Bootstrap Tests for Multivariate Event Studies

Statistical tests for multivariate event studies—exact or asymptotic—are derived based on multivariate normality. As it has been previously documented that the performances of these tests are not satisfactory, because stock returns are far from normally distributed (especially for daily returns), this paper proposes the use of bootstrap methods, which are free from any specific distributional assumption, to provide better approximations to the sampling distributions of test statistics in multivariate event studies. The Monte Carlo experiments based on real daily returns data show that the bootstrap tests outperform the traditional tests by having close rejection rates to the nominal significance levels. The traditional tests, in contrast, tend to reject the null hypotheses too often.     

Testing weak exogeneity in multiplicative error models

Empirical market microstructure literature widely employs the non-linear and non-Gaussian multiplicative error class of models (MEMs) in modelling the dynamics of trading duration and financial marks. It routinely maintains the weak exogeneity of duration vis-à-vis marks in estimations. However, microstructure theory states that trade duration, volume and transaction prices are simultaneously determined. We propose Lagrange-multiplier (LM) tests for weak exogeneity for the MEMs. Our LM tests are extensions of the weak exogeneity tests applicable to VAR or VECM models with Gaussian distributions. Empirical assessments show that (i) weak exogeneity is widely rejected by the data in the MEMs and (ii) the failure of weak exogeneity seriously biases parameter estimates. We hope our tests will be of interest in future empirical applications.     

An analysis of “exactly distributed” test statistics for multivariate event tests

In this paper we derive a rule that identifies when exact tests may be used in the context of the multivariate regression model. Our derivation extends distribution theory reported in Rao (1973) and leads to the specification of exact tests for several event study hypothesis forms of interest to accounting and finance researchers. For tests where the event parameter is constrained to be equal across firms, we show that an infinite set of exact tests is available, of which the well known portfoliot-test is a special case. We conduct simulations using data from the CRSP Daily Returns file, and find that several test statistics, including exactly distributed statistics derived using the multivariate regression model, significantly over-reject the hypotheses examined.     

A Monte Carlo investigation of the accuracy of multivariate CAPM tests

In a multivariate regression model relating individual returns to the market return, CAPM implies non-linear restrictions on the parameters. Several asymptotically valid tests of these restrictions have been suggested. The existing Monte Carlo evidence shows that some of these tests are unreliable for reasonable sample sizes, but does not indicate well which tests are reliable. This paper reports the results of an extensive Monte Carlo experiment. Shanken's CSR test and Jobson and Korkie's corrected likelihood ratio test are quite accurate in all cases we consider.     

Some curious power properties of long-horizon tests

Based on simulations and asymptotic results, I highlight three distinct properties of long-horizon predictive tests. (i) The asymptotic power of long-horizon tests increases only with the sample size relative to the forecasting horizon. Keeping this ratio fixed as the sample size increases does not lead to any power gains asymptotically. (ii) Although the power of long-horizon tests increases with the magnitude of the slope coefficient for alternatives close to the null hypothesis, there are no gains in power as the slope coefficient grows large. That is, the power curve is asymptotically horizontal when viewed as a function of the slope coefficient. (iii) For endogenous regressors—i.e., when the innovations to the regressand are contemporaneously correlated with the innovations to the regressor—traditional tests based on the standard long-run OLS estimator result in power curves that are sometimes decreasing in the magnitude of the slope coefficient.     

Which Extreme Values Are Really Extreme?

We define the extreme values of any random sample of size nfrom a distribution function F as the observations exceedinga threshold and following a type of generalized Pareto distribution(GPD) involving the tail index of F. The threshold is the orderstatistic that minimizes a Kolmogorov-Smirnov statistic betweenthe empirical distribution of the corresponding largest observationsand the corresponding GPD. To formalize the definition we usea semiparametric bootstrap to test the corresponding GPD approximation.Finally, we use our methodology to estimate the tail index andvalue at risk (VaR) of some financial indexes of major stockmarkets.     

Monotonicity in asset returns: New tests with applications to the term structure,the CAPM,and portfolio sorts

Many theories in finance imply monotonic patterns in expected returns and other financial variables. The liquidity preference hypothesis predicts higher expected returns for bonds with longer times to maturity; the Capital Asset Pricing Model (CAPM) implies higher expected returns for stocks with higher betas; and standard asset pricing models imply that the pricing kernel is declining in market returns. The full set of implications of monotonicity is generally not exploited in empirical work, however. This paper proposes new and simple ways to test for monotonicity in financial variables and compares the proposed tests with extant alternatives such as t-tests, Bonferroni bounds, and multivariate inequality tests through empirical applications and simulations.     

The BDS Test as a Test for the Adequacy of a GARCH(1,1) Specification: A Monte Carlo Study

In this study we examine the widely used Brock, Dechert, andScheinkman (BDS) test when applied to the logarithm of the squaredstandardized residuals of an estimated GARCH(1,1) model as atest for the adequacy of this specification. We review the conditionsderived by De Lima (1996; Econometric Reviews 15, 237–259)for the nuisance-parameter-free property to hold and addressthe issue of their necessity, using the flexible framework offeredby the GARCH(1,1) model in terms of moment, memory, and timeheterogeneity properties. By means of Monte Carlo simulations,we show that the BDS test statistic still approximates the standardnull distribution even for mildly explosive processes that violatethe majority of the conditions. Thus the test performs reasonablywell, its empirical size being rather close to the nominal one.As a by-product of this study, we also shed light on the relatedissue of the consistency of the QML estimators of the conditionalvariance parameters under various parameter configurations andalternative distributional assumptions on the innovation process.     

A Markov switching analysis of contagion in the EMS

This paper applies the multivariate version of the Forbes and Rigobon (2002) contagion test, as proposed by Dungey et al. (2005a), to detect contagion effects in the Exchange Rate Mechanism (ERM) of the European Monetary System (EMS). Crisis and non-crisis observations are determined endogenously via a Markov-switching vector autoregression (MSVAR). We show that the MSVAR is suitable for this purpose, as it does particularly well in identifying the 11 realignments of the ERM. We examine whether Denmark’s rejection of the Maastricht Treaty and Italy’s competitiveness problems affected other EMS participants and find evidence for contagion.     

The effect of limited information and estimation risk on optimal portfolio diversification

This paper analyzes the optimal portfolio choice problem when security returns have a joint multivariate normal distribution with unknown parameters. For the case of limited, but sufficient (sample plus prior) information, we show that for a general family of conjugate priors, the optimal portfolio choice is obtained by the use of a mean-variance analysis that differs from traditional mean-variance analysis due to estimation risk. We also consider two illustrative cases of insufficient sample information and minimal prior information and show that in these cases it is asymptotically optimal for an investor to limit diversification to a subset of the securities. These theoretical results corroborate observed investor behavior in capital markets.     

Equality of Real Returns on Canadian and US Treasury Bills: A Fractional Cointegration Analysis

This paper empirically analyzes the long memory relationship between the real returns on Canadian and US Treasury bills. A fractional cointegration approach, instead of conventional integer integration (unit root) and cointegration approaches, is used in analyzing the relationship. The advantage of fractionally integrated models is that they allow a smooth transition from a stationary process to a unit-root process. Furthermore, such models embody unit-root models as a special case. The models are therefore more general and appropriate for empirical analysis. By using fractionally integrated models, one also resolves the problems of an inconsistency in test results associated with using unit root and cointegration approaches. Briefly, it is found that the real returns on Canadian and US Treasury bills are fractionally integrated and the order of integration is significantly less than unity. Furthermore, the difference between the real returns follows a stationary process. This indicates that the Canadian and the US capital markets as well as product markets are well integrated. Furthermore, the domestic monetary authorities will not be able to influence the domestic real interest rate independent of the other market in the long-run.     

Aggregation of Nonparametric Estimators for Volatility Matrix

An aggregated method of nonparametric estimators based on time-domainand state-domain estimators is proposed and studied. To attenuatethe curse of dimensionality, we propose a factor modeling strategy.We first investigate the asymptotic behavior of nonparametricestimators of the volatility matrix in the time domain and inthe state domain. Asymptotic normality is separately establishedfor nonparametric estimators in the time domain and state domain.These two estimators are asymptotically independent. Hence,they can be combined, through a dynamic weighting scheme, toimprove the efficiency of volatility matrix estimation. Theoptimal dynamic weights are derived, and it is shown that theaggregated estimator uniformly dominates volatility matrix estimatorsusing time-domain or state-domain smoothing alone. A simulationstudy, based on an essentially affine model for the term structure,is conducted, and it demonstrates convincingly that the newlyproposed procedure outperforms both time- and state-domain estimators.Empirical studies further endorse the advantages of our aggregatedmethod.     

The monetary model in the presence of I(2) components: long-run relationships, short-run dynamics and forecasting of the Greek drachma

This paper re-examines the long-run properties of the monetary exchange rate model using data for the drachma–dollar and drachma–mark exchange rates under the hypothesis that the system contains variables that are I(2). Using the recent I(2) test by Paruolo (On the determination of integration indices in I(2) systems. J. Economet. 72 (1996) 313–356) to examine the presence of I(2) and I(1) components in a multivariate context we find that the system contains two I(2) variables in both cases and this finding is reconfirmed by the estimated roots of the companion matrix (Do purchasing power parity and uncovered interest rate parity hold in the long-run? An example of likelihood inference in a multivariate time-series model. Juselius, J. Economet. 69 (1995) 211–240). The I(2) component led to the transformation of the estimated model by imposing long-run but not short-run proportionality between domestic and foreign money. Two statistically significant cointegrating vectors were found and, by imposing linear restrictions on each vector as suggested by Johansen and Juselius (Identification of the long-run and the short-run structure: an applicaion to the ISLM model. J. Economet. 63 (1994) 7–36) and Johansen (Identifying restrictions of linear equations with applications to simultaneous equations and cointegration. J. Economet. 69 (1995b) 111–132), the order and rank conditions for identification are satisfied, but the test for overidentifying restrictions was not significant only for the case of the drachma/mark rate. The main findings suggest that we reject the forward-looking version of the monetary model for the drachma/dollar case but not when the drachma/mark rate is used, a result that is attributed to the monetary and exchange rate policy followed by the Greek authorities since Greece's joining of the European Union. Furthermore, we test for parameter stability using the tests developed by Hansen and Johansen (Recursive estimation in cointegrated VAR-models. Working paper (1993) University of Copenhagen) and it is shown that the dimension of the cointegration rank is sample independent while the estimated coefficients do not exhibit instabilities in recursive estimations. Finally, it is shown that the monetary model outperforms the random walk model in an out-of-sample forecasting contest.     

Periodic Stochastic Volatility and Fat Tails

This article provides a comprehensive analysis of the size andstatistical significance of the day of the week, month of theyear, and holiday effects in daily stock index returns and volatility.We employ data from the Dow Jones Industrial Average (DJIA),the S&P 500, the S&P MidCap 400, and the S&P SmallCap600 in order to test whether the seasonal patterns of mediumand small firms are similar to those of large firms. Using formalhypothesis tests based on bootstrapping, we demonstrate thatthere are more significant calendar effects in volatility thanin expected returns, especially for the two large cap indices.More importantly, we introduce the periodic stochastic volatility(PSV) model for characterizing the observed seasonal patternsof daily financial market volatility. We analyze the interactionbetween seasonal heteroskedasticity and fat tails by comparingthe performance of Gaussian PSV and fat-tailed PSVt specificationsto the plain vanilla SV and SVt benchmarks. Consistent withour model-free results, we find strong evidence of seasonalperiodicity in volatility, which essentially eliminates theneed for a fat-tailed conditional distribution, and is robustto the exclusion of the crash of 1987 outliers.     

On measuring synchronization of bulls and bears: The case of East Asia

This paper implements estimation and testing procedures for comovements of stock market “cycles” or “phases” in Asia. We extend the Harding and Pagan [Harding, D., Pagan, A.P., 2006. Synchronization of cycles. Journal of Econometrics 132 (1), 59–79] test for strong multivariate nonsynchronization (SMNS) between business cycles to a test that allows for an imperfect degree of multivariate synchronization between stock market cycles. Moreover, we propose a test for endogenously determining structural change in the bivariate and multivariate synchronization indices. Upon applying the technique to five Asian stock markets we find a significant increase in the cross country comovements of Asian bullish and bearish periods in 1997. A power study of the stability test suggests that the detected increase in comovement is more of a sudden nature (i.e. contagion or “Asian Flu”) instead of gradual (i.e. financial integration). It is furthermore argued that stock market cycles and their propensity toward (increased) synchronization contain useful information for both investors, policy makers and financial regulators.     

Tests for explosive financial bubbles in the presence of non-stationary volatility

This paper studies the impact of permanent volatility shifts in the innovation process on the performance of the test for explosive financial bubbles based on recursive right-tailed Dickey–Fuller-type unit root tests proposed by Phillips, Wu and Yu (2011). We show that, in this situation, their supremum-based test has a non-pivotal limit distribution under the unit root null, and can be quite severely over-sized, thereby giving rise to spurious indications of explosive behaviour. We investigate the performance of a wild bootstrap implementation of their test procedure for this problem, and show it is effective in controlling size, both asymptotically and in finite samples, yet does not sacrifice power relative to an (infeasible) size-adjusted version of their test, even when the shocks are homoskedastic. We also discuss an empirical application involving commodity price time series and find considerably less emphatic evidence for the presence of explosive bubbles in these data when using our proposed wild bootstrap implementation of the Phillips, Wu and Yu (2011) test.     

Nonnormalities and Tests of Asset Pricing Theories

The robustness of the multivariate test of Gibbons, Ross, and Shanken (1986) to nonnormalities in the residual covariance matrix is examined. After considering the relative performance of various tests of normality, simulation techniques are used to determine the effects of nonnormalities on the multivariate test. It is found that, where the sample nonnormalities are severe, the size and/or power of the test can be seriously misstated. However, it is also shown that these extreme sample values may overestimate the population parameters. Hence, we conclude that the multivariate test is reasonably robust with respect to typical levels of nonnormality.     

Statistical inference in multiperiod event studies

We examine several event-study test statistics that can be used to detect abnormal performance during amultiperiod event window. We demonstrate that one of the most commonly used test statistics does not, under the assumptions made, have the distribution claimed (standard normal), and thus tests using it will be biased. The magnitude of that bias is shown to increase with the length of the event window and can generally be expected to lead to excessive rejection of the null hypothesis. We also compare the relative power of alternative test statistics that are normally distributed and are straightforward to apply.     

Structural Breaks and Predictive Regression Models of Aggregate U.S. Stock Returns

In this article we examine the structural stability of predictiveregression models of U.S. quarterly aggregate real stock returnsover the postwar era. We consider predictive regressions modelsof S&P 500 and CRSP equal-weighted real stock returns basedon eight financial variables that display predictive abilityin the extant literature. We test for structural stability usingthe popular Andrews SupF statistic and the Bai subsample procedurein conjunction with the Hansen heteroskedastic fixed-regressorbootstrap. We also test for structural stability using the recentlydeveloped methodologies of Elliott and Müller, and Baiand Perron. We find strong evidence of structural breaks infive of eight bivariate predictive regression models of S&P500 returns and some evidence of structural breaks in the threeother models. There is less evidence of structural instabilityin bivariate predictive regression models of CRSP equal-weightedreturns, with four of eight models displaying some evidenceof structural breaks. We also obtain evidence of structuralinstability in a multivariate predictive regression model ofS&P 500 returns. When we estimate the predictive regressionmodels over the different regimes defined by structural breaks,we find that the predictive ability of financial variables canvary markedly over time.     

The long memory of the forward premium during the 1920s’ float: evidence from the European foreign exchange market

This paper investigates the fractional dynamics of the foreign exchange forward premium during the floating period of the 1920s. We apply weekly exchange rates of the currencies from Belgium, France, Germany, Holland, Italy and the USA against the British pound from February 1921 to May 1925 and employ two different definitions of the forward premium. The German data are for the period ranging from February 1921 to December 1922. This period includes the German hyperinflation era. The empirical investigation is conducted by means of two different fractional integration methods: the Geweke and Porter-Hudak and the Robinson tests. The results provide some evidence of long memory, mostly in the case of Belgium, Holland and Italy. Many of the forward premiums during the 1920s may have become non-stationary as markets began to anticipate the UK's return to gold at its pre-war parity. In the case of Germany, it may have been due to market failure. The varying results presented could be due to the wide differences in the microeconomic and macroeconomic fundamentals and political setups of the countries during the 1920s.