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Multivariate Models III
Structural VARs
Review of Methodologies:
Univariate models:
AR, MA, ARIMA models. Must be stationary to model them as ARMA.
Nonstationary models – spurious regressions. Nonstationarity tests (ADF, PP, etc.).
Multivariate models single equation models:
Stationary variables – OLS
Nonstationary variables
–cointegration: residual based nonstationarity tests, ECM.
Multivariate models: VAR
Stationary variables – OLS or UVAR in levels.
Nonstationary variables –cointegration and Johansen MLE method, VECM (CVAR). If
there is more than 1 CI vector, you cannot identify them, must impose identification
restrictions on  .
How do we choose between UVAR, SVAR, CVAR (VECM)?
 If the goal is inference, estimate of parameters (e.g. quantifying SR responses),
must worry about nonstationarity, use VECM. If variables are I(1) and there is no
CI vector, model may be misspecified, respecify it. If there is no theoretical
model that provides a CI vector, then first-difference the variables. You should
also use VECM if your goal is the LR implications of the model.
 If the goal is forecasting or impulse responses (e.g., policy analysis: how variables
respond to policy shocks given policy constraints, CB’s reaction function, how
much i must change to to offset the rise in unemployment rate? ) no need to worry
about nonstationarity. Add sufficient number of lags to remove serial correlation
and make the errors I(0) and proceed to the analysis. Not adding the CI term will
make you lose efficiency, but this will not affect the forecasting or the impulse
responses.
 If variables are correlated with each other, the error terms in the UVAR
will be correlated across equations. To identify shocks, you must make
them orthogonal  uncorrelated between equations. There are two
possibilities:
(i)
Use a recursive VAR (Cholesky decomposition). But this is ad
hoc and results are order-dependent.
(ii)
Impose contemporaneous SR restrictions on SVAR on levels
whether the variables are I(1) or I(0). Add enough lags to get
I(0) errors. With these “identifying assumptions” , correlations
can be interpreted causally. Ex: a Taylor rule sets the interest
rate equal to lagged inflation and unemployment
(=instrumental variable regression) and is the interest rate
equation in the VAR.
(iii)
Impose LR restrictions (Blanchard and Quah). Then you must
have at least one I(1) variable and all series in VAR must be
I(0). First-difference the I(1) variable to run the VAR with the
other I(0) variables (must make sure that first-differencing
makes sense theoretically).
For further details on VAR, UVAR and SVAR see: Stock and Watson,
Vectorautoregressions, JEP (2001).
Ref: Enders Ch.5, Favero Ch.6, Bernanke and Mihov (1998, QJE), Blanchard and Quah
(1989, AER)
Recall the VAR(1) model with 2 variables:
yt  b10  b12 zt  c11 yt 1  c12 zt 1   yt
(1)
SVAR or the Primitive System
(2)
zt  b20  b21 yt  c21 yt 1  c22 zt 1   zt
 it ~ i.i.d (0,  2i ) and cov( y ,  z )  0
Or in matrix form:
(3)
 1 b12   yt  b10   c11 c12   yt 1   yt 
b
      
    
 21 1   zt  b20  c21 c22   zt 1   zt 
More simply:
(4)
BX t  0  1 X t 1   t
To get the VAR in standard form: multiply the equation by inverse B:
B1BXt  B10  B11 X t 1  B1 t ,
(5)
X t  A0  A1 X t 1  et
or:
(6)
 yt   a10   a11
 z   a   a
 t   20   21
a12   yt 1   e1t 

a22   zt 1  e2t 
Error terms are composites of the structural innovations from the primitive system.
(7)
 b12   yt 
 e1t 
 1
1
et  B 1 t    

1   zt 
e2t  (1  b21b12 )  b21
Or
e1t 
 yt  b12 zt

e2t 
b21 yt   zt

where   1  b21b12
The var/covar matrix of the VAR shocks:
  2  12 
.
 1
2
 21  2 
Identification
To get the effect of a structural innovation on the dependent variables we recover the
parameters for the primitive system from the estimated system.
VAR: 9 parameters; SVAR: 10 parameters, underidentified.
A triangular Cholesky decomposition makes the SVAR exactly identified: impose the 0
condition on b12 or b21 .
Problem with this identification restrictions:
 impulse responses depend on the ordering chosen. If the correlation between the
variables is low, it doesn’t matter but unlikely in time series.
 How do we decide on which b should we impose the restrictions? Can be very
counterintuitive.
Sims (1986), Bernanke (1986) proposed using theory to derive the identifying restrictions
in B:
Estimation gives the error terms e defined as et  B 1 t so
 t  Bet .
Consider a VAR(1) with n variables:
BX t  0  1 X t 1   t where B and 1 are (nxn) and 0 is an (nx1) matrices.
X t  B 10  B 11 X t 1  B 1 t  A0  A1 X t 1  et .
To know whether we have an identified VAR or not, count #knowns and #unkowns
Exact identification: order condition
Known estimates: distinct elements of the var/covar matrix:
 12  12


 22
Eee'     21
 .
.

 n1  n 2
.  1n 

.  2n 
n(n  1)
n2  n
n
altogether
distinct elements.
2
2
. . 

.  n2 
Unknown parameters: elements of B and var/cov matrix of structural shocks (pure
shocks):
 Diagonal elements of B are 1 and known, so unknown parameters in B =
n2  n

Pure structural shock’s covariances are zero.
 21 0

0  22
E '    
 .
.

0
 0
. 0 

. 0 
 altogether n unknown elements in cov matrix.
. . 

.  2n 
n2  n
Total number of unknowns = n 2  n  n  n 2 
2
Additional restrictions needed to identify the system:
n2  n n2  n

2
2
n2 
Necessary condition for exact identification
More formally:
et  B 1 t  Eet e't  EB1 t  't ( B1 )'  B 1E( t  't )(B 1 )'
Hence:
  B 1  ( B 1 )'
( nxn)
n2  n
2
knowns
( nxn) ( nxn) ( nxn)
B:
 :
n
n2  n
unkowns unknowns
Ex: Cholesky decomposition:
All elements above the diagonal are zero, thus restricted:
i.e.
n(n  1) n 2  n

2
2
additional restrictions. Thus the system becomes exactly identified.
Ex: Decomposition with a two-variable VAR(1)
t
E (eit )
1
2
3 4
5
1
-0.5 0 -1
0.5 0
0.5 -1
0 -0.5 1
0
e1t
e2 t
Identification restrictions: n=1, because known=2(1+2)/2=3, unknown=22=4. We need
one more restriction. To see this numerically:
E (e1t ) 2
 0.5   22
5
E (e1t e2t )

 0.4   21
5
 12 
 12
 2  12  0.5 0.4
 1


2
 21  2  0.4 0.5
0 
Var ( y )
and   
Var ( z )
 0
We have the relation
  B1 ( B 1 )' or
 1 b12 
.
b21 1 
  BB'
where B  
Hence
0   1 b12  0.5 0.4  1 b21 
Var ( y )

 0
Var ( z ) b21 1  0.4 0.5 b12 1 

4 equations and 4 unknowns: Var ( y ) , Var( z ) , b12 , b21 .
But actually there are 3 equations since the diagonal elements of  are identical.
Var( y )  0.5  0.8b12  0.5b122
0  0.4  0.5b12  0.5b21  0.4b12b21
0  0.4  0.5b12  0.5b21  0.4b12b21
2
Var ( z )  0.5  0.8b21  0.5b21
The 2nd and the third equations are identical. So we need another restriction to solve the
relations:
 yt  e1t  b12e2t
 zt  b21e1t  e2t
(i) Cholesky decomposition: b12  0
We saw before that this leads to an exact identification. Number of knowns=2(3)/2=3.
# unknowns = 22-1=3. The restriction corresponds to a recursive system, where the most
endogenous variable is placed last.
Numerically:
Var ( y )  0.5
 b21  0.8  Var( z )  0.18
0  0.4  0.5b21
Var ( z )  0.5  0.8b21  0.5b
2
21
Since  t  Bet we can now identify the structural shocks:
 yt  e1t
0  e1t 
 yt   1
hence
    0.8 1 e 
 zt  0.8e1t  e2t
  2t 
 zt  
We can then trace back the structural shocks at every point in time:
2
3 4
-0.5 0 -1
E ( it )
5
0.5 0
t
 yt  e1t
1
1
 zt
-0.3 -0.6 0 0.3 0.6 0
(ii) Structural models with contemporaneous coefficient restriction:
ex. b12  1
We had:
 1 b12   yt  b10   c11 c12   yt 1   yt 
b
      
    
 21 1   zt  b20  c21 c22   zt 1   zt 
Justification for the restriction:
Suppose zt /  yt  1 / b12  1 thus b12  1 and we can pin down the first structural shock:
 yt  e1t  e2t
Then we get the remaining unknowns by substituting in:
Var ( y )  0.5  0.8  0.5  1.8
 b21  1   zt
0  0.4  0.5  0.5  0.4b21
 e1t  e2t
Var( z )  0.5  0.8(1)  0.5  0.2
and
 yt   1 1  e1t 
    1 1 e 
  2t 
 zt  
(iii) Symmetry restriction ex: b12  b21 then multiple solution
a. b12  b21  0.5
 0.5  e1t 
   1
  yt   
1  e2t 
 zt   0.5
b. b12  b21  2
   1  2  e1t 
  yt   
 
 zt   2 1  e2t 
(iv) VAR restriction on variances of structural shocks: multiple solutions for coefficients
of B.
ex: Var ( yt )  1.8
Var( y )  0.5  0.8b12  0.5b122  1.8
a. b12  1  b21  1
 b12  (1,2.6)
   1 1  e1t 
  yt   
 
 zt   1 1 e2t 
b. b12  2.6  b21  5 / 3
 2.6  e1t 
   1
  yt   
1  e2t 
 zt   5 / 3
(v) Model restrictions: overidentification
If #restrictions > (n2-n)/2. This does not affect the estimation of the VAR coefficients.
Steps to follow to test the significance of restrictions:
a) Run VAR without additional restrictions and get the var-covar matrix  .
b) Impose the additional restrictions and maximize the likelihood function, get (r ) -restricted—
c) Calculate (r )   , which will be distributed as  2 ( R)
where R=#restrictions-(n2-n)/2. If calculated test statistics less than tabulated values,
do not reject the null (of restrictions).
How did the VARs enter the monetary policy analyses?
After the collapse of confidence in large macro models to analyze policy, Sims (1980)
proposed a methodology free of a structural model. He introduced VAR analysis to study
the monetary transmission mechanism between money and GDP.
VAR with logs of GDP, P level and money (M1 or M2).
Choleski ordering:
zt  ( yt , pt , mt )
Although there is no formal model, the VAR is based on a standard theoretical model :
y is exogenous, only affected by its own shocks (output equation)
p is affected contemporaneously by its own shocks but also by y shocks and it is not
affected by m shocks (from an inflation equation such as PC).
m is affected by its own and the two other shocks (money demand equation).
Subsequent additions to the model:
interest rate i (Fedfund rate), e.g. Leeper, Sims and Zha (1997) estimate a level VAR with
the order:
zt  ( pt , yt , it , mt )
Despite the appeal of this methodology, researchers had to deal with several persistent
counterintuitive results in both closed economy and open economy VAR MODELS:
Closed economy
Liquidity Puzzle: a typical finding was not to find the familiar liquidity effect (an inverse
relation between a money market shock and interest rates). A money market shock
increases the interest rate, instead of decreasing it.
Literature background
Early research: increases in MS reduce interest rates (Gibson 1968, QJE, Cagan
and Gandolfi, 1968 AER P&P).
Research in 1980s no good news: this effect disappears in the 1970s. Tests
started to focus on unanticipated changes in policy –Rational expectations
revolution—.
Findings: Correlations between innovations in M and i.r. positive or
insignificant; announcements of MS larger than expected associated with
increases in ST i.r.
Recent literature mixed:
Findings: eliminating high frequency movements in MS and i.r.
reestablished some LE for subsamples (Cochrane, 89). LE is supported if
(i) there is LR identification consisting of M neutrality (Gali, 92); (ii)
emphasis on bank reserves and the FF rate (Strongin, 95). But an equal
number of studies show that evidence supports the a positive relation
between liquidity and i.r.
Possible reasons for the puzzle:
 If you are using growth of money supply, then a rise in M growth will
increase inflationary expectations, which will push nominal rates up
(Fisher equation). So you must be careful between the level and the
change in the MS.
 This shock can be interpreted as a Money demand shock. Then XD in the
MM will lead to a rise in the interest rate.
 If you are using a too wide definition of M beyond the Fed’s, it may
involve the behavior of the banking sector. So use a narrow monetary
aggregate.
Price Puzzle: a positive interest rate shock (contractionary monetary policy) leads to
persistent rise in the price level. A tightening of MP generally is expected to reduce the P
level, not increase it. Result holds for other countries as well and is especially
pronounced if short-term interest rate is taken to be the policy instrument rather than a
monetary aggregate.
Possible reasons for puzzle:
 Misspecification of the model. Sims (1992, EER): a factor may be
missing in the reaction function of the CB. If the Fed is reacting to a
leading indicator (e.g. commodity price index) and increasing the rates,
then we observe the simultaneous rise in i.r. and P.
Christiano, Eichenbaum, Evans (1996), Sims (1996): price puzzle
disappears.
Hanson (2004, JME) compares various commodity P indices: (i) little
correlation between those indices that forecast best the P inflation and
those that resolve the puzzle; (ii) the indices that solve the puzzle best are
broad ones or financial data as opposed to individual commodity price
indices.

Puzzle may be due to using detrended output instead of output gap
(Giordani, 2004, JME). The monetary policy shock gets contaminated by
other shocks if detrended output is used. The VAR becomes triangular if
output gap is taken.
Non-neutrality of money in the LR: a wide consensus in the literature is that money has
real effects in the short run but it is neutral in the long run, and changes in the MS do not
have a LR effect on the real variables such as output, employment, real interest rates, real
money balances.
Both old and recent studies including VAR analysis (1960s-1990s) finds that changes in
MP lead to long, protracted response in real variables.
Open economy
In addition to the usual closed economy variables, the crucial variable included to the list
is the bilateral exchange rate along with the foreign variables.
Forward-discount puzzle. According to the Uncovered Interest Rate Parity:
it  it * ( Et et 1  et ) , where i=domestic rate, i*=foreign rate, e=spot exchange rate (defined
as the dollar price of the foreign currency, $/for.curr; a rise means depreciation of the
dollar) and Et et 1 expected future spot rate, approximated as the forward rate or the longterm value of e. A positive innovation in i should lead to an impact appreciation (with a
fall in e>fall in the future rate, thus expected depreciation since the term in parentheses
will go up, also called forward discount on the dollar if positive) followed by spot
depreciation over time as i will converge towards i*.
IRs show that a restrictive US monetary shock (interest rate shock) continuously
appreciates the dollar, and increases the spread between i and i* (the term in parentheses
is persistently positive) for two years. This means that there is persistent profit
opportunity and no arbitrage taking in international capital markets. (does this relate to
the carry-trade phenomenon?).
Exchange-rate puzzle: Theory predicts the higher the return on foreign investment, the
higher should be the demand for the currency, which should appreciate and, therefore, the
dollar should depreciate.
VAR results show the opposite: A restrictive foreign monetary policy (i* shock) leads to
an appreciation of the US$.
A. Closed Economy models:
I. SR Model Based Identification
Bernanke and Mihov (1998)
They examine the controversies around the liquidity effect (LE) and LR neutrality of M
(LRN), simultaneously using an SVAR.
Finding: problems occur when innovations to MP are identified with innovations to
monetary aggregates. The existence of LE depends on the identification scheme
employed.
Consider the model with
Y=vector of macroeconomic variables (real GDP, GDP price deflator, PGDP, commodity
prices, PCOM, and real MS, M2/PGDP)
and
P=MP variables (total reserves, TR, nonborrowed reserves, NBR, Fed fund rate, FFR).
Yt  ik0 BiYt i  ik1 Ci Pt i  A y vty
Pt   DiYt i   Gi Pt i  A v
k
i 0
k
i 0
p
p
t
where Y ' gdp
pgdp
pcom m2 / pgdp
P'  TR NBR FF

The vector of policy indicators have information about policy but also affected by macro
variables: P depends on current and lagged values of Y and P and disturbances, one of
them being a MS shock v s . Y depends on its current and lagged values and on lagged
values of P. Note that there is a block exogeneity assumption, in the sense that P does not
enter Y during the same period, while Y (and P) does enter P during the period. This
means that innovations to policy variables do not feedback to the economy
contemporaneously. The v y and v p are the unobservable, structural shocks, which we
want to retrieve. In particular v p  [v d vb v s ] , representing TR demand shock,
disturbance to the borrowing function and shock to the stance of MP, respectively.
We are looking for a way to measure the dynamic responses of variables to v s .
Write the system as a UVAR. Then the policy innovation from UVAR can be written:
utp  ( I  G0 ) 1 A p vtp
or without subscript and superscripts:
u=Gu+Av
This is in SVAR system that relates the observed VAR-based residuals u to unobserved
structural shocks v, including v s . We can identify the system and recover the structural
shocks, in particular MS shock, using the impulse response functions.
For identification, B&M consider a model of the market for commercial bank reserves
and Fed’s operating procedures expressed in innovation form:
(8)
uTR   uFF  v d
Commercial banks’ total demand for reserves: innovation in the demand for total reserves
(TR) falls with innovation in the Fed fund rate (FF) and rises with a demand disturbance.
(9)
uBR   uFF  vb
Portion of reserves banks choose to borrow at the discount window: Innovation in the
demand for BR rises with innovations in the Fed fund rate and a disturbance term.
(10) uNBR   d v d   bvb  v s
Behavior of FRB: The Fed sets the innovation to the supply of NBR. In doing so, it
observes the structural innovations to the demand for TR and BR and responds to
structural shocks to TR, BR and MS.
Moreover, uNBR  uTR  uBR . We want to identify the MS shock.
Rewrite the system after substituting for BR:

1


0



1
 FF
   u 
0   u TR 
1  u NBR 

1

1
0
1
0


 0 1
 b  d



0 vb 
 
0  v d 
1   v s 


We can invert the relation
 u FF   0
 TR  
u   0
u NBR   b




0
1
d
 1 /(   )   v b 
 
 /(   )  v d  (check)
 v s 
1
 
and get: v s  ( d   b )uTR  (1   b )uNBR  (d   b )uFF
# knowns = (n2+n)/2=12/2=6 (estimated from covariances of u’s, the policy block UVAR
residuals)
# unkowns = 4 parameters + 3 structural variances = 7.
The model is underidentified.
1. Overidentifying SR restrictions
(i) Federal funds targeting
The Fed fully offsets shocks to TR demand and BR demand:  b  1 and  d  1
 u FF   0 0  1 /(   )   v B 
 D 
 TR  
 u    0 1  /(   )  v 
 v S 
u NBR    1 1
1
 

 
Thus: v s  (   )uFF . MP shock is proportional to the FF rate shock.
(ii) NBR targeting (Christiano and Eichenbaum)
NBR respond only to policy shocks:  b  0 and  d  0
 u FF   0 0  1 /(   )   v B 
 D 
 TR  
 u    0 1  /(   )  v 
 v S 
u NBR   0 0
1
 

 
Then MP shock equals the NBR shock v s  uNBR
(iii) Strongin identification (1994): orthogonalized NBR
Shocks to reserves are purely demand shocks and the Fed accommodates them (thru
OMO or discount window). Also the Fed does not react to innovations in BR:  b  0 and
 0
 u FF   0 0
 TR  
 u   0 1
u NBR   0  d

 
 1 /  v B 
 
0  v D 
1   v S 
Innovations to MP are: v s   d uTR  uNBR
(iv) BR control (Cosimano Sheehan, 1994)
The Fed was targeting BR for some periods:  b   /  and  d  1 
 u FF   0
 TR  
u   0
u NBR    / 

 
v s
0
u FF
0  1 /(   )   v B 
 
1  /(   )  v D 
 v S 
1
1
 
Innovations to MP: v s  (1   /  )(uTR  u NBR )  (1   /  )uBR are proportional to the
negative of BR innovations.
2. Just identifying SR restriction:   0
Strongin: demand for TR is inelastic to FFR in the SR 
 u FF   0 0
 TR  
u   0 1
u NBR    b  d

 
uTR
0
u FF
 1 /  v B 
 
0  v D 
1   v S 
v s  ( d   b )uTR  (1   b )u NBR   buFF
The previous restrictions were overidentifying the model. Their test is done according to
the overidentifying restriction.
With a just-identified model, you can check how well the parameters estimated compare
with the predictions of the alternative models.
II. LR Model based Identification:
Blanchard and Quah (1989) decomposition
This is along the lines of Beveridge-Nelson decomposition of real GDP into permanent
and temporary components. With a univariate model, BN decomposition was not unique.
B&Q show how to decompose real GDP and recover two pure shocks (identify VARs) by
assuming that the LR demand shocks have zero impact on output.
For this, the identifying restriction is that the cumulative impulse response of output
growth to demand innovation is zero.
Start with a VAR:
yt  ip1 Ai yt i  Bvt
where y  {LY ,UN } growth rate of GDP and unemployment, both I(0) and v={v1, v2}
where v1 is an AD shock and v2 is an AS shock.
B&Q decompose the shock into a temporary and a permanent one by assuming that the
effect of the demand shock on output is temporary (dies out in the LR) and the effect of
supply shock is permanent. But both the demand and the supply shocks are not
observable. Hence, they propose to recover them from the VAR estimation.
Derive the LR effect of the structural shocks on the variables.
yt  ( I  ip1 Ai L) 1Bvt   1 Bvt
Get the coefficients of  from estimates and impose LR restrictions. Note that for  to
be invertible VAR must have stationary variables.
B&Q used quarterly data, 8 lags, trend in UN:
LYt 
 LYt 1 
 LYt 8 
1  b11 b12  v1t 
 
 UN   A1 UN   ...  A8 UN   A9    b
t 
t 1 
t 8 
 t   21 b22  v2t 





b  v 
1  b
 I  ip1 Ai L  11 12  1t 
 b21 b22  v2t 
k  b
b  v 
k
  11 12  11 12  1t 
 k21 k22  b21 b22  v2t 
B&Q identification: the LR impact of demand shocks on output are zero.
If v1t is demand shock, then:
k11b11  k12b21  0 and L=1.
LR  The sum of all coefficients at every lag is zero.
 Cholesky: No restrictions on dynamics, but a restriction on the contemporaneous
coefficients + and B is an identity matrix.
Application
Data: B&Q series, Favero Ch.6 bq.wks
LY~I(1) and USUNRATE~I(0)
Run VAR with DLY, USUNRATEE
Estimate an UVAR with 2 variables 8 lags, a constant and a trend.
Proc Estimate Structural Factorization
Eviews assumes that structural innovations are orthonormal (covar matrix is an identity matrix)
Identifying restrictions: you can choose SR or LR restrictions, not both. You can specify by text or
matrix. For the latter you need to create a “pattern” matrix and set the coefficients values. Do this if you
have a large model with lots of constraints. Otherwise go with “by text”.
By text
We have:
LYt 
 k11 k12  b11 b12  v1t 

 
 UN     k
t 
 21 k22  b21 b22  v2t 

 C C2  v1t 
 
  1
 C4 C3  v2t 
In Eviews notation,
X=Cv
Av = Bu
(Note that A and B are switched compared to our notation:
X t  B 10  B 11 X t 1  B 1 t  e  B A where A is identity matrix)
1
Here v is the vector of observable errors, u is the vector of pure shocks and
C  ( I  A1  A2  ....) A1B  (1) A1B is the estimated accumulated responses to
observed shocks. Long-run restrictions are imposed by setting elements of C to zero.
We want to set the LR response of the first variable to the first structural shock (demand
shock) to zero: C11  0 (here: C1  0 ).
In the text window, type:
@lr1(@u1)=0
@lr# : the symbol # stands for the response variable that we want to restrict. If you wanted to restrict the
response of the 2nd variable to the first shock then write: @lr2(@u1)
(@u&) : the impulse & that you want to restrict the effect on the variable specified in #.
=0 : to how much you want to restrict the response of # to an impulse in &.
Structural VAR Estimates
Date: 05/03/07 Time: 09:32
Sample (adjusted): 1953Q2 1990Q4
Included observations: 151 after adjustments
Estimation method: method of scoring (analytic derivatives)
Convergence achieved after 7 iterations
Structural VAR is just-identified
Model: Ae = Bu where E[uu']=I
Restriction Type: long-run text form
Long-run response pattern:
0
C(2)
C(1)
C(3)
C(1)
C(2)
C(3)
Log likelihood
Estimated A matrix:
1.000000
0.000000
Estimated B matrix:
0.845159
-0.248859
Coefficient
Std. Error
z-Statistic
Prob.
4.677249
0.441142
-1.585484
0.269145
0.025385
0.391411
17.37815
17.37815
-4.050694
0.0000
0.0000
0.0001
-186.6746
0.000000
1.000000
0.286935
0.154010
In the literature, it is common to assume that structural innovations have a diagonal covar matrix. To
compare these results to the literature, divide each column of B by the element on its diagonal.
b12 / b22 
 1

B  
1 
 b21 / b11
Impulse response function
Consider a SVAR(p) with m variables:
A0 yt  ip1 Ai yt i  Bvt
Get the MA representation to look at the impulse response function, historical
decomposition and forecasting error variance decomposition
A0  A( L)yt  Bvt
assuming that the LHS matrix is invertible, we get:
yt  C ( L)vt where C( L)  [ A0  A( L)]1 B
1
yt  C0vt  C1vt 1  ...  Cs vt s where C0  A0 B
yt  s
 Cs
vt
IRF: The effect of a shock to jth element on the ith element when all the other shocks are
kept constant.
yi ,t  s
v j ,t
 Ci , j , s
This can be calculated if the shocks are orthogonal, such as the structural shocks. Not
possible with the VAR shocks, which are correlated (must impose an identification
scheme such as Choleski).
B. Identification in Open Economy VAR models
Eichenbaum and Evans (1995)
zt  ( yt , pt , nbrx or it , yt *, pt *, it *, et or qt ) were i=FF, nbrx=nonborrowed/total reserves.
Use i for MP shocks if the CB is interest rate targeting, nbrx if it is targeting the MS.
A triangular VAR:
A0 zt  ip1 Ai zt i  Bvt
The B matrix is diagonal and A is lower triangular.
Results:




Forward-discount puzzle.
Same response of US P and Y to monetary shocks as in closed economy.
Exchange-rate puzzle.
The response of q to US and foreign MP similar to the response of e,
suggesting sluggishness in prices.
Problems with the model structure:
 Missing commodity p level series  same misspecification problem as
before.
 Recursive structure is unsatisfactory. It assumes that there is not
contemporaneous change in any other variables to e.r. shock. Wrong if
the countries are intervening in the FX markets.
Kim and Roubini (1998):
Impose structural identification to resolve the last problem, include oil prices to resolve
the first one.
Model for a foreign economy:
z t  (opwt , ff t , y t *, pt *, mt *, it *, et ) , opw=world index of oil prices, ff=fed funds rate, m=M0
or M1, p and y log of CPI and IP.
Monthly data 1974-1992.
A0 z t  ip1 Ai z t i  Bv t
B is a diagonal matrix,
1
a
 21
a 31

A0  a 41
 0

a 61
a
 71
0
1
0
0
0
0
a 72
0
0
1
a 43
a 53
0
a 73
0
0
0
1
a 54
0
a 74
0
0
0
0
1
a 65
a 75
0
0
0
0
a 56
1
a 76
0 
0 
0 

0  and
0 

a 67 
1 
z t  (opwt , ff t , y t *, pt *, mt *, it *, et )
Model is overidentified but passes the identifying restrictions.
Identifying restrictions:
The US econ is exogenous. The US Y and P not in the VAR.
The e.r. does not enter the Fed’s reaction function.
Feedback between the US and foreign MP: US MP enters foreign MP through its
interaction with the e.r.
The e.r. contemporaneously reacts to all other shocks.
Results:
Puzzles disappear.
Most of the closed economy results are replicated by impulse response functions.
But questions remain: only two parameters are precisely estimated (a56=dm*/dy* and
a72=de/dff), and the meaning of a56 is not clear (S shock or D shock?). Hanson’s
comments for closed economy and liquidity puzzle still remains: what resolves the
puzzle?
Cushman and Zha (1997, JME) introduce the trade sector for Canadian economy.
Smets (1997 BIS WP) exclude US variables and commodity prices but impose BQ
identification scheme: in the LR the effect of demand shocks on output is zero.
Other questions/controversies:
Which choice of policy instrument? i, m or other?
Bernanke and Blinder (1992, AER): fedfund rate (=market interest rate for overnight US$
loans by banks).
Eichenbaum (1992, EER): M0 because it has a less significant price-puzzle.
There are also non-VAR measures of MP shocks
Info from financial markets. Ex:
shocks=FFrate at t – 30-day FF future at t-1
Availability of data problems; shocks may reflect both policy surprises and info
surprises.
Shocks: change in 3-mo interest rates at the same time as MP announcements occurred.
C. Identification in CVAR (Skip)
Some examples: Mellander, Vredin and Warne (1993), Vredin (199?), Favero, Giavazzi,
Spaventa(1996)
yt  yt 1  Bvt
where   ' and Bvt  ut .
Identification of the CI vector is different from identification of the structural shocks.
Suppose   (1  1) .
yt   11 
 yt 1   b11 b12  v1t 
 
x     1  1 x    b
 t   21 
 t 1   21 b22  v2t 
Redefine y as y-x, and impose identification constraints distinguishing between transitory
and permanent shocks on the LR responses (ex. assume the v2 is the transitory shock and
proceed as in B&Q).
Application: An open economy VAR
Favero Ch 6., workfile: berlin.wf
We will first look at the UVAR, see that there are the puzzles. We will then add an
omitted variable and dummies for structural breaks, and show that the puzzles disappear.
Finally, we will impose some short-run restrictions and estimate an SVAR.
1. UVAR
VAR (varopen0):
We will run a VAR on levels using US real GDP, USCPI, German real GDP, German
call money rate, exchange rate (DM per 1$US: a rise is an appreciation of the $). All
variables are in logs, except the interest rates:
y (lusy), p (luscp), i (usff), y* (lgery), p* (lgercp), i* (gercmr), e (lusdm)
Lags: 6
UVAR
Exogenous variable: constant.
zt  ( yt , pt , nbrx or it , yt *, pt *, it *, et or qt )
Impulse response functions:
Periods: 50
Response standard errors: analytic
Impulse definition: Cholesky, degree of freedom adjusted.
Graph: impresvaropen0
We are interested in the impulse responses to 1 standard deviation US monetary policy
shock and the German monetary policy shock (i.e. responses to usff and gercmr).
Findings similar to the literature:
 A restrictive US monetary shock (interest rate shock) appreciates the
dollar instantaneously as the theory predicts but instead of depreciating it
immediately, it keeps on appreciating it for the following couple of
periods. The appreciation in our case picks of later, increasing the spread
between i and i* in the long-run= forward-discount puzzle.
 Price puzzle in both the US and Germany: an interest rate shock leads to
an increase in domestic price level.
 A restrictive German MP (i* shock) leads to an appreciation of the US$ =
exchange-rate puzzle (although the magnitude is quite small here).
 Same U-shaped response of US output to a US MP shock.
 We also see that German variables (i* and P*) are highly responsive to the
US monetary policy but the opposite is not true. The US MP is exogenous
to foreign policies.
Response to Cholesky One S.D. Innovations ± 2 S.E.
Response of LUSY to GERCMR
Response of LUSY to USFF
.008
.008
.006
.006
.004
.004
.002
.002
.000
.000
-.002
-.002
-.004
-.004
-.006
-.006
5
10
15
20
25
30
35
40
45
5
50
Response of LUSCP to USFF
10
15
20
25
30
35
40
45
50
Response of LUSCP to GERCMR
.004
.004
.002
.002
.000
.000
-.002
-.002
-.004
-.004
5
10
15
20
25
30
35
40
45
5
50
10
15
20
25
30
35
40
45
50
Response of USFF to GERCMR
Response of USFF to USFF
.5
.5
.4
.4
.3
.3
.2
.2
.1
.1
.0
.0
-.1
-.1
-.2
-.2
-.3
-.3
5
5
10
15
20
25
30
35
40
45
10
15
20
25
30
35
40
45
50
50
Response of LGERY to USFF
Response of LGERY to GERCMR
.020
.020
.015
.015
.010
.010
.005
.005
.000
.000
-.005
-.005
-.010
-.010
5
10
15
20
25
30
35
40
45
5
50
Response of LGERCP to USFF
15
20
25
30
35
40
45
50
Response of LGERCP to GERCMR
.004
.004
.003
.003
.002
.002
.001
.001
.000
.000
-.001
-.001
-.002
-.002
-.003
10
-.003
-.004
5
10
15
20
25
30
35
40
45
50
-.004
5
10
15
20
25
30
35
40
45
50
Response of GERCMR to GERCMR
Response of GERCMR to USFF
.3
.3
.2
.2
.1
.1
.0
.0
-.1
-.1
-.2
-.2
-.3
5
-.3
5
10
15
20
25
30
35
40
45
10
15
20
25
30
35
40
45
50
50
Response of LUSDM to USFF
Response of LUSDM to GERCMR
.03
.03
.02
.02
.01
.01
.00
.00
-.01
-.01
-.02
-.02
-.03
-.03
5
5
10
15
20
25
30
35
40
45
50
10
15
20
25
30
35
40
45
50
Do results change when we include missing variables?
 raw material price index
 structural breaks
Add pcm first in the Cholesky ordering and three break points.
lpcm lusy luscp usff lgery lgercp gercmr lusdm
Lags: 6
UVAR: varopen
Most of the puzzles disappear:
 A restrictive US monetary shock (interest rate shock) appreciates the
dollar instantaneously and starts to depreciate it immediatel as the theory
predicts. Forward-discount puzzle disappears.
 Price puzzle disappears also in both the US and in particular Germany.
 Although we can still see it in the short run, the exchange rate puzzle
disappears in the long-run.
Response to Cholesky One S.D. Innovations ± 2 S.E.
Response of LPCM to USFF
Response of LPCM to GERCMR
.02
.02
.01
.01
.00
.00
- .01
- .01
- .02
5
10
15
20
25
30
35
40
45
50
- .02
5
Response of LUSY to USFF
.00 4
10
15
20
25
30
35
40
45
50
Response of LUSY to GERCMR
.00 4
.00 3
.00 3
.00 2
.00 2
.00 1
.00 1
.00 0
.00 0
- .00 1
- .00 1
- .00 2
- .00 2
- .00 3
- .00 3
- .00 4
5
10
15
20
25
30
35
40
45
50
- .00 4
5
Response of LUSCP to USFF
.00 3
.00 3
.00 2
.00 2
.00 1
.00 1
.00 0
.00 0
- .00 1
- .00 1
- .00 2
- .00 2
- .00 3
10
15
20
25
30
35
40
45
50
Response of LUSCP to GERCMR
- .00 3
- .00 4
- .00 4
5
10
15
20
25
30
35
40
45
50
5
Response of USFF to USFF
10
15
20
25
30
35
40
45
50
Response of USFF to GERCMR
.4
.4
.3
.3
.2
.2
.1
.1
.0
.0
- .1
- .1
- .2
- .2
- .3
5
10
15
20
25
30
35
40
45
50
- .3
5
Response of LGERY to USFF
.00 8
10
15
20
25
30
35
40
45
50
Response of LGERY to GERCMR
.00 8
.00 4
.00 4
.00 0
.00 0
- .00 4
- .00 4
- .00 8
5
10
15
20
25
30
35
40
45
50
Response of LGERCP to USFF
- .00 8
5
15
20
25
30
35
40
45
50
Response of LGERCP to GERCMR
.00 2
.00 2
.00 1
.00 1
.00 0
.00 0
- .00 1
- .00 1
- .00 2
10
- .00 2
5
10
15
20
25
30
35
40
45
50
Response of GERCMR to USFF
5
10
15
20
25
30
35
40
45
50
Response of GERCMR to GERCMR
.3
.3
.2
.2
.1
.1
.0
.0
- .1
- .1
- .2
- .2
- .3
- .3
- .4
5
10
15
20
25
30
35
40
45
5
50
Response of LUSDM to USFF
.02
10
15
20
25
30
35
40
45
50
Response of LUSDM to GERCMR
.02
.01
.01
.00
.00
- .01
- .01
- .02
5
10
15
20
25
30
35
40
45
50
- .02
5
10
15
20
25
30
35
40
45
50
2. Short-Run restrictions: SVAR
After running the UVAR(6) with sample 1970.1-2004.12 and dummies, click
proc
structural factorization
then type in your restrictions for each equation (7 restrictions here), such as
@e1=c(1)*@u1 (innovations to pcm are exogenous: they are not affected by the other
innovations)
@e1=the first elements of the e vector
@e2=the second element of the e vector
@u1=the first element of the u vector … so on.
Imposing SR restrictions a la Bernanke and Mihov:
@e1=c(1)*@u1
@e2=-c(2)*@e1+c(3)*@u2
@e3=-c(4)*@e1-c(5)*@e2+c(6)*@u3
@e4=-c(7)*@e1-c(8)*@e2-c(9)*@e3+c(10)*@u4
@e5=-c(11)*@e1+c(12)*@u5
@e6=-c(13)*@e1-c(14)*@e5+c(15)*@u6
@e7=-c(16)*@e1-c(17)*@e4-c(18)*@e5-c(19)*@e6+c(20)*@u7
@e8=-c(21)*@e1-c(22)*@e2-c(23)*@e3-c(24)*@e4-c(25)*@e5-c(26)*@e6c(27)*@e7 +c(28)*@u8
Structural VAR Estimates
Date: 12/18/07 Time: 14:26
Sample (adjusted): 1976M01 1997M11
Included observations: 263 after adjustments
Estimation method: method of scoring (analytic derivatives)
Convergence achieved after 18 iterations
Structural VAR is over-identified (8 degrees of freedom)
Model: Ae = Bu where E[uu']=I
Restriction Type: short-run text form
@e1=c(1)*@u1
@e2=-c(2)*@e1+c(3)*@u2
@e3=-c(4)*@e1-c(5)*@e2+c(6)*@u3
@e4=-c(7)*@e1-c(8)*@e2-c(9)*@e3+c(10)*@u4
@e5=-c(11)*@e1+c(12)*@u5
@e6=-c(13)*@e1-c(14)*@e5+c(15)*@u6
@e7=-c(16)*@e1-c(17)*@e4-c(18)*@e5-c(19)*@e6+c(20)*@u7
@e8=-c(21)*@e1-c(22)*@e2-c(23)*@e3-c(24)*@e4-c(25)*@e5-c(26)*@e6-c(27)*@e7
+c(28)*@u8
where
@e1 represents LPCM residuals
@e2 represents LUSY residuals
@e3 represents LUSCP residuals
@e4 represents USFF residuals
@e5 represents LGERY residuals
@e6 represents LGERCP residuals
@e7 represents GERCMR residuals
@e8 represents LUSDM residuals
C(2)
C(4)
C(5)
C(7)
C(8)
C(9)
C(11)
C(13)
C(14)
C(16)
C(17)
C(18)
C(19)
C(21)
C(22)
C(23)
C(24)
C(25)
C(26)
C(27)
C(1)
C(3)
C(6)
C(10)
C(12)
C(15)
C(20)
C(28)
Coefficient
Std. Error
z-Statistic
Prob.
-0.018207
-0.005520
-0.027026
3.395368
-26.49765
-30.85571
0.058473
-0.013780
-0.012357
0.256565
-0.017476
-2.672097
25.21627
0.236355
-0.687508
-0.485800
-0.006826
-0.010322
-1.808124
-0.008052
0.019031
0.006298
0.002037
0.596943
0.015340
0.002430
0.300670
0.023614
0.020406
0.006611
0.019948
1.939606
5.864707
18.06631
0.049702
0.007893
0.009767
0.985718
0.029691
1.212285
7.630252
0.077822
0.240832
0.718622
0.002441
0.096085
0.611578
0.004843
0.000830
0.000275
8.88E-05
0.026028
0.000669
0.000106
0.013110
0.001030
-0.892228
-0.834976
-1.354834
1.750545
-4.518153
-1.707914
1.176474
-1.745789
-1.265141
0.260282
-0.588614
-2.204182
3.304776
3.037143
-2.854720
-0.676017
-2.796606
-0.107430
-2.956490
-1.662621
22.93469
22.93469
22.93469
22.93469
22.93469
22.93469
22.93469
22.93469
0.3723
0.4037
0.1755
0.0800
0.0000
0.0877
0.2394
0.0808
0.2058
0.7946
0.5561
0.0275
0.0010
0.0024
0.0043
0.4990
0.0052
0.9144
0.0031
0.0964
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
Log likelihood
5137.579
LR test for over-identification:
Chi-square(8)
13.85457
Estimated A matrix:
1.000000
-0.018207
0.000000
1.000000
Probability
0.000000
0.000000
0.000000
0.000000
0.0856
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
-0.005520
3.395368
0.058473
-0.013780
0.256565
0.236355
Estimated B matrix:
0.019031
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
-0.027026
-26.49765
0.000000
0.000000
0.000000
-0.687508
1.000000
-30.85571
0.000000
0.000000
0.000000
-0.485800
0.000000
1.000000
0.000000
0.000000
-0.017476
-0.006826
0.000000
0.000000
1.000000
-0.012357
-2.672097
-0.010322
0.000000
0.000000
0.000000
1.000000
25.21627
-1.808124
0.000000
0.000000
0.000000
0.000000
1.000000
-0.008052
0.000000
0.006298
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.002037
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.596943
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.015340
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.002430
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.300670
0.000000
The LR test for over-identification is
LR  2(lu  lr )  T (trP  log P  k ) where
P  A' B T B 1 A
The null hypothesis: restrictions are valid
LR ~  2 (q  k ) where q = # restrictions.
Here we got  2 (10) = 13.85 < 18.31 at 5% do not reject the null. But many coefficients
are not significant. So you can go back and refine the restrictions.