1 / 53

Chapter 13 Vector Error Correction and Vector Autoregressive Models

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

star
Download Presentation

Chapter 13 Vector Error Correction and Vector Autoregressive Models

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. 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. 13.1b 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. 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. 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. 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. 13.2 Estimating a Vector Error Correction Model

  14. 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. 13.2 Estimating a Vector Error Correction Model FIGURE 13.1 Real gross domestic products (GDP = 100 in 2000) 13.2.1 Example

  16. 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. 13.2 Estimating a Vector Error Correction Model FIGURE 13.2 Residuals derived from the cointegrating relationship

  18. 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. 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. 13.10

  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. 13.11a Eq. 13.11b

  24. 13.4 Impulse Responses and Variance Decompositions

  25. 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 • Impulse response functions show the effects of shocks on the adjustment path of the variables 13.4.1 Impulse Response Functions

  27. 13.4 Impulse Responses and Variance Decompositions • 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: 13.4.1a The Univariate Case

  28. 13.4 Impulse Responses and Variance Decompositions • 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 13.4.1a The Univariate Case

  29. 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. 13.4 Impulse Responses and Variance Decompositions • Consider an impulse response function analysis with two time series based on a bivariate VAR system of stationary variables: 13.4.1b The Bivariate Case Eq. 13.12

  31. 13.4 Impulse Responses and Variance Decompositions • 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 13.4.1b The Bivariate Case

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

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

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

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

  36. 13.4 Impulse Responses and 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 13.4.2 Forecast Error Variance Decompositions

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

  38. 13.4 Impulse Responses and Variance Decompositions • 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 13.4.2a Univariate Analysis

  39. 13.4 Impulse Responses and Variance Decompositions • 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: 13.4.2b Bivariate Analysis

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

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

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

  43. 13.4 Impulse Responses and Variance Decompositions • 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: 13.4.2b Bivariate Analysis

  44. 13.4 Impulse Responses and Variance Decompositions • 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: 13.4.2b Bivariate Analysis

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

  46. Key Words

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

  48. Appendix

  49. 13A 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. 13A The Identification Problem • More compactly, BYt = AYt-1 + Et where:

More Related