1. VARs, SVARs and VECMs Ibrahim Stevens Joint HKIMR/CCBS Workshop
Advanced Modelling for Monetary Policy in the Asia-Pacific Region
May 2004

2. Contents What is a VAR? Advantages and disadvantages of VARs
Number of lags
Identification
Impulse response functions
Variance decomposition
SVARs
VECMs

3. 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)

4. 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)

5. 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

6. 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)

7. 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:

8. Structural system of interest
Or,

9. Structural system of interest
Or,

10. Structural system of interest
Where,
Are polynomials in the lag operator L

11. Big Problem 1 We can’t estimate the structural system directly
Instead of:
We estimate the reduced form:

12. 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

13. 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

14. 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

15. 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

16. 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?

17. 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

18. 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

19. 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

20. 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??

21. 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.

22. Example
The g equation of,
is,

23. 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

24. 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

25. Cholesky Decomposition
Take the system:
Then:

26. 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)

27. 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

28. 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

29. 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

30. 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:
Orthogonality of structural disturbances
Macroeconomic variables do not react to changes in the monetary variables
Restrictions on the monetary block

31. Bernanke and Mihov – semi-structural VAR
For point 3, Bernanke y Mihov assume:

32. Bernanke and Mihov –semi-structural VAR
To identify the structural disturbances:

33. Bernanke and Mihov – semi-structural VAR
One still needs to make other restrictions which depend on theory

34. 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

35. Blanchard-Quah As a VMA,
Take the structural VAR we are interested in ( Blanchard-Quah use GDP and unemployment)
As a VMA,

36. 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)

37. 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

38. 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

39. 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

40. 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

41. 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

42. Example of going from AR to MA
Take a VAR with one lag

43. 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

44. 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

45. 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’

46. 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

47. VECMs Vector error correction mechanisms are generalisations of ECMs
zt are the equilibrium relationships

48. 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)

49. 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

50. 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

51. Johansen with 2 variables
If there is an equilibrium relationship between x and z
The are the ‘loadings’ and gives the equilibrium relationship

52. 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

53. 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

54. Estimating VECM: technicalities
Imposing restrictions is ‘easy’. These can be imposed as follows: There are r cointegrating vectors (found in matrix ), then
H : =(H11, H22,…,Hrr)
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)

55. Estimating VECM: technicalities
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`(H11, H22,…,Hr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.

56. 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

57. IDENTIFICATION
AY(t)=BY(t-1)+CE(t) where E(t) is white noise with Mean 0 and variance-covariance I
Y(t)=A -1BY(t-1)+ A -1CE(t) is the atheoretical VAR
C is diagonal Orthogonal Shocks (N(N-1))/2
A is triangular Causal Ordering (N(N-1))/2
A -1B=P Long run
(I-A -1B)-1A -1C=Q Blanchard and Quah
A -1B=I+R Johansen

58. 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
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
Quah, Danny; Vahey, Shaun P (1995) ‘Measuring Core Inflation’, Economic Journal; 105(432), September 1995, pages
Others
Faust, John and John Rogers, (2003?), ‘Monetary Policy’s Role in Exchange Rate Behaviour’, Journal of Monetary Economics, forthcoming