Geert Dhaene

Unit root tests for panel data with AR(1) errors and small T

We propose unit root tests for panel data with a small number of time periods, T, and increments that follow an AR(1) process under the null. The model is a fixed-effect panel version of the augmented Dickey–Fuller regression of order 1. Individual-specific linear trends may also be included. The test statistics are t-type statistics based on least-squares estimates from which the Nickell bias is removed. Their limiting distributions (for an increasing number of independent cross-section units, N, and fixed T) are standard normal. Our test generalizes the panel unit root test of Harris and Tzavalis, which is based on an unaugmented Dickey–Fuller regression. As an illustration, we examine whether the Law of One Price holds in the European car market since the start of stage three of the EMU in 1999. We find strong evidence of price convergence in the EMU countries.

Sequential reciprocity in two-player, two-stage games: An experimental analysis

We experimentally test Dufwenberg and Kirchsteiger’s (2004) theory of sequential reciprocity in a sequential prisoner’s dilemma (SPD) and a mini-ultimatum game (MUG). Data on behavior and first- and second-order beliefs allow us to classify each subject’s behavior as a material best response, a reciprocity best response, both, or none. We found that in both games the behavior of about 80% of the firstmovers was a material best response, a reciprocity best response, or both. The remaining 20% of first-movers almost always made choices that were “too kind” according to the theory of reciprocity. Secondmover behavior, in both games, was fully in line with the predictions of the theory. The average behavior and beliefs across subjects were compatible with a sequential reciprocity equilibrium in the SPD but not in the MUG. We also found first- and second-order beliefs to be unbiased in the SPD and nearly unbiased in the MUG.

LIKELIHOOD INFERENCE IN AN AUTOREGRESSION WITH FIXED EFFECTS

We calculate the bias of the profile score for the regression coefficients in a multistratum autoregressive model with stratum-specific intercepts. The bias is free of incidental parameters. Centering the profile score delivers an unbiased estimating equation and, upon integration, an adjusted profile likelihood. A variety of other approaches to constructing modified profile likelihoods are shown to yield equivalent results. However, the global maximizer of the adjusted likelihood lies at infinity for any sample size, and the adjusted profile score has multiple zeros. We argue that the parameters are local maximizers inside or on an ellipsoid centered at the maximum likelihood estimator.

Median-based estimation of dynamic panel models with fixed effects☆

Outlier-robust estimators are proposed for linear dynamic fixed-effect panel data models where the number of observations is large and the number of time periods is small. In the simple setting of estimating the AR(1) coefficient from stationary Gaussian panel data, the estimator is (a linear transformation of) the median ratio of adjacent first-differenced data pairs. Its influence function is bounded under contamination by independent or patched additive outliers. The influence function and the gross-error sensitivity are derived. When there are independent additive outliers, the estimator is asymptotically biased towards 0, but its sign remains correct and it has a reasonably high breakdown point. When there are patched additive outliers with point mass distribution, the asymptotic bias is upward in nearly all cases; breakdown towards 1 can occur; and the associated breakdown point increases with the patch length.

Best affine unbiased response decomposition

Given two linear regression models y1 = X1β1 + u1 and y2 = X2β2 + u2 where the response vectors y1 and y2 are unobservable but the sum y = y1 + y2 is observable, we study the problem of decomposing y into components ŷ1 and ŷ2, intended to be close to y1 and y2, respectively. We develop a theory of best affine unbiased decomposition in this setting. A necessary and sufficient condition for the existence of an affine unbiased decomposition is given. Under this condition, we establish the existence and uniqueness of the best affine unbiased decomposition and provide an expression for it.

Mixed-Frequency Multivariate GARCH

We introduce and evaluate mixed-frequency multivariate GARCH models for forecasting low-frequency (weekly or monthly) multivariate volatility based on high-frequency intra-day returns (at five-minute intervals) and on the overnight returns. The low-frequency conditional volatility matrix is modelled as a weighted sum of an intra-day and an overnight component, driven by the intra-day and the overnight returns, respectively. The components are specified as multivariate GARCH (1,1) models of the BEKK type, adapted to the mixed-frequency data setting. For the intra-day component, the squared high-frequency returns enter the GARCH model through a parametrically specified mixed-data sampling (MIDAS) weight function or through the sum of the intra-day realized volatilities. For the overnight component, the squared overnight returns enter the model with equal weights. Alternatively, the low-frequency conditional volatility matrix may be modelled as a single-component BEKK-GARCH model where the overnight returns and the high-frequency returns enter through the weekly realized volatility (defined as the unweighted sum of squares of overnight and high-frequency returns), or where the overnight returns are simply ignored. All model variants may further be extended by allowing for a non-parametrically estimated slowly-varying long-run volatility matrix. The proposed models are evaluated using five-minute and overnight return data on four DJIA stocks (AXP, GE, HD, and IBM) from January 1988 to November 2014. The focus is on forecasting weekly volatilities (defined as the low frequency). The mixed-frequency GARCH models are found to systematically dominate the low-frequency GARCH model in terms of in-sample fit and out-of-sample forecasting accuracy. They also exhibit much lower low-frequency volatility persistence than the low-frequency GARCH model. Among the mixed-frequency models, the low-frequency persistence estimates decrease as the data frequency increases from daily to five-minute frequency, and as overnight returns are included. That is, ignoring the available high-frequency information leads to spuriously high volatility persistence. Among the other findings are that the single-component model variants perform worse than the two-component variants; that the overnight volatility component exhibits more persistence than the intra-day component; and that MIDAS weighting performs better than not weighting at all (i.e., than realized volatility).

Testing Futures Returns Predictability: Implications for Hedgers

The predictability of futures returns is investigated using a semiparametric approach where it is assumed that the expected returns depend non parametrically on a combination of predictors. We first collapse the forecasting variables into a single index variable where the weights are identified up to scale, using the average derivative estimator proposed by Stoker (1986). We then use the Nadaraya-Watson kernel estimator to calculate (and visually depict) the relation between the estimated index and the expected futures returns. An application to four agricultural commodity futures illustrates the technique. The results indicate that for each of the commodities considered, the estimated index contains statistically significant information regarding the expected futures returns. Economic implications for a non-infinitely risk averse hedger are also discussed.

Encompassing tests for the linear model

The study of linear models together with all the extensions and ramifications has a long history in econometrics, and in statistics in general. Many tools for statistical inference have been designed for, or first been applied to, linear models. This is partly for reasons of analytical and computational tractability, although the latter becomes less and less of a problem. Furthermore, new ideas, unless by nature they pertain to nonlinear models, are more easily introduced and absorbed in the context of linear models. Non-nested hypotheses testing and encompassing are no exception to this rule. The normal linear regression model has indeed been the workhorse in this area.

Pseudo-true values

This preliminary chapter discusses the issue of approximating the distributions of a parametric family F = {F α |α є Ω F ⊂ R m } by those of another parametric family g = {G β |β є Ωg ⊂ R n }. The definition of encompassing will be based on such approximations, which are therefore discussed at some length here. The general problem consists of finding the mapping of F into g which associates with each distribution F α є F the distribution G β є g closest to it according to some criterion. The adoption of the Kullback-Leibler [1951] Information Criterion (KLIC) as a distance measure between distributions defines such a mapping. The mapping which results from this choice will be the sole object of study in this chapter. In line with Sawa [1978], we call the values obtained under this mapping pseudo-true distributions, and their associated parameter vectors pseudo-true (parameter) values. Although only implicitly, pseudo-true values seem to have appeared first in the pioneering work of Cox [1961, 1962] in connection with non-nested hypothesis testing.

Testing the encompassing hypothesis

The statistical inference concerning the encompassing relation is the main object of this chapter. In general, theoretical considerations of the real world phenomena under study should narrow the scope to plausible models. If then we are left with a number of competing models which all have a sound theoretical basis, it is only natural to conduct statistical inference in order to assess the relative merits of these models. The significance of the encompassing relation for empirical model building justifies this interest.