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Estimating the parameters of DSGE models

Introduction

Dynamic stochastic general equilibrium (DSGE) models are used in macroeconomics to model the joint behavior of aggregate time series like inflation, interest rates, and unemployment. They are used to analyze policy, for example, to answer the question, “What is the effect of a surprise rise in interest rates on inflation and output?” To answer that question we need a model of the relationship among interest rates, inflation, and output. DSGE models are distinguished from other models of multiple time series by their close connection to economic theory. Macroeconomic theories consist of systems of equations that are derived from models of the decisions of households, firms, policymakers, and other agents. These equations form the DSGE model. Because the DSGE model is derived from theory, its parameters can be interpreted directly in terms of the theory.

In this post, I build a small DSGE model that is similar to models used for monetary policy analysis. I show how to estimate the parameters of this model using the new dsge command in Stata 15. I then shock the model with a contraction in monetary policy and graph the response of model variables to the shock. Read more…

Categories: Statistics Tags: ,

Long-run restrictions in a structural vector autoregression

\(\def\bfA{{\bf A}}
\def\bfB{{\bf }}
\def\bfC{{\bf C}}\)Introduction

In this blog post, I describe Stata’s capabilities for estimating and analyzing vector autoregression (VAR) models with long-run restrictions by replicating some of the results of Blanchard and Quah (1989). Read more…

Structural vector autoregression models

\(\def\bfy{{\bf y}}
\def\bfA{{\bf A}}
\def\bfB{{\bf B}}
\def\bfu{{\bf u}}
\def\bfI{{\bf I}}
\def\bfe{{\bf e}}
\def\bfC{{\bf C}}
\def\bfsig{{\boldsymbol \Sigma}}\)In my last post, I discusssed estimation of the vector autoregression (VAR) model,

\begin{align}
\bfy_t &= \bfA_1 \bfy_{t-1} + \dots + \bfA_k \bfy_{t-k} + \bfe_t \tag{1}
\label{var1} \\
E(\bfe_t \bfe_t’) &= \bfsig \label{var2}\tag{2}
\end{align}

where \(\bfy_t\) is a vector of \(n\) endogenous variables, \(\bfA_i\) are coefficient matrices, \(\bfe_t\) are error terms, and \(\bfsig\) is the covariance matrix of the errors.

In discussing impulse–response analysis last time, I briefly discussed the concept of orthogonalizing the shocks in a VAR—that is, decomposing the reduced-form errors in the VAR into mutually uncorrelated shocks. In this post, I will go into more detail on orthogonalization: what it is, why economists do it, and what sorts of questions we hope to answer with it. Read more…

Vector autoregressions in Stata

Introduction

In a univariate autoregression, a stationary time-series variable \(y_t\) can often be modeled as depending on its own lagged values:

\begin{align}
y_t = \alpha_0 + \alpha_1 y_{t-1} + \alpha_2 y_{t-2} + \dots
+ \alpha_k y_{t-k} + \varepsilon_t
\end{align}

When one analyzes multiple time series, the natural extension to the autoregressive model is the vector autoregression, or VAR, in which a vector of variables is modeled as depending on their own lags and on the lags of every other variable in the vector. A two-variable VAR with one lag looks like

\begin{align}
y_t &= \alpha_{0} + \alpha_{1} y_{t-1} + \alpha_{2} x_{t-1}
+ \varepsilon_{1t} \\
x_t &= \beta_0 + \beta_{1} y_{t-1} + \beta_{2} x_{t-1}
+ \varepsilon_{2t}
\end{align}

Applied macroeconomists use models of this form to both describe macroeconomic data and to perform causal inference and provide policy advice.

In this post, I will estimate a three-variable VAR using the U.S. unemployment rate, the inflation rate, and the nominal interest rate. This VAR is similar to those used in macroeconomics for monetary policy analysis. I focus on basic issues in estimation and postestimation. Data and do-files are provided at the end. Additional background and theoretical details can be found in Ashish Rajbhandari’s [earlier post], which explored VAR estimation using simulated data. Read more…