VARs, SVARs and VECMs

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 • 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 onD(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 ECM: VECM: 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