1. Economics Department seminar, KIMEP
Convergence of distributions arising in autocorrelation hypothesis testing
Kairat Mynbaev International School of Economics Kazakh-British Technical University
Economics Department seminar, KIMEP
January 27, 2012
(Presented at 2011 World Congress of Engineering and Technology, October 30, 2011, Paper ID 22492)

2. Linear regression with correlated errors
Null hypothesis: Alternative:
Assumption is positive definite for
and

3. Example (Le Sage, J. & R.K. Pace, Introduction to Spatial Econometrics, Taylor & Francis, 2009, p.10)

4. R1 R2 R3 R4
CBD R5 R6 R7
West Highway East R1 R2 R3 R4
CBD R5 R6 R7
Seven regions in the city; CBD is the Central Business District.
Population density decreases and travel times increase with the distance to CBD

5. First-order contiguity matrix

6. Problem and history
Krämer, W., (2005) J. Stat. Plan. Inf. 128,
Martellosio, F. (2010) Econometric Theory 26,
Theorem (Martellosio) Under Assumption 1 consider an invariant test and let the density of u be continuous and unimodal at the origin. Then

7. Cliff-Ord test:
Assumption 2. The density g of ε is spherically symmetric:
Assumption 3. (a) The matrix A(ρ) satisfies Assumption 1 and
Other eigenvalues have positive limits.
(b)

8. Assumption 4. The density g satisfies
Theorem 2. Let Assumptions 1-4 hold and suppose that the inclusion

9. Distributions escaping to infinity
Example 1. Let g be a density on R and put
Example 2 (stretching-out). g1
g1/2
g1/3

10. Theorem 3. If
Theorem 4. Let
exists, then

11. Application to the theory of characteristic functions
F is a distribution function of a random variable X.
j(x) is the jump of F at point x.
φ(t) is the characteristic function of X.
Corollary 1. If g is summable and even, then
where the sum on the right is over all jump points of F.
Lukacs, E. (1970) Characteristic Functions, Griffin, Theorems and 3.3.4

12. Application to the theory of almost-periodic functions
φ is an almost-periodic function on the real line.
Corollary 2. If g is integrable and even, then
See H. Bohr’s theorem in: Akhiezer, N.I. & I.M. Glazman (1993) Theory of Linear Operators in Hilbert Spaces. Dover Publications.