Category A COMPANION TO Theoretical Econometrics

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...

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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

Подпись: (29.30)

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:...

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Granger-causality analysis

The concept

The causality concept introduced by Granger (1969) is perhaps the most widely discussed form of causality in the econometrics literature. Granger defines a variable y1t to be causal for another time series variable y2t if the former helps predicting the latter. Formally, denoting by y2,t+h|iif the optimal h-step predictor of y2t at origin t based on the set of all the relevant information in the universe Qt, y1t may be defined to be Granger-noncausal for y2t if and only if

y2,t+hQt = y2,t+h Qt{y1s s<t}, h — 1, 2, . . . . (32.18)

Here Qt A denotes the set containing all elements of Q. t which are not in the set A. In other words, y1t is not causal for y2t if removing the past of y1t from the information set does not change the optimal forecast for y2t at any forecast hori­zon...

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Uniform linear hypothesis in multivariate regression models

Multivariate linear regression (MLR) models involve a set of p regression equa­tions with cross-correlated errors. When regressors may differ across equations, the model is known as the seemingly unrelated regression model (SUR or SURE; Zellner, 1962). The MLR model can be expressed as follows:

Подпись:Y = XB + U,

where Y = (Y1,… ,Yp) is an n x p matrix of observation on p dependent variables, X is an n x k full-column rank matrix of fixed regressors, B = [pv…, Pp] is a k x p matrix of unknown coefficients and U = [Uv…, Up] = [Q1,…, Un]’ is an n x p matrix of random disturbances with covariance matrix X where det (X) Ф 0. To derive the distribution of the relevant test statistics, we also assume the following:

A = W, i = 1,…, n, (23.19)

where the vector w = vec(W1,…, Wn) has a known distribution...

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Differencing the data

A question that arises in practice is whether to difference the data prior to con­struction of a forecasting model. This arises in all the models discussed above, but for simplicity it is discussed here in the context of a pure AR model. If one knows a priori that there is in fact a unit autoregressive root, then it is efficient to impose this information and to estimate the model in first differences. Of course, in practice this is not known. If there is a unit autoregressive root, then estimates of this root (or the coefficients associated with this root) are generally biased towards zero, and conditionally biased forecasts can obtain...

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Higher order cointegrated systems

The statistical theory of I(d) systems with d = 2, 3,…, is much less developed than the theory for the I(1) model, partly because it is uncommon to find time series, at least in economics, whose degree of integration higher than two, partly because the theory is quite involved as it must deal with possibly multicointegrated cases where, for instance, linear combinations of levels and first differences can achieve stationarity. We refer the reader to Haldrup (1999) for a survey of the statistical treatment of I(2) models, restricting the discussion in this chapter to the basics of the CI(2, 2) case.

Assuming, thus, that yt ~ CI(2, 2), with Wold representation given by

(1 – L)2 y = C(L)e f, (30.23)

then, by means of a Taylor expansion, we can write C(L) as C(L) = C(1) – C *(1)(1 – L) + C(L)(...

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