The performance of unit root tests under level-dependent heteroskedasticity

Several financial variables exhibit level-dependent conditional heteroskedasticity. This may cause severe distortions in conventional unit root tests. Given the absence of theoretical results, we conduct Monte Carlo investigation to assess the performance of the standard Dickey–Fuller tests, a nonparametric alternative, and a heteroskedastic-robust extension of the Dickey–Fuller t-test. While these procedures have approximately correct size, we find strong distortions in the power of the standard Dickey–Fuller tests.     

Autocorrelation and heteroskedasticity in equations describing the demand for money during rapid inflations

Estimating econometric equations with linear autoregressive error terms is standard if the covariance matrix of these error terms are homoskedastic. However, if heteroskedasticity prevails, this heteroskedasticity should be taken into account in order to obtain efficient estimates. In this paper, the well-known maximum-likelihood estimation method due to Beach and MacKinnon (1978) is extended to the heteroskedastic case and then applied to equations describing the demand for money in advanced inflations. Since this method isnot a straightforward generalization of the Beach-MacKinnon procedure, it not necessarily leads to estimates which are to be preferred to those obtained under the assumption of homoskedasticity.     

Bootstrap LM tests for higher-order spatial effects in spatial linear regression models

This paper first extends the methodology of Yang (J Econom 185:33–59, 2015) to allow for non-normality and/or unknown heteroskedasticity in obtaining asymptotically refined critical values for the LM-type tests through bootstrap. Bootstrap refinements in critical values require the LM test statistics to be asymptotically pivotal under the null hypothesis, and for this we provide a set of general methods for constructing LM and robust LM tests. We then give detailed treatments for two general higher-order spatial linear regression models: namely the \(\mathtt{SARAR}(p,q)\) model and the \(\mathtt{MESS}(p,q)\) model, by providing a complete set of non-normality robust LM and bootstrap LM tests for higher-order spatial effects, and a complete set of LM and bootstrap LM tests robust against both unknown heteroskedasticity and non-normality. Monte Carlo experiments are run, and results show an excellent performance of the bootstrap LM-type tests.     

Specification and Goldfeld money demand function

Despite intensive investigation of the temporal stability of the Goldfield formulation of the money demand function, a clear consensus on its stability has yet to emerge. This paper builds a statistical case supporting the first difference of log-levels specification, as opposed to the more commonly used log-levels specification, of the Goldfeld equation and then examines the stability of both specifications. Formal stability tests proposed by Cooley and Prescott, Farley and Hinich, and Brown, Durbin, and Evans are employed; the out-of-sample predictive performance is examined as well. These tests strongly support the first difference specification over thelog-levels specification.     

Small sample properties of Canonical Cointegrating Regressions

Monte Carlo simulations are performed to examine small sample properties of Canonical Cointegrating Regressions (CCR). The first data generation process is designed to generate both cointegrated and non-cointegrated systems with normal disturbances. If the near-observational equivalence of the stationary and the integrated processes is not significant, both powers and empirical sizes of CCR tests are acceptable. The second data generation process is based on the error correction model. Cointegrated systems with various fat-tailed disturbances are generated and analyzed. The empirical sizes of CCR tests with studentt disturbances and GARCH disturbances are found to be reasonable under certain restrictions. The last data generation process is a generalized least squares (GLS) process that incorporates heteroskedasticity into the error correction model. Again, the empirical sizes of CCR tests are reasonable.     

Does Purchasing Power Parity hold? New evidence from wild-bootstrapped nonlinear unit root tests in the presence of heteroskedasticity

In spite of the extensive research which has already been undertaken, the issue as to whether Purchasing Power Parity (PPP) empirically holds, continues to be strongly debated. Existing studies have been criticized for their reliance on unit root tests which are deemed to suffer from certain weaknesses such as the size distortion bias arising from heteroskedasticity. In this paper, we provide new evidence on PPP based on a new methodology that overcomes this problem. We use the widely accepted KSS (Kapetanios et al., 2003) non-linear unit root tests which we, however, wild bootstrapped. Through Monte Carlo simulation, we demonstrate that the wild-bootstrapped KSS is robust to heteroskedasticity-induced size distortion problem. We apply this method to test PPP across 61 countries over the period 1994 to 2012 — a period characterized by a number of crises such as the Asian Financial Crisis, Russian Crisis, dotcom crisis, Global Financial Crises, among others, and therefore, intense heteroskedasticity. Our results provide strong evidence against PPP. This paper contributes to both the international financial economics and econometrics literatures.     

Comments on “Testing for nonlinear structure and chaos in economic time series”

This short paper is a comment on “Univariate tests for nonlinear structure” by Catherine Kyrtsou and Apostolos Serletis. We summarize their main results and discuss some of their conclusions concerning the role of outliers and noisy chaos. In particular, we include some new simulations to investigate whether economic time series may be characterized by low-dimensional noisy chaos.     

The Chinese Stock Market Does not React to the Japanese Market: Using Intraday Data to Analyse Return and Volatility Spillover Effects

In this paper, we use high‐frequency data to explore the effects of return and volatility spillover during periods in which trading hours in China and Japan overlap. Specifically, we utilize 5‐min returns to estimate fractionally integrated asymmetric power autoregressive conditional heteroskedasticity and fractionally integrated exponential generalized autoregressive conditional heteroskedasticity models, then use the models' standardized residuals to employ a cross‐correlation function approach that tests for the degree to which the Chinese and Japanese markets affect each other. Results indicate a unidirectional influence of the Chinese stock market on Japanese markets in terms of return. This result is likely attributable to restrictions on foreign investment in the Chinese market and the lack of diversified international portfolios among individual Chinese investors.     

The link between economic growth and growth volatility

This paper investigates the relationship between economic growth and growth volatility through simultaneous equations system. By employing the identification through heteroskedasticity method of Rigobon (Rev Econ Stat 85:777–792, 2003) and using a panel of 158 countries over the period 1960–2010, we find that output volatility is detrimental to economic growth, suggesting that stabilization policies to mitigate short-run economic fluctuations contribute to long-run economic growth. And economic growth accelerates output variability, supporting the feedback effects from growth to the volatility. The evidence is robust to a number of sensitivity tests.     

German stock market dynamics

This article estimates generalized ARCH (GARCH) models for German stock market indices returns, using weekly and monthly data, various GARCH specifications and (non)normal error densities, and a variety of diagnostic checks. German stock return series exhibit significant levels of second-order dependence. Our results clearly demonstrate that for both weekly as well as monthly return series the Student-t distribution is superior to the standard normal distribution. In particular, the estimated GARCH-t models appear to be reasonably successful in accounting for both observed leptokurtosis and conditional heteroskedasticity from German stock return movements.     

Heteroskedasticity testing through a comparison of Wald statistics

This paper shows that a test for heteroskedasticity within the context of classical linear regression can be based on the difference between Wald statistics in heteroskedasticity-robust and nonrobust forms. The test is asymptotically distributed under the null hypothesis of homoskedasticity as chi-squared with one degree of freedom. The power of the test is sensitive to the choice of parametric restriction used by the Wald statistics, so the supremum of a range of individual test statistics is proposed. Two versions of a supremum-based test are considered: the first version does not have a known asymptotic null distribution, so the bootstrap is employed to approximate its empirical distribution. The second version has a known asymptotic distribution and, in some cases, is asymptotically pivotal under the null. A simulation study illustrates the use and finite-sample performance of both versions of the test. In this study, the bootstrap is found to provide better size control than asymptotic critical values, namely with heavy-tailed, asymmetric distributions of the covariates. In addition, the use of well-known modifications of the heteroskedasticity consistent covariance matrix estimator of OLS coefficients is also found to benefit the tests’ overall behaviour.     

Constancy of equilibrium interest rates for power utility functions and stochastic constant returns to scale technologies

Metron (1969, 1971) examined, inter alia the consumption and portfolio rules in a partial equilibrium context wherein the preferences are represented by a time additive power utility function. In these papers he exogenously assumed a constant instantaneously risk free rate of interest. In this paper we naturally embellish the results of Merton in a closed production economy where there are a number of stochastic constant returns to scale technologies with correlated outputs and show that the endogenously determined interest rate is a constant. While this result is recognized by researchers, explicit analytical characterization of equilibrium interest rate obtained in this paper is a new result. This paper establishes the internal consistency of the assumptions found in Metron (1969, 1971).     

Pricing and hedging options with GARCH-stable proxy volatilities

This article considers modelling nonnormality in return with stable Paretian (SP) innovations in generalized autoregressive conditional heteroskedasticity (GARCH), exponential generalized autoregressive conditional heteroskedasticity (EGARCH) and Glosten-Jagannathan-Runkle generalized autoregressive conditional heteroskedasticity (GJR-GARCH) volatility dynamics. The forecasted volatilities from these dynamics have been used as a proxy to the volatility parameter of the Black–Scholes (BS) model. The performance of these proxy-BS models has been compared with the performance of the BS model of constant volatility. Using a cross section of S&P500 options data, we find that EGARCH volatility forecast with SP innovations is an excellent proxy to BS constant volatility in terms of pricing. We find improved performance of hedging for an illustrative option portfolio. We also find better performance of spectral risk measure (SRM) than value-at-risk (VaR) and expected shortfall (ES) in estimating option portfolio risk in case of the proxy-BS models under SP innovations.

Abbreviation: generalized autoregressive conditional heteroskedasticity (GARCH), exponential generalized autoregressive conditional heteroskedasticity (EGARCH) and Glosten-Jagannathan-Runkle generalized autoregressive conditional heteroskedasticity (GJR-GARCH)     

An omnibus test for heteroskedasticity

Tests for heteroskedasticity in the classical linear regression framework typically require the researcher to specify the form of variance heterogeneity. This paper proposes a simulation-based test with power to detect unspecified forms of heteroskedasticity.     

Optimal shill bidding in the VCG mechanism

This paper studies shill bidding in the Vickrey?CClarke?CGroves (VCG) mechanism applied to combinatorial auctions. Shill bidding is a strategy whereby a single decision-maker enters the auction under the guise of multiple identities (Yokoo et?al. Games Econ Behav, 46?pp. 174?C188, 2004). I formulate the problem of optimal shill bidding for a bidder who knows the aggregate bid of her opponents. A key to the analysis is a subproblem??the cost minimization problem (CMP)??which searches for the cheapest way to win a given package using shills. An analysis of the CMP leads to several fundamental results about shill bidding: (i) I provide an exact characterization of the aggregate bids b such that some bidder would have an incentive to shill bid against b in terms of a new property Submodularity at the Top; (ii) the problem of optimally sponsoring shills is equivalent to the winner determination problem (for single minded bidders)??the problem of finding an efficient allocation in a combinatorial auction; (iii) shill bidding can occur in equilibrium; and (iv) the problem of shill bidding has an inverse, namely the collusive problem that a coalition of bidders may have an incentive to merge (even after competition among coalition members has been suppressed). I show that only when valuations are additive can the incentives to shill and merge simultaneously disappear.     

Multivariate tests of the capm under heteroskedasticity: A note

This note extends the multivariate testing procedure to the case where heteroskedasticity is present. Previous tests of the CAPM relied on the market model. However, a substantial body of literature indicates that the error term in the market model is heteroskedastic. Failing to correct for heteroskedasticity can lead to biased estimates of the variance-covariance matrix and hence, incorrect statistics for hypothesis testing. In this note, a Wald test with a variance-covariance matrix corrected for heteroskedasticity is derived to test the CAPM. Using monthly data from 1926 to 1994, the adjusted test overwhelmingly rejects the zerobeta version of the CAPM for the 14 subperiods and for the entire sample period.     

TAIL FORECASTING WITH MULTIVARIATE BAYESIAN ADDITIVE REGRESSION TREES

We develop multivariate time-series models using Bayesian additive regression trees that posit nonlinearities among macroeconomic variables, their lags, and possibly their lagged errors. The error variances can be stable, feature stochastic volatility, or follow a nonparametric specification. We evaluate density and tail forecast performance for a set of U.S. macroeconomic and financial indicators. Our results suggest that the proposed models improve forecast accuracy both overall and in the tails. Another finding is that when allowing for nonlinearities in the conditional mean, heteroskedasticity becomes less important. A scenario analysis reveals nonlinear relations between predictive distributions and financial conditions.     

On maximum likelihood estimation of functional form and heteroskedasticity

The maximum-likelihood procedure has been developed to estimate the Box-Cox functional form parameter and the degree of heteroskedasticity simultaneously. By using an illustrative example, it is shown the likelihood ratio tests conditional on homoskedastic prior can sometimes yield misleading results.     

Panel LM unit root tests with level and trend shifts

This paper considers a Lagrange multiplier (LM) based panel unit root test that allows for heterogeneous structural breaks in both the intercepts and slopes of a series. We note that many popular time series variables are likely to exhibit changing means and/or trends over time. Given that the usual tests will depend on the nuisance parameters indicating the locations of the trend breaks, we adopt a transformation procedure that makes our new panel unit root tests invariant to the nuisance parameters. To illustrate the importance of the power gain provided by our test, we examine the convergence hypothesis using relative ratios of per capita health care expenditures in 20 OECD countries. Our results provide evidence that the convergence hypothesis is supported.     

Algebraic aggregation theory

An aggregation procedure merges a list of objects into a representative object. This paper considers the problem of aggregating n rows in an n-by-m matrix into a summary row, where every entry is an element in an algebraic field. It focuses on consistent aggregators, which require each entry in the summary row to depend only on its column entries in the matrix and to be the same as the column entry if the column in constant. Consistent aggregators are related to additive, linear and projective aggregators.