19
[y’, x’]’ is
Σ = |
' Λy [BΦB' + Ψ] Λ'y + Θγ |
I ΛyBΦΛ'x ’ | ||
ΛxΦB' Λ'y |
I ΛχΦΛ'χ + Θδ |
- | ||
(4.7)
Assuming that the latent explanatory variables (ξ) equal the observed (x), thus ξ = x, then
Θδ = 0 and Λx = I, and equation (4.7) simplifies to14)
Σ = |
' Λy [BΦB' + Ψ] Λ'y + Θγ |
I ΛyBΦ ■ | ||
ΦB' Λ'y |
I φ |
- | ||
(4.8)
The parameters occuring in Σ (Λy, B, Φ, Ψ, Θγ) are estimated on the basis of the matrix S
of second sample moments of x and y. In order to identify all parameters, additional
restrictions on the parameters have to be imposed. Given these restrictions and the
structure that equation (4.8) imposes on the data, LISREL computes FIML estimates of the
parameters when [y’, x’] is normally distributed, i.e. when the following criterion is
minimized
ln I ∑ + tr [SΣ-1] (4.9)
To be able to identify all parameters of the model, we have made the following two
additional restrictions:
(i) λy3 = 1, which implies that the latent optimal degree of central bank independence
(η) has the same unit of measurement as the observed legal index of Eijffinger
and Schaling (ES_M);15) and
14) So, we make only a distinction between the latent optimal degree of central bank independence (η) and
the observed actual degree (y) measured by the legal indices of central bank independence. Thus, the
optimal degree of central bank independence is derived from the covariances of the four legal indices.
15) It is, however, also possible to choose as the unit of measurement for the latent optimal degree one of the
other observed legal indices (λy1 =1,λy2 =1orλy4 = 1). In principal, this choice will not make a
difference regarding the identification of the parameters.
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