Vector Autoregressions IEmpirical Macroeconomics - Lect 1

Dr. Ana Beatriz Galvao

Queen Mary University of London

January 2012

Vector Autoregressions

"This simple framework provides a systematic way tocapture rich dynamics in multiple time series, and thestatistical toolkit that came with VARs was easy to use andinterpret. As Sims (1980) and other argued, VARs held outthe promise of providing a coherent and credible approach todata description, forecasting, structural inference andpolicy analysis" (Stock and Watson, 2001).

All empirical results of this lecture are from Stock and Watson(2001). US Data, 1960-2000, unemployment, ination and fedrate.

Vector Autoregressive Model I

A VAR(p) model represents the process of the m 1 vectorof time series yt = (y1t, y2t, ..., ymt) with autoregressiveorder p:

yt = c+A1yt1 + ...+Apytp + t

where Ai are mm coefcients matrices and c is a m 1 ofintercepts.

t is a m 1 vector of disturbances that have the followingproperties:

E(t) = 0 (mean zero);E(tt) = (full variance-covariance matrix);E(ts) = 0 for s = t (no serial correlation).

Vector Autoregressive Model II

For example, for p = 2 and m = 3: y1ty2t

y3t

=

c1c2

c3

+

a1,11 a1,12 a1,13a1,21 a1,22 a1,23

a1,31 a1,32 a1,33

y1t1y2t1

y3t1

+

a2,11 a2,12 a2,13a2,21 a2,22 a2,23

a2,31 a2,32 a2,33

y1t2y2t2

y3t2

+

1t2t

3t

The implied dynamic mechanism of a VAR(1) I

Suppose that p = 1, so yt = c+A1yt1 + t, the dynamicmechanism starting at t = 1 up to t = t is:

y1 = c+A1y0 + 1y2 = c+A1y1 + 2 = c+A1(c+A1y0 + 1) + 2

= (Im +A1)c+A21y0 +A11 + 2...

yt = (Im +A1 + ...+At11 )c+At1y0 +

t1i=0 A

i1ti

If all eigenvalues of A1have module less than 1, then whent ,

(Im +A1 + ...+At11 )c (Im A1)1c = ,At1 0,

and we can write the RHS sum as an innite sum.

The implied dynamic mechanism of a VAR(1) II

This means that the VAR(1) process can be represented as:

yt = +i=0 Ai1ti

where = E(yt) is the unconditional mean computed as(Im A1)1c.

yt can be decomposed between a deterministic term ()and a stochastic term. The stochastic term is the weightedsum of all past disturbances.

The MA representation of a VAR(p): I

Write a VAR(p) as:

yt = c+ (A1L+A2L2 + ...+ApLp)yt + t.

or:

A(L) = Im A1LA2L2 ...ApLpA(L)yt = c+ t.

Now dene a operator (L) = i=0iLi such that:

(L)A(L) = Im

that is, (L) = A(L)1. This operator only works if theinverse of A(L) exists that requires thatdet(Im A1z ...Apzp) = 0, which also implies that theVAR is stable.

The MA representation of a VAR(p): II

Multiplying the previous VAR(p) representation by (L):

yt = (L)c+(L)t= (i=0i) c+

i=0iti

which is the MA() representation of a VAR(p). Algebra can be used to show that how to compute (L)

out of A(L):

0 = Imi = ij=1ijAj = (Im A1 ...Ap)1c.

And that the MA representation can be simplied to:

yt = + t +1t1 +2t2 + ...

The MA representation of a VAR(p): III

For example, suppose a VAR with p = 2 and m = 2:

yt = c+[

.5 .1

.4 .5

]yt1 +

[0 0.25 0

]yt2 + t.

The coefcients of the MA representation are:

1 = A1 =[

.5 .1

.4 .5

]

2 = 1A1 +A2 = A21 +A2 =[

.29 .1

.65 .29

]

3 = 2A1 +1A2 = A31 +A2A1 +A1A2 =[

.21 .079.566 .21

]

...i = i1A1 +i2A2

Wold Decomposition I

Macroeconomists like to use VARs to represent the empiricaljoint empirical process of macroeconomic time series. Animportant support for this is the Wold decomposition.

The stochastic process of yt is (covariance) stationary if itsmean and variance do not depend on time, that is, they areconstant over time.

A stable VAR(p) process is stationary.Any stationary process xt can be written as a sum of adeterministic (perfectly predictable) and a news component,that is:

yt = t +i=0iti,

where t is white noise and it is called news because it is theerror in forecasting yt using all past information up to t 1:

t yt E(yt|yt1, ...).

Wold Decomposition II

When using a VAR(p), the disturbances can be alsointerpreted as forecast errors, that is, they can also beinterpreted as news.

This means that the VAR(p) is way of obtaining the Wolddecomposition of a time series.

Note that t has m processes that are contemporaneouscorrelated. This means that to name/identify a specicnews (shock), the disturbances need to be transformedsuch that they are not contemporaneously correlated.

The Impulse-Response Function I

The coefcients of the MA() representation of the VAR(1)have the interesting interpretation:

yt+st

= s,

that is, s,ij is the element of s that measures the effect ofone unit increase in the jth variables news at date t for thevalue of the ithvariable at date t+ s for all s 0.

If m = 2, the effect of a shock on 2t at y1t+3 is 3,12. If onewants to compute the cumulative effect up to t+ 3:0,12 + 1,12 + 2,12 + 3,12.

The Impulse-Response Function II

The problem is that t, which are the disturbances of thereduced-form VAR, are contemporaneously correlated. We arenormally interested in shocks that are orthogonal to each other.For uncorrelated shocks, we can compute the effect ofunexpected changes (news) in the variable yj,t on the vector yt+scomputed conditional to all information up to t, that is, theeffect of changes in yj,t on E(yt+s|yjt, yj1t, y1t, yt1, ...) for s 0. A matrix P can be used to create shocks vt from t such that:

vt P1twhere P is a (mm) lower triangular matrix with positiveelements in the diagonal.

The Impulse-Response Function III Matrix P is computed using the Cholesky decomposition

of = PP. The matrix P can be decomposed as:

P AD1/2

where D is a diagonal matrix that has the values of thediagonal of P, and A is a lower triangular matrix that has1s in the diagonal and the remaining elements of P.

The structural shocks ut are dened as:

ut A1t. By denition = ADA, so we can show that:

var(ut) = E(utut) = D,

that is, the shocks ut are not contemporaneous correlatedand their variances are in the diagonal of matrix of matrixD. Note also that var(vt) = Im.

The Impulse-Response Function IV Using m = 3, we can explicitly write:

A =

1 0 0a21 1 0

a31 a32 1

D1/2 =

u1 0 00 u2 0

0 0 u3

D = diag(2u1 , 2u2,

2u3)

P =

u1 0 0a21u1 u2 0

a31u1 a32u2 u3

where P comes from the Cholesky decomposition of

=

21

cov(1, 2) cov(1, 3)cov(1, 2) 22 cov(2, 3)cov(1, 3) cov(2, 3) 23

.

The Impulse-Response Function V Recursive VAR I

To understand better the difference between the structuralshocks ut and the reduced form shocks t, write (m = 3):

Aut = t 1 0 0a21 1 0

a31 a32 1

u1tu2t

u3t

=

1t2t

3t

.

The implied equations (that allow the computation of areuts) are:

u1t = 1tu2t = 2t a21u1tu3t = 3t a31u1t a32u2t.

Recursive VAR II

These equations show that this recursive identicationimplies that y1t, the variable of the rst equation, is notaffected by shocks at other equations at time t (recall0 = Im). The variable of the third equation is affected byshocks on y1t at time t (impact of a31 on y3t) and shocks ony2t (impact of a32). While the second variable is affected byshocks at y1t.

Impulse responses of the recursive VAR I

The response of one-unit shock on ujt on yt+s (the responsemeasures changes in the predictive values of yt at horizons) is:

saj

where aj is a vector taking from column j of matrix A. The effect of a shock on y1t on y2t is:

t = 0; a21t = 1; 1,21a21t = 2; 2,21a21

...

Impulse responses of the recursive VAR II

The problem with unit responses is that depending on thescale of the variable, one-unit (1%, 1 pound) represents adifferent sized shock. A solution is the use of one-standarddeviation shock:

saj

var(uj) = spj

where pj is a column taking from the matrix P (Choleskydecomposition).

Forecast errors I

VAR(p) forecasts can be computed as:

yt+1 = c+A1yt + ...+Apytp1yt+2 = c+A1yt+1 + ...+Apytp2

...yt+s = c+A1yt+s1 + ...+Apytsp.

The sum of the forecasts errors for each step from 1 to s areequivalent to the MA specication of the VAR, that is:

yt+s yt+s = t+s +1t+s1 +2t+s2 + ...+s1t+1.

Forecast errors II

The mean squared error of using yt+s as forecast is:

MSE(yt+s) = E[(yt+s yt+s)(yt+s yt+s)]= E[(t+s +1t+s1 + ...+s1t+1)

(t+s +1t+s1 + ...+s1t+1)]= +11 +2

2 + ...+s1

s1

Variance Decomposition I

Similarly to impulse responses, the variancedecomposition is the effect of a structural shock on thetotal variance of yt+s, or the contribution of uj for thevariation of yt+s for values of s 0

We can write the MSE(yt+s) using ut instead of t. Recall that:

t = Aut = a1u1t + a2u1t + ...+ amumt

where, as before, aj is a vector taking from column j ofmatrix A.

Using the previous expression:

= a1a1var(u1t) + a2a2var(u2t) + ...+ ama

mvar(umt).

Variance Decomposition II

Using this representation, we can compute the MSE(yt+s)using ujt :

MSE(yt+s) = mj=1{var(ujt)[ajaj +1ajaj1 +2ajaj

2 + ...+s1aja

js1]}.

If we want to see the effect of a specic shock j on the totalvariance of yt+s, we can use:

var(ujt)[ajaj +1ajaj1 +2aja

j2

+...+s1ajajs1].

Econometric software normally computes the proportionof the forecast error variance MSE(yt+s) that is explainedby each of the structural shocks (ujt for j = 1, ...,m) for eachhorizon s.

Variance Decomposition III

For example, if m = 3, the proportion of the variation of y2texplained by u1t is:

t = 0; a221(var(u1t))/MSE(y1t)t = 1; (1,21a21)

2(var(u1t))/MSE(y1t+1)

t = 2; (2,21a21)2(var(u1t))/MSE(y1t+2)

...