Asymptotic Distribution of a Unit Root Process Under Double Truncation

Abstract

We prove that a doubly-truncated random walk process asymptotically converges to a uniform distribution. The result is then extended for a general unit root process with a stationary ARMA error term. We show how this property can be applied for diagnosing unit root behavior of the time series within upper and lower limits. As an example we use Russian ruble–U.K. pound exchange rate constrained within the bounds of gold points for the period 1897–1904.

Keywords:

• Double truncation
• Unit root

Notes

^aFor censored models, however, there is a problem with point masses.

^bProof is available from authors on request.

^cSometimes also called reflected Brownian Motion if there is no drift term.

^dThis model can be generalized to random walk with drift.

^eAssumption of normality will be relaxed later.

^fNote that both u_t and ε _t are truncated since .

^gNote that Hamilton (1994) does not prove the Proposition 7.11 and instead refers to Anderson (1971, p. 429).

^hThe historical literature describes how the exchange rates were bounded by either gold points which measure the costs of shipping gold from one country to another or were forced directly by government interventions to stay within certain limits. Einzig (1970) and Officer (1996) provide good surveys of functioning of the gold standard system.

ⁱThe data were collected from two sources in the Moscow archives: Ezhegodnik Minesterstva Finansov (EMF) (1898–1905) and Kashkarov (1898). After 1904 the data was missing in the publications of EMF, as a result the collected monthly data for St. Petersburg exchange rate on London consists of n = 96 observations.

^jThese data are from Capie and Webber (1985).