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

2. 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

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

4. =>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

5. 13.1 VEC and VAR Models (13.4) (13.5a)
Principles of Econometrics, 3rd Edition

6. 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

7. 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

8. 13.1 VEC and VAR Models (13.5b) (13.5c)
Principles of Econometrics, 3rd Edition

9. 13.2 Estimating a Vector Error Correction Model
(13.6b)
Principles of Econometrics, 3rd Edition

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

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

12. Example
(13.9)
Principles of Econometrics, 3rd Edition

13. 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

14. 13.3 Estimating a VAR Model (13.10)
Principles of Econometrics, 3rd Edition

15. 13.3 Estimating a VAR Model (13.11a) (13.11b)
Principles of Econometrics, 3rd Edition

16. 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

17. 13.4 Impulse Responses and Variance Decompositions
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

18. 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

19. 13.4.1b The Bivariate Case
(13.12)
Principles of Econometrics, 3rd Edition

20. 13.4.1b The Bivariate Case
Principles of Econometrics, 3rd Edition

21. 13.4.1b The Bivariate Case
Principles of Econometrics, 3rd Edition

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

23. 13.4.2 Forecast Error Variance Decompositions
13.4.2a The Univariate Case
Principles of Econometrics, 3rd Edition

24. 13.4.2 Forecast Error Variance Decompositions
13.4.2b The Bivariate Case
Principles of Econometrics, 3rd Edition

25. 13.4.2 Forecast Error Variance Decompositions
13.4.2b The Bivariate Case
Principles of Econometrics, 3rd Edition

26. 13.4.2 Forecast Error Variance Decompositions
13.4.2b The Bivariate Case
Principles of Econometrics, 3rd Edition

27. 13.4.2 Forecast Error Variance Decompositions
13.4.2b The Bivariate Case
Principles of Econometrics, 3rd Edition

28. 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

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

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

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

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