Operational Algebra and Regression t-Tests
Data reduction involves a physical transition from sample data to econometric estimator and test statistic. This transition induces a mapping on the probability law of the sample, whose image is the distribution of the statistic of interest. At a general level, the mapping can often be captured by means of an operational algebra. Some methods than employ nonlinear functions of diﬀerential operators are suggested which can perform this task. The methods are related to pseudodiﬀerential operator techniques that are used in abstract mathematics to solve systems of partial diﬀerential equations. They also generalize the fractional calculus methods developed by the author in earlier work (1984, 1985). Two examples are studied in detail. One of these deals with the feasible generalized least squares estimator and its regression t -statistic in the linear model with a non scalar error covariance matrix whose elements are functions of a ﬁnite dimensional vector of nuisance parameters. This includes a wide class of models such as general SUR systems and models with serially dependent or heterogeneous errors.
Phillips, Peter C.B., "Operational Algebra and Regression t-Tests" (1990). Cowles Foundation Discussion Papers. 1191.