An Introduction to Macroeconometrics: VEC and VAR Models Modified JJ Vera Tabakova, East Carolina University

Chapter 13: An Introduction to Macroeconometrics: VEC and VAR Models 13.2 Estimating a Vector Error Correction model 13.3 Estimating a VAR Model 13.4 Impulse Responses and Variance Decompositions The next slide shows: (13.2): VAR(1) for stationary x_t and y_t (13.3):VAR(1) in first differences when x_t, y_t are nonstationary and are not cointegrated

13.1 VEC and VAR Models (13.2) (13.3) Principles of Econometrics, 3rd Edition

=>error correction: in parentheses is e_(t-1) The next slide introduces the ECM model: the first equation is the long-run equilibrium between x_t and y_t. The departure from LT equilibrium at t is error e_t which is stationary and has expected value 0: E(e_t)=0 The changes in x and y, written as their first differences , are supposed to eliminate at t the last departure from LT equilibrium at t-1 =>error correction: in parentheses is e_(t-1) Principles of Econometrics, 3rd Edition

13.1 VEC and VAR Models (13.4) (13.5a) Principles of Econometrics, 3rd Edition

Coefficients alpha_11 and alpha_21 are the speed of adjustment coefficients. They multiply the past error e_(t-1) to determine the change in y and x that is needed to reduce the gap between the series at time t In applied research these coefficients HAVE to be of opposite signs and at least one has to be statistically significant Principles of Econometrics, 3rd Edition

By multiplying all terms by their coefficients as it is done on the next slide, you obtain the VEC model defined in my lecture notes on the last-but-one page. The VEC model has first differences of x and y on the lhs. Variables on the rhs are x_t and y_t plus perhaps their lagged first differences. VEC looks like a multivariate ADF model. Principles of Econometrics, 3rd Edition

13.1 VEC and VAR Models (13.5b) (13.5c) Principles of Econometrics, 3rd Edition

13.2 Estimating a Vector Error Correction Model (13.6b) Principles of Econometrics, 3rd Edition

Figure 13.1 Real Gross Domestic Products (GDP) 13.2.1 Example Figure 13.1 Real Gross Domestic Products (GDP) Principles of Econometrics, 3rd Edition

13.2.1 Example (13.7) (13.8) Principles of Econometrics, 3rd Edition

13.2.1 Example (13.9) Principles of Econometrics, 3rd Edition

Figure 13.2 Real GDP and the Consumer Price Index (CPI) 13.3 Estimating a VAR Model Figure 13.2 Real GDP and the Consumer Price Index (CPI) Principles of Econometrics, 3rd Edition

13.3 Estimating a VAR Model (13.10) Principles of Econometrics, 3rd Edition

13.3 Estimating a VAR Model (13.11a) (13.11b) Principles of Econometrics, 3rd Edition

The inference on VAR(VEC) includes impulse response analysis: how a one-time shock of (one st.dev) affects future values of x and y, and how fast the shock effect dissipates (think stimulating the economy) and: Forecast error variance decomposition: how much of variation in x is explained by y and vice-versa, to determine dependence between variables. No dependence=“exogenous” x and y Principles of Econometrics, 3rd Edition

13.4 Impulse Responses and Variance Decompositions 13.4.1 Impulse Response Functions 13.4.1a The Univariate Case The series is subject it to a shock of size ν in period 1. Principles of Econometrics, 3rd Edition

13.4.1a The Univariate Case Figure 13.3 Impulse Responses for an AR(1) model (y = .9y(–1)+e) following a unit shock Principles of Econometrics, 3rd Edition

13.4.1b The Bivariate Case (13.12) Principles of Econometrics, 3rd Edition

13.4.1b The Bivariate Case Principles of Econometrics, 3rd Edition

13.4.1b The Bivariate Case Principles of Econometrics, 3rd Edition

Figure 13.4 Impulse Responses to Standard Deviation Shock 13.4.1b The Bivariate Case Figure 13.4 Impulse Responses to Standard Deviation Shock Principles of Econometrics, 3rd Edition

13.4.2 Forecast Error Variance Decompositions 13.4.2a The Univariate Case Principles of Econometrics, 3rd Edition

13.4.2 Forecast Error Variance Decompositions 13.4.2b The Bivariate Case Principles of Econometrics, 3rd Edition

13.4.2 Forecast Error Variance Decompositions 13.4.2b The Bivariate Case Principles of Econometrics, 3rd Edition

13.4.2 Forecast Error Variance Decompositions 13.4.2b The Bivariate Case Principles of Econometrics, 3rd Edition

13.4.2 Forecast Error Variance Decompositions 13.4.2b The Bivariate Case Principles of Econometrics, 3rd Edition

13.4.2 Forecast Error Variance Decompositions 13.4.2c The General Case The example above assumes that x and y are not contemporaneously related and that the shocks are uncorrelated. There is no identification problem and the generation and interpretation of the impulse response functions and decomposition of the forecast error variance are straightforward. In general, this is unlikely to be the case. Contemporaneous interactions and correlated errors complicate the identification of the nature of shocks and hence the interpretation of the impulses and decomposition of the causes of the forecast error variance. Principles of Econometrics, 3rd Edition

Keywords Dynamic relationships Error Correction Forecast Error Variance Decomposition Identification problem Impulse Response Functions VAR model VEC Model Principles of Econometrics, 3rd Edition

Chapter 13 Appendix Appendix 13A The Identification Problem Principles of Econometrics, 3rd Edition

Appendix 13A The Identification Problem Principles of Econometrics, 3rd Edition

Appendix 13A The Identification Problem Principles of Econometrics, 3rd Edition