TIME SERIES REGRESSION: COINTEGRATION

1
TIME SERIES REGRESSION COINTEGRATION

• COINTEGRATION
• COINTEGRATION APPROACHES - Single equation -
  Multivariate VEC model

2
THE PROBLEM WITH NONSTATIONARY TIME SERIES

• Non-stationary time series give rise to spurious
  regression
• Traditional approaches to handling non-stationary
  time series
• Detrending, but not possible with
  stochastic trends (I(1), I(2), )
• Differencing, but sensitive to short-term
  noise components biased estimates when series
  have long-run equilibrium relationship ? error
  correction mechanism

3
THE ERROR CORRECTION PROBLEM
With differencing yt xt ?t becomes ?yt ?xt
?t. But ?t ?t - ?t-1 and ?t and ?t-1 are
not independent when a long-run equilibrium
between y and x exists. Future ys cannot
merely grow along with x (?yti ?xti) when
an equilibrium relationship in levels exists. The
gap between y and x must be closed over time
error correction mechanism.Engle/Granger (1987)
Assumeyt a xt ?txt xt-1 ??1 gt 0?j
0 all j
when ?y ?x
y a x
x
t1 time
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COINTEGRATION
For example, yt and xt are both I(1), but if yt
? ? xt ?t and zt ( ?t) yt ? ? xt
I(0), then yt and xt are said to be
cointegrated. A linear combination of 2 (or more)
integrated variables y and x can result in a
stationary error term z.If the error term is
stationary, econometrics can develop new
estimation methods Economic significance Cointegr
ation can be viewed as the statistical expression
of the nature of long-run equilibrium
relationships. If y and x are linked by some
long-run relationship, from which they can
deviate in the short run but must return to in
the long run, residuals will be stationary. If
variables diverge without bound (i.e.
non-stationary residuals) we must assume no
equilibrium relationship exists.
5
COMMON STOCHASTIC TRENDS
Different names for basically the same thing. One
interpretation of cointegrated variables is that
they share common stochastic trends (Stock/Watson
1988). E.g. yt ?yt ?yt with ?yt ?yt-1
?yt xt ?xt ?xt with ?xt ?xt-1 ?xt If
yt and xt are cointegrated we can write some
linear combination ?1 yt ?2 xt (?1 ?yt ?2
?xt) ?1 ?yt ?2 ?xt
must be stationionary Must be that ?1 ?yt ?2
?xt c or ?yt c /?1 (?2/?1) ?xt then xt
?xt ?xt yt (?2/?1) ?xt c /?1 ?yt
Up to some scalar (?2/?1) the two variables
have the same stochastic trend ?xt.
6
COINTEGRATION IN ECONOMICS

• Many examples of possible cointegration
  equilibrium relationships always assuming that
  certain coefficients/parameters are constant
• consumption, income and wealth
• stock prices and dividends
• exchange rate and domestic, foreign price
  levels
• long-term and short-term interest rates
• money, price level and income

7
ERROR-CORRECTION MODEL
Cointegration always implies an error-correction
model (Granger representation theorem) ECM is
superior to modeling integrated data in
first-differences or in levels (Engle/Granger,
1987). For example, simple dynamic (short-run)
model of y and x in levels yt ?0 ?1 yt-1
?0 xt ?1 xt-1 ?t (1) Long-run or
steady-state solution (ytyt-1, xtxt-1) is yt
?0 ?1 xt with ?0 ?0/(1 ?1), ?1
(?0?1)/(1 ?1) (2) Alternatively, rearranging
dynamic model yields ?yt ?0 ?xt (1 ?1)
yt-1 ?0 ?1 xt-1 ?t ?y, ?x, yt-1 ?0
?1 xt-1 are all stationary variables clear
long-run component y ?0 ?1 x if
cointegration exists (1- ?1) measures speed of
adjustment to long-run equilibrium
8
ERROR-CORRECTION MODEL (Cont)
General vector-error-correction model (VECM) ?yt
?1(L) ?yt-1 ?1(L) ?xt-1 ?1 zt-1 ?1t ?xt
?2(L) ?yt-1 ?2(L) ?xt-1 ?2 zt-1 ?2t y, x
represent vectors of possibly more than 1
variablesi.e. x x1, x2, xk (short run)
dynamic structure is captured by the difference
terms and ECM. error-correction term zt-1
captures the levels (long run) information.
using lags to circumvent simultaneous equation
problem(see VAR models, and the link between
reduced form and structural models).
appropriate number of lags (L) of the variables
is unknown, but can be determined by examining
the data.
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COINTEGRATION APPROACHES

• SINGLE EQUATION APPROACHES
• Engle-Granger (1987) 2-step approach
• Engle-Granger-Yoo (1991) 3-step approach
• Stock-Watson (1993), Saikonnen (1991) Dynamic OLS
  (DOLS) approach
• Pesaran-Shin-Smith (1998, 2001) AutoRegressive
  Distributed Lag (ARDL) approach
• Other
• MULTIVARIATE VAR MODEL (more than 1 cointegration
  equation expected)
• Johansen (1988) VECM methodology
• Other

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ENGLE-GRANGER METHOD

• The Engle-Granger two-step procedure
• Test the variables to see if they are
  non-stationary (e.g. use ADF)
• Given that 2 or more are non-stationary, test
  whether variables are cointegrated
• Step 1 - use OLS to regress one variable (y)
  against the others (x1, x2, ) to obtain an
  estimate of the cointegrating vector, or
  cointegrating equation, or long run equation.
• e.g. yt ?0 ?1 xt
• OLS estimates provide superconsistent
  coefficients of long-run model
• but OLS standard errors of the coefficients
  are unreliable

11
ENGLE-GRANGER METHOD (Cont 2)

• Step 2 - test for cointegration, using one of
  several alternatives.See next.
• Residuals of long-run model can be used in
  dynamic, short-run error-correction model, for
  example
• ?yt ?1(L) ?yt-1 ?1(L) ?xt ?1 yt-1 ?0
  ?1 xt-1 ?1t
• all variables are stationary
• normal regression OLS

12
RESIDUAL BASED COINTEGRATION TESTS

• UNIVARIATE RESIDUAL UNIT ROOT TESTS
• Test the estimated cointegrating residual (zt
  yt - ?0 - ?1 xt) for unit root.
• If the residual is stationary conclude in favor
  of cointegration.
• Alternatives
• Cointegration ADF test (CADF) ?zt (?-1) zt-1
  ?k1p ?zt-k ?t
• No use of constant and/or trend except special
  case
• Special critical values ADF-cointegration test
  (MacKinnon, 1991, 1996) because values of z
  are not observed but estimated
• Cointegration PP-testspecial critical values
  Phillips-Ouliaris, 1990
• Cointegration KPSS-testspecial critical values
  Shin, 1994

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RESIDUAL BASED COINTEGRATION TESTS

• ECM COINTEGRATION TEST
• ECM test more powerful than cointegration ADF
  test, because it does not impose a so-called
  common factor constraint on short-term dynamics
  (Kremers et al., 1992)
• Basic ECM cointegration test
• ECM test equation ?yt ?1(L) ?yt-1 ?1(L)
  ?xt ?1 zt-1 ?1t
• estimate using OLS test significance of ECM
  variable zt-1 i.e. test ?1lt0 Standard t-test
  allowed as approx. using normal distribution
  (Banerjee et al., 1993) Alternatively, special
  tables calculated for various assumptions

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RESIDUAL BASED COINTEGRATION TESTS

• Alternative ECM cointegration test
• ECM could be estimated directly (one-step method)
  reducing errors-in-regressors bias
• ECM test equation ?yt ?1(L) ?yt-1 ?1(L) ?xt
  ?1 yt-1 ?1 ?0 ?1 ?1 xt-1 ?1t
• estimate using OLS
• test significance of coefficient yt-1 i.e.
  ?1lt0
• coefficients, except the constant term, can be
  recovered/solved from the estimated equation
• NoteUsing non-linear OLS (NLS) you can estimate
  all the coefficients directlyCorrect estimation
  may need additional leads of ?xt (Banerjee et
  al., 1998)

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ENGLE-GRANGER METHOD (3)

• EG approach weaknesses
• All X variables are assumed exogenous.
• You can do the cointegrating regression in
  different ways (LHS, RHS variables)! Different
  results for each alternative. Should we test all
  combinations?
• No statistical tests possible on cointegration
  vector (long-run model) coefficients because
  standard errors are unreliable.
• Long-run model estimate suffers from
  small-sample bias.
• Rules out multiple cointegration vectors between
  more than 2 variables.
• Step-wise procedure of testing implies the
  compounding of errors.
• Univariate cointegration test with residuals
  imposes possibly invalid restrictions on short
  run behavior

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STOCK-WATSON DOLS METHOD

• Improvement over E-G approach, taking into
  account possible endogeneity of the regressors
• Test the variables to see if they are
  non-stationary (e.g. use ADF)
• Given that 2 or more are non-stationary, test
  whether variables are cointegrated Step 1
  Estimate using OLS
• k1 and k2 denote leads (future) and lags
  (past), usually k1k2 select lags using
  AIC/SIC/HIC OLS estimates of long-run
  coefficients ? are superconsistent correct
  standard errors for serial correlation in ? use
  Newey-West HAC-errors or DGLS
  (Cochrane-Orcutt) normal t-tests allowed on
  cointegration model yt ?0 ?i1n ?i xi,t for
  example to determine which variables are important

17
STOCK-WATSON DOLS METHOD (Cont 2)
Step 2 Test for cointegration as in E-G, using
one of residual based cointegration tests (see
earlier)zt yt - ?0 - ?i1n ?i xi,t
Univariate residual unit root test ECM
cointegration test Notes Some studies employ
first EG-OLS to test for cointegration and then
second apply DOLS to estimate and test final
cointegration coefficients. Stock-Watson (1988)
developed a special cointegration test. Other
studies use the Johansen test for cointegration
first.
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JOHANSEN VECM PROCEDURE
Multivariate case in Vector AutoRegression form
(VAR, n variables) all the variables must be
assumed endogenous between n variables there
may be n-1 separate cointegration relations VAR
yt A(L) yt-1 ?t A(L) A1 A2 L
AkLk-1 a series of coefficient matrices for
all the lagged variables t-1 to t-k Rewritten in
VECM ?yt ?(L) ?yt-1 ? yt-k ?t ?i
(1 A1 Ai), i 1,, k-1 ? (1
A1 Ak) or ? ? ? where ? speeds of
adjustment, and ? is matrix of long-run
coefficients such that ? yt-k represent the
multiple cointegration relationships
Alternative but equivalent representation is?yt
?(L) ?yt-1 ? yt-1 ?t with same ? but
different ?
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JOHANSEN PROCEDURE (Cont 2)
RANK Everything for cointegration depends on the
rank of matrix ?. Matrix ? can give rise to a
number of possibilities ? yt-k must be I(0) (i)
Rank r n all variables in y are I(0), not an
interesting case to start with. (ii) Rank r 0
there are no linear combinations of y that are
I(0), no cointegration exists, and ? is full of
zeros. (iii) Rank r ? (n-1) up to (n-1)
cointegration relationships ? yt-k. I.e. r ?
(n-1) rows of ? form r linearly independent
combinations of variables in y, each of which is
I(0) alternatively (n-r) nonstationary vectors
forming I(1) stochastic trends. Note that in
practice every (trend-)stationary, I(0) variable
in the model also creates its own cointegration
relationship
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JOHANSEN PROCEDURE (3)
Independent cointegration equations For
example Vector variables m p y i are assumed to
be I(1). Theory suggests long-run money demand
equation m p c ? y ? i Thus in equation
for m m p ? y ? i c ? I(0) But
also in equation for y y 1/? m 1/? p ?/?
i c/? ?/? I(0) Similarly for p and
I However, the 4 variables (n4) have only 1
linearly independent cointegration vector (r 1
lt 4).
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JOHANSEN PROCEDURE (4)
COINTEGRATION TEST STATISTICSThe number of
significant characteristic roots (eigenvalues) of
a matrix determine its rank. Two test statistics
for the number of roots are called Trace
statistic Maximum eigenvalue statistics(dont
mind their calculation, EViews does it for you)
Trace statistic considered more robust to
deviations from normality (Cheung/Lai, 1993)
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JOHANSEN PROCEDURE (5)

• Steps in the Johansen procedure
• Test the variables to see if they are
  non-stationary (e.g. use ADF).Standard Johansen
  method allows only use of I(1) - and I(0) -
  variables. I(2) variables require different
  method.
• Given that 2 or more variables are I(1), consider
  whether they are cointegrated.
• (i) Select the VEC model- number of lags in VAR
  (various methods Sims LR, AIC/SIC/HIC) -
  constant terms trends in CE and VAR (requires
  some economic reasoning)
• (ii) Test for number of cointegration
  relationships (rank r)- Trace statistic
  Maximum eigenvalue statistic

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JOHANSEN PROCEDURE (5b)

• Re-estimate the VECM with the predetermined r
  cointegration equations and impose restrictions2
  types of restrictions - identifying restrictions
  (which are the LHS variables) - binding
  restrictions (test coefficient values RHS
  variables)
• Continue with other VAR/VEC analysis -
  forecasting - impulse response functions
  (IRFs) - forecast error decomposition (FEDVs) -
  causality tests

24
JOHANSEN PROCEDURE (6)

• Johansen approach weaknesses
• Method sensitive to variables selection and
  number of lags included.
• Method does not perform very well in small
  samples

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EXAMPLE COINTEGRATION
As before Monthly observations on FTA All
Share index, FTA Dividend index, yield on
20 year UK Gilts, 91 day Treasury
bills January 1965 to December 1995 (372
months) Unit root tests for all variables, see
previous results
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EXAMPLE COINTRATION
Cointegration model?Theoretical foundations
p log share priced log dividendsr constant
one-period returng constant long run dividend
growth
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Engle-Granger cointegration approach
E-G Step 1 Estimate long-run model using OLS
Dependent Variable LPRICEMethod Least
SquaresSample(adjusted) 196501
199512Included observations 372 after
adjusting endpoints Variable Coefficient Std.
Error t-Statistic Prob. C 3.472363 0.048990 70.
87869 0.0000 LDIV 1.065223 0.010775 98.85835 0.00
00 R20 -0.049613 0.005175 -9.587883 0.0000 RS -0
.010390 0.004478 -2.320065 0.0209 R-squared 0.97
0405 Mean dependent var 5.936924 Adjusted
R-squared 0.970164 S.D. dependent
var 0.970096 S.E. of regression 0.167567
Akaike info criterion -0.724178Sum squared
resid 10.33290 Schwarz criterion -0.682040 L
og likelihood 138.6972 F-statistic 4022.170
Durbin-Watson stat 0.120788
Prob(F-statistic) 0.000000
OLS standard t-values are unreliable with I(1)
regressors and cannot be used!
28
E-G cointegration test residual unit root test
E-G Step 2 Save residuals from long-run model
as ECM and test for stationarity (ADF)
Null Hypothesis ECM_EG has a unit
rootExogenous NoneLag Length 0 (Automatic
based on SIC, MAXLAG16) t-Statistic
Prob. Augmented Dickey-Fuller test
statistic -3.544133 0.0004 Test critical
values 1 level -2.571210 5
level -1.941680 10 level -1.616127 MacKinn
on (1996) one-sided p-values. Augmented
Dickey-Fuller Test EquationDependent Variable
D(ECM_EG) Method Least SquaresSample(adjusted)
196502 199512Included observations 371 after
adjusting endpoints Variable Coefficient Std.
Error t-Statistic Prob. ECM_EG(-1) -0.06
2978 0.017770 -3.544133
0.0004 R-squared 0.032748 Mean dependent
var 0.000547Adjusted R-squared 0.032748
S.D. dependent var 0.058077S.E. of
regression 0.057118 Akaike info
criterion -2.884705Sum squared resid 1.207110
Schwarz criterion -2.874150 Log
likelihood 536.1129 Durbin-Watson
stat 1.933098
McKinnon (1991, Table 1) cointegration test
critical values N4, T371, no trend1
-4.696065 -4.1291210 -3.83346Conclusion
ECM is not stationary at 10 significance level,
no evidence that cointegration relation exists
Original ADF values cannot be used in
cointegration test!
29
E-G cointegration test ECM approach
E-G Step 2Estimate short-run dynamic model with
ECM and test significance of ECMGeneral-to-specif
ic modeling strategy remove all insignificant
lagged variables, impose restrictions where
possible
Dependent Variable DLPRICEMethod Least
SquaresSample(adjusted) 196505
199512Included observations 368 after
adjusting endpoints Variable Coefficient Std.
Error t-Statistic Prob. C 0.010044 0.003399
2.954730 0.0033 D(DLPRICE(-1))
0.147663 0.038114 3.874195 0.0001 DLPRICE(-3)
0.135877 0.051167 2.655553 0.0083 DLDIV(-3) -0.
423932 0.214837 -1.973279 0.0492 DR20(-1) -0.030
075 0.009418 -3.193466 0.0015 DRS(-1)
0.012719 0.006347 2.003798 0.0458 ECM(-1) -0.06
5163 0.018570 -3.509101 0.0005 R-squared 0.10778
4 Mean dependent var 0.007884 Adjusted
R-squared 0.092955 S.D. dependent
var 0.061379 S.E. of regression 0.058457
Akaike info criterion -2.822212 Sum squared
resid 1.233617 Schwarz criterion -2.747874 L
og likelihood 526.2870 F-statistic 7.268410
Durbin-Watson stat 2.001159
Prob(F-statistic) 0.000000
Banerjee et al (1993) normal t-test
allowed Conclusion ECM(-1) is significant and
negative, evidence cointegration relation exists
30
DOLS cointegration equation (1)
AIC/SIC results actually suggest 0 leads/lags,
but using some and longer lags is suggested
Dependent Variable LPRICEMethod Least
SquaresSample (adjusted) 1965M03
1995M11Included observations 369Newey-West HAC
Standard Errors Covariance (lag
truncation5) Variable Coefficient Std.
Error t-Statistic Prob.  C 3.494944 0.173231 20.1
7500 0.0000 LDIV 1.062994 0.026420 40.23384 0.000
0 R20 -0.050569 0.014613 -3.460608 0.0006 RS -0.
012182 0.008101 -1.503843 0.1335 DLDIV(-1) 0.2979
49 0.659400 0.451848 0.6517 DLDIV 0.410475 0.6765
50 0.606718 0.5444 DLDIV(1) 1.034190 0.669289 1.5
45205 0.1232 DR20(-1) -0.028571 0.027568 -1.03635
6 0.3007 DR20 -0.005637 0.022831 -0.246908 0.8051
DR20(1) -0.017509 0.029950 -0.584602 0.5592 DRS
(-1) 0.029615 0.015429 1.919404 0.0557 DRS 0.0151
79 0.012727 1.192647 0.2338 DRS(1) 0.014788 0.016
648 0.888258 0.3750 R-squared 0.971129    
 Mean dependent var 5.939261Adjusted
R-squared 0.970156     S.D. dependent
var 0.965898 S.E. of regression 0.166864     Akaik
e info criterion -0.708679 Sum squared
resid 9.912331     Schwarz criterion -0.570901Log
likelihood 143.7513     F-statistic 997.8838Dur
bin-Watson stat 0.135411     Prob(F-statistic) 0.0
00000
Compare EG-OLSVariable CoefficientC 3.472363 LD
IV 1.065223 R20 -0.049613RS -0.010390
RS variable not significant
31
DOLS cointegration equation (2)
Insignificant RS omitted from long run equation
Dependent Variable LPRICEMethod Least
SquaresSample 1965M03 1995M11 Included
observations 369 Newey-West HAC Standard
Errors Covariance (lag truncation5)
Variable Coefficient Std.
Error t-Statistic Prob.   C 3.529679 0.169372 20.
83983 0.0000LDIV 1.049880 0.024834 42.27509 0.000
0R20 -0.061125 0.010253 -5.961875 0.0000DLDIV(-1
) 0.284710 0.653013 0.435994 0.6631 DLDIV 0.435271
0.673649 0.646139 0.5186DLDIV(1) 1.129331 0.6708
57 1.683416 0.0932DR20(-1) -0.002880 0.021125 -0.
136328 0.8916DR20 0.006509 0.017247 0.377389 0.70
61DR20(1) -0.007961 0.024488 -0.325111 0.7453
R-squared 0.970251     Mean dependent
var 5.939261Adjusted R-squared 0.969590     S.D.
dependent var 0.965898S.E. of regression 0.168437
    Akaike info criterion -0.700426Sum squared
resid 10.21353     Schwarz criterion -0.605040Log
likelihood 138.2285     F-statistic 1467.679Dur
bin-Watson stat 0.126380     Prob(F-statistic) 0.0
00000
All variables significant Economic model ok
Coeff LDIV close to 1 Coeff R20 negative
32
DOLS cointegration test residual unit root test
DOLS Step 2 Save residuals from long-run model
as ECM and test for stationarity (ADF)
Null Hypothesis ECM_DOLS has a unit
rootExogenous None Lag Length 0 (Automatic
based on SIC, MAXLAG16) t-Statistic   Prob.
Augmented Dickey-Fuller test
statistic -3.544718  0.0004Test critical
values 1 level -2.571210 5
level -1.941680 10 level -1.616127 MacKinn
on (1996) one-sided p-values. Augmented
Dickey-Fuller Test Equation Dependent
Variable D(ECM_DOLS) Method Least
Squares Sample (adjusted) 1965M02
1995M12 Included observations 371 after
adjustments Variable Coefficient Std.
Error t-Statistic Prob.  ECM_DOLS(-1) -0.062373 0.
017596 -3.544718 0.0004 R-squared 0.032699
    Mean dependent var 0.000710 Adjusted
R-squared 0.032699     S.D. dependent
var 0.058119 S.E. of regression 0.057161     Akai
ke info criterion -2.883188 Sum squared
resid 1.208943     Schwarz criterion -2.872632 Lo
g likelihood 535.8313     Durbin-Watson
stat 1.961328
McKinnon (1991, Table 1) cointegration test
critical valuesN3, T371, no trend1
-4.335615 -3.7655110 -3.46864Conclusion
ECM is stationary at 10 significance level, weak
evidence cointegration relation exists
Original ADF values cannot be used in
cointegration test!
33
Johansen cointegration test

• VEC Model4 variables p, d, rl, rsExpected
  cointegration dividend discount model p,
  d, r term structure interest rates rl,
  rs
• VECM options
• number of VAR lags is set at 4(should be
  tested)
• intercept (no trend) in CE - no intercept in VAR
  (EViews option 2)using theory and logic
  economic model has constant for linear approx and
  dividend growth rate none of observed variables
  has deterministic trend unobserved variables
  (risk premium) no trend VAR average growth
  rate dividends and stock price equal
  average change interest rates zero, or equal

34
Johansen cointegration test
VECM Step 1 reduced rank tests
Sample(adjusted) 196506 199512Included
observations 367 after adjusting endpointsTrend
assumption No deterministic trend (restricted
constant)Series LPRICE LDIV R20 RSLags
interval (in first differences) 1 to
4 Unrestricted Cointegration Rank
Test Hypothesized Trace 5 Percent 1
PercentNo. of CE(s) Eigenvalue Statistic Critica
l Value Critical ValueNone 0.116620
79.97545 53.12 60.16At most 1 0.043900
34.46768 34.91 41.07At most 2 0.029442
17.99217 19.96 24.60At most 3 0.018959
7.024876 9.24 12.97() denotes
rejection of the hypothesis at the 5(1)
levelTrace test indicates 1 cointegrating
equation(s) at both 5 and 1 levels Hypothesized
Max-Eigen 5 Percent 1 PercentNo. of
CE(s) Eigenvalue Statistic Critical Value
Critical ValueNone 0.116620 45.50777
28.14 33.24At most 1 0.043900 16.47551
22.00 26.81At most 2 0.029442 10.96729
15.67 20.20At most 3 0.018959 7.024876
9.24 12.97() denotes rejection of the
hypothesis at the 5(1) levelMax-eigenvalue
test indicates 1 cointegrating equation(s) at
both 5 and 1 levels
Rank tests indicate 1 significant cointegrating
equation H0 of no CE (r0) is rejected H0 of 1
CE (r?1) is not rejected Trace test and Max-
Eigenvalue test agree
35
Johansen cointegration equation(s) estimate
VECM Step 2 Unrestricted estimate assuming 1 CE
1 Cointegrating Equation(s) Log likelihood
1232.818 Normalized cointegrating coefficients
(std.err. in parentheses)LPRICE LDIV R20 RS C 1
.000000 -0.335611 0.517688 0.548403 -9.241250
(0.89692) (0.43026) (0.38088)
(4.06584) Adjustment coefficients (std.err. in
parentheses)D(LPRICE) 0.001722 (0.00066)
D(LDIV) 0.000947 (0.00016) D(R20) -0.006053
(0.00394) D(RS) -0.013216 (0.00596)
Additional EViews output for cases of 2 and 3 CEs
is not relevant given our previous test result of
1 CE
coefficients are negative/positive forLPRICE
0.335.. LDIV 0.517.. R20 0.548.. RS 9.241
I(0)but LPRICE 0.335.. LDIV 0.517.. R20
0.548.. RS 9.241 variables R20 and RS are found
to be insignificant for LR model, coefficient for
LDIV not significantly different from 1 (or from
0)These restrictions can be imposed in the
restricted VEC
Adjustment speed coefficients for 1 CE ECM in
each of the 4 VAR equations
36
Johansen CE estimate with coefficient restrictions
restriction B(1, -1, 0, 0) is rejected (not
shown) restrictions B(1, -1, x, 0) and (1, -1,
0, x) (not shown) are not rejectedbut
restriction on R20 (1, -1, 0, x) produces
implausible coefficient for RS
Vector Error Correction EstimatesSample(adjusted)
196506 199512Included observations 367
after adjusting endpointsStandard errors in ( )
t-statistics in Cointegration
Restrictions B(1,1)1,B(1,2)-1,B(1,4)0
Convergence achieved after 355 iterations.
Restrictions identify all cointegrating
vectorsLR test for binding restrictions (rank
1)Chi-square(2) 3.859979Probability
0.145150
Restrictions, not estimates
Cointegrating Eq CointEq1LPRICE(-1)
1.000000LDIV(-1) -1.000000R20(-1)
0.362298 (0.10549) 3.43455 RS(-1)
0.000000C -4.686244 (1.09448) -4.28172
Formula based on 2 restrictions, excl.
normalization
ECM wrong sign !
Error Correction D(LPRICE) D(LDIV)
D(R20) D(RS) CointEq1 0.004676
0.002768 -0.013684 -0.023992 (0.00194)
(0.00045) (0.01151) (0.01744) 2.41040
6.08611 -1.18838 -1.37538Cut
ECM significant coeff correct sign
37
OTHER USE OF VECM IRFs (36 months)subject to
weaknesses of SVAR
38
Mehra (1998) The bond rate and actual future
inflation

• Economic relevanceFisher hypothesis suggests
  coefficient inflation for nominal interest rate
  equals 1 in cointegration equationShort run
  relationships are complex, because of changes in
  monetary policy over time
• Cointegration estimates (plus causality
  tests)Unit root testsJohansen method and DOLS
• Data United States10-year government bond,
  federal funds rate, CPI inflation, output
  gapQuarterly, 1959Q1-1996Q4

39
Mehra (1998) The bond rate and actual future
inflation
40
Mehra (1998)
Note typing error
41
Mehra (1998)
DOLS standard errors not reported
42
CONCLUSIONS

• Unit root and cointegration tests are essential
  in modern empirical research
• Economics has many (theoretical) equilibrium
  relationships
• Cointegration requires some (economic)
  parameters to be constant / stationary over
  timeHowever, structural breaks and regime shifts
  are most likely
• Adjustment processes in economics usually
  require long adjustment periodsCointegration /
  ECM tests require long time periods span rather
  than many observations daily vs annual

43
VAR LAG LENGTH SELECTION
Unit root variables, use 1st differences
VAR Lag Order Selection Criteria Endogenous
variables DLDIV DLPRICE DR20 DRS Exogenous
variables CSample 196501 199512Included
observations 359 Lag LogL LR FPE AIC SC HQ 0
1102.694 NA 2.58E-08 -6.120858 -6.077589 -6.103
651 1 1157.779 108.6361 2.08E-08 -6.338603 -6.
122262 -6.252572 2 1174.263 32.14157
2.07E-08 -6.341299 -5.951885 -6.186445 3
1184.995 20.68635 2.13E-08 -6.311950 -5.749463 -
6.088271 4 1188.681 7.022523
2.28E-08 -6.243347 -5.507787 -5.950844 5
1196.543 14.80367 2.39E-08 -6.198008 -5.289376 -
5.836681 6 1210.511 25.99237
2.42E-08 -6.186693 -5.104988 -5.756542 7
1217.982 13.73406 2.54E-08 -6.139175 -4.884397 -
5.640200 8 1233.620 28.40201
2.54E-08 -6.137161 -4.709310 -5.569362 9
1240.182 11.77085 2.68E-08 -6.084580 -4.483656 -
5.447956 10 1248.197 14.19882
2.81E-08 -6.040094 -4.266097 -5.334646 11
1254.971 11.84952 2.96E-08 -5.988695 -4.041625 -
5.214423 12 1264.552 16.54766
3.07E-08 -5.952938 -3.832796 -5.109842
indicates lag order selected by the criterionLR
sequential modified LR test statistic (each test
at 5 level)FPE Final prediction
error AIC Akaike information
criterion SC Schwarz information
criterion HQ Hannan-Quinn information
criterion
3112 372 1 for 1st diff 12 for max lag
44
LAG LENGTH AND RESIDUAL CORRELATION1 LAG MODEL
VAR Residual Portmanteau Tests for
AutocorrelationsH0 no residual autocorrelations
up to lag hSample 196501 199512Included
observations 370Lags Q-Stat Prob. Adj
Q-Stat Prob. df 1 2.322186 NA
2.328479 NA NA 2 32.02764 0.0099 32.19538
0.0094 16 3 52.35899 0.0130 52.69292
0.0121 32 4 65.63168 0.0462 66.11067
0.0425 48 5 85.36840 0.0384 86.11776
0.0341 64 6 109.9050 0.0149 111.0588
0.0124 80 7 120.7004 0.0449 122.0624
0.0375 96 8 150.5627 0.0088 152.5847
0.0065 112 9 164.6290 0.0161 167.0016
0.0117 128 10 184.7106 0.0125 187.6410
0.0085 144 11 194.8653 0.0315 198.1069
0.0218 160 12 207.7349 0.0511 211.4079
0.0352 176 The test is valid only for lags
larger than the VAR lag order. df is degrees of
freedom for (approximate) chi-square distribution
3112 372 1 for 1st diff - 1 lag model
Q-Stat B-PAdj Q-Stat L-B
Residual correlation remains with 1 lag VAR model
45
LAG LENGTH AND RESIDUAL CORRELATION2 LAG MODEL
VAR Residual Portmanteau Tests for
AutocorrelationsH0 no residual autocorrelations
up to lag hSample 196501 199512Included
observations 369Lags Q-Stat Prob. Adj
Q-Stat Prob. df 1 0.582023 NA
0.583605 NA NA 2 1.794387 NA
1.802575 NA NA 3 18.77958 0.2803 18.92699
0.2725 16 4 32.45501 0.4443 32.75229
0.4299 32 5 49.08332 0.4295 49.60901
0.4089 48 6 70.06338 0.2815 70.93585
0.2576 64 7 82.40645 0.4048 83.51760
0.3720 80 8 106.4813 0.2182 108.1260
0.1872 96 9 122.4393 0.2354 124.4829
0.1979 112 10 141.1596 0.2012 143.7246
0.1619 128 11 150.9869 0.3284 153.8539
0.2719 144 12 163.9779 0.3983 167.2816
0.3306 160 The test is valid only for lags
larger than the VAR lag order. df is degrees of
freedom for (approximate) chi-square distribution
3112 372 1 for 1st diff - 2 lags model
Residual correlation is removed with 2 lag VAR
model