Union-Intersection and Sample-Split Methods in Econometrics with Applications to MA and SURE Models ∗

In this paper, we develop inference procedures (tests and confidence sets) for two apparently distinct classes of situations: first, problems of comparing or pooling information from several samples whose stochastic relationship is not specified; second, problems where the distributions of standard test statistics are difficult to assess (e.g., because they involve unknown nuisance parameters), while it is possible to obtain more tractable distributional results for statistics based on appropriately chosen subsamples. A large number of econometric models lead to such situations, such as comparisons of regression equations when the relationship between the disturbances across equations is unknown or complicated: seemingly unrelated regression equations (SURE), regressions with moving average (MA) errors, etc. To deal with such problems, we propose a general approach which uses union-intersection techniques to combine tests (or confidence sets) based on different samples. In particular, we make a systematic use of Boole-Bonferroni inequalities to control the overall level of the procedure. This approach is easy to apply and transposable to a wide spectrum of models. In addition to being robust to various misspecifications of interest, the approach studied turns out to have surprisingly good power properties with respect to other available techniques (e.g., various asymptotically motivated methods and other bounds procedures). Applications to inference in SURE and regressions with MA(q) errors are discussed in detail. In the latter case, we also present an extensive Monte Carlo study, demonstrating the advantages of the sample-split approach. Finally, the methods proposed are applied to a demand system for inputs, a multivariate return to schooling model, and a time series model of Canadian per capita GDP.

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