VARs, SVARs and VECMs

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VARs, SVARs and VECMs

Ibrahim StevensJoint HKIMR/CCBS Workshop

Advanced Modelling for Monetary Policy in the Asia-Pacific Region

May 2004

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Contents

• What is a VAR?

• Advantages and disadvantages of VARs• Number of lags• Identification• Impulse response functions• Variance decomposition• SVARs• VECMs

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What is a VAR? • Every stationary univariate process has an autoregressive

representation (see Wold decomposition):

• Useful if the process can be modelled well using few lags, eg

• must be small relative to xt and be ‘well-behaved’ (eg white noise)

......2211 ntnttt xxxx

tttt xxx 2211

t

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What is a VAR? • Let {xt} be a sequence of vectors rather than scalars and you

have a VAR!

• The B’s are now matrices of coefficients and is a vector of constants. (It is useful to work with few lags)

•Again we wish to be small relative to xt and to be ‘well-behaved’ (eg white noise)

......2211 ntnttt xBxBxBx

tttt xBxBx 2211

t

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Why use VARs

• VARs treat all variables in the system as endogenous

• (Might) avoid ‘incredible identification’ problems • Model the dynamic response to shocks • Can aid identification of shocks - including

monetary policy shocks• Often considered good for short-term forecasting

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Problems with using VARs

• Identification is a big issue– different identification schemes can give very

different results– ditto different lag lengths

• Tend to be over-fitted (too many variables and lags)

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An Example Of Economic Model

• Firms set free prices (f) on the basis of current overall price levels;

• Government sets administered prices (g) on the basis of current overall price level;

• The overall price level is a weighted sum of free and administered prices

So:

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Structural system of interest

ttttt fbfagbg 111212111

ttttt fbgbgaf 212212121

tttpBpA

1

Or,

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tttpBpA

1

t

tt

t

t

t f

gp

bb

bbB

a

aA

2

1

2221

1211

21

12 , , ,1

1

Or,

ttLDpLC )()(


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Where,

ttLDpLC )()(


ILDBLALC )( ,)(

Are polynomials in the lag operator L

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Big Problem 1

We can’t estimate the structural system directly

Instead of:

We estimate the reduced form:ttt

pBpA 1

tttApBAp 1

1

1

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Estimation issue 1

• Which estimation method should we choose?• We can use OLS because all variables are the same

on the RHS for each equation• As long as we use the same number of lags for each

equation• [If not, more efficient to use SUR]

– Near-VAR models

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Estimation issue 2

• Should the variables in the VAR be stationary?• Non-stationary variables might lead to spurious results• But Chris Sims argues that, by differencing or

detrending, a lot of information is lost• And that we are interested in the inter-relationships

between variables, not the coefficients themselves

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Estimation issue 3

• How many lags should we use?• More lags improve the fit …• … but reduce the degrees of freedom and increase the

likelihood of over-fitting

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Overfitting • For a stationary process

• But this does not imply that you improve the model by increasing the number of lags• A big problem is noisy data. By increasing the number of explanatory variables you may be ‘explaining’ noise, not the underlying DGP

0lim where,n1

tt

n

iitit xx

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Estimation issue 3

Two ways round this problem:• Economic theory might suggest appropriate lag

length• Use lag length selection criteria: trade-off parsimony

against fit• Think about the results

– do you believe them?– do they suggest you should seasonally adjust?

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Big Problem 2

• We can move easily from the structural model to the reduced form

• But not the other way• In this example we estimate 7 parameters but the

structural model has 8• Therefore there are an infinite number of ways of

moving from reduced form to the unobserved structural model

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Big Problem 2

• So, 8 unknowns and 7 estimates, hmm• We need to make one of the unknowns known• ie we need to set the value of (at least) one coefficient

[make an identifying restriction]• If we make 1 identifying restriction the system is just

identified• If we make more than 1, it is over identified: we can

then test restrictions

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Unrestricted VARs and SVARs

To summarise: • We estimate an unrestricted (or reduced-form) VAR• By imposing suitable theoretical restrictions we can

recover the restricted (or structural) VAR• The unrestricted VAR is a statistical description of the

data, the SVAR adds some economics

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Unrestricted VARs vs SVARs

• Unrestricted VAR compatible with a lot of theories: produces good short-term forecast

• Structural VAR tied to a particular theory: provides a better interpretation of forecast.

• Long-term forecast is a combination of both• Structural VAR is better for a Central Bank??

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Identifying restrictions

1. Recursive/Wold chain/Cholesky decomposition. Sims’ lag of data availability

2. Coefficient restriction: eg a12=1

3. Variance restriction: eg Var(1)=3

4. Symmetry restriction : eg a12=a21

5. Long-run restrictions, eg nominal shocks don’t affect real variables.

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Example

The g equation of,

is,

tttApBAp 1

1

1

]

[1

1

12121122212

12112112112

ttt

tt

afbba

gbabaa

g

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Example

If a12 =0 we have,

ie the reduced-form error on the equation is the same as the structural error• This is the Cholesky decomposition• We are assuming a recursive ordering of the transmission of fundamental shocks in the system

tttt fbgbg 1112111

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Cholesky decomposition In order to recover the fundamental shocks to the system we restrict the A matrix to be triangular (zeroes above or below the main diagonal)

• We need to be confident about the ordering of the impact of shocks• The frequency of the data becomes a big issue• Nonetheless, probably the most common form of identification

tttpBpA

1

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Cholesky DecompositionTake the system:

Then:

Mt

Yt

t

t

t

t

v

vB

M

YLC

M

YA

1

1)(

nn

ii

nnn

b

b

b

B

aa

aA

00

00

00

1.

.1..

001

000111

11

21

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Monetary Policy

“There is little hope that economists can evaluate alternative theories of monetary policy transmission, or obtain quantitative estimates of the impact of monetary policy changes on various sectors in the economy, if there exists no reasonably objective means of determining the direction and size of changes in policy stance”

Bernanke and Mihov (QJE, 1998)

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Using the Cholesky decomposition

• Huge literature on identifying monetary policy shocks

• Typically use recursive identification schemes• Note that if we are interested in just one of the structural shocks we don’t have to identify the whole system• Nonetheless, controversies abound

– eg what is the monetary policy instrument– what is the information set of the policy maker

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Assessing the stance of MP

• Bernanke and Mihov (1998) use an SVAR to determine the stance of monetary policy• Essentially they estimate an MCI but try to get round the problem of shock identification• They claim they do this by identifying fundamental shocks to policy

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Bernanke and Mihov - method

• They use a block-recursive identification scheme with two blocks: policy and non-policy• They assume that policy has no effect on the non-policy variables in the initial period• They identify shocks to the policy variables (FFR, exchange rate, term spread)• But do not identify the fundamental shocks to the non-policy part of the system

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Bernanke and Mihov – semi-structural VARs

• Take the VAR with variables GDP (Y), CPI (P), price of raw materials (Pcm), Federal Funds Rate (FF), total bank reserves (TR) and non-borrowed reserves (NBR)• They assume:

1. Orthogonality of structural disturbances

2. Macroeconomic variables do not react to changes in the monetary variables

3. Restrictions on the monetary block

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Bernanke and Mihov – semi-structural VAR

For point 3, Bernanke y Mihov assume:

SBBDDNBR

BFFBR

DFFTR

vvvu

vuu

vuu

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Bernanke and Mihov –semi-structural VARTo identify the structural disturbances:

S

D

B

NP

NP

NP

DB

NBR

TR

FF

Pcm

P

Y

v

v

v

v

v

v

u

u

uu

u

u

a

a

a

a

a

aaa

a

a

aa

3

2

1

11

11

63

53

62

52

61

514342

32

41

3121

1

0

00

0

10

0 0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

00010

0001

1

0

00

0

0

1

0 0

0

0

1

01001

0001

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Bernanke and Mihov – semi-structural VAR

• One still needs to make other restrictions which depend on theory

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Blanchard-Quah

• Alternative way of identifying the VAR is to think about variables interaction in the long run• Blanchard and Quah (AER, 1989) advocate an identification scheme where

– supply shocks have permanent effects on real variables

– demand shocks only have temporary effects on real variables

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Blanchard-Quah• Take the structural VAR we are interested in ( Blanchard-Quah use GDP and unemployment)

•As a VMA,

10

01

)var(),cov(

),cov()var(

,,)( 1

tu

tu

ty

tu

ty

ty

u

yzBzLAz

tu

ty

tt

ttttt

tt BLAz 1)(1

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Blanchard-Quah• The VAR we estimate (reduced form) in VMA form

•We must impose restrictions on D(L) to obtain A(L) and B. Assume (1-A(L))-1B=C(L)

tt eLDz )(

)()(

)()()(,)(

2221

1211

LCLC

LCLCLCLCz tt

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Blanchard-Quah• We can see

• If we assume that shocks to GDP are demand shocks and shocks to unemployment are supply shocks, we can assume that GDP shocks will not have a permanent effect on GDP, that is

C11(L) yt=0

• Under this assumption one can recover the demand and supply shocks from the reduced VAR

tu

ty

t

t

LCLC

LCLC

u

y

)()(

)()(

2221

1211

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Using Blanchard-Quah

• Quah and Vahey (EJ, 1995) use the BQ technique to identify core inflation• They define core inflation as that part of inflation that has persistent effects on the price level• So core inflation is like a real variable in the BQ paper and transitory inflation like a nominal variable

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Faust and Rogers (2000) • Investigate:

– How delayed is the overshooting of the exchange rate following a nominal shock

– Test to see how well UIP fares– Ask what proportion of the volatility of the exchange rate can be

explained by MP shocks• Find:

– delay of overshooting is sensitive to assumptions– UIP performs miserably– between 2 and 30% of the volatility is explained by MP shocks

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Faust and Rogers - Technique • Under identify the VAR arguing:

– few variables make identification easier but you have omitted variables

– but you can only be confident about a few restrictions with bigger systems

• Then run the VARs using other possible restrictions• If the impulse responses are similar: fantastic!• Otherwise choose the impulse response you think best fits reality

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Impulse Response Functions

• Perhaps the most useful product of VARs

• Use the moving average representation of a VAR to show the dynamic response of variables to fundamental shocks• Often used to determine the lags of the monetary transmission mechanism, etc

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Example of going from AR to MA

Take a VAR with one lag

nA

AxAx

AxAx

Axx

Axx

i it

i

n

i it

i

nt

n

t

tttt

ttt

ttt

as

....

0

1

0

12

2

121

1

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Impulse response for an AR(1)

• The effect on {xt} of a one-period shock is given by the MA representation• In this case• Here

0 t

tx

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

8.0

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Variance decomposition

• Analyses the relative ‘importance’ of variables

• More precisely, the proportion over time of the variance of a variable due to each fundamental shocks• eg, in the Faust and Rogers paper, they look at what proportion of the volatility of the exchange rate is explained by the various shocks

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General principles of identifying VARs

• Identify what you are interested in

– eg if you simply want a forecast, do you need to identify it at all?

• Think about the economics• Test for robustness• Compare impulse responses - do you believe them?

– eg the ‘price puzzle’

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Cointegration • Recall, a vector of I(n) variables are cointegrated if

there is a (non-trivial) linear combination of them which is I(m) where m < n• Normally we are thinking about combinations of I(1) variables which are I(0)• If they are cointegrated there is an error-correction representation of the variables• In economic terms, variables are cointegrated if there is a long-run equilibrium relationship between them

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VECMs • Vector error correction mechanisms are generalisations of

ECMs

ECM:

VECM:

zt are the equilibrium relationships

111 ttttt xyyxy

11)( ttt CyyLByA

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VECMs

• Like ECMs, VECMs combine short-run (dynamic) information with long-run (static) information• They are like a combination of an unrestricted VAR (the dynamic part) and a structural VAR (the long-run is (should be) consistent with theory)

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Identification Issues

• As with VARs you estimate it with no current dependent variables on the RHS• But theory might suggest that the structural model should include them• Finding the cointegrating vectors (and even knowing how many there are) is a fragile business• If there are n variables in the system there are (n-1) possible cointegrating vectors

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Johansen

• You are testing the rank of the matrix A-1C in the equation below• If it has less than full rank there is at least one cointegrating vector

11)(

ttt CyAyLDy

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Johansen with 2 variables

• If there is an equilibrium relationship between x and z

• The are the ‘loadings’ and gives the equilibrium relationship

1

1

2

11

1 1t

tt z

xCyA

s'

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Estimating a VECM: technicalities

• Lag length of VECM: - Gaussian residuals • Worth considering exogenous short-run variables (eg

dummies, other economic variables)• Inclusion of deterministic components (constant, trend) -

Johansen (1992) suggests:

- Begin with estimation of most restrictive model (no constant, no trends) and estimate less and less restrictive variants until reach least restrictive (constant, trends)

- Compare trace stats to their critical level

-Stop the first time null is not rejected

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Estimating VECM: technicalities

•Identification of cointegrating vectors:

- Johansen test tells you the number of cointegrating vectors; but it does NOT tell you if these are unique

- If one cointegrating vector, then the vector is identified

- If more than one cointegrating vector, we must impose restrictions

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•Imposing restrictions is ‘easy’. These can be imposed as follows: There are r cointegrating vectors (found in matrix ), then

H : =(H11, H22,…,Hrr)

where Hi i=1,…,r are matrices representing the linear economic relationships to be tested and i i=1,…,r are vectors with parameters to be estimated (for each of the cointegrating relationships)

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•If we choose a matrix Ri orthogonal to Hi (Ri`Hi=0) then testing a restriction on the first vector is equivalent to:

rank(R1`( 1, 2,…, r))

= rank(R1`(H11, H22,…,Hrr))

= r-1

ie identification is achieved if applying the restrictions of the first vector to the other r-1 vectors results in a matrix of rank r-1.

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Gonzalo and Ng (2001)

• The ‘loadings’ identify the effects of the disequilibrium terms on the endogenous variables• Show that the orthogonal complement of loadings and cointegrating vectors:

can be used to define Permanent and Transitory shocks• Apply Choletsky, can do variance decomposition

'

'

G

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Textbooks• Hamilton, James D, (1994) ‘Time Series Analysis’, Princeton University Press• Enders, Walter, (1995) ‘Applied econometric time series’, Wiley• Harris, Richard (1995), ‘Cointegration Analysis in Econometric Modelling’, Harvester Wheatsheaf

Identifying Monetary Policy Shocks• Christiano, Lawrence J.; Eichenbaum, Martin; Evans, Charles L, (1999) ‘Monetary Policy Shocks: What

Have We Learned and to What End?’ in Taylor, John B.; Woodford, Michael, eds. Handbook of Macroeconomics. Volume 1A. Handbooks in Economics, vol. 15, North Holland

• Leeper, Eric M.; Sims, Christopher A.; Zha, Tao, (1996) ‘What Does Monetary Policy Do?’, Brookings Papers on Economic Activity; 0(2), 1996, pages 1 63

• Bernanke, Ben S.; Mihov, Ilian (1998) ‘Measuring Monetary Policy’, Quarterly Journal of Economics; 113(3), August 1998, pages 869 902.

Blanchard Quah• Blanchard, Olivier Jean; Quah, Danny. (1989) ‘The Dynamic Effects of Aggregate Demand and Supply

Disturbances’, American Economic Review; 79(4), September 1989, pages 655 73• Quah, Danny; Vahey, Shaun P (1995) ‘Measuring Core Inflation’, Economic Journal; 105(432), September

1995, pages 1130 44

Others• Faust, John and John Rogers, (2003?), ‘Monetary Policy’s Role in Exchange Rate Behaviour’, Journal of

Monetary Economics, forthcoming

Documents

VARs, SVARs and VECMs