend that all tests should be carried with a fixed scheme to have best power. This ensures a maximum difference between the fitted in and out-of-sample means of the losses and avoids contamination issues under the rolling and recursive schemes. With a fixed scheme, Giacomini and Rossi’s (GR) test is simply a Wald test for a one-time change in the mean of the total (the in-sample plus out-of-sample) losses at a known break date, say m, the value that separates the in and out-of-sample periods. To alleviate this problem, we consider a variety of tests: maximizing the GR test over values of m within a prespecified range; a Double sup-Wald (DSW) test which for each m performs a sup-Wald test for a change in the mean of the out-of-sample losses and takes the maximum of such tests over some range; we also propose to work directly with the total loss series to define the Total Loss sup-Wald and Total Loss UDmax (TLUD) tests. Using theoretical analyses and simulations, we show that with forecasting models potentially involving lagged dependent variables, the only tests having a monotonic power function for all data-generating processes considered are the DSW and TLUD tests, constructed with a fixed forecasting window scheme. Some explanations are provided and empirical applications illustrate the relevance of our findings in practice. Supplementary materials for this article are available online.

Inference for Factor-Augmented Forecasting Regressions Under Structural Instability

This paper introduces a factor-augmented forecasting regression model in the presence of threshold effects. We consider least squares estimation of the regression parameters, and establish asymptotic theories for estimators of both slope coefficients and the threshold parameter. Prediction intervals are also constructed for factor-augmented forecasts. Moreover, we develop a likelihood ratio statistic for tests on the threshold parameter and a sup-Wald test statistic for tests on the presence of threshold effects, respectively. Simulation results show that the proposed estimation method and testing procedures work very well in finite samples. Finally, we demonstrate the usefulness of the proposed model through applications to forecasting stock market returns and the annual growth rate of industrial production, respectively.

Change-in-mean tests in long-memory time series: a review of recent developments

It is well known that standard tests for a mean shift are invalid in long-range dependent time series. Therefore, several long-memory robust extensions of standard testing principles for a change-in-mean have been proposed in the literature. These can be divided into two groups: those that utilize consistent estimates of the long-run variance and self-normalized test statistics. Here, we review this literature and complement it by deriving a new long-memory robust version of the sup-Wald test. Apart from giving a systematic review, we conduct an extensive Monte Carlo study to compare the relative performance of these methods. Special attention is paid to the interaction of the test results with the estimation of the long-memory parameter. Furthermore, we show that the power of self-normalized test statistics can be improved considerably by using an estimator that is robust to mean shifts.

Threshold Cointegration and Nonlinear Causality test between Inflation Rate and Repo Rate

The current study investigated a cointegration and nonlinear causality relationships between inflation and repo rates of South Africa using the data spanning the period of January 2002 to March 2016. We used a threshold vector error correction model (TVECM) and nonlinear Granger frameworks causality to carry out the analysis. Preliminary analysis of data revealed the expected properties of the data such as nonlinearity, non-stationarity and co-movement of the variables. The two variables confirmed to be moving together in the long-run according to the observed supWald test statistic. Finally, the Diks-Panchenko nonlinear Causality test revealed a strong bidirectional nonlinear causal relationship between repo rate and inflation rate. The results imply that the use of repo rate to target the inflation rate during the target period did not address the financial problem in South Africa. Consequently, the study concluded that repo rate may not be a good measure to use for controlling inflation rates of South Africa.

Threshold Cointegration and Nonlinear Causality test between Inflation Rate and Repo Rate

The current study investigated a cointegration and nonlinear causality relationships between inflation and repo rates of South Africa using the data spanning the period of January 2002 to March 2016. We used a threshold vector error correction model (TVECM) and nonlinear Granger frameworks causality to carry out the analysis. Preliminary analysis of data revealed the expected properties of the data such as nonlinearity, non-stationarity and co-movement of the variables. The two variables confirmed to be moving together in the long-run according to the observed supWald test statistic. Finally, the Diks-Panchenko nonlinear Causality test revealed a strong bidirectional nonlinear causal relationship between repo rate and inflation rate. The results imply that the use of repo rate to target the inflation rate during the target period did not address the financial problem in South Africa. Consequently, the study concluded that repo rate may not be a good measure to use for controlling inflation rates of South Africa.

Common threshold in quantile regressions with an application to pricing for reputation

The paper develops a systematic estimation and inference procedure for quantile regression models where there may exist a common threshold effect across different quantile indices. We first propose a sup-Wald test for the existence of a threshold effect, and then study the asymptotic properties of the estimators in a threshold quantile regression model under the shrinking threshold effect framework. We consider several tests for the presence of a common threshold value across different quantile indices and obtain their limiting distributions. We apply our methodology to study the pricing strategy for reputation through the use of a data set from Taobao.com. In our economic model, an online seller maximizes the sum of the profit from current sales and the possible future gain from a targeted higher reputation level. We show that the model can predict a jump in optimal pricing behavior, which is considered as “reputation effect” in this paper. The use of threshold quantile regression model allows us to identify and explore the reputation effect and its heterogeneity in data. We find both reputation effects and common thresholds for a range of quantile indices in seller’s pricing strategy in our application.

Panel kink regression with an unknown threshold

In this paper, we extend the kink regression model with an unknown threshold in Hansen (2017) to the panel data framework, where the cross-sectional dimension (N) goes to infinity and the time period (T) is fixed. Following the literature of threshold regressions, we propose an estimator based on the within-group transformation. Under fixed threshold effect assumption, we establish that the slope and threshold estimators are jointly normally distributed with the same convergence rate OpN−1∕2 and a non-zero asymptotic covariance. We also suggest a sup-Wald test for the presence of kink effect, and derive its limiting distribution. A bootstrap procedure is proposed to obtain the bootstrap p-values to improve the finite sample performance of the test. Monte Carlo simulations show that the FE estimator and the sup-Wald test perform quite well in estimating the unknown parameters and testing for kink effect, respectively.

Tests for Linearity in Star Models: Supwald and Lm‐Type Tests

This article investigates approximation and supremum approaches for testing linearity in smooth transition autoregressive (STAR) models. We show that since the approximation of STAR models by Taylor series expansions may not accurately describe the specific transition dynamic when the process is away from the null, LM‐type tests may fail to detect the form of nonlinearity for which they are designed for. Investigating a supremum approach, the article provides the asymptotic distribution of a SupWald test that is obtained by taking the supremum of a Wald statistic over the Cartesian product of the spaces for the transition and threshold parameters. Simulated asymptotic critical values for the resulting tests are provided for a wide range of autoregressive orders and shown to differ across exponential and logistic STAR (ESTAR and LSTAR) models. Monte Carlo experiments show that SupWald tests for ESTAR and LSTAR models outperform LM‐type tests, compares well relative to the recently developed score‐based tests and each SupWald statistic performs the best against the true alternative for which it is formed. SupWald tests also provide results that are consistent with the findings from (independently) estimating and diagnostic testing of STAR models in real exchange rate data.

On the Usefulness or Lack Thereof of Optimality Criteria for Structural Change Tests

Elliott and Müller (2006) considered the problem of testing for general types of parameter variations, including infrequent breaks. They developed a framework that yields optimal tests, in the sense that they nearly attain some local Gaussian power envelop. The main ingredient in their setup is that the variance of the process generating the changes in the parameters must go to zero at a fast rate. They recommended the so-called qL̂L test, a partial sums type test based on the residuals obtained from the restricted model. We show that for breaks that are very small, its power is indeed higher than other tests, including the popular sup-Wald (SW) test. However, the differences are very minor. When the magnitude of change is moderate to large, the power of the test is very low in the context of a regression with lagged dependent variables or when a correction is applied to account for serial correlation in the errors. In many cases, the power goes to zero as the magnitude of change increases. The power of the SW test does not show this non-monotonicity and its power is far superior to the qL̂L test when the break is not very small. We claim that the optimality of the qL̂L test does not come from the properties of the test statistics but the criterion adopted, which is not useful to analyze structural change tests. Instead, we use fixed-break size asymptotic approximations to assess the relative efficiency or power of the two tests. When doing so, it is shown that the SW test indeed dominates the qL̂L test and, in many cases, the latter has zero relative asymptotic efficiency.

WALD TESTS FOR DETECTING MULTIPLE STRUCTURAL CHANGES IN PERSISTENCE

This paper considers the problem of testing for multiple structural changes in the persistence of a univariate time series. We propose sup-Wald tests of the null hypothesis that the process has an autoregressive unit root throughout the sample against the alternative hypothesis that the process alternates between stationary and unit root regimes. We derive the limit distributions of the tests under the null and establish their consistency under the relevant alternatives. We further show that the tests are inconsistent when directed against the incorrect alternative, thereby enabling identification of the nature of persistence in the initial regime. We also propose hybrid testing procedures that allow ruling out of stable stationary processes or ones that are subject to only stationary changes under the null, thereby aiding the researcher in interpreting a rejection as emanating from a switch between a unit root and stationary regime. The computation of the test statistics as well as asymptotic critical values is facilitated by the dynamic programming algorithm proposed in Perron and Qu (2006, Journal of Econometrics134, 373–399) which allows imposing within- and cross-regime restrictions on the parameters. Finally, we present Monte Carlo evidence to show that the proposed procedures perform well in finite samples relative to those available in the literature.

Quantile Regression on Quantile Ranges

Motivated by the fact that a linear specification in a quantile regression setting is unable to describe the non-linear relations among economic variables, well documented in the empirical econometrics literature, we formulate a threshold quantile regression model for one, known and unknown threshold value. Our modeling framework is more general than that of Hansen (2000), Galvao et. al. (2010) in that we consider covariates that do not exhibit threshold effects themselves, but are allowed to be affected by the covariate subject to regime-changes. We are also dealing with different inference problems. We derive the asymptotic properties of the model parameters as well as the threshold value(s) and develop inferencial procedures to identify heterogeneous effects of different covariate quantile ranges on different quantiles of the response. We conduct inference by developing a sup-Wald test that converges to a two-parameter Gaussian process. In addition, we derive the limiting distribution of the threshold value(s) for the single and multiple threshold effects settings under two asymptotic frameworks: one assuming fixed and another assuming shrinking shifts. Furthermore, we construct confidence intervals for the unknown threshold value via a likelihood-ratio-type test statistic and conduct a simulation study to investigate the coverage probability of the confidence intervals for different quantiles. Our asymptotic results complement those found in the existing literature on threshold regression models.

Evolving Perceptions of Central Bank Credibility: The European Central Bank Experience

What are the preferences of a central bank over inflation and output-gap stabilization objectives, and what is its preferred long-run inflation rate? While statements of priorities and goals are important, the credibility of such statements and the market perception of the policy reaction function of a central bank play a key role in determining economic outcomes. This point, early on described as the “credibility” of central bank policies, is a standard theoretical result with recent interpretation in the New Keynesian paradigms. It is also received wisdom among practitioners. The importance of the market perception of the central bank’s acceptable trade-offs between inflation and output goals as well its specific targets naturally leads to the question of how the market acquires this perception and whether and how it evolves over time. One view is that establishing an appropriate institutional structure is the key element in insulating the monetary authority from political pressure and thereby convincing markets that a central bank has a strong and unvarying aversion to inflation. A second, more dynamic, view focuses on the role that actual policy conduct plays in building the reputation of a central bank. These two different views have distinct implications for the relative importance of the institutional structure of a central bank, as compared to its conduct, for attaining and maintaining its credibility. A survey of the heads of central banks and prominent monetary economists reflects a belief that the credibility of a central bank is based more on its past actions than on institutional structures that afford it independence by insulating it from political concerns, although there is also a consensus that structurematters (Blinder 2000). Empirical research has found that institutional features related to central bank independence are associated with economic performance in cross sections of

Testing for structural breaks in factor loadings: An application to international business cycle

This paper proposes the implementation of the SupWald test of Andrews (1993) to detect structural breaks in the loadings of a static factor model. The procedure is illustrated by testing for structural breaks in the common factors of GDP growth series for a sample of advanced countries from 1950 until 2006.

Testing for Multiple Structural Changes in Cointegrated Regression Models

We consider testing for multiple structural changes in cointegrated systems and derive the limiting distribution of the sup-Wald test under mild conditions on the errors and regressors for a variety of testing problems. We show that even if the coefficients of the integrated regressors are held fixed but the intercept is allowed to change, the limit distributions are not the same as would prevail in a stationary framework. We also propose a sequential procedure that permits consistent estimation of the number of breaks present. We show via simulations that our tests maintain the correct size in finite samples and are much more powerful than the commonly used LM tests, which suffer from important problems of nonmonotonic power in the presence of serial correlation in the errors.

Causality in quantiles and dynamic stock return–volume relations

This paper investigates the causal relations between stock return and volume based on quantile regressions. We first define Granger non-causality in all quantiles and propose testing non-causality by a sup-Wald test. Such a test is consistent against any deviation from non-causality in distribution, as opposed to the existing tests that check only non-causality in certain moment. This test is readily extended to test non-causality in different quantile ranges. In the empirical studies of three major stock market indices, we find that the causal effects of volume on return are usually heterogeneous across quantiles and those of return on volume are more stable. In particular, the quantile causal effects of volume on return exhibit a spectrum of (symmetric) V-shape relations so that the dispersion of return distribution increases with lagged volume. This is an alternative evidence that volume has a positive effect on return volatility. Moreover, the inclusion of the squares of lagged returns in the model may weaken the quantile causal effects of volume on return but does not affect the causality per se.

Generic consistency of the break-point estimators under specification errors in a multiple-break model

Summary This paper considers the estimation of multiple-structural-break models under specification errors. A common example in economics is that the true model is measured in level, but a linear-log model is estimated. We show that, under specification errors, if there are more than one break points and if a single-break model is estimated, the estimated break point is consistent for one of the true break points. This consistency result applies to models with multiple regressors where some or all of the regressors are misspecified. Another important contribution of this paper is that we have constructed a Sup-Wald test whose limiting distribution is not affected by model misspecification. Using this robust test, we show that the break points can be estimated sequentially one at a time. Simulation evidence and an empirical application are provided.