# An empirical Ω

## Abstract

We hope to arrive at powerful tests using test statistics of the quadratic ratio type (2.1) with significance points that can be tabulated. For this purpose, we developed the *BLUF* estimator **w** of **J**′**u**; see (3.2). The estimator depends on the following matrices: the *n* × *p* matrix **J**, the *p* × *p* matrix **Q**, the *p* × *p* matrix Ω = **KK** ′, the *n* × *n* matrix Г, the *n* × *k* matrix **X**, and the *n*-element vector y. Both **X** and y are specified by observation and it is assumed that y ~* n*(

**Xβ**, σ

^{2}Г), where Г = Г

_{ 0 }under the null hypothesis ℋ

_{ 0 }. Hence, Г in (3.2) is specified by ℋ

_{ 0 }. One is free to choose

**J**and

**Q**. The only practical specifications we know are

**J**=

**I**

_{(n)}or

**J**is an

*n*× (

*n−k*) submatrix of

**I**

_{(n)}, like

**J**in the

*BLUS*vectors, and

**Q**=

**I**or

**Q**=

**Г**

^{−1}. In this chapter we take

**J**=

**I**, so that

*p*=

*n*. Because most current tests have Г =

**I**as the null hypothesis, the most important specifications for practical application would seem to be Г=

**Q**=

**J**=

**I**

_{(n)}.

## Keywords

Scalar Multiplication Cipal Component Selection Device Economic Time Series Regression Space## Preview

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