Abstract
Refined asymptotic methods are used to produce degrees-of-freedom- adjusted Edgeworth and Cornish-Fisher size corrections of the t and F testing procedures for the parameters of a S.U.R. model with serially correlated errors. The corrected tests follow the Student-t and F distributions, respectively, with an approximation error of order
Appendix
In this appendix, we provide proofs of the main results of the paper. To prove these results, we rely on a number of lemmas, which are presented with their proofs in a Technical Appendix, available at the …. This appendix is structured as follows: First, we introduce the stochastic order ω(·), which measures the approximation error of the asymptotic expansions given in the paper. Then, using the lemmas in the Technical Appendix, we provide the proofs of the theorems.
The Order ω(·)
Following Magdalinos (1992, 344), let I be a given set of indexes which, without loss of generality, can be considered to belong to the open interval (0,1). For any collection of real-valued stochastic quantities (scalars, vectors, or matrices) Yτ (τ∊I), we write Yτ = ω(τi), if for any given n > 0, there exists a 0 < ε < ∞ such that
as τ → 0, where the ||·|| is the Euclidean norm. If eq. [69] is valid for any n > 0, we write Yτ = (ω). The use of this order of magnitude is motivated by the fact that, if two stochastic quantities differ by a quantity of order ω(τi), then, under general conditions, the distribution function of the one provides an asymptotic approximation of the distribution function of the other, with an error of order O(τi). Furthermore, orders ω(·) and O(·) have similar operational properties (Magdalinos (1992).
Asymptotic Expansions of Size Corrected Tests: Proofs of Theorems
Given the lemmas in the Technical Appendix, next we derive the proofs of the theorems presented in the main text. These are based on known expansions of standard normal and chi-square distributed tests. We derive new expansions of the degrees- of-freedom-adjusted versions of these tests, by inverting their characteristic functions. These degrees-of-freedom-adjusted approximations of distribution functions are proved to be locally exact.
Approximation eq. [42] of Theorem 1 can be proved following the steps of the proof in Rothenberg (1988). The quantities in eq. [40] can be obtained by expanding the corresponding quantities given by Rothenberg and retaining the first term in each of these expansions. The approximation eq. [44] of Theorem 2 follows from the approximation eq. [42] and the following asymptotic approximations of the Student-t distribution and density functions, which are given in terms of the standard normal distribution and density functions, respectively (see Fisher (1925)):
Note that approximation eq. [44] of Theorem 2 is locally exact. This can be easily seen as follows: If parameter vector
Then, the analytic formulas of pi and p2, given in eq. [43], become
This result implies that, with an error of order O(τ3), approximation eq. [44] becomes the Student-t distribution function with MT – n degrees of freedom. □
Proof of Theorem 3: We begin the proof by noticing that, under null hypothesis eq. [36], the t statistic, given by eq. [37], admits a stochastic expansion of the form
where the first term in the expansion is given as
The result given by eq. [73] implies that the Cornish-Fisher corrected statistic t*, given by eq. [47], admits a stochastic expansion of the form
where
Let s be an imaginary number, and ψ(s) and ϕ(s) denote the characteristic functions of the t statistic, given by eq. [37], and a standard normal random variable, respectively. Using eq. [74] and the relationships:
we can show that the characteristic function of the Cornish-Fisher corrected statistic t*, denoted as ψ *(s), can be approximated as follows:
Dividing ψ*(s) by –s, applying the inverse Fourier transform, and using Theorem 2, we can show that
The last result means that the Cornish-Fisher corrected statistic t* is distributed as a Student-t random variable with MT – n degrees of freedom. □
Approximation eq. [58] of Theorem 4 can be proved following the steps of the proof in Rothenberg (1984b). The quantities in eq. [56] can be obtained by expanding the corresponding quantities given by Rothenberg and retaining the first term in each of these expansions. Approximation eq. [60] of Theorem 5 follows from approximation eq. [58] and the following asymptotic approximations of the F distribution and density functions, which are given in terms of the chi-square distribution and density functions, respectively:
Note that approximation eq. [60] of Theorem 5 can be easily seen to be locally exact. By using eqs [71], [59], and [61], we can show that
This result means that, with an error of order O(τ3), approximation eq. [60] becomes the F distribution function with m and MT – n degrees of freedom. □
To begin the proof, we first notice that, under null hypothesis eq. [48], the F statistic, given by eq. [50], admits a stochastic expansion of the form
where the first term in the expansion is
Equation [78] Implies that the Cornish-Fisher corrected statistic F*, given by eq. [64], admits a stochastic expansion of the form
where
Let s be an imaginary number, and ψ(s) and ϕ(s) now denote the characteristic functions of the F statistic, given by eq. [50], and a chi-square random variable with m degrees of freedom, respectively. Using eq. [79] and the following relationships:
we can show that the characteristic function of the Cornish-Fisher corrected statistic F*, denoted as ψ*(s), can be approximated as follows:
For the chi-square density/m(x), the following results can be shown:
Dividing eq. [80] by –s, applying the inverse Fourier transform, and using Theorem 5 and the results of eqs [76] and [81], we can show that
The last result implies that the Cornish-Fisher corrected statistic
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