1. Vector Error Correction and Vector Autoregressive Models
Chapter 13
Vector Error Correction and Vector Autoregressive Models
Walter R. Paczkowski
Rutgers University

2. 13.2 Estimating a Vector Error Correction Model
Chapter Contents
13.1 VEC and VAR Models
13.2 Estimating a Vector Error Correction Model
13.3 Estimating a VAR Model
13.4 Impulse Responses and Variance

3. A priori, unless we have good reasons not to, we could just as easily have assumed that yt is the independent variable and xt is the dependent variable
Our models could be:
Eq. 13.1a
Eq. 13.1b

4. For Eq. 13. 1a, we say that we have normalized on y whereas for Eq. 13
For Eq. 13.1a, we say that we have normalized on y whereas for Eq. 13.1b we say that we have normalized on x

5. We want to explore the causal relationship between pairs of time-series variables
We will discuss the vector error correction (VEC) and vector autoregressive (VAR) models

6. Terminology
Univariate analysis examines a single data series
Bivariate analysis examines a pair of series
The term vector indicates that we are considering a number of series: two, three, or more
The term ‘‘vector’’ is a generalization of the univariate and bivariate cases

7. 13.1
VEC and VAR Models

8. Consider the system of equations:
13.1
VEC and VAR Models
Consider the system of equations:
Together the equations constitute a system known as a vector autoregression (VAR)
In this example, since the maximum lag is of order 1, we have a VAR(1)
Eq. 13.2

9. 13.1
VEC and VAR Models
If y and x are nonstationary I(1) and not cointegrated, then we work with the first differences:
Eq. 13.3

10. 13.1
VEC and VAR Models
Consider two nonstationary variables yt and xt that are integrated of order 1 so that:
Eq. 13.4

11. The coefficients α11, α21 are known as error correction coefficients
13.1
VEC and VAR Models
The VEC model is:
which we can expand as:
The coefficients α11, α21 are known as error correction coefficients
Eq. 13.5a
Eq. 13.5b

12. Let’s consider the role of the intercept terms
13.1
VEC and VAR Models
Let’s consider the role of the intercept terms
Collect all the intercept terms and rewrite Eq. 13.5b as:
Eq. 13.5c

13. Estimating a Vector Error Correction Model
13.2
Estimating a Vector Error Correction Model

14. Estimating a Vector Error Correction Model
13.2
Estimating a Vector Error Correction Model
There are many econometric methods to estimate the error correction model
A two step least squares procedure is:
Use least squares to estimate the cointegrating relationship and generate the lagged residuals
Use least squares to estimate the equations:
Eq. 13.6a
Eq. 13.6b

15. Estimating a Vector Error Correction Model
13.2
Estimating a Vector Error Correction Model
FIGURE 13.1 Real gross domestic products (GDP = 100 in 2000)
13.2.1
Example

16. Estimating a Vector Error Correction Model
13.2
Estimating a Vector Error Correction Model
To check for cointegration we obtain the fitted equation (the intercept term is omitted because it has no economic meaning):
A formal unit root test is performed and the estimated unit root test equation is:
Eq. 13.7
Eq. 13.8

17. Estimating a Vector Error Correction Model
13.2
Estimating a Vector Error Correction Model
FIGURE 13.2 Residuals derived from the cointegrating relationship

18. Estimating a Vector Error Correction Model
13.2
Estimating a Vector Error Correction Model
The estimated VEC model for {At, Ut} is:
Eq. 13.9

19. 13.3
Estimating a VAR Model

20. 13.3
Estimating a VAR Model
The VEC is a multivariate dynamic model that incorporates a cointegrating equation
We now ask: what should we do if we are interested in the interdependencies between y and x, but they are not cointegrated?

21. 13.3
Estimating a VAR Model
FIGURE 13.3 Real personal disposable income and real personal consumption expenditure (in logarithms)

22. The test for cointegration for the case normalized on C is:
13.3
Estimating a VAR Model
The test for cointegration for the case normalized on C is:
This is a Case 2 since the cointegrating relationship contains an intercept term Eq

23. 13.3
Estimating a VAR Model
For illustrative purposes, the order of lag in this example has been restricted to one
In general, we should test for the significance of lag terms greater than one
The results are:
Eq a
Eq b

24. Impulse Responses and Variance Decompositions
13.4
Impulse Responses and Variance Decompositions

25. Impulse Responses and Variance Decompositions
13.4
Impulse Responses and Variance Decompositions
Impulse response functions and variance decompositions are techniques that are used by macroeconometricians to analyze problems such as the effect of an oil price shock on inflation and GDP growth, and the effect of a change in monetary policy on the economy

26. 13.4
Impulse Responses and Variance Decompositions
13.4.1
Impulse Response Functions
Impulse response functions show the effects of shocks on the adjustment path of the variables

27. Impulse Responses and Variance Decompositions
13.4
Impulse Responses and Variance Decompositions
13.4.1a
The Univariate Case
Consider a univariate series: yt = ρyt-1 + vt
The series is subject to a shock of size v in period 1
At time t = 1 following the shock, the value of y in period 1 and subsequent periods will be:

28. Impulse Responses and Variance Decompositions
13.4
Impulse Responses and Variance Decompositions
13.4.1a
The Univariate Case
The values of the coefficients {1, ρ, ρ2, ...} are known as multipliers and the time-path of y following the shock is known as the impulse response function

29. Impulse Responses and Variance Decompositions
13.4
Impulse Responses and Variance Decompositions
FIGURE 13.4 Impulse responses for an AR(1) model yt = 0.9yt-1 + et following a unit shock
13.4.1a
The Univariate Case

30. Impulse Responses and Variance Decompositions
13.4
Impulse Responses and Variance Decompositions
13.4.1b
The Bivariate Case
Consider an impulse response function analysis with two time series based on a bivariate VAR system of stationary variables: Eq

31. Impulse Responses and Variance Decompositions
13.4
Impulse Responses and Variance Decompositions
13.4.1b
The Bivariate Case
The mechanics of generating impulse responses in a system is complicated by the facts that:
one has to allow for interdependent dynamics (the multivariate analog of generating the multipliers)
one has to identify the correct shock from unobservable data
Together, these two complications lead to what is known as the identification problem

32. Impulse Responses and Variance Decompositions
13.4
Impulse Responses and Variance Decompositions
13.4.1b
The Bivariate Case
Consider the case when there is a one–standard deviation shock (alternatively called an innovation) to y:

33. Impulse Responses and Variance Decompositions
13.4
Impulse Responses and Variance Decompositions
13.4.1b
The Bivariate Case

34. Impulse Responses and Variance Decompositions
13.4
Impulse Responses and Variance Decompositions
FIGURE 13.5 Impulse responses to standard deviation shock
13.4.1b
The Bivariate Case

35. Impulse Responses and Variance Decompositions
13.4
Impulse Responses and Variance Decompositions
13.4.1b
The Bivariate Case

36. 13.4
Impulse Responses and Variance Decompositions
13.4.2
Forecast Error Variance Decompositions
Another way to disentangle the effects of various shocks is to consider the contribution of each type of shock to the forecast error variance

37. Impulse Responses and Variance Decompositions
13.4
Impulse Responses and Variance Decompositions
13.4.2a
Univariate Analysis
Consider a univariate series: yt = ρyt-1 + vt
The best one-step-ahead forecast (alternatively the forecast one period ahead) is:

38. Impulse Responses and Variance Decompositions
13.4
Impulse Responses and Variance Decompositions
13.4.2a
Univariate Analysis
In this univariate example, there is only one shock that leads to a forecast error
The forecast error variance is 100% due to its own shock
The exercise of attributing the source of the variation in the forecast error is known as variance decomposition

39. Impulse Responses and Variance Decompositions
13.4
Impulse Responses and Variance Decompositions
13.4.2b
Bivariate Analysis
We can perform a variance decomposition for our special bivariate example where there is no identification problem
Ignoring the intercepts (since they are constants), the one–step ahead forecasts are:

40. Impulse Responses and Variance Decompositions
13.4
Impulse Responses and Variance Decompositions
13.4.2b
Bivariate Analysis
The two–step ahead forecast for y is:

41. Impulse Responses and Variance Decompositions
13.4
Impulse Responses and Variance Decompositions
13.4.2b
Bivariate Analysis
The two–step ahead forecast for x is:

42. Impulse Responses and Variance Decompositions
13.4
Impulse Responses and Variance Decompositions
13.4.2b
Bivariate Analysis
The corresponding two-step-ahead forecast errors and variances are:

43. Impulse Responses and Variance Decompositions
13.4
Impulse Responses and Variance Decompositions
13.4.2b
Bivariate Analysis
This decomposition is often expressed in proportional terms
The proportion of the two step forecast error variance of y explained by its ‘‘own’’ shock is:
The proportion of the two-step forecast error variance of y explained by the ‘‘other’’ shock is:

44. Impulse Responses and Variance Decompositions
13.4
Impulse Responses and Variance Decompositions
13.4.2b
Bivariate Analysis
Similarly, the proportion of the two-step forecast error variance of x explained by its own shock is:
The proportion of the forecast error of x explained by the other shock is:

45. Impulse Responses and Variance Decompositions
13.4
Impulse Responses and Variance Decompositions
13.4.2c
The General 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

46. Key Words

47. dynamic relationships error correction
Keywords
dynamic relationships
error correction
forecast error variance decomposition
identification problem
impulse response functions
VAR model
VEC model

48. Appendix

49. The Identification Problem
A bivariate dynamic system with contemporaneous interactions (also known as a structural model) is written as:
In matrix terms:
Eq. 13A.1

50. The Identification Problem
More compactly, BYt = AYt-1 + Et where:

51. The Identification Problem
A VAR representation (also known as reduced-form model) is written as:
Eq. 13A.2

52. The Identification Problem
In matrix form, this is Yt = Cyt-1 + Vt where:

53. The Identification Problem
There is a relationship between Eq. 13A.1 and Eq. 13A.2 with: