Transcript VARs, SVARs y VECMs
VARs, SVARs and VECMs
Ibrahim Stevens Joint HKIMR/CCBS Workshop Advanced Modelling for Monetary Policy in the Asia-Pacific Region May 2004 © Bank of England
Contents
• What is a VAR ?
• Advantages and disadvantages of VARs • Number of lags • Identification • Impulse response functions • Variance decomposition • SVARs • VECMs © Bank of England
What is a VAR?
• Every stationary univariate process has an autoregressive representation (see Wold decomposition):
x t
1
x t
1 2
x t
2 ...
n x t
n
...
• Useful if the process can be modelled well using few lags, eg
x t
1
x t
1 2
x t
2
t
• must be small relative to
t
(eg white noise)
x t
and be ‘well-behaved’ © Bank of England
What is a VAR?
• Let
{x
t
}
be a sequence of vectors rather than scalars and you have a VAR!
x t
B
1
x t
1
B
2
x t
2 ...
B n x t
n
...
• The
B’s
vector of constants. (It is useful to work with few lags)
x t
B
1
x t
1
B
2
x t
2
t
‘well-behaved’ (eg white noise)
x t
t
and to be © Bank of England
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 © Bank of England
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) © Bank of England
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: © Bank of England
Structural system of interest
g t
b
11
g t
1
a
12
f t
b
12
f t
1 1
t f t
a
21
g t
b
21
g t
1
b
22
f t
1 2
t
Or,
A p t
B p t
1
t
© Bank of England
Structural system of interest
A p t
B p t
1
t A
1
a
21
a
12 1 ,
B
b
11
b
21
b
12
b
22 ,
p t
g t f t
,
t
1
t
2
t
Or,
C
(
L
)
p t
D
(
L
)
t
© Bank of England
Structural system of interest
C
(
L
)
p t
D
(
L
)
t
Where,
C
(
L
)
A
BL
,
D
(
L
)
I
Are polynomials in the lag operator
L
© Bank of England
Big Problem 1
We can’t estimate the structural system directly Instead of:
A p
B p
t t
1
t
We estimate the reduced form:
p t
A
1
B p t
1
A
1
t
© Bank of England
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 © Bank of England
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 © Bank of England
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 © Bank of England
Overfitting
• For a stationary process
x t
i n
1
i x t
i
t
, where lim
n
t
0
• 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 © Bank of England
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?
© Bank of England
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 © Bank of England
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 © Bank of England
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 © Bank of England
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??
© Bank of England
Identifying restrictions
1. Recursive/Wold chain/Cholesky decomposition. Sims’ lag of data availability 2. Coefficient restriction: eg
a
12 =1 3. Variance restriction: eg Var( 1 )=3 4. Symmetry restriction : eg
a
12 =
a
21 5. Long-run restrictions, eg nominal shocks don’t affect real variables.
© Bank of England
Example
The
g
equation of,
p t
A
1
B p t
1
A
1
t
is,
g t
1 1
a
12
a
21 [
b
11
a
12
b
22
a
12
b
21
g t
1
b
12
f t
1
a
12 2
t
1
t
] © Bank of England
Example
If
a 12 =0
we have,
g t
b
11
g t
1
b
12
f t
1 1
t
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 © Bank of England
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)
A p t
B p t
1
t
• 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 © Bank of England
Cholesky Decomposition
Take the system:
A
Y M t t
C
(
L
)
Y M t
1
t
1
B
v t Y v t M
Then:
A
a a
.
1 21
n
1 .
.
0 1 0 0 1
a nn
1
B
b
0 0 11 0
b ii
0 0 0
b nn
© Bank of England
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) © Bank of England
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 © Bank of England
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 © Bank of England
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 © Bank of England
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 © Bank of England
Bernanke and Mihov – semi-structural VAR
For point 3, Bernanke y Mihov assume:
u TR u BR
u FF
v D
u FF
v B u NBR
D v D
B v B
v S
© Bank of England
Bernanke and Mihov –semi-structural VAR
To identify the structural disturbances: 1
a
21
a
31
a
41
a
51
a
61 1 0 0 0 0 0 0 1 0 0 0 0 0 1
a
32
a
42
a
52
a
62 0 0 1 0 0 0 0 1 0
a
43
a
53
a
63 0 0 0 1 0
B
0 0 0 1 0 1 0 0 0 0
D
0 0 0 1 1 0 1 0 0 0 0 1
u
u
u u u u Y P Pcm FF TR NBR
0 0 1 0 0 1
v v
1
v
2 3
v v v NP NP NP B D S
© Bank of England
Bernanke and Mihov – semi-structural VAR
• One still needs to make other restrictions which depend on theory © Bank of England
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 © Bank of England
Blanchard-Quah
• Take the structural VAR we are interested in ( Blanchard-Quah use GDP and unemployment)
z t
A
(
L
)
z t
1
B
t
,
z t
u t y t
,
t
y t u t
var( cov( •As a VMA,
y
t
,
y t
)
u t
) cov( var(
y t
,
u t
)
u t
) 1 0 0 1
z t
1
A
(
L
) 1
B
t
© Bank of England
Blanchard-Quah
• The VAR we estimate (reduced form) in VMA form
z t
D
(
L
)
e t
•We must impose
restrictions on D(L)
A(L) and B. Assume (1-A(L)) -1 B=C(L) to obtain
z t
C
(
L
)
t
,
C
(
L
)
C C
11 21 ( (
L L
) )
C
12
C
22 ( (
L L
) ) © Bank of England
Blanchard-Quah
• We can see
u t y t
C
11 (
C
21 (
L
)
L
)
C
12
C
22 ( (
L
)
L
)
y u t t
• 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
C 11 (L)
y t =0
• Under this assumption one can recover the demand and supply shocks from the reduced VAR © Bank of England
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 © Bank of England
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 © Bank of England
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 © Bank of England
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 © Bank of England
Example of going from AR to MA
Take a VAR with one lag
x t
Ax x t
1
t
1
t Ax t
2
x t
A
2
x t
2
t
1
A
t
1
t
....
x t
i
0
A n x t
n
n i
1 0
A i
t
i A i
t
i
as
n
© Bank of England
Impulse response for an AR(1)
• The effect on {
x t
} of a one-period shock is given by the MA representation • In this case • Here
x t
0 .
8
t
0 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 © Bank of England
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 © Bank of England
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’ © Bank of England
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 © Bank of England
VECMs
• Vector error correction mechanisms are generalisations of ECMs ECM:
y t
x t
y t
1
y t
1
x t
1 VECM:
A
y t
B
(
L
)
y t
1
Cy t
1
z t
are the equilibrium relationships © Bank of England
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) © Bank of England
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 © Bank of England
Johansen
• You are testing the rank of the matrix
A -1 C
in the equation below • If it has less than full rank there is at least one cointegrating vector
y t
D
(
L
)
y t
A
1
Cy t
1 © Bank of England
Johansen with 2 variables
• If there is an equilibrium relationship between
x
and
z A
1
Cy t
1 1 2 1 relationship
x t z t
1 1 © Bank of England
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 © Bank of England
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 © Bank of England
Estimating VECM: technicalities
•Imposing restrictions is ‘easy’. These can be imposed as follows: There are
r
matrix ), then cointegrating vectors (found in H : =(H 1 1 , H 2 2 ,…,H r r ) where H i
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) © Bank of England
Estimating VECM: technicalities
•If we choose a matrix R i orthogonal to H i (R i
`
H i =0) then testing a restriction on the first vector is equivalent to: rank(R 1
`
( 1 , 2 ,…, r )) = rank(R 1
`
(H 1 1 , H 2 2 ,…,H r r )) =
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.
© Bank of England
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:
G
' ' can be used to define Permanent and Transitory shocks • Apply Choletsky, can do variance decomposition © Bank of England
• • • 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 © Bank of England