Chapter 13 Vector Error Correction and Vector Autoregressive Models

Principles of Econometrics, 4th Edition Chapter 13: Vector Error Correction and Vector

Autoregressive Models

Chapter 13Vector Error Correction and

Vector Autoregressive Models

Walter R. Paczkowski Rutgers University



13.1 VEC and VAR Models13.2 Estimating a Vector Error Correction Model13.3 Estimating a VAR Model13.4 Impulse Responses and Variance

Chapter Contents



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:

210 11 , ~ (0, )y y

t t t t yy x e e N 2

20 21 , ~ (0, )x xt t t t xx y e e N

Eq. 13.1a

Eq. 13.1b



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



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



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



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)

13.1VEC and VAR

Models

10 11 1 12 1

20 21 1 22 1

yt t t t

xt t t t

y y x v

x y x v

Eq. 13.2



If y and x are nonstationary I(1) and not cointegrated, then we work with the first differences:

13.1VEC and VAR

Models

11 1 12 1

21 1 22 1

yt t t t

xt t t t

y y x v

x y x v

Eq. 13.3



Consider two nonstationary variables yt and xt that are integrated of order 1 so that:

13.1VEC and VAR

Models

Eq. 13.4 0 1t t ty x e



The VEC model is:

which we can expand as:

The coefficients α11, α21 are known as error correction coefficients

13.1VEC and VAR

Models

Eq. 13.5a10 11 1 0 1 1

20 21 1 0 1 1

( )

( )

yt t t t

xt t t t

y y x v

x y x v

10 11 1 11 0 11 1 1

20 21 1 21 0 21 1 1

( 1)

( 1)

yt t t t

xt t t t

y y x v

x y x v

Eq. 13.5b



Let’s consider the role of the intercept terms– Collect all the intercept terms and rewrite

Eq. 13.5b as:

13.1VEC and VAR

Models

10 11 0 11 1 11 1 1

20 21 0 21 1 21 1 1

( ) ( 1)

( ) ( 1)

yt t t t

xt t t t

y y x v

x y x v

Eq. 13.5c



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:

13.2Estimating a Vector Error Correction

Model

Eq. 13.6a 10 11 1ˆ yt t ty e v

20 21 1ˆ xt t tx e v Eq. 13.6b




Model

13.2.1Example

FIGURE 13.1 Real gross domestic products (GDP = 100 in 2000)



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:


Model

Eq. 13.7

Eq. 13.8

ˆ 0.985t tA U

1ˆ ˆ.128

( ) ( 2.889)t te e

tau




ModelFIGURE 13.2 Residuals derived from the cointegrating relationship



The estimated VEC model for {At, Ut} is:


Model

Eq. 13.9

1

1

ˆ0.492 0.099( ) (2.077)

ˆ0.510 0.030( ) (0.789)

t t

t t

A et

U et



13.3 Estimating a VAR Model



The VEC is a multivariate dynamic model that incorporates a cointegrating equationWe now ask: what should we do if we are interested in the interdependencies between y and x, but they are not cointegrated?

13.3Estimating a VAR

Model




ModelFIGURE 13.3 Real personal disposable income and real personal consumption expenditure (in logarithms)



The test for cointegration for the case normalized on C is:

– This is a Case 2 since the cointegrating relationship contains an intercept term


Model

Eq. 13.10

1 1

ˆ 0.404 1.035

ˆ ˆ ˆ0.088 0.299Δ( ) ( 2.873)

t t t

t t t

e C Y

e e etau



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:


Model

Eq. 13.11a

Eq. 13.11b

1 1

1 1

ˆΔ 0.005 0.215Δ 0.149Δ

6.969 2.884 2.587ˆΔ 0.006 0.475Δ 0.217Δ

6.122 4.885 2.889

t t t

t t t

C C Y

t

Y C Y

t



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

13.4Impulse Responses

and Variance Decompositions



Impulse response functions show the effects of shocks on the adjustment path of the variables



13.4.1Impulse Response

Functions



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:



13.4.1aThe Univariate Case

1 0 1

2 1

23 2 1

2

1, 2,

3, ( )...

the shock is , , ,

t y y v vt y y v

t y y y v

v v v



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









FIGURE 13.4 Impulse responses for an AR(1) model yt = 0.9yt-1 + et following a unit shock



Consider an impulse response function analysis with two time series based on a bivariate VAR system of stationary variables:



13.4.1bThe Bivariate Case

10 11 1 12 1

20 21 1 22 1

yt t t t

xt t t t

y y x v

x y x v

Eq. 13.12



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






Consider the case when there is a one–standard deviation shock (alternatively called an innovation) to y:









1

1 1

1 1

2 11 1 12 1 11 12 11

2 21 1 22 1 21 22 21

3 11 2 12 2 11 11 12 21

Let , 0 for 1, 0 for all :

1

02 0

0

3

y y xy t t

yy

x

y y

y y

y y

v v t v t

t y v

x vt y y x

x y x

t y y x

3 21 2 22 2 21 11 22 21

11 11 11 12 21

21 21 11 22 21

...impulse response to on : {1, , , }

impulse response to on : {0, , , }

y y

y

y

x y x

y y

y x






FIGURE 13.5 Impulse responses to standard deviation shock






1

1 1

1

2 11 1 12 1 11 12 12

2 21 1 22 1 21 22 22

Now let , 0 for 1, 0 for all :

1 0

2 0

0...impulse response to on : {0,

x x yx t t

y

xt x

x x

x x

x

v v t v t

t y v

x vt y y x

x y x

x y

12 11 12 12 22

22 21 12 22 22

, , }

impulse response to on : {1, , , }xx x





13.4.2Forecast 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





13.4.2aUnivariate Analysis Consider a univariate series: yt = ρyt-1 + vt

– The best one-step-ahead forecast (alternatively the forecast one period ahead) is:

1

1 1

1 1 1 1

22 1 2 1 2

22 2 2 1 2

[ ]

[ ]

[ ] [ ( ) ]

[ ]

t t t

Ft t t t

t t t t t t

Ft t t t t t t t t

t t t t t t t

y y v

y E y v

y E y y y v

y E y v E y v v y

y E y y y v v





13.4.2aUnivariate 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





13.4.2bBivariate 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:

1 11 12 1 11 12

1 21 22 1 21 22

21 1 1 1 1

21 1 1 1 1

[ ]

[ ]

[ ] ; var( )

[ ] ; var( )

F yt t t t t t t

F xt t t t t t t

y y yt t t t y

x x xt t t t x

y E y x v y x

x E y x v y x

FE y E y v FE

FE x E x v FE





13.4.2bBivariate Analysis

The two–step ahead forecast for y is:

2 11 1 12 1 2

11 11 12 1 12 21 22 1 2

11 11 12 12 21 22

[ ]

[ ]

F yt t t t t

y x yt t t t t t t t

t t t t

y E y x v

E y x v y x v v

y x y x






The two–step ahead forecast for x is:

2 21 1 22 1 2

21 11 12 1 22 21 22 1 2

21 11 12 22 21 22

[ ]

[ ( ) ( ) ]

( ) ( )

F xt t t t t

y x xt t t t t t t t

t t t t

x E y x v

E y x v y x v v

y x y x






The corresponding two-step-ahead forecast errors and variances are:

2 2 2 11 1 12 1 2

2 2 2 2 22 11 12

2 2 2 21 1 22 1 2

2 2 2 2 22 21 22

[ ] [ ]

var( )

[ ] [ ]

var( )

y y x yt t t t t t

yy x y

x y x xt t t t t t

xy x x

FE y E y v v v

FE

FE x E x v v v

FE






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:

2 2 2 2 2 2 2 211 11 12y y y x x

2 2 2 2 2 2 212 11 12x y x x






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:

2 2 2 2 2 2 2 222 21 22x x y x x

2 2 2 2 2 2 221 21 22y y x x





13.4.2cThe 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



Key Words



Keywords

dynamic relationshipserror correctionforecast error variance decompositionidentification problem

impulse response functionsVAR modelVEC model



Appendix



A bivariate dynamic system with contemporaneous interactions (also known as a structural model) is written as:

– In matrix terms:

13AThe

Identification Problem

Eq. 13A.11 1 1 2 1

2 3 1 4 1

yt t t t t

xt t t t t

y x y x e

x y y x e

1 2 11

3 4 12

11

yt t t

xt t t

y y ex x e



More compactly, BYt = AYt-1 + Et where:

13AThe


1 21

3 42

1

1

yt t

xt t

y eY B A E

x e



A VAR representation (also known as reduced-form model) is written as:

13AThe


1 1 2 1

3 1 4 1

yt t t t

xt t t t

y y x v

x y x v

Eq. 13A.2



In matrix form, this is Yt = Cyt-1 + Vt where:

13AThe


1 2

3 4

ytxt

vC V

v



There is a relationship between Eq. 13A.1 and Eq. 13A.2 with:

13AThe


1 1, t tC B A V B E

Chapter 13 Vector Error Correction and Vector Autoregressive Models

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Transcript of Chapter 13 Vector Error Correction and Vector Autoregressive Models