Category A COMPANION TO Theoretical Econometrics

Testing complex unit roots

Before proceeding to the examination of the procedure proposed by Hylleberg et al. (1990) it will be useful to consider some of the issues related to testing complex unit roots, because these are an intrinsic part of any SI(1) process.

The simplest process which contains a pair of complex unit roots is

Vt = – У t-2 + ut, (31.25)

with ut ~ iid(0, о2). This process has S = 2 and, using the notation identifying the season s and year n, it can be equivalently written as

Vsn = – Vs, n-1 + Usn s = 1, 2. (31.26)

Notice that the seasonal patterns reverse each year. Due to this alternating pat­tern, and assuming y0 = y-1 = 0, it can be seen that

n-1

Vt = S*n = X H)4n-; = – S*n-1 + uSn, (31.27)

1=0

where, in this case, n = [у1]...

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The ACD model

This model was introduced by Engle and Russell (1998) to represent the dynam­ics of durations between trades on stock or exchange rate markets. Typically, intertrade durations are generated by a computerized order matching system which automatically selects trading partners who satisfy elementary matching criteria. Therefore, the timing of such automatically triggered transactions is a priori unknown and adds a significant element of randomness to the trading process. From the economic point of view, research on intertrade durations is motivated by the relevance of the time varying speed of trading for purposes
such as strategic market interventions. Typically the market displays episodes of accelerated activity and slowdowns which reflect the varying liquidity of the asset...

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Tests for stochastic dominance

In the area of income distributions and tax analysis, it is important to look at Lorenz curves and similar comparisons. In practice, a finite number of ordinates of the desired curves or functions are compared. These ordinates are typically represented by quantiles and/or conditional interval means. Thus, the distribu­tion theory of the proposed tests are typically derived from the existing asymp­totic theory for ordered statistics or conditional means and variances. A most up-to-date outline of the required asymptotic theory is Davidson and Duclos

(1998) . To control for the size of a sequence of tests at several points the union intersection (UI) and Studentized Maximum Modulus technique for multiple comparisons is generally favored in this area...

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Weak convergence of random functions

In order to establish the limiting distribution of (29.14), and other asymptotic results, we need to extend the well known concept of convergence in distribution of random variables to convergence in distribution of a sequence of random functions. Recall that for random variables Xn, X, Xn ^ X in distribution if the distribution function Fn(x) of Xn converges pointwise to the distribution function F(x) of X in the continuity points of F(x). Moreover, recall that distribution func­tions are uniquely associated to probability measures on the Borel sets,5 i. e. there exists one and only one probability measure ц„(В) on the Borel sets B such that F„(x) = ц„((- <ж, x]), and similarly, F(x) is uniquely associated to a probability measure ц on the Borel sets, such that F(x) = ц((-^, x])...

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Forecasting VAR processes

Neglecting deterministic terms and exogenous variables the levels VAR form (32.1) is particularly convenient to use in forecasting the variables yt. Suppose the ut are generated by an independent rather than just uncorrelated white noise process. Then the optimal (minimum MSE) one-step forecast in period T is the conditional expectation,

yT+1|T = E(yT+1 | yT, yT-1, . . . ) = A1yT + … + ApyT+1-p. (32.14)

Forecasts for larger horizons h > 1 may be obtained recursively as

yT+h | T = A1 yT+h-1|T + … + ApyT+h-p|T, (32.15)

where yT+;-|T = yT+j for j < 0. The corresponding forecast errors are

yT+h — yT+h | T = UT+h + ®1UT+h-1 + … + ФУ—1uT+1, (32.16)

where it is easy to see by successive substitution that Ф8 = j Aj (s = 1, 2,…)

with Ф0 = IK and Aj = 0 for j > p (see Lutkepohl, 1991, sec...

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Normality tests

Let us now consider the fundamental problem of testing disturbance normality in the context of the linear regression model:

Y = Xp + u, (23.12)

where Y = (y1, …, yn)’ is a vector of observations on the dependent variable, X is the matrix of n observations on k regressors, P is a vector of unknown coefficients and u = (u1, …, un)’ is an n-dimensional vector of iid disturbances. The problem consists in testing:

H0 : f(u) = ф (u; 0, о), о > 0, (23.13)

where f(u) is the probability density function (pdf) of ui, and ф (u; p, о) is the normal pdf with mean p and standard deviation о. In this context, normality tests are typically based on the least squares residual vector

й = y – xp = Mxu, (23.14)

where p = (XX)-1 X’y and Mx = In – X(XX)-1X’...

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