Economic Models - Convex Optimization

54 Dipak R. Basu and Alexis Lazaridis
3.5. Stability of the Model
Characteristic roots (real) of the system transition matrix
− 0.0000008
− 0.0000008
− 0.1213610
0.2442519
0.3087129
0.5123629
0.7824260
0.0000001
0.0000001
All the roots are real and they are less than unity, so the system is
stable. [In the equivalent control system, the rank of the controllability
matrix is the same as the dimension of the reduced state vector so that the
system is controllable and observable, i.e., the system parameters can be
identified.]
We can, however, transform our model to the following form:
Y t = ρY t−1 + e t , t = 1, 2,...,n (24)
where ρ is a real number and (e t ) is a sequence of normally distributed
random variables with mean zero and variance σt 2 . Box and Pierce (1970)
suggested the following test statistic
where
Q m = n
m∑
rk 2 (25)
k=1
/(
n∑
n∑
r k = ê t ê t−k
t=k+1 t=1
ê 2 t
)
(26)
n = the number of observations, m = n − k, where k = the number of
parameters estimated, and ê t are the residuals from the fitted model.
If (Y t ) satisfies the system, then under the null hypothesis, Q m is
distributed as a chi-squared random variable with m degrees of freedom.
The null hypothesis is that ρ = 1 where ê t = Y t − Y t−1 and thus k = 0.