Category A COMPANION TO Theoretical Econometrics

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)...

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...

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 m

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...

Definitions and tests

Let X and Y be two income variables at either two different points in time, before and after taxes, or for different regions or countries. Let X1, X2,…, Xn be n not necessarily iid observations on X, and Y1, Y2,…, Ym be similar observations on Y. Let U1 denote the class of all utility functions u such that u’ > 0, (increasing). Also, let U2 denote the subset of all utility functions in U1 for which u" < 0 (strict concavity), and U3 denote a subset of U2 for which u”’ > 0. Let X0 and Y^ denote the ith order statistics, and assume F(x) and G(x) are continuous and monotonic cumulative distribution functions (cdfs) of X and Y, respectively. Let the quantile functions X(p) and Y(p) be defined by, for example, Y(p) = inf{y : F(y) > p}.

Proposition 1...

The Gaussian AR(1) Case with Intercept under the Alternative of Stationary

If under the stationarity hypothesis the AR(1) process has an intercept, but not under the unit root hypothesis, the AR(1) model that covers both the null and the alternative is:

Ayt = a 0 + a 1 yt-1 + ut, where a 0 = – ca1. (29.28) If -2 < a 1 < 0, then the process yt is stationary around the constant c:

hence E( yt) = c2 + (1 – (1 + a1)2) 1o2, E(ytyt-1) = c2 + (1 + a 1)(1 – (1 + a 1)2) 1o2, and  which approaches zero if c2/о2 ^ «>. Therefore, the power of the test p0 will be low if the variance of ut is small relative to [E(yf)]2. The same applies to the f-test x0. We should therefore use the OLS estimator of a1 and the corresponding f-value in the regression of Ayt on yf-1 with intercept.

Denoting y_1 = (1/п)Щ=1 У f-1, й = (1/„)’L„=1u f, the OLS estimator of a 1 is:...