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Chapter 13 Vector Error Correction and Vector Autoregressive Models Walter R. Paczkowski Rutgers University Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 1 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 Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 2 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 yt 10 11xt ety , ety ~ N (0, 2y ) Eq. 13.1b xt 20 21 yt etx , etx ~ N (0, 2x ) Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 3 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 Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 4 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 Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 5 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 Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 6 13.1 VEC and VAR Models Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 7 13.1 VEC and VAR Models Consider the system of equations: yt 10 11 yt 1 12 xt 1 vty Eq. 13.2 xt 20 21 yt 1 22 xt 1 vtx – 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) Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 8 13.1 VEC and VAR Models If y and x are nonstationary I(1) and not cointegrated, then we work with the first differences: yt 11yt 1 12 xt 1 vty Eq. 13.3 Principles of Econometrics, 4th Edition xt 21yt 1 22 xt 1 vtx Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 9 13.1 VEC and VAR Models Consider two nonstationary variables yt and xt that are integrated of order 1 so that: Eq. 13.4 Principles of Econometrics, 4th Edition yt 0 1 xt et Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 10 13.1 VEC and VAR Models The VEC model is: yt 10 11 ( yt 1 0 1 xt 1 ) vty Eq. 13.5a xt 20 21 ( yt 1 0 1 xt 1 ) vtx which we can expand as: yt 10 (11 1) yt 1 110 111 xt 1 vty Eq. 13.5b xt 20 21 yt 1 210 ( 211 1) xt 1 vtx The coefficients α11, α21 are known as error correction coefficients Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 11 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: yt (10 110 ) (11 1) yt 1 111 xt 1 vty Eq. 13.5c xt ( 20 210 ) 21 yt 1 ( 211 1) xt 1 vtx Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 12 13.2 Estimating a Vector Error Correction Model Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 13 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 yt 10 11eˆt 1 vty Eq. 13.6b xt 20 21eˆt 1 vtx Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 14 13.2 Estimating a Vector Error Correction Model FIGURE 13.1 Real gross domestic products (GDP = 100 in 2000) 13.2.1 Example Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 15 13.2 Estimating a Vector Error Correction Model Eq. 13.7 To check for cointegration we obtain the fitted equation (the intercept term is omitted because it has no economic meaning): Aˆ 0.985U t t A formal unit root test is performed and the estimated unit root test equation is: Eq. 13.8 Principles of Econometrics, 4th Edition eˆt .128eˆt 1 (tau ) ( 2.889) Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 16 13.2 Estimating a Vector Error Correction Model FIGURE 13.2 Residuals derived from the cointegrating relationship Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 17 13.2 Estimating a Vector Error Correction Model The estimated VEC model for {At, Ut} is: At 0.492 0.099eˆt 1 (t ) (2.077) Eq. 13.9 U t 0.510 0.030eˆt 1 (t ) Principles of Econometrics, 4th Edition (0.789) Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 18 13.3 Estimating a VAR Model Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 19 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? Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 20 13.3 Estimating a VAR Model FIGURE 13.3 Real personal disposable income and real personal consumption expenditure (in logarithms) Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 21 13.3 Estimating a VAR Model The test for cointegration for the case normalized on C is: eˆt Ct 0.404 1.035Yt Eq. 13.10 eˆt 0.088eˆt 1 0.299Δeˆt 1 (tau ) ( 2.873) – This is a Case 2 since the cointegrating relationship contains an intercept term Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 22 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 Principles of Econometrics, 4th Edition ΔCˆ t 0.005 0.215ΔCt 1 0.149ΔYt 1 t 6.969 2.884 2.587 ΔYˆt 0.006 0.475ΔCt 1 0.217ΔYt 1 t 6.122 4.885 Chapter 13: Vector Error Correction and Vector Autoregressive Models 2.889 Page 23 13.4 Impulse Responses and Variance Decompositions Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 24 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 Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 25 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 Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 26 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: t 1, y1 y0 v1 v t 2, y2 y1 v t 3, y3 y2 (y1 ) 2 v ... the shock is v, v, 2 v, Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 27 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 Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 28 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 Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 29 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: yt 10 11 yt 1 12 xt 1 vty Eq. 13.12 xt 20 21 yt 1 22 xt 1 vtx Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 30 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: 1. one has to allow for interdependent dynamics (the multivariate analog of generating the multipliers) 2. one has to identify the correct shock from unobservable data Together, these two complications lead to what is known as the identification problem Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 31 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: Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 32 13.4 Impulse Responses and Variance Decompositions 13.4.1b The Bivariate Case Let v1 y y , vty 0 for t 1, vt x 0 for all t: t 1 y1 v1y y x1 v1x 0 t2 y2 11 y1 12 x1 11 y 12 0 11 y x2 21 y1 22 x1 21 y 22 0 21 y t 3 y3 11 y2 12 x2 1111 y 12 21 y x3 21 y2 22 x2 2111 y 22 21 y ... impulse response to y on y: y {1, 11 , 1111 12 21 , } impulse response to y on x: y {0, 21 , 2111 22 21 , } Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 33 13.4 Impulse Responses and Variance Decompositions FIGURE 13.5 Impulse responses to standard deviation shock 13.4.1b The Bivariate Case Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 34 13.4 Impulse Responses and Variance Decompositions 13.4.1b The Bivariate Case Now let v1x x , vtx 0 for t 1, vty 0 for all t: t 1 y1 v1y 0 x1 vtx x t2 y2 11 y1 12 x1 11 0 12 x 12 x x2 21 y1 22 x1 21 0 22 x 22 x ... impulse response to x on y: x {0, 12 , 1112 12 22 , } impulse response to x on x: x {1, 22 , 2112 22 22 , } Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 35 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 Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 36 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: yt yt 1 vt ytF1 Et [yt vt 1 ] yt 1 Et [ yt 1 ] yt 1 yt vt 1 ytF 2 Et [yt 1 vt 2 ] Et [(yt vt 1 ) vt 2 ] 2 yt yt 2 Et [ yt 2 ] yt 2 2 yt vt 1 vt 2 Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 37 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 Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 38 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: ytF1 Et [11 yt 12 xt vty1 ] 11 yt 12 xt xtF1 Et [21 yt 22 xt vtx1 ] 21 yt 22 xt FE1y yt 1 Et [ yt 1 ] vty1 ; var( FE1y ) 2y FE1x xt 1 Et [ xt 1 ] vtx1 ; var( FE1x ) 2x Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 39 13.4 Impulse Responses and Variance Decompositions 13.4.2b Bivariate Analysis The two–step ahead forecast for y is: ytF 2 Et [11 yt 1 12 xt 1 vty 2 ] Et [11 11 yt 12 xt vty1 12 21 yt 22 xt vtx1 vty 2 ] 11 11 yt 12 xt 12 21 yt 22 xt Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 40 13.4 Impulse Responses and Variance Decompositions 13.4.2b Bivariate Analysis The two–step ahead forecast for x is: xtF 2 Et [21 yt 1 22 xt 1 vtx 2 ] Et [21 (11 yt 12 xt vty1 ) 22 ( 21 yt 22 xt vtx1 ) vtx 2 ] 21 (11 yt 12 xt ) 22 ( 21 yt 22 xt ) Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 41 13.4 Impulse Responses and Variance Decompositions 13.4.2b Bivariate Analysis The corresponding two-step-ahead forecast errors and variances are: FE2y yt 2 Et [ yt 2 ] [11vty1 12 vtx1 vty 2 ] 2 2 2 var( FE2y ) 11 y 12 2x 2y FE2x xt 2 Et [ xt 2 ] [21vty1 22 vtx1 vtx 2 ] var( FE2x ) 2212y 222 2x 2x Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 42 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: 2 2 2 2 2 2 2 2 11 y y 11 y 12 x x – The proportion of the two-step forecast error variance of y explained by the ‘‘other’’ shock is: 2 2 2 2 2 2 2 12 x 11 y 12 x x Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 43 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: 2 2 2 2 2 2 2 2 22 x x 21 y 22 x x The proportion of the forecast error of x explained by the other shock is: 2 2 2 2 2 2 2 21 y 21 y 22 x x Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 44 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 Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 45 Key Words Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 46 Keywords dynamic relationships error correction forecast error variance decomposition identification problem Principles of Econometrics, 4th Edition impulse response functions VAR model VEC model Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 47 Appendix Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 48 13A The Identification Problem A bivariate dynamic system with contemporaneous interactions (also known as a structural model) is written as: yt 1 xt 1 yt 1 2 xt 1 ety Eq. 13A.1 xt 2 yt 3 yt 1 4 xt 1 etx – In matrix terms: 1 1 yt 1 2 yt 1 ety 1 x x x 4 t 1 2 t 3 et Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 49 13A The Identification Problem More compactly, BYt = AYt-1 + Et where: yt 1 1 1 Y B A 2 1 xt 3 Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models 2 ety E x 4 et Page 50 13A The Identification Problem A VAR representation (also known as reducedform model) is written as: Eq. 13A.2 Principles of Econometrics, 4th Edition yt 1 yt 1 2 xt 1 vty xt 3 yt 1 4 xt 1 vtx Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 51 13A The Identification Problem In matrix form, this is Yt = Cyt-1 + Vt where: 1 C 3 Principles of Econometrics, 4th Edition 2 vty V x 4 vt Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 52 13A The Identification Problem There is a relationship between Eq. 13A.1 and Eq. 13A.2 with: C B1 A, Vt B1Et Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector Autoregressive Models Page 53