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Posts Tagged ‘time series’

Dynamic stochastic general equilibrium models for policy analysis

What are DSGE models?

Dynamic stochastic general equilibrium (DSGE) models are used by macroeconomists to model multiple time series. A DSGE model is based on economic theory. A theory will have equations for how individuals or sectors in the economy behave and how the sectors interact. What emerges is a system of equations whose parameters can be linked back to the decisions of economic actors. In many economic theories, individuals take actions based partly on the values they expect variables to take in the future, not just on the values those variables take in the current period. The strength of DSGE models is that they incorporate these expectations explicitly, unlike other models of multiple time series.

DSGE models are often used in the analysis of shocks or counterfactuals. A researcher might subject the model economy to an unexpected change in policy or the environment and see how variables respond. For example, what is the effect of an unexpected rise in interest rates on output? Or a researcher might compare the responses of economic variables with different policy regimes. For example, a model might be used to compare outcomes under a high-tax versus a low-tax regime. A researcher would explore the behavior of the model under different settings for tax rate parameters, holding other parameters constant.

In this post, I show you how to estimate the parameters of a DSGE model, how to create and interpret an impulse response, and how to compare the impulse response estimated from the data with an impulse response generated by a counterfactual policy regime. Read more…

Categories: Statistics Tags:

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…

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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…

Categories: Statistics Tags:

Cointegration or spurious regression?

$$\newcommand{\betab}{\boldsymbol{\beta}}$$Time-series data often appear nonstationary and also tend to comove. A set of nonstationary series that are cointegrated implies existence of a long-run equilibrium relation. If such an equlibrium does not exist, then the apparent comovement is spurious and no meaningful interpretation ensues.

Analyzing multiple nonstationary time series that are cointegrated provides useful insights about their long-run behavior. Consider long- and short-term interest rates such as the yield on a 30-year and a 3-month U.S. Treasury bond. According to the expectations hypothesis, long-term interest rates are determined by the average of expected future short-term rates. This implies that the yields on the two bonds cannot deviate from one another over time. Thus, if the two yields are cointegrated, any influence to the short-term rate leads to adjustments in the long-term interest rate. This has important implications in making various policy or investment decisions.

In a cointegration analysis, we begin by regressing a nonstationary variable on a set of other nonstationary variables. Suprisingly, in finite samples, regressing a nonstationary series with another arbitrary nonstationary series usually results in significant coefficients with a high $$R^2$$. This gives a false impression that the series may be cointegrated, a phenomenon commonly known as spurious regression.

In this post, I use simulated data to show the asymptotic properties of an ordinary least-squares (OLS) estimator under cointegration and spurious regression. I then perform a test for cointegration using the Engle and Granger (1987) method. These exercises provide a good first step toward understanding cointegrated processes. Read more…

Categories: Statistics Tags:

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…

Categories: Statistics Tags:

Unit-root tests in Stata


In most observed series, however, the presence of a trend component results in the series being nonstationary. Furthermore, the trend can be either deterministic or stochastic, depending on which appropriate transformations must be applied to obtain a stationary series. For example, a stochastic trend, or commonly known as a unit root, is eliminated by differencing the series. However, differencing a series that in fact contains a deterministic trend results in a unit root in the moving-average process. Similarly, subtracting a deterministic trend from a series that in fact contains a stochastic trend does not render a stationary series. Hence, it is important to identify whether nonstationarity is due to a deterministic or a stochastic trend before applying the proper transformations.

ARMA processes with nonnormal disturbances

Autoregressive (AR) and moving-average (MA) models are combined to obtain ARMA models. The parameters of an ARMA model are typically estimated by maximizing a likelihood function assuming independently and identically distributed Gaussian errors. This is a rather strict assumption. If the underlying distribution of the error is nonnormal, does maximum likelihood estimation still work? The short answer is yes under certain regularity conditions and the estimator is known as the quasi-maximum likelihood estimator (QMLE) (White 1982).

In this post, I use Monte Carlo Simulations (MCS) to verify that the QMLE of a stationary and invertible ARMA model is consistent and asymptotically normal. See Yao and Brockwell (2006) for a formal proof. For an overview of performing MCS in Stata, refer to Monte Carlo simulations using Stata. Also see A simulation-based explanation of consistency and asymptotic normality for a discussion of performing such an exercise in Stata.

Simulation

Vector autoregression—simulation, estimation, and inference in Stata


Simulation

Let’s begin by simulating a bivariate VAR(2) process using the following specification,

$\begin{bmatrix} y_{1,t}\\ y_{2,t} \end{bmatrix} = \mub + {\bf A}_1 \begin{bmatrix} y_{1,t-1}\\ y_{2,t-1} \end{bmatrix} + {\bf A}_2 \begin{bmatrix} y_{1,t-2}\\ y_{2,t-2} \end{bmatrix} + \epsb_t$

where $$y_{1,t}$$ and $$y_{2,t}$$ are the observed series at time $$t$$, $$\mub$$ is a $$2 \times 1$$ vector of intercepts, $${\bf A}_1$$ and $${\bf A}_2$$ are $$2\times 2$$ parameter matrices, and $$\epsb_t$$ is a $$2\times 1$$ vector of innovations that is uncorrelated over time. I assume a $$N({\bf 0},\Sigmab)$$ distribution for the innovations $$\epsb_t$$, where $$\Sigmab$$ is a $$2\times 2$$ covariance matrix.

I set my sample size to 1,100 and Read more…