Category A COMPANION TO Theoretical Econometrics

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’...

Univariate forecasts are made solely using past observations on the series being forecast. Even if economic theory suggests additional variables that should be useful in forecasting a particular variable, univariate forecasts provide a simple and often reliable benchmark against which to assess the performance of those multivariate methods. In this section, some linear and nonlinear univariate fore­casting methods are briefly presented. The performance of these methods is then illustrated for the macroeconomic time series in Figures 27.1-27.5.

2.1 Linear models

One of the simplest forecasting methods is the exponential smoothing or ex­ponentially weighted moving average (EWMA) method. The EWMA forecast is,

Dt+h 11 = aDt+h-i t-i + (1 – a)yt, (27.4)

where a is a parameter chosen by the for...

Although the discussion in the previous sections has been confined to the pos­sibility of cointegration arising from linear combinations of I(1) variables, the literature is currently proceeding in several interesting extensions of this stan­dard setup. In the sequel we will briefly outline some of those extensions which have drawn a substantial amount of research in the recent past.

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).

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, . . ...

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...

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...