# Category A COMPANION TO Theoretical Econometrics

## Instrumental regressions

Consider the limited information (LI) structural regression model:

 y = Ye + Xayі + u = Z5 + u, (23.3) Y = Xana + X2n2 + V, (23.4)

where Y and Xj are n x m and n x k matrices which respectively contain the observations on the included endogenous and exogenous variables, Z = [Y, Xa], 5 = (P’, y1)’ and X2 refers to the excluded exogenous variables. If more than m variables are excluded from the structural equation, the system is said to be overidentified. The associated LI reduced form is:

п 1 = ПіР + y1/ п2 = П2р.

The necessary and sufficient condition for identification follows from the relation п2 = П2р. Indeed P is recoverable if and only if

rank(n2) = m. (23.7)

To test the general linear hypothesis R5 = r, where R is a full row rank q x (m ...

## Salient Features of US Macroeconomic Time Series Data

The methods discussed in this chapter will be illustrated by application to five monthly economic time series for the US macroeconomy: inflation, as measured by the annual percentage change in the consumer price index (CPI); output growth, as measured by the growth rate of the index of industrial production; the unem­ployment rate; a short-term interest rate, as measured by the rate on 90-day US Treasury bills; and total real manufacturing and trade inventories, in logarithms.1 Time series plots of these five series are presented as the heavy solid lines in Figures 27.1-27.5.

 1960 1965 1970 1975 1980 1985 1990 1995 2000 Year Figure 27.1 US unemployment rate (heavy solid line), recursive AR(BIC)/unit root pretest forecast (light solid line), and neural network forecast (dotted line)

## Common trends representation

As mentioned above, there is a dual relationship between the number of cointegrating vectors (r) and the number of common trends (n – r) in an n – dimensional system. Hence, testing for the dimension of the set of "common trends" provides an alternative approach to testing for the cointegration order in a VAR//VECM representation. Stock and Watson (1988) provide a detailed study of this type of methodology based on the use of the so-called Beveridge-Nelson (1981) decomposition. This works from the Wold representation of an I(1) system, which we can write as in expression (30.11) with C(L) = Xj= 0 Cp, C0 = In. As shown in expression (30.12), C(L) can be expanded as C(L) = C(1) + C(L)(1 – L), so that, by integrating (30.11), we get

yt = C(1)Yt + +, (30.21)

where +t = C(L)et can be shown to b...

## Duration Variables

In this section we introduce basic concepts in duration analysis and present the commonly used duration distributions.

2.1 Survivor and hazard functions

Let us consider a continuous duration variable Y measuring the time spent in a given state, taking values in R+. The probabilistic properties of Y can be defined either by:

the probability density (pdf) function f(y), assumed strictly positive, or the cumulative distribution (cdf) function F( y) = Pn f (u)du, or the survivor function S( y) = 1 – F( y) = /Jf(u)du.

The survivor function gives the probability of survival to y, or otherwise, the chance of remaining in the present state for at least y time units. Essentially, the survivor function concerns the future.

In many applications the exit time has an economic meaning and may signify a ...

## Bayesian inference

In order to define the sampling model,5 we make the following assumptions about and zi for i = 1 … N:

1. p(vi | h_1) = f N(vi |0, h_1) and the vis are independent;

2. vi and z; are independent of one another for all i and l;

3. p(zt | ^-1) = fG(zi |1, ^-1) and the zis are independent.

The first assumption is commonly made in cross-sectional analysis, but the last two require some justification. Assumption 2 says that measurement error and inefficiency are independent of one another. Assumption 3 is a common choice for the nonnegative random variable, zi, although others (e. g. the half-normal) are possible. Ritter and Simar (1997) show that the use of very flexible one-sided distributions for zi such as the unrestricted gamma may result in a problem of weak identification...

## Autoregressive Models: Univariate

The objective of this section is to provide a brief overview of the most commonly used time series model, the AR(1), from both, the traditional and the probabilistic reduction (PR) perspectives.