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

Linear regression with correlated errors Null hypothesis: Alternative: Assumption 1. is positive definite for and

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

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

First-order contiguity matrix

Problem and history Krämer, W., (2005) J. Stat. Plan. Inf. 128, 489-496. Martellosio, F. (2010) Econometric Theory 26, 152-186. Theorem (Martellosio) Under Assumption 1 consider an invariant test and let the density of u be continuous and unimodal at the origin. Then

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)

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

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

Theorem 3. If Theorem 4. Let exists, then

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 3.2.3 and 3.3.4

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.