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Inferential Statistics 377 critical region ΔC. The probability of this statistic having a value in the critical region is indica ...

378 The Basics of financial economeTrics to us, no matter what we use the test for. Consequently, we attempt to avoid this type ...

Inferential Statistics 379 that the test statistic might become even more extreme than t(x). This prob- ability is equal to the ...

380 The Basics of financial economeTrics Quality Criteria of a Test So far, we have learned how to construct a test for a given ...

Inferential Statistics 381 P(δ(X) = d 1 ), computed for all values θ in the set Θ 1 corresponding to the alternative hypothesis. ...

382 The Basics of financial economeTrics Uniformly most powerful (UMP) test of size α. A test δ* of size α is uniformly most pow ...

Inferential Statistics 383 PII(δ(X)) ≤ 1 – α for any θ ∈ Θ 1 So if a test is unbiased, we reject the null hypothesis when it is ...

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385 Appendix d Fundamentals of Matrix Algebra i n financial econometrics, it is useful to consider operations performed on order ...

386 The Basics of financial economeTrics Here are two examples. Suppose that we let wn be a risky asset’s weight in a portfolio. ...

Fundamentals of Matrix Algebra 387 Note that the first subscript indicates rows while the second subscript indicates columns. Th ...

388 The Basics of financial economeTrics Given a square n × n matrix A, the matrix dgA is the diagonal matrix extracted from A. ...

Fundamentals of Matrix Algebra 389 where the sum is extended over all permutations (j 1 ,... , jn) of the set (1,2,... , n) and ...

390 The Basics of financial economeTrics formed by adding to the coefficient matrix a column formed with the con- stant term—is ...

Fundamentals of Matrix Algebra 391 Linear Independence and Rank Consider an n × m matrix A. A set of p columns extracted from th ...

392 The Basics of financial economeTrics vectors, and matrices to produce new scalars, vectors, or matrices. The notion of opera ...

Fundamentals of Matrix Algebra 393 Multiplication We define two types of multiplication: (1) multiplication of a scalar and a ve ...

394 The Basics of financial economeTrics The transpose of A, denoted AT or A' is the m × n matrix whose ith row is the ith colum ...

Fundamentals of Matrix Algebra 395 Multiplication of a matrix by a scalar is associative with respect to matrix addition: cc()AB ...

396 The Basics of financial economeTrics of the matrix A. The adjoint of the matrix A, denoted as Adj(A), is the fol- lowing mat ...