System Reduction from 3 to 2 Equations Galí (2015), Chapter 3, derives the basic New Keynesian model (BNKM) represented in 3 endogenous equations and 3 exogenous stochastic shock processes, as follows. The 3 equations determining in equilibrium the 3 endogenous variables in the system, the output gap, inflation and the nominal interest rate set by the central bank according to a contemporaneous feedback, or current-looking, interest rate rule (CLIRR) are, respectively: 𝑦̃𝑡 = 𝐸𝑡𝑦̃𝑡+1 − 1 𝜎 (𝑖𝑡 − 𝐸𝑡𝜋𝑡+1 − 𝑟𝑡 𝑛) 𝜋𝑡 = 𝛽𝐸𝑡𝜋𝑡+1 + 𝜅𝑦̃𝑡 𝑖𝑡 = 𝜌 + 𝜑𝜋𝜋𝑡 + 𝜑𝑦𝑦̃𝑡 + 𝜑𝑦𝑦̂𝑡 𝑛 + 𝜐𝑡 where 𝑟𝑡 𝑛 ≡ 𝜌 + 𝜎(1 − 𝜌𝑎 )𝜓𝑦𝑎𝑎𝑡 + (1 − 𝜌𝑧 )𝑧𝑡 is the natural rate of interest, with 𝜓𝑦𝑎 ≡ 1+𝜑 𝜎(1−𝑎)+𝜑+𝑎 , 𝑦𝑡 𝑛 ≡ 𝜓𝑦𝑎𝑎𝑡 + 𝜓𝑦, with 𝜓𝑦 ≡ (1−𝑎)[𝜇−𝑙𝑛(1−𝑎)] 𝜎(1−𝑎)+𝜑+𝑎 , is the natural (log-)level of output, 𝑦̃𝑡 ≡ 𝑦𝑡 − 𝑦𝑡 𝑛 is the “(New Keynesian) theoretical” output gap, 𝑦̂𝑡 ≡ 𝑦𝑡 − 𝑦 is the “empirical” output gap (expressed as deviation from some empirical measure of the stead𝑦𝑡 𝑛y state y, such as a constant linear or drifting nonlinear trend), and 𝑦̂𝑡 𝑛 ≡ 𝑦𝑡 𝑛 − 𝑦.
The BNKM is assumed to be driven by 3 exogenous stochastic processes, for technology, 𝑎𝑡 , preferences, 𝑧𝑡 , and a “surprise” or “imperfect instrument control” component of monetary policy, 𝜐𝑡 , all assumed AR(1) in log-levels: 𝑎𝑡 = 𝜌𝑎𝑎𝑡−1 + 𝜀𝑡 𝑎 𝑧𝑡 = 𝜌𝑧𝑧𝑡−1 + 𝜀𝑡 𝑧 𝜐𝑡 = 𝜌𝜐𝜐𝑡−1 + 𝜀𝑡 𝜐 , where the 𝜌𝑖 ’s measure persistence and the 𝜀𝑡 𝑖 ’s are zero mean i.i.d. innovations to the respective shock processes.
(a) Show algebraically (in detailed steps) that the BNKM can be alternatively re-written as a system from 3 into 2 equations, in matrix form, and the reduced form (RF) of the equilibrium conditions of the BNKM is then [ 𝑦̃𝑡 𝜋𝑡 ] = Ω[ 𝜎 1 − 𝛽𝜑𝜋 𝜎𝜅 𝜅 + 𝛽(𝜎 + 𝜑𝑦) ] ⏟ 𝐸𝑡 [ 𝑦̃𝑡+1 𝜋𝑡+1 ]+Ω [ 1 𝜅 ] ⏟ 𝑢𝑡 ≡ 𝑨𝑇 ≡ 𝑩𝑇 where the composite shock 𝑢𝑡 is defined as 𝑢𝑡 ≡ 𝑟̂𝑡 𝑛 − 𝜑𝑦𝑦̂𝑡 𝑛 − 𝜐𝑡 = −𝜓𝑦𝑎[𝜑𝑦 + 𝜎(1 − 𝜌𝑎 )]𝑎𝑡 + (1 − 𝜌𝑧 )𝑧𝑡 − 𝜐𝑡 , and Ω ≡ 1 𝜎 + 𝜑𝑦 + 𝜅𝜑𝜋 is a constant.
(b) Why is the reduced form of the BNKM in (a) convenient, that is, what key condition regarding the properties of the model equilibrium can analytically be derived from it in a quite straightforward way? (You do not have to show this derivation, due to Bullard and Mitra (2002).)
(c) What does the system tell us about macro dynamics? More precisely, through which matrix of coefficients do materialized current-period shocks and expectation changes during the current period affect the contemporaneous values of the endogenous variables, the output gap and inflation?
2. Bayesian Model Comparison
(a) Bayesian Model Uncertainty and Marginal Likelihood for a Model. In Bayesian statistics, a model is formally defined by (i) a likelihood function 𝑝(𝑦|𝜃 𝑖 , 𝑀𝑖) for the data (set or sample) 𝑦 given the parameter vector 𝜃 𝑖 and (ii) a prior 𝑝(𝜃 𝑖 |𝑀𝑖) for the parameter vector – but a researcher may have in mind 𝑚 different models, indexed as 𝑀𝑖 , 𝑖 = 1, … , 𝑚, all aiming to explain 𝑦; hence, (iii) the posterior for the parameters computed when model 𝑀𝑖 is taken as the basis can be written in explicit model-conditioning notation 𝑝(𝜃 𝑖 |𝑦, 𝑀𝑖) = 𝑝(𝑦|𝜃 𝑖 , 𝑀𝑖 )𝑝(𝜃 𝑖 |𝑀𝑖 ) 𝑝(𝑦|𝑀𝑖 ) (1) Furthermore, the posterior model probability 𝑝(𝑀𝑖 |𝑦) can be used to assess the degree of support for a model 𝑀𝑖 just using Bayes Rule – in effect, by reversing the conditioning in 𝑝(𝑀𝑖 |𝑦) = 𝑝(𝑦|𝑀𝑖 )𝑝(𝑀𝑖 ) 𝑝(𝑦) (2) into an expression for the marginal likelihood for the data 𝑦 given model 𝑖, 𝑝(𝑦|𝑀𝑖 ), obtained by integrating (1) and rearranging. Show this in two explicit steps of algebra and briefly interpret, beginning with a dense summary (in words, as we did in class) of how Bayesians think about model uncertainty and model comparison.
(b) Bayesian Model Comparison as an Analogy to Hypothesis Testing in Frequentist Econometrics. Using the definitions and results in part (a), explain why – following Geweke (1999) and the notation in our lectures – Bayesian model comparison can be viewed as an analogy to hypothesis testing in frequentist econometrics. Define the related concepts of posterior odd ratios (PostORs), Bayes factors (BFs), and associated Bayes critical values (BCVs).
(c) Kass and Raftery (1995) Criterion: Meaning and Limitations. Define the Kass-Raftery (1995) criterion and discuss its limitations in Bayesian model comparison, as in the lectures.