# Multivariate VAR Models II

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```					                              Vector Error Correction Models
Johansen FIML Approach

The first part of this lecture draws from K. Juselius online lecturenotes at:
http://www.econ.ku.dk/okokj
The second part is from Favero Chapter 2

A VECM is more appropriate to model macro and several financial data. It distinguishes
between stationary variables with transitory (temporary) effects and nonstationary
variables with permanent (persistent) effects.

The dynamics part of the model describes the SR effects;
The CI relation describes the LR relation between the variables.

US CPI inflation (yoy percent change in the CPI)
Source: Global Financial Data

Since 1975:
.14

.12

.10

.08

.06

.04

.02

.00
1975        1980      1985      1990       1995      2000     2005

USCPIINFL

Since WWII
.20

.16

.12

.08

.04

.00

-.04
45   50   55   60   65   70    75   80   85   90   95   00   05

USCPIINFL
For the last century
.3

.2

.1

.0

-.1

-.2
1825    1850      1875      1900        1925    1950      1975      2000

USCPIINFL

In this lesson, we will look at:
 Derivation of the VECM for VAR
 Johansen FIML procedure
 Testing for the number of CI relations
 Decomposition of the components of CI models
 Identification problem in the CI relation.

Johansen Full Information Maximum Likelihood (FIML) procedure and higher
order systems

Consider a system of equations where y represents a vector of variables with k=n and
p=4.
yt  A1 yt 1  A2 yt 2  ...  A4 yt 4  ut

Reparameterize the VAR:

Add and subtract A4 yt 3 from RHS:
yt  A1 yt 1  A2 yt 2  A3 yt 3  ( A4  A4 ) yt 3  A4 yt 4  ut
yt  A1 yt 1  A2 yt 2  ( A3  A4 ) yt 3  A4yt 3  ut

Add and substract (A3  A4 )yt 2 from RHS
yt  A1 yt 1  A2 yt 2  ( A3  A4 ) yt 3  ( A3  A4 ) yt 2  ( A3  A4 ) yt 2  A4yt 3  ut
yt  A1 yt 1  ( A2  A3  A4 ) yt 2  ( A3  A4 )yt 2  A4yt 3  ut

Add and subtract ( A2  A3  A4 ) yt1 from the RHS
yt  A1 yt 1  ( A2  A3  A4 ) yt 2  ( A2  A3  A4 ) yt 1  ( A2  A3  A4 ) yt 1  ( A3  A4 )yt 2  A4yt 3  ut
yt  ( A1  A2  A3  A4 ) yt 1  ( A2  A3  A4 )yt 1  ( A3  A4 )yt 2  A4yt 3  ut

Subtract yt 1 from LHS and RHS
yt  (I  A1  A2  A3  A4 ) yt 1  ( A2  A3  A4 )yt 1  ( A3  A4 )yt 2  A4yt 3  ut

Sum the Ai’s:
yt   j 1 ( I  Aj ) yt 1   j 2 Aj yt 1   j 3 Aj yt 2   j 4 Aj yt 3  ut
4                         4                   4                   4

yt  yt 1  1yt 1  2yt 2  3 yt 3  ut

Substitute n=4 and sum the y’s:
yt  yt 1  i 1 i yt i  ut
n 1
(1)

where   I  in1 Ai  and                                 
i    j i 1 Aj = -A*(L)
n

If we had started the substitutions from yt 1 we would have a slightly different
n 1
expression: yt  yt  n  i 1 i yt  i  ut (e.g. Favero)
Here  is placed at yt  n . This changes the interpretation of  i coefficients (they
measure the cumulative LR effects instead of pure transitory effects as in (1)) but the
definition of  remains unchanged.

Estimation of the VECM (equation (1))
The rows of this matrix are not linearly independent if the variables are cointegrated.
Each variable appearing in VECM is I(0) either because of first-differencing or to taking
linear combinations of variables, which are stationary.

Geometric interpretation

The Johansen approach is based on the relationship between the rank of a matrix and its
characteristic roots.

The rank of a matrix = #characteristic roots  0 (i.e. s  1 )= # of cointegrating vectors.
 All s  1 (roots=0)
 All s  1
 Some s  1

Three cases:
1. Rank () =0: There are no cointegrating variables, all rows are linearly
dependent, and the system is nonstationary. in Ai  I . First-difference all the
variables to remove nonstationarity, then standard inference applies (based on t, F
and  2 ). We can thus write the VECM as a simple VAR in first differences:
yt  in11i yt i  ut

2. Rank () =k (# variables), full rank, hence is nonsingular: all rows (columns) are
linearly independent (all variables are stationary, i.e., yt ~I(0)), all roots are in the
unit circle with modulus<1, and hence the system is stationary and the levels of
variables have stationary means. Estimating with unrestricted OLS the level VAR
and the VECM will give identical results.

3. Rank () = r < k. The system is nonstationary but there are r cointegrating
relations among the variables (r rows are linearly independent, thus r linearly
independent combinations of the { yit } sequence are stationary). The y vector
may be I(1) or higher and the CI relation is determined by
   ' ,
o  =a (k x r) matrix of weights, the loading matrix, which measures
the average speed of convergence towards LR equilibrium
o  = a (k x r) matrix of parameters determining the cointegrating
vectors.
 ' yt 1  0 is the long-run equilibrium error. The RHS of the VECM contains r
cointegrating variables.

Geometric representation---

Example
Consider a bivarite process with 2 lags again.
yt  A1 yt 1  A2 yt 2  Dt  ut  ( I p  A1 L  A2 L2 ) yt  yt  Dt  ut   where D is an exogenous
dummy or exogenous variable.

Reparameterize it and express it as an ECM:
yt  yt 1  ( A1  A2  I p ) yt 1  A2 ( yt 2  yt 1 )  Dt  ut
yt  yt 1  1yt 1  Dt  ut

If the characteristic polynomial ( )  I p  1  2 2  (1   ) I p    1 (1   ) --see ex. 2 in
J. LN, Multiv Models I-- (or the companion matrix) has unit root, then ( )  0 for   1 and
(1)     '

And the ECM model becomes:
yt   ' yt 1  1yt 1  Dt  ut

A VECM or a CVAR models several effects:
    ij coefficients show the LR equilibrium relationships between levels of
variables.
    ij coefficients show the amount of changes in the variables to bring the
system back to equilibrium.
    ij coefficients show the SR changes occurring due to previous changes in
the variables.
     ij coefficients show the effect on the dynamics of external events.

Testing the number of CI vectors (rank of  )
First estimate the parameters of the matrix  and then get the associated eigenvalues.
Order the eigenvalues of (1) as 1  2  ..  k and determine the number of characteristic
roots that are significantly different than zero.
Ex:
If rk ()  0 , then all characteristic roots=0, ln(1  i )  0  i  1 i .
If rk ()  1 , then 0  1  1  ln(1  1 )  0 and ln(1  2 )  ln(1  3 )  ...  0 .
Etc.

Two test statistics are used to test the number of cointegrating vectors, based on the
characteristic roots. For both the null is at most r cointegrating vectors:

 The trace statistics:                                     ˆ
trace(r )  T ikr 1 ln(1  i ) . The alternative: at most k CI
vectors.
It looks at the trace of A(1) = the sum of eigenvalues. If there is no cointegration,
ˆ
then all 1  i are zero and trA(1)=0. Run the test in sequence: start from the null
of at most 0 CI vectors upto at most k CI vectors against the alternative.

ˆ
 Lambda max statistics: max (r , r  1)  T ln(1  r 1 ) . The alternative: at most r+1 CI
ˆ
vectors. It tests rank r+1 by testing if r 1 is zero.
Critical values are provided by Johansen.
Note: If the results of the two test statistics are not consistent, go with the trace statistics.

Series: LOG(RGDP) CPIINFY LOG(M2SAREAL) M2OWN TBILL3
Unrestricted Cointegration Rank Test (Trace)

Hypothesized                                   Trace           0.05
No. of CE(s)           Eigenvalue             Statistic   Critical Value        Prob.**

None *               0.248081              133.7973      69.81889             0.0000
At most 1 *            0.189194              73.06545      47.85613             0.0000
At most 2             0.111896              28.39365      29.79707             0.0719
At most 3             0.013944              3.117806      15.49471             0.9613
At most 4             0.000595              0.126846      3.841466             0.7217

The VAR model and the MA representation: Duality between CI and common
trends
Consider again the ex: yt  A1 yt 1  A2 yt 2  Dt  ut  ( I p  A1L  A2 L2 ) yt  yt  Dt  ut .
Take   0 for simplicity: ( I p  A1L  A2 L2 ) yt  yt  ut

When the characteristic polynomial  has a unit root, it is of reduced rank and the
determinant   0 for   1 , so it cannot be inverted to express it as a MA.

We can express  in two components:
()  (1  ) * () where  * is now stationary (no unit root).

We can now write the VAR as:
( I p  L) * ( L) yt  ut  (1  L) yt  [ * ( L)]1 ut

and express  [ * (L)]1 as a MA
[ * (L)]1  C0  C1L  C2 L2  ....  C(L)

The Beveridge-Nelson decomposition says that C(L)  C(1)  (1  L)C * (L) . So we can
rewrite the VAR as
yt  C(1)ut  C * ( L)(1  L)ut .

Integrate both sides (divide by (1-L))
yt  C(1) zt  C * ( L)ut  y0 where zt  ut and y0 is the initial conditions of integration that
we can set =0.
Y is decomposed into a stationary stochastic component C*(L)u and a stochastic trend
C(1)z (common trends=TR).
yt  C ti1 ui  C * ( L)ut .

The nonstationary component involves restrictions due to CI. We can write it as:

yt  C * ( L)ut    (α' β  )1 ( ' zt )

Note: an important concept for reduced rank matrices such as  and  is
that they have orthogonal complements   and   such that   '  0
and  '   0 . The   and    matrices are (p x p) with full rank p.

The deterministic components in the CI model: intercepts in CI relations and
growth rates (H-J. LN II)

When two or more series have the same stochastic and deterministic trends, we can find a
linear combination that is trend-free even though the variables are trending.
 Include a trend in the cointegration space (  detrending a series). The same argument
can be made about major changes e.g. policy or exogenous shocks, which can be
captured by “intervention” dummies.
Caution with dummies: in each case, the asymptotic distributional properties of the tests are
modified. The Johansen procedure provides different critical values for deterministic trends, but
not for the dummies. Must generate the critical values yourself (procedure in RATS).

Consider the VAR(1) system:
(2)    yt   ' yt 1  1yt 1  y0  ut

We will decompose the intercept y0 into the mean of the CI relation and a growth
component.

  ' yt 1 is I(0) since it has r cointegrating relations, each with a constant mean:
E[ ' yt 1 ]   . This describes an (r x 1) vector of intercepts in the CI relations.

 y t is I(0) and therefore it has a constant mean:
E[yt ]   . This describes a (p x 1) vector of growth rates.

Take the expectations of (2) and replace the means:
( I p  1 ) E (yt )  E (  ' yt 1 )  y0
 )
( I p  1 )    y0  y0  ( I  1                               

p                           mean of CI relation
growthcomponent

So, the constant term in the VAR has both components: linear growth rates in the data
and the mean values (=intercepts) of the CI relations.

Substitute the two components into the VAR:

Trends and intercepts in a CVAR model
Rewrite (2) with an explicit trend component assuming 1 = 0

(3)       yt   ' yt 1  y0  t  ut

Decompose y0 and  into two vectors each related to the mean value of the CI relation
(,  ) and the growth rates ( , ) :
y0     (constant) growth rate of y
     (constant) growth rate of trend
Substitute them back into (3):

yt   ' yt 1      t  t  ut

we can write the model with the decomposition of the deterministic terms as:
yt   (  ' yt 1   '  ' t )         (  t )             ut
                           
CI relation( EC term)      deterministict terms
outside the CI relation

  ' yt 1  y0  t  ut where               y0     and      and ut ~ NIID(0, )

Case 5: y0 and  unrestricted (no restrictions on  ,  ,  , ),
yt   ( ' yt 1   '  ' t )  ( t )
Linear trends in the CI relations (I(0) components) and quadratic trends in the level data
yt (the I(1) components). Linear trends in the differenced series  quadratic trends in the
level yt . Unless there is a good theoretical reason to include quadratic trends, such as to
capture population growth, it is not advisable to include this case. Instead, you should
change the model, include more lags, etc.

Case 4:   0 , No restrictions on  , ,  , yt   ( ' yt 1   '  ' t )  
Both the CI relations and the level data yt have linear trends. The model excludes
quadratic trends. The model contains trend-stationary variables, or an equilibrium relation
including a linear trend.
Case 3:   0 No restrictions on  ,  , yt   ( ' yt 1   ' )   because   0  t  t .
Linear trend in the data but no trends in the CI regressions.

Case 2:     0 , yt   ( ' yt 1   ' )
No linear trends in the level data nor in the CI relations. The only deterministic
components=intercepts in the CI relations.

Case 1:     0 , yt   ' yt 1
No deterministic components in the level data yt , all intercepts in the CI relations are
zero. Too restrictive, don’t use unless there are good reasons.

Identification problem:
The matrices  and  are not unique, there are many such matrices that contain the
cointegration relation(s). Only the cointegration space is estimated consistently but not
the cointegration parameters.
To see this, consider a nonsingular matrix B. We can get a new loading matrix B1 and a
cointegrating matrix B'  that still gives the same LR relationship:
  B 1B ' =  '
The new version of the model is simply a linear transformation of the original model.
This means that it has the same time series properties. But the CI vector that we get may
or may not have any economic meaning. See Wickens (1996 EJ). To define  and 

1. Assigning an identity matrix to the first part of  (default in softwares):
 '  [ I r :  '( K r ) x r ] . For r=1, this means normalizing the first coefficient to 1. Economic
theory should guide this choice. You cannot choose an arbitrary variable to normalize,
since it may not be in the CI vector. This type of restrictions does not change the CI
relations and does not affect the value of the likelihood function.

2. Drawing identifying restrictions from theory: these change the value of the likelihood
function.
(i) Same restrictions to all equations (ex: whether the dummies are significant or
not). These are not identifying restrictions.
(ii) Over-identifying restrictions.

Example: The MD equation (Favero Ch 2):
mt  a0  a1mt 1  a2 mt 2  a3 yt 1  a4 yt 2  a5 Rm,t 1  a6 R m,t 2 a7 Rb,t 1  a8 Rb,t 2  ut

where m=ln M/P, Rm, Rb=nominal returns on money and bonds.

In a VAR(2) framework with k=4
mt           mt 1       mt 2  u1,t 
                                    
 yt   A  A  yt 1   A  yt 2   u2,t 
 Rm,t   0   1
Rm,t 1   2
Rm,t 2  u3,t 
                                    
 Rb ,t 
              Rb,t 1 
             Rb,t 2  u4,t 
           

or:
xt  A0  A1 xt 1  A2 xt 2  ut ,     where xt '  (mt , yt , Rm,t , Rb,t ) .

Reparameterized as VECM:
xt  A0  A(1) xt 1  A2 xt 2  ut

or:
xt  0  xt 1  1xt 2  ut

Assume that there are two cointegration relationships drawn from theory:
 m-y (velocity of circulation of money from the QTM)
    Rm  bRb (banks’ markup on safe asset yield).
Since there are two CI vectors, the rk(  )=2. One CI vector = (1,-1) and the other is
(1, -b):

1  1 0 0 
   '   thus  '             .
0 0 1  b

We want  to have non zero elements for the money market equilibrium condition and
the interest rates. Income elasticity should be 1. Everything else can be random. Thus
the LR matrix should look like:

 11   11  13  14        b 
0                  0  0
11   11   12     12
0     0                               0 
                                0     0          
0      0     33   33   0   0     32   32b
                                               
 0     0     0     0  0       0     0      0 

To see how we get this we need to answer the following questions:

1. How do we impose the CI restrictions?
 11 12 
      
Consider the general representation of the CI vector    21 22  ,
  31  32 
           
  41  42 
 11 0 
    0 
which in our case is    21      with specific values for the betas.
 0  32 
        
 0  42 

Given our constraints, we want to test whether the null hypothesis is
H 0 : 11  1, 21  1, 32  1, 42  b and 12   22   31   41  0 .
Thus the CI matrix has the form:

1  1 0 0 
 '           .
0 0 1  b

We know that
11 12                   11            11 12      12b 
          1 1 0 0                                     22b 
 21  22               21

  21  22             . Comparing this (4x4) matrix
 31  32  0 0 1  b   31
                             31  32      32b 
                                                              
 41  42                  41           41  42      42b 

with  we get:
 21   22   31   41   42  0 .

11 12 
0    0 
         
 0  32 
        
0    0 

Get back to the VECM model:
mt          11 12                 mt 1  u1,t 
y                                                
 t   0            0  1  1 0 0   yt 1  u2,t 
                          
Rm ,t    0
 0  32  0 0 1  b   Rm ,t 1  u3,t 
          
                                                 
Rb ,t 
             0       0                Rb ,t 1  u4,t 
            
11 12                           u1,t 
0        m  y                    
t 1 
u2,t
 
     0           t 1
 0 
 0  32   Rm ,t 1  bRb ,t 1  u3,t 
                        
        
0    0                            u4,t 
 

We have two CI vectors and the specification of the loading matrix says that if 11 <0 and
12 >0 (a hypothesis we can test)
(i)    money demand adjusts to deviations from the velocity of money (first
CI relation) by 11 and to deviations from interest spread (second CI
relation) by 12 . It increases with deviations from equilibrium when
velocity is too high (m<y) and when the opportunity cost of holding
money is too low (Rb<Rm)
(ii)   Interest rates on bonds and income do not react to disequilibrium in the
velocity of money or interest rate spread.
(iii) Interest rate on money reacts to disequilibrium in the spread by an
amount  32 .

An Application: Leeper, Sims, Zha, Hall, Bernanke (BPEA, 1996)
Data file: Favero, lszusa.xls

We will see the interaction of variables from a typical macro model with the US data:
All data is monthly.
Endogenous variables
log(M2SAreal): log of real money supply S.A.
CPIinf: yoy CPI inflation rate (%)
Log(RGDP): log of real GDP
Tbill3: 3-month Treasury bill rate.

Exogenous variables:
Constant, trend, PCMinf (PPI for crude materials inflation), break dummies for various
periods, taking the value 1 for that period and zero elsewhere.

Objective:
specify a VAR model,
specify and test for LR CI vectors,
specify and test for LR restrictions.

1. Specification of the standard VAR model (no CI)
 Decide on the variables that enter the VAR: need a model for this. If the VAR is
misspecified because of missing variables, it will create an omitted variable(s)
problem and be reflected in serially correlated error terms.
 Number of lags. We need to include the optimal number of lags.
Note : increasing the number of lags does not solve the residual correlation if
there are omitted variables.
 Even if there is no omitted variables and we include the optimal number (or
reasonable #) of lags, residuals can still reflect a problem caused by structural
breaks. At this stage we will control for them by determining the break dates
exogenously.
Note: inclusion of any dummies requires generation of new critical values for the
dummies. But we will not do it here.
(We will talk more about the specification issues in the section on VARs)

1. VAR Specification
We start estimating a VAR(15) with a constant and trend:
xt  a0  a1t  141 Ai Li xt  ut
i                where x't  { yt ,  t , mt , Rtm , Rtb } .
Although the sample is 1959.7-2001.12 we use the 1960.1-79.6 subsample because after
that the Fed changed the implementation of monetary policy from money targeting to
interest rate targeting. This would introduce substantial instability in the system.

Run the UVAR with lag length = 15, set of exog variables = {c, trend}.
var varbaseline.ls 1 15 log(rgdp) cpiinfy log(m2sareal) m2own tbill3 @ c @trend

VAR Stability Condition Check (lag structure-AR roots):
No roots lie outside the unit circle but there are two quasi-unit roots (modulus>0.98).

Residuals from the baseline VAR: VAR-Residuals-Graph

LOG(RGDP) Residuals                                CPIINFY Residuals
.024                                                 .8
.020                                                 .6
.016                                                 .4
.012
.2
.008
.0
.004
-.2
.000
-.4
-.004
-.008                                                -.6

-.012                                                -.8
62   64   66   68   70   72   74   76   78         62   64   66   68   70   72   74   76   78

LOG(M2SAREAL) Residuals                                 M2OW N Residuals
.006                                                 .3

.004
.2
.002

.1
.000

-.002
.0

-.004
-.1
-.006

-.008                                                -.2
62   64   66   68   70   72   74   76   78         62   64   66   68   70   72   74   76   78

TBILL3 Residuals
.8

.6

.4

.2

.0

-.2

-.4

-.6
62   64   66   68   70   72   74   76   78

Many outliers. Normality test is rejected. Misspecification? If so, we need to improve
the model. Check first if variables are mismeasured, model is misspecified. Other
potential problems are:
 Potential Omitted Variables:
Many outliers around the 1st oil price shock.
(i) Include a PPI for crude materials (PCM) as an exogenous variable since none of
the endog variables is likely to affect it. Include contemporaneous and 6 lags of
PCM.
(ii) Monthly dummies for periods around the oil price increases: 1 for the month of a
year, 0 elsewhere.

Generate a series D70 that takes the value 1 for all observation from 1970:1:
series d70=0
d70.fill(o=1970:1,l) 1

The first command line generates a series d70 that is filled with zeros. The second is
the fill command objectname.fill(options). We use the option to fill o=[date,integer],
where date is 1970:1, the starting date from which the program starts filling the series,
and l is the command to loop over the list of values as many times as it takes to fill
the series. The number 1 shows what the program should fill the series with.

You do the same thing for all the dummies you want to create. Then estimate the
VAR (var01), with 15 lags, the dependent variables and the series of independent
variables, which start after the @ sign, including the constant, trend, lagged oil price
inflation, and all the dummies:
var var01.ls 1 15 log(rgdp) cpiinfy log(m2sareal) m2own tbill3 @ c @trend pcminfy(0 to -6)
d7306 d7307 d7308 d7310 d7311 d7312 d7402 d7403 d7407 d7408 d7409 d7501 d7505
d7806 d7808 d7811 d7904

Two more points to check:

 Parameter constancy
Are there structural shocks to the system due to war, major policy change, oil price
changes?
We truncated the sample to avoid breaks due to policy change.
We included control dummies to account for breaks caused by oil price changes.

 Increase lag length
We already started with a large lag length.
Residuals from VAR01
LOG(RGDP) Residuals                                 CPIINFY Residuals
.020                                                  .6

.015                                                  .4

.010
.2
.005
.0
.000
-.2
-.005

-.010                                                 -.4

-.015                                                 -.6
62   64   66   68   70   72   74   76   78          62   64   66   68   70   72   74   76   78

LOG(M2SAREAL) Residuals                                  M2OWN Residuals
.004                                                .10

.08
.002
.06
.000
.04

-.002                                                .02

.00
-.004
-.02
-.006
-.04

-.008                                                -.06
62   64   66   68   70   72   74   76   78          62   64   66   68   70   72   74   76   78

TBILL3 Residuals
.6

.4

.2

.0

-.2

-.4

-.6
62   64   66   68   70   72   74   76   78

There are considerably fewer outliers and the fluctuation bands are smaller. Normality
tests are still rejected (though less strongly) but now it is due to kurtosis. Skewness of the
series is not significantly different from a normal distribution.
Statistical inference is sensitive to parameter instability, serial correlation in residuals and
residual skewness. It is somewhat robust to residual kurtosis and heteroscedasticity.
Thus we have acceptable results with this model.

Lag length
Run a LR test to compare between different lengths of lags (more on this on the lesson on
VARs). Preferably select 15 vs 12 for ex., 12 vs 8, etc.
Following Favero and LSZ we select lag=6

2. VECM Specification
Johansen (1996) tabulated different critical values depending on the specification of the
deterministic components.

(i) Rank of 
Eviews:
View-Cointegration Test

Specify the model you would like to estimate based on deterministic components.
Specify the exogenous variables (except c and trend).

Here: choose 4 because level data has linear trends and CI may have too.
For #lags:
use #(lags for VAR)-1. Here it should be 5 since the VAR is in first-difference.

Output:
You get the results for Trace Test and Maximum Eigenvalue (lambda-max) Test.
Trend assumption: Linear deterministic trend
Series: LOG(RGDP) CPIINFY LOG(M2SAREAL) M2OWN TBILL3
Exogenous series: PCMINFY(0 TO -6) D7306 D7307 D7308 D7310 D7311 D7312 D7402 D7403 D7407 D7408 D7409
D7501 D7505 D7806 D7808 D7811 D7904
Warning: Critical values assume no exogenous series
Lags interval (in first differences): 1 to 5
Unrestricted Cointegration Rank Test (Trace)

Hypothesized                                    Trace                 0.05
No. of CE(s)            Eigenvalue            Statistic        Critical Value    Prob.**

None *                0.248081             133.7973            69.81889       0.0000
At most 1 *              0.189194             73.06545            47.85613       0.0000
At most 2               0.111896             28.39365            29.79707       0.0719
At most 3               0.013944             3.117806            15.49471       0.9613
At most 4               0.000595             0.126846            3.841466       0.7217

Trace test indicates 2 cointegrating eqn(s) at the 0.05 level
* denotes rejection of the hypothesis at the 0.05 level
**MacKinnon-Haug-Michelis (1999) p-values

Unrestricted Cointegration Rank Test (Maximum Eigenvalue)

Hypothesized                                  Max-Eigen               0.05
No. of CE(s)            Eigenvalue            Statistic        Critical Value    Prob.**

None *                0.248081             60.73185            33.87687       0.0000
At most 1 *              0.189194             44.67180            27.58434       0.0001
At most 2 *              0.111896             25.27584            21.13162       0.0123
At most 3               0.013944             2.990960            14.26460       0.9473
At most 4               0.000595             0.126846            3.841466       0.7217

Max-eigenvalue test indicates 3 cointegrating eqn(s) at the 0.05 level
* denotes rejection of the hypothesis at the 0.05 level
**MacKinnon-Haug-Michelis (1999) p-values

Unrestricted Cointegrating Coefficients (normalized by b'*S11*b=I):

LOG(RGDP)                 CPIINFY           LOG(M2SAREAL)          M2OWN          TBILL3
-16.17005              -1.407204             2.187633            3.314965      2.460214
38.66775               0.744846            -22.60465            6.195358      -2.872095
4.651794               0.616725            -4.988204            -1.418077     0.789988
-15.68292               0.353712            -0.768622            3.128172      -0.489658
26.27699               0.083358            -13.25400            0.911053      -0.873680

D(LOG(RGDP))            0.000153             -0.001046            -0.001295       0.000261    1.77E-05
D(CPIINFY)             0.078769             -0.007659            -0.023121       -0.017939   0.001529
D(LOG(M2SAREA
L))                -0.000217              0.000416            -0.000350       -2.29E-05   -1.90E-05
D(M2OWN)             -0.007806             -0.006534                0.000286    -0.002289   -5.22E-05
D(TBILL3)            -0.038561              0.030892            -0.015293       -0.007610   0.004174

1 Cointegrating Equation(s):                 Log likelihood             2398.550

Normalized cointegrating coefficients (standard error in parentheses)
LOG(RGDP)              CPIINFY          LOG(M2SAREAL)                M2OWN        TBILL3
1.000000             0.087025             -0.135289            -0.205007       -0.152146
(0.01226)            (0.09442)                (0.06191)   (0.02114)

Adjustment coefficients (standard error in parentheses)
D(LOG(RGDP))           -0.002471
(0.00653)
D(CPIINFY)            -1.273703
(0.28751)
D(LOG(M2SAREA
L))                 0.003505
(0.00213)
D(M2OWN)              0.126221
(0.03585)
D(TBILL3)             0.623528
(0.27049)

2 Cointegrating Equation(s):                 Log likelihood             2420.886

Normalized cointegrating coefficients (standard error in parentheses)
LOG(RGDP)              CPIINFY          LOG(M2SAREAL)                M2OWN        TBILL3
1.000000             0.000000             -0.712307                0.264042    -0.052140
(0.06597)                (0.04189)   (0.01135)
0.000000             1.000000              6.630462            -5.389799       -1.149161
(1.55543)                (0.98773)   (0.26769)

Adjustment coefficients (standard error in parentheses)
D(LOG(RGDP))           -0.042927             -0.000994
(0.01656)            (0.00063)
D(CPIINFY)            -1.569873             -0.116549
(0.74480)            (0.02829)
D(LOG(M2SAREA
L))                 0.019596              0.000615
(0.00535)            (0.00020)
D(M2OWN)             -0.126423              0.006118
(0.09039)            (0.00343)
D(TBILL3)           1.818045           0.077272
(0.69365)           (0.02635)

Trace test indicates 2, rank test 3 CI vectors. We will go with the trace test results.

We need to impose and test our own restrictions.

Testing of the theories:
In each CI relation, we will look for two sets of relations to test inflation targeting: by
controlling MS growth and interest rates.
(i)     The money demand equation + the markup of nominal rates over T-bill rates.
(ii)    Interest rate reaction function (where authorities adjust nominal interest rates
according to inflation and real money balances) + markup of nominal rates.

   '
Leave  unrestricted. Impose restrictions on 

1st set of identifying restrictions: Money market

Theoretical model:
A traditional money demand and a relation between interest rate on money and bond and
inflation (Inflation targeting via the control of money growth).
11 yt  13Rtm  14Rtb  (m  p)  16T  0
22Inft  23Rtm  Rtb  0

11yt  0Inft  13Rtm  14Rtb  (m  p)  16T  0
0 yt  22Inft  23Rtm  Rtb  0(m  p)  0T  0

Remember the CI matrix with 2 CI vectors:
 11     21 
        22 
 12          
   13     23 
             
 14     24 
 15
         25 


We test whether the null hypothesis
H 0 : 12  0, 15  1,  21   25   26  0,  24  1
thus the CI matrix has the form:
 11 0             13 14 1
 ' 
 0  22             23 1 0

2nd set of identifying restrictions: interest rate reaction function
Inflation targeting via control of interest rates: nominal rates respond to inflation, output,
and a linear trend + relation between interest rates and inflation.
11 yt  Inft  Rtb  16T  0
22Inft  23Rtm  Rtb  0

11yt  Inft  0Rtm  14Rtb  0(m  p)  16T  0
0 yt  22Inft  23Rtm  Rtb  0(m  p)  0T  0

We test whether the null hypothesis (ignore trend)
H 0 : 12  1, 13  0, 15  0,  21   25  0,  24  1 and thus the CI matrix has the form:

11  1         0     14 0
' 
 0  22         32     1    0


Application
Run the VECM model with 2 CI relations:
Estimate:VEC, lags 5-Cointegration:#CI vectors 2, model 4-VEC restrictions :

impose restrictions :
Restrictions may be placed on the coefficients B(r,k) of the r-th
cointegrating relation:

B(r,1)*LOG(RGDP) + B(r,2)*CPIINFY + B(r,3)*M2OWN +
B(r,4)*TBILL3 + B(r,5)*LOG(M2SAREAL)

Enter the first set of identifying restrictions in Eviews:
B(1,2)=0, B(1,5)=1, B(2,1)=0, B(2,4)=1,B(2,5)=0

Sample: 1961M10 1979M06
Included observations: 213
Standard errors in ( ) & t-statistics in [ ]

Cointegration Restrictions:
B(1,2)=0, B(1,5)=1, B(2,1)=0, B(2,4)=1,B(2,5)=0
Convergence achieved after 61 iterations.
Restrictions identify all cointegrating vectors
LR test for binding restrictions (rank = 2):
Chi-square(1)                               0.082124          <3.84 hence not   reject the null H
Probability                                 0.774440
Cointegrating Eq:      CointEq1       CointEq2

LOG(RGDP(-1))          -0.816764      0.000000
(0.23149)
[-3.52830]

CPIINFY(-1)          0.000000      -0.686935
(0.09151)
[-7.50642]

M2OWN(-1)            -0.268589      3.079197
(0.03742)      (0.58406)
[-7.17853]    [ 5.27202]

TBILL3(-1)          0.062989       1.000000
(0.01104)
[ 5.70367]

LOG(M2SAREAL(-1))        1.000000       0.000000

@TREND(59M07)          -0.002725     -0.028931
(0.00086)      (0.00684)
[-3.17698]    [-4.23064]

C              1.122244      -7.980040

D(LOG(M2SAREA
Error Correction:    D(LOG(RGDP))    D(CPIINFY)    D(M2OWN)        D(TBILL3)          L))

CointEq1           0.036138       1.232691       0.145519      -1.069758      -0.019164
(0.01416)      (0.63262)      (0.07717)     (0.59866)       (0.00450)
[ 2.55195]    [ 1.94854]     [ 1.88563]     [-1.78692]     [-4.25777]

CointEq2           0.001014       0.157233      -0.009180      -0.074209      -0.000834
(0.00078)      (0.03485)      (0.00425)     (0.03298)       (0.00025)
[ 1.29978]    [ 4.51127]     [-2.15916]     [-2.24995]     [-3.36202]

D(LOG(RGDP(-1)))        -0.154737      2.348891       0.482651      2.134618        0.008240

Not reject the null hypothesis about the constraints on all variables.
Estimates highly significant.
The 1st CI relation is consistent with the traditional MD equation. Income elasticity close
to 1, semi-elasticities of interest rates are also consistent with the theory.
Weights of the CI vector: very small reaction by MS to a deviation from the LR in the 1st
CI relation, although highly significant. The Tbill rate reacts much more strongly to
disequilibrium but it has a larger variance.
2nd set of identifying restrictions:
B(1,2)=-1, B(1,3)=0,B(1,5)=0, B(2,1)=0, B(2,4)=1,B(2,5)=0

Standard errors in ( ) & t-statistics in [ ]

Cointegration Restrictions:
B(1,2)=-1, B(1,3)=0,B(1,5)=0, B(2,1)=0, B(2,4)=1,B(2,5)=0
Convergence achieved after 25 iterations.
Restrictions identify all cointegrating vectors
LR test for binding restrictions (rank = 2):
Chi-square(2)                               10.69273       >3.84   Reject the null
Probability                                 0.004765

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