Consider the AR(1) model without intercept, rewritten as3
Ayt = a0yt-1 + ut, where ut is iid N(0, a2), (29.2)
and y t is observed for t = 1, 2,…, n. For convenience I will assume that
yt = 0 for t < 0. (29.3)
This assumption is, of course, quite unrealistic, but is made for the sake of transparency of the argument, and will appear to be innocent.
The OLS estimator of a 0 is:
X yt-iAyt X yt-iut
7 о = = a о + І=П
X y2-i X y2-i
If -2 < a 0 < 0, so that yt is stationary, then it is a standard exercise to verify that л/n (a0 – a0) ^ N(0, 1 – (1 + a0)2) in distribution. On the other hand, if a0 = 0, so that yt is a unit root process, this result reads: a0 ^ N(0, 0) in distribution,
hence plimn^^/n a 0 = 0... Read More
When an adequate model for the DGP of a system of variables has been found it may be used for forecasting and economic analysis. Different tools have been proposed for the latter purpose. For instance, there has been an extensive discussion of how to analyze causal relations between the variables of a system of interest. In this section forecasting VAR processes will be discussed first. Forecasting in more general terms is discussed in Chapter 27 by Stock in this volume. In subsection 5.2 the concept of Granger-causality will be introduced which is based on forecast performance. It has received considerable attention in the theoretical and empirical literature. In subsection 5.3 impulse responses are considered... Read More
Consider the limited information (LI) structural regression model:
y = Ye + Xayі + u = Z5 + u,
Y = Xana + X2n2 + V,
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 ... Read More
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
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)
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... Read More
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 ... Read More