Category A COMPANION TO Theoretical Econometrics

Monte Carlo tests based on pivotal statistics

In the following, we briefly outline the MC test methodology as it applies to the pivotal statistic context and a right-tailed test; for a more detailed discussion, see Dufour (1995) and Dufour and Kiviet (1998).

Подпись: GN(S0) Подпись: 1 N N X -Ьн (Sj - So), N j_1 Подпись: 1, if z Є A 0, if z € A .

Let S0 denote the observed test statistic S, where S is the test criterion. We assume S has a unique continuous distribution under the null hypothesis (S is a continuous pivotal statistic). Suppose we can generate N iid replications, Sj, j = 1,…, N, of this test statistic under the null hypothesis. Compute

In other words, NGn(S0) is the number of simulated statistics which are greater or equal to S0, and provided none of the simulated values Sj, j = 1, …, N, is equal to S0, PN(S0) = N – NGn(S0) + 1 gives the rank of S0 among the variables S0, S1, . . ...

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Duration dependence

The duration dependence describes the relationship between the exit rate and the time already spent in the state. Technically it is determined by the hazard func­tion, which may be a decreasing, increasing, or constant function of y. Accord­ingly, we distinguish (i) negative duration dependence; (ii) positive duration dependence; and (iii) absence of duration dependence.

Negative duration dependence

The longer the time spent in a given state, the lower the probability of leaving it soon. This negative relationship is found for example in the job search analysis. The longer the job search lasts, the less chance an unemployed person has of finding a job.

Positive duration dependence

The longer the time spent in a given state, the higher the probability of leaving it soon...

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There are many ways of extending the previous model. For instance, we could allow for different distributions for zi (see Koop et al., 1995) or for many outputs

to exist (see Fernandez, Koop and Steel, 2000). Here we focus on two other extensions which are interesting in and of themselves, but also allow us to dis­cuss some useful Bayesian techniques.

Explanatory variables in the efficiency distribution

Consider, for instance, a case where data are available for many firms, but some are private companies and others are state owned. Interest centers on investigat­ing whether private companies tend to be more efficient than state owned ones...

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AR(1): the probabilistic reduction perspective

The probabilistic reduction perspective has been developed in Spanos (1986). This perspective begins with the observable process {yt, t Є T} and specifies the statistical model exclusively in terms of this process. In particular, it contemplates the DGM (28.3) from left to right as an orthogonal decomposition of the form:

yt = E(ytc(Y0-i)) + ut, t Є T, (28.9)

where Yt-i := (yt-i, yt-2,…, Уо) and Ut = yt – E(yt o(Yt-i)), with the underlying

statistical model viewed as a reduction from the joint distribution of the under­lying process {yt, t Є T}. The form of the autoregressive function depends on:

f(Уо, yv У2,…, yr, v^ for all (yо, Уl, У2,…, Ут) є

in the sense of Kolmogorov (1933)...

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The Hylleberg-Engle-Granger-Yoo test

It is well known that the seasonal difference operator As = 1 – Ls can always be factorized as

1 – Ls = (1 – L)(1 + L + L2 + … + Ls-1). (31.39)

Hence, (31.39) indicates that an SI(1) process always contains a conventional unit root and a set of § – 1 seasonal unit roots. The approach suggested by Hylleberg et al. (1990), commonly known as HEGY, examines the validity of A§ through exploiting (31.39) by testing the unit root of 1 and the § – 1 separate nonstationary roots on the unit circle implied by 1 + L + … + L§-1. To see the implications of this factorization, consider the case of quarterly data (§ = 4) where

1 – L4 = (1 – L)(1 + L + L2 + L3)

= (1 – L)(1 + L)(1 + L2). (31.40)

Thus, A4 = 1 – L4 has four roots on the unit circle,2 namely 1 and -1 which occur at the 0 and n frequencie...

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The SVD model

This model represents dynamics of both the conditional mean and variance in duration data. In this way it allows for the presence of both conditional under – and overdispersion in the data. Technically, it shares some similarities with the stochastic volatility models used in finance. The main difference in SVD

Подпись: 0 5000 10,000 15,000 20,000 Figure 21.5 (Under) Overdispersion of intertrade durations

specification compared to ACD is that it relies on two latent factor variables which are assumed to follow autoregressive stochastic processes. Note that despite the fact that the conditional variance in the ACD model is stochastic it is entirely determined by past durations. The introduction of additional random terms enhances the structure of the model and improves significantly the fit...

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