Economic Models - Convex Optimization

26 Alexis Lazaridis
where
⎛ ⎞
j∑
Q j = ⎝ A k − I⎠ .
k=1
3. The Singular Value Decomposition
Matrix in Eq. (5) may be augmented to accommodate, as an additional
column, the vector δ, µ or z. It is also possible to accommodate all these
vectors in the following way:
˜ = [ .δµz ] (6)
In this case, ˜ is defined on E n × E m , with m>n.
For conformability, the vector x i−p in Eq. (5) should be augmented
accordingly by using the linear advance operator L (such that L k y i = y i+k ),
i.e.,
⎡ ⎤
x i−p
1
˜x i−p = ⎢
⎣ L p ⎥
t i−p ⎦ .
L p d i−p
Hence, Eq. (5) takes the form:
p−1
∑
x i = Q j x i−j + ˜ ˜X i−p + w i . (7)
j=1
We can compute the unique generalized inverse (or pseudo-inverse) of ˜,
denoted by ˜ + (Lazaridis and Basu, 1981), from:
˜ + = VF ∗ U ′ (8)
where
V is of dimension (m×m), and its columns are the orthonormal eigenvectors
of ˜ ′ ˜.
U is (n × m), and its columns are the orthonormal eigenvectors of ˜ ˜ ′ ,
corresponding to the largest m eigenvalues of this matrix so that:
U ′ U = V ′ V = VV ′ = I m (9)
F is diagonal (m × m), with elements f ii f i , called the singular values
of ˜, which are the non-negative square roots of the eigenvalues of ˜ ′ ˜.