Palgrave Handbook of Econometrics: Applied Econometrics

Tommaso Proietti 411

Table 9.2 ML estimation results for bivariate models

of quarterly US log GDP (yt) and the consumer price

inflation rate (pt), 1950:1–2006:4

Bivariate Great Moderation

Parameter Std. error Parameter Std. error

ytequation

ση^2 0.33 0.14

ση^2 a 0.58 0.27

ση^2 b 0.13 0.05

σκ^2 0.38 0.15

σκ^2 a 0.47 0.24

σκ^2 b 0.06 0.04

φ 1 1.47 0.06 1.55 0.06

φ 2 −0.54 0.10 −0.60 0.09

ptequation

σp^2 ε 0.11 0.03 0.12 0.03

στη^2 0.05 0.02 0.05 0.02

θψ 0 0.12 0.05 0.12 0.06

θψ 1 −0.10 0.05 −0.10 0.06

Wald tests of restrictionθψ( 1 )= 0

2.00 1.68

loglik −447.79 −415.53

ergodic Markov chain whose invariant distribution is the target density (see Chib,
2001, and the references therein).
This is achieved by the following iterative scheme. Specify an initial value

α(^1 ),(^1 ). Fori=1, 2,...,M:

1. Generateα(i)∼f(α|(i−^1 ),y)using the simulation smoother (see Appendix C,
   section 9.7.4)


2. Generate(i)∼f((i)|α(i),y). This block is divided into smaller components,
   whose full conditional distribution is available for sampling. In particular:
   (a) Generate(φ( 1 i),φ 2 (i))′from the full conditional(φ 1 ,φ 2 )′|ψ,σκ^2 (i−^1 )(this distri-
   bution is conditionally independent ofy, givenψ). Assuming a Gaussian
   prior distribution, N(mφ 0 ,φ 0 ),(φ 1 ,φ 2 )′|ψ,σκ^2 (i−^1 )∼N(mφ 1 ,φ 1 )where,
   denotingχt− 1 =(ψt(−i− 11 ),ψt(i−− 21 ))′,

φ 1 =

⎛

⎝φ− 01 +^1

σκ^2 (i−^1 )

∑

t

χt− 1 χ′t− 1

⎞

⎠

− 1

,

mφ 1 =φ 1

⎛

⎝φ− 01 mφ 0 +^1

σκ^2 (i−^1 )

∑

t

χt− 1 ψt

⎞

⎠.