V3' VECM

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V-3. VECM

• Johansens MLE
• Test of cointegration

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Why Johansens Procedure?

• Engle and Grangers two step procedure and the
  1-step ECM have concentrated on a single
  equation, with one variable designated as the
  dependent variable, explained by other variables
  that are assumed to be weakly exogenous for the
  parameters of interest.
• Do you need to model a system?
• Does the CI vector feed into more than one
  equation?
• If yes, this is sufficient to violate weak
  exogeneity since CI parameters are inherently
  cross linked.
• If no, you may legitimately focus on the one
  valid ECM either estimated by OLS or IV.

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Why Johansens Procedure?

• Johansens approach allows us to deal with models
  with several endogenous variables and has a
  number of other advantages
• The procedure begins with an unrestricted VAR
  involving potentially non-stationary variables.
• A key aspect of the approach is isolating and
  identifying the r cointegrating combinations
  among a set of k integrated variables and
  incorporating them into an empirical model.

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More generally, a system-based method (of which
Johansen's is the most popular) can provide
several advantages

• (1) Flexibility
• to capture a rich dynamic structure and
  interactions
• (2) Robustness
• can deal with I(0) and I(1) variables avoiding
  much of the pre-testing problem
• can cope with testing for and estimating multiple
  cointegrating vectors
• can capture a wide range of DGPs
• (3) Ability to test hypotheses
• can test restricted versions of vectors and
  speeds of adjustment

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Johansen Approach

• The appropriate estimation procedure is
• Step 1 Determining the cointegrating rank
• Step 2 Determining the factorization P ab
  Estimating the matrix of cointegrating vectors, b
  and the weighting matrix a.
• Step 3 Estimating the VAR, incorporating the
  cointegrating relations from the previous step.
• Most attractive approach is the MLE proposed by
  Johansen

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Johansen Approach

• Johansen's approach is based on MLE of the VECM,
  by step-wise concentrating the parameters out
• i.e., maximizing the likelihood function over a
  subset of parameters, treating the other
  parameters as known,
• given the number r of cointegrating vectors,
  where the matrix b is the last to be concentrated
  out.
• L(r , b) concentrated likelihood, given r and
  b,
• br maximum likelihood estimator of b given r ,

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Johansen Approach

• Johansen also proposes a likelihood ratio test of
  parametric restrictions on b of the form b Hf,
  where H is a given q s matrix of rank s?r and f
  is an unrestricted s r matrix. For example, in
  the case r 1, q 2, one might wish to test
  whether bT is proportional to (1, -1) HT.
• The likelihood ratio test statistic
• has a limiting c2 null distribution with r(q-s)
  degrees of freedom.

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Procedures for cointegration test

• Step 1 Regress
• Regress
• Step 2 compute S00, S0p, Sp0, and Spp
• Step 3 solve the equation
• Find roots or eigenvalues of the polynomial
  equation in l the solution yields the
  eigenvalues l1gtl2gtgt lk and associated
  eigenvector vi,
• Step 4 for each l, compute the LR statistic

residuals
If rankr ltn, the first r eigenvectors are the
coint vectors the columns of b
H0 at most r cointegrating vectors
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Two cointegration tests

• 1. The trace test
• Sequential tests
• i. H0 r0, cannot be rejected ?stop
• (at most zero coint) rejected ?next test
• ii. H0 rlt1, cannot be rejected ?stop?r1
• (at most one coint) rejected ?next test
• iii. H0 rlt2, cannot be rejected ?stop?r2
• (at most two coint) rejected ?next test
• Johansen has shown that the first r estimated
  eigenvectors v1, v2,,vr are the MLE of the
  columns of b, the cointegrating vectors.

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Johansens procedure

• 2. The lmax test

• H0 r0 vs. H1 r1 if reject H0 then
• H1 r1 vs. H2 r2 if reject H1 then
• H2 r2 vs. H3 r3
• Hk-1 rk-1 vs. Hk rk

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Step 5 choosing the appropriate table of
critical values

• Tables depend on k and deterministic terms
  including intercepts and time trends

Including m in the VECM has two meanings
confining m to the b when data show no evidence
of linear trend
Constants in the b
Linear trends in the data
Linear trend in the b
quadratic trends in the data
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Which table should we choose?

• Ruling out impractical models
• d m0 too restrictive models since at least a
  constant will usually be included in b
• d ? m ? 0 quadratic trend in the data occurs
  relatively infrequently
• Linear trend in the data? Check the graph
• Linear trend in the cointegrating space?
• See Tables 14.2 14.3 and 14.4

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Sample 1951 2001 Test assumption Linear
deterministic trend in the data Series
REAL_C01 REAL_Y01 Lags interval 1 to
2 Likelihood 5 Percent 1
Percent Hypothesized Eigenvalue Ratio Critical
Value Critical Value No. of CE(s) 0.228632
12.74889 15.41 20.04 None 0.005994
0.28855 3.76 6.65
At most 1 () denotes rejection of the
hypothesis at 5(1) significance level
L.R. rejects any cointegration at 5 significance
level
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Unnormalized Cointegrating Coefficients REAL
_C01 REAL_Y01 -3.98E-05 2.58E-05
2.88E-06 2.03E-06 Normalized Cointegrating
Coefficients 1 Cointegrating Equation(s) REA
L_C01 REAL_Y01 C 1.000000 -0.648540
11903.71 (0.02592) Log
likelihood -875.8804
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Conflicting test results

• In practice, the results of the two formal tests
  can conflict. Why might this happen?
• the tests use different information
• alternative hypotheses differ the maximum
  eigenvalue test has a sharper alternative .
• Conclusions from Monte Carlo study by Gregory
  (1994)
• both tests display some size distortion ie. a
  tendency to over reject H0, most likely due to
  overfitting the VAR.
• size is better for max eigenvalue test (which
  uses just 1 eigenvalue) than for the trace test
  (which uses all eigenvalues).
• If the results conflict, put more weight on max
  eigenvalue test, but also look at the
  implications of both.

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Example money demand

• Hendry and Ericsson (1991) k5 r2
• Money, m, price index, p, real income, y, own
  interest rate on money, Rm, and opportunity cost
  of holding money, Rb.
• Two cointegration relationships
• (m-p-y) stationarity of velocity of circulation
  of money
• (Rm-b22Rb) the interest spread behavior of
  banking sector set the interest rate on money as
  a mark-down of the opportunity cost of holding
  money

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Example money demand

• The baseline VAR ?VECM

Md adjusts to both deviations from equilibrium in
velocity and disequilibrium in interest spread
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Identification of multiple cointegrating vectors

• The Johansen procedure allows us to identify the
  number of b. But it lefts us a further problem
• Identification problem
• b and bg are two observational equivalent bases
  of the cointegrating space.
• Before solving the identification problem, no
  meaningful economic interpretation of coeff in
  cointegrating vector can be proposed.
• We need to impose sufficient number of
  restrictions on parameters such that the matrix
  satisfying such constraints in the cointegrating
  space is unique.

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Identification of multiple cointegrating vectors

• Any structure of linear constraint can be
  represented as
• A necessary and sufficient condition for i-th
  cointegration vector rank (Ribi)r-1. If number
  of cointegrating vector r, there must be at
  least r-1 independent restrictions of the form
  Ribi0 placed on each cointegrating vector

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Money demand (m-p, y, Rm, Rb)
General representation of matrix ß
Our constraints implying the following matrices
Ri
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Hypothesis testing

• The Johansen procedure allows for testing the
  validity of restricted forms of cointegrating
  vectors. The validity of restrictions
  (over-identifying restrictions) in addition to
  those necessary to identify the long-run
  equilibria can be tested.
• Intuition when there are r cointegrating
  vectors, only these r linear combinations of
  variables are stationary.
• Test statistics involve comparing the number of
  cointegrating vectors under the null and the
  alternative hypotheses.

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Testing restrictions on r identified
cointegrating vectors b

• Let be the ordered
  eigenvalues of the P matrix in the unrestricted
  model, and the
  ordered eigenvalues of the P matrix in the
  restricted model (null).
• Test statistic

si of freely estimated coeff.
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Testing restrictions on r identified
cointegrating vectors b

• Johansen (1992) shows that the statistic has a
  c2-distribution with degree of freedom equal to
  the number of over-identifying restrictions.
• The smaller values of with respect to
  imply a reduction of the rank of P when the
  restrictions are imposed and hence the rejection
  of null hypothesis.

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Review

• Engle-Granger 2-step
• Johansen
• VAR

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Testing and Estimation of the Cointegrating Vector

• Step 1 Estimate by OLSand check that ?t is
  stationary with a unit root test.
• Step 2 Construct the zt from step 1 and estimate
  the Error Correction Model the estimated
  parameters are consistent.

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Full Information, Maximum Likelihood Analysis of
Cointegrated Systems

• Directly produces the number of cointegrating
  relations
• Jointly estimates the cointegrating relations and
  the VAR in ECM form.
• The intuition behind the producer is to test the
  rank of the matrix P in
• Given the rank of P then we know that the VECM
  becomes
• The rank of P tells us how many cointegrating
  relations there are.

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A comment on the specifics of the Johansen test
and Eviews

• We have seen that the cointegrating vectors are
  not uniquely identified.
• Eviews normalizes these vectors by solving for
  the first h variables in Yt (i.e., assigns a
  value of 1 to the first variable, then a 1 to the
  second variable and so on).

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Deterministic Trend Assumptions

• Intercepts and trends affect the distribution of
  the Johansen test. Therefore, in conducting the
  test, care needs to be taken. Eviews allows the
  following 5 possibilities
• 1. No deterministic trends in the VAR and the ECM
  has no intercept.Where Xt are deterministic
  variables.
• 2. No deterministic trends in Y and intercepts in
  the EMC

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Deterministic Trend Assumptions

• Y has linear trends and the ECM has
  interceptwhere B? is the orthogonal to matrix
  B.
• Both Y and ECM have liner trends
• Y has a quadratic trend and ECM has a trend

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Modeling a Possibly Non-Stationary VAR in Practice

• Reference Hamilton pg. 651The particular
  technique to use when modeling a VAR depends
  heavily on the objective pursued.
• Forecasting in this scenario, we typically have
  little concern for hypothesis testing of
  particular parameter values. Estimating the VAR
  in levels is a good strategy. Choose the VAR lag
  length with an information criterions, such as
  AIC. Then, estimate the VAR. This places the
  least restrictions on the coefficient estimates.
  Test statistics will typically have non-standard
  distributions.

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Modeling a Possibly Non-Stationary VAR in Practice

• Structural Analysis this is often the situation
  we are most concerned with.
• We have seen that blindly differencing the data
  may cause omitted variable bias if there is
  cointegration.
• A good approach is to check for the stationarity
  of the data with unit root tests. Then the more
  standard procedure is to use a Johansen test and
  model the VECM. Most parameter tests of interest
  will have standard distributions.

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Supplementary Reading

• An excellent text book treatment of Johansen
  estimation (and VAR analysis more generally) is
  provided by
• Walter Enders "Applied Econometric Time Series",
  Ch. 6.
• For an application of this methodology to
  estimating a money demand equation see
• David Hendry Dynamic Econometrics 1995 OUP,
  Chapter 16 Econometrics in Action
• Further examples, including an application to
  modeling imports, can be found in
• Kerry Patterson An Introduction to Applied
  Econometrics 2000 Macmillan, Chapters 8 and Ch14.