DSGE Modeling

Estimated dynamic stochastic general equilibrium (DSGE) models are now widely used by academics to conduct empirical research macroeconomics as well as by central banks to interpret the current state of the economy, to analyze the impact of changes in monetary or fiscal policy, and to generate predictions for key macroeconomic aggregates. The term DSGE model encompasses a broad class of macroeconomic models that span the real business cycle models of Kydland and Prescott (1982) and King, Plosser, and Rebelo (1988) as well as the New Keynesian models of Rotemberg and Woodford (1997) or Christiano, Eichenbaum, and Evans (2005), which feature nominal price and wage rigidities and a role for central banks to adjust interest rates in response to inflation and output fluctuations. A common feature of these models is that decision rules of economic agents are derived from assumptions about preferences and technologies by solving intertemporal optimization problems. Moreover, agents potentially face uncertainty with respect to aggregate variables such as total factor productivity or nominal interest rates set by a central bank. This uncertainty is generated by exogenous stochastic processes that may shift technology or generate unanticipated deviations from a central bank's interest-rate feedback rule.

The focus of this book is the Bayesian estimation of DSGE models. Conditional on distributional assumptions for the exogenous shocks, the DSGE model generates a likelihood function, that is, a joint probability distribution for the endogenous model variables such as output, consumption, investment, and inflation that depends on the structural parameters of the model. These structural parameters characterize agents' preferences, production technologies, and the law of motion of the exogenous shocks. In a Bayesian framework, this likelihood function can be used to transform a prior distribution for the structural parameters of the DSGE model into a posterior distribution. This posterior is the basis for substantive inference and decision making. Unfortunately, it is not feasible to characterize moments and quantiles of the posterior distribution analytically. Instead, we have to use computational techniques to generate draws from the posterior and then approximate posterior expectations by Monte Carlo averages.

In Section 1.1 we will present a small-scale New Keynesian DSGE model and describe the decision problems of firms and households and the behavior of the monetary and fiscal authorities. We then characterize the resulting equilibrium conditions. This model is subsequently used in many of the numerical illustrations. Section 1.2 briefly sketches two other DSGE models that will be estimated in subsequent chapters.

1.1 A Small-Scale New Keynesian DSGE Model

We begin with a small-scale New Keynesian DSGE model that has been widely studied in the literature (see Woodford (2003) or Gali (2008) for textbook treatments). The particular specification presented below is based on An and Schorfheide (2007). The model economy consists of final goods producing firms, intermediate goods producing firms, households, a central bank, and a fiscal authority. We will first describe the decision problems of these agents, then describe the law of motion of the exogenous processes, and finally summarize the equilibrium conditions. The likelihood function for a linearized version of this model can be quickly evaluated, which makes the model an excellent showcase for the computational algorithms studied in this book.

1.1.1 Firms

Production takes place in two stages. There are monopolistically competitive intermediate goods producing firms and perfectly competitive final goods producing firms that aggregate the intermediate goods into a single good that is used for household and government consumption. This two-stage production process makes it fairly straightforward to introduce price stickiness, which in turn creates a real effect of monetary policy.

The perfectly competitive final good producing firms combine a continuum of intermediate goods indexed by j [member of] [0, 1] using the technology

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

The final good producers take input

prices Pt([j]) and output prices Pt as given. The revenue from the sale of the final good is PtYt and the input costs incurred to produce Yt are ∫10Pt(j)Yt (j)dj. Maximization of profits

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

with respect to the inputs Yt]([j]) implies that the demand for intermediate good j is given by

Yt(j) = (Pt(j) Pt-11/ν Yt (1.3)

Thus, the parameter 1/v represents the elasticity of demand for each intermediate good. In the absence of an entry cost, final good producers will enter the market until profits are equal to zero. From the zero-profit condition, it is possible to derive the following relationship between the intermediate goods prices and the price of the final good:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

Intermediate good j is produced by a monopolist who has access to the following linear production technology:

Y(t(t) = AtNt(j), (1.5)

where At is an exogenous productivity process that is common to all firms and Nt](j) is the labor input of firm j. To keep the model simple, we abstract from capital as a factor or production for now. Labor is hired in a perfectly competitive factor market at the real wage Wt.

In order to introduce nominal price stickiness, we assume that firms face quadratic price adjustment costs

ACt(j) = φ/2 (Pt(j)/Pt-1 - π)2 Yt(j), (1.6)

where / governs the price rigidity in the economy and π is the steady state inflation rate associated with the final good. Under this adjustment cost specification it is costless to change prices at the rate π. If the price change deviates from π, the firm incurs a cost in terms of lost output that is a quadratic function of the discrepancy between the price change and π. The larger the adjustment cost parameter /, the more reluctant the intermediate goods producers are to change their prices and the more rigid the prices are at the aggregate level. Firm j chooses its labor input Nt(j) and the price Pt(j) to maximize the present value of future...