Continuity or discontinuity of probability density functions of data often plays a fundamental role in empirical economic analysis. For example, for identification and inference of causal effects in regression discontinuity designs it is typically assumed that the density function of a conditioning variable is continuous at a cutoff point that determines assignment of a treatment. Also, discontinuity in density functions can be a parameter of economic interest, such as in analysis of bunching behaviors of taxpayers. In order to facilitate researchers to conduct valid inference for these problems, this paper extends the binning and local likelihood approaches to estimate discontinuity of density functions and proposes empirical likelihood-based tests and confidence sets for the discontinuity.In contrast to the conventional Wald-type test and confidence set using the binning estimator, our empirical likelihood-based methods (i) circumvent asymptotic variance estimation to construct the test statistics and confidence sets; (ii) are invariant to nonlinear transformations of the parameters of interest; (iii) offer confidence sets whose shapes are automatically determined by data; and (iv) admit higher-order refinements, so-called Bartlett corrections. First-and second-order asymptotic theories are developed. Simulations demonstrate the superior finite sample behaviors of the proposed methods. In an empirical application, we assess the identifying assumption of no manipulation of class sizes in the regression discontinuity design studied by Angrist and Lavy (1999).

In this paper we apply the recently developed fractionally cointegrated vector autoregressive (FCVAR) model to analyze price discovery in the spot and futures markets for five non-ferrous metals (aluminium, copper, lead, nickel, and zinc). The FCVAR model allows for long memory (fractional integration) in the equilibrium errors, and, following Figuerola-Ferretti and Gonzalo (2010), we allow for the existence of long-run backwardation or contango in the equilibrium as well, i.e. a non-unit cointegration coefficient. Price discovery can be analyzed in the FCVAR model by a relatively straightforward examination of the adjustment coefficients. In our empirical analysis we use the data from Figuerola-Ferretti and Gonzalo (2010), who conduct a similar analysis using the usual (non-fractional) CVAR model. Our first finding is that, for all markets except copper, the fractional integration parameter is highly significant, showing that the usual, non-fractional model is not appropriate. Next, when allowing for fractional integration in the long-run equilibrium relations, fewer lags are needed in the autoregressive formulation, further stressing the usefulness of the fractional model. Compared to the results from the non-fractional model, we find slightly more evidence of price discovery in the spot market. Specifically, using standard likelihood ratio tests, we do not reject the hypothesis that price discovery takes place exclusively in the spot (futures) market for copper, lead, and zinc (aluminium and nickel).JEL Codes: C32, G13.

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. (aluminium, copper, lead, nickel, and zinc). To this end, we first extend the FCVAR model to accommodate deterministic trends in the levels of the processes. The methodological contribution is to provide representation theory for the FCVAR model with deterministic trends, where we show that the presence of the deterministic trend in the process induces both restricted and unrestricted constant terms in the vector error correction model. The consequences for the cointegration rank test are also briefly discussed. In our empirical application we use the data from FiguerolaFerretti and Gonzalo (2010), who conduct a similar analysis using the usual (non-fractional) cointegrated VAR model. The main conclusion from the empirical analysis is that, when using the FCVAR model, there is more support for the cointegration vector (1, −1) in the long-run equilibrium relationship between spot and futures prices, and hence less evidence of long-run backwardation, compared to the results from the non-fractional model. Specifically, we reject the hypothesis that the cointegration vector is (1, −1) using standard likelihood ratio tests only for the lead and nickel markets. Terms of use: Documents inJEL codes: C32, G14.

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Abstract Based on recent evidence of fractional cointegration in commodity spot and futures markets, we investigate whether a fractionally cointegrated model can provide statistically and/or economically signi…cant forecasts of commodity returns. Speci…cally, we propose to model and forecast commodity spot and futures prices using a fractionally cointegrated vector autoregressive model that generalizes the more well-known cointegrated vector autoregressive model to allow fractional integration. We derive the best linear predictor forecast for this model and perform an out-of-sample forecast comparison with forecasts from the more standard (non-fractional) model. In our empirical analysis to daily data on 15 commodity spot and futures markets, the fractional model is found to be superior in terms of in-sample …t and also out-of-sample forecasting based on statistical metrics of forecast comparison. We then analyze the economic signi…cance of the forecasts through a dynamic trading strategy based on a portfolio with weights derived from a mean-variance utility function. This analysis leads to statistically signi…cant and economically meaningful pro…ts in the commodity markets, and also shows that the fractional model generates higher pro…ts on average compared with the non-fractional model. Terms of use: Documents in

This paper proposes empirical likelihood based inference methods for causal effects identified from regression discontinuity designs. We consider both the sharp and fuzzy regression discontinuity designs and treat the regression functions as nonparametric. The proposed inference procedures do not require asymptotic variance estimation and the confidence sets have natural shapes, unlike the conventional Wald-type method. These features are illustrated by simulations and an empirical example which evaluates the effect of class size on pupils' scholastic achievements. Furthermore, for the sharp regression discontinuity design, we show that the empirical likelihood statistic admits a higher-order refinement, so-called the Bartlett correction. Bandwidth selection methods are also discussed.