Palgrave Handbook of Econometrics: Applied Econometrics

Tommaso Proietti 409

whereFtis the information set at timet. Basistha and Nelson (2007) estimate a
bivariate model of output and inflation where the output equation features the
MNZ decomposition with correlated components, and in the inflation equation,
survey-based expectations replace E(pt+ 1 |Ft).

9.3.2 A bivariate quarterly model of output and inflation for the US

This section is devoted to the estimation of a bivariate model for US quarterly real
GDP and the quarterly rate of inflationpt, whereptis the logarithm of quarterly
CPI for the US, using data from the first quarter of 1950 to the fourth quarter
of 2006. The KPSS test conducted on the inflation series leads to the rejection of
the null of stationarity against a random walk for all values of the lag truncation
parameter up to 5; if a linear trend is considered and stationarity is tested against
a random walk with drift, then the null is also rejected for much higher values of
the lag truncation parameter. In the sequel, inflation will be taken to be integrated
of order one. The model has the following specification:

yt = μt+ψt, t=1,...,n,

μt = μt− 1 +βt+ηt, ηt∼NID(0,ση^2 )

ψt = φ 1 ψt− 1 +φ 2 ψt− 2 +κt, κt∼NID(0,σκ^2 )

pt = τt+εpt εpt∼NID(0,σp^2 ε)

τt = τt− 1 +θψ(L)ψt+ητt ητt∼NID(0,στη^2 ),

(9.22)

whereηt,κt,εpt, andηptare mutually independent.
The output equation is the usual decomposition into orthogonal components;
the inflation equation is a decomposition into a core component,τt, and a tran-
sitory one. The changes in the core component are driven by the output gap and
by the idiosyncratic disturbancesητt. The lag polynomialθψ(L)=θψ 0 +θψ 1 Lcan
be rewritten asθψ( 1 )−θψ 1 , which enables us to isolate the level effect of the gap
from the change effect, which we expect to be positive, that is we expectθψ 1 <0.
Ifθψ( 1 )=0, the inflation equation can be rewritten aspt=τt∗−θψ 1 ψt+εt, with

τt∗=ητt, so that output and inflation would share a common cycle.
We also extend the specification of model (9.22) to take into account an impor-
tant stylized fact, known as the “Great Moderation” of the business cycle, and
which consists of a substantive reduction in the volatility of GDP growth. This
feature is visible from the plot ofytin Figure 9.1. The date when the structural
break in volatility occurred is identified as the first quarter of 1984 (see Kim and
Nelson, 1999a; McConnell and Perez Quiros, 2000; Stock and Watson, 2003).
LetStdenote an indicator variable which takes the value 1 in the high volatility
state (which we label regimea) and 0 in the low volatility state (regimeb). The trend
and cycle disturbance variances are time varying and the model will be specified
as in (9.22) with:

ηt∼N

(

0,Stση^2 a+( 1 −St)ση^2 b

)

, κt∼N

(

0,Stσκ^2 a+( 1 −St)σκ^2 b

)

.