An Estimated DSGE Model of the Indian Economy.

2.5.1 Bayesian Methods

Bayesian estimation entails obtaining the posterior distribution of the model’s parameters,

say θ, conditional on the data. Using the Bayes’ theorem, the posterior distribution is

where p(θ) denotes the prior density of the parameter vector θ, L(Y^T∣θ/) is the likelihood of
the sample Y^T with T observations (evaluated with the Kalman filter) and L(Y^T ∣θ)p(θ) is
the marginal likelihood. Since there is no closed form analytical expression for the posterior,
this must be simulated.

One of the main advantages of adopting a Bayesian approach is that it facilitates a formal
comparison of different models through their posterior marginal likelihoods, computed using
the Geweke (1999) modified harmonic-mean estimator. For a given model mi ∈ M and
common data set, the marginal likelihood is obtained by integrating out vector θ,

(YT∣m_i} = L L (YT∣θ,m_i} p (θmi) dθ
Θ

where pi (θ∣mi) is the prior density for model mi, and L Y^T∣mi is the data density for
model mi given parameter vector θ. To compare models (say, mi and mj ) we calculate
the posterior odds ratio which is the ratio of their posterior model probabilities (or Bayes
Factor when the prior odds ratio, ^p(mi), is set to unity):
in terms of the log-likelihoods. Components (37) and (38) provide a framework for com-
paring alternative and potentially misspecified models based on their marginal likelihood.
Such comparisons are important in the assessment of rival models, as the model which
attains the highest odds outperforms its rivals and is therefore favoured.

Given Bayes factors, we can easily compute the model probabilities p 1 ,p2, ∙ ∙ ∙,p_n for n
models. Since ɪɪ ∣ pi = 1 we have that pl = ∑n=₂ BF_i, 1, from which p 1 is obtained. Then
pi = p1B F(i, 1) gives the remaining model probabilities.

2.5.2 Data, Priors and Calibration

To estimate the system for each economy, we use four macroeconomic observables at quar-
terly frequency. For the US, we use real GDP, real investment, the GDP deflator and
the nominal interest rate. All data are taken from the FRED Database available through
the Federal Reserve Bank of St.Louis and the sample period is 1980:1-2006:4. We use the

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