Testing Purchasing Power Parity (PPP) between India and the US
Part II: Cointegration Analysis of Long-run PPP
The analysis in Seminars 6 and 7 involved univariate tests of PPP based on an analysis of the stability of the (log) real exchange rate. In this context the series was: 代写留学生论文tested for non-stationarity/unit roots using ADF, P-P and KPSS tests; and tested for mean reversion using the GPH long memory test. The results from this analysis suggested conclusively that PPP does not hold in the sample for India and the US.
The main objective in Seminar 8 is to re-assess the evidence for PPP using cointegration techniques applied to the same data-set (ppp_ind_us.wf1). This analysis involves:
• Testing for cointegration using the Engle-Granger 2-Step estimator.
• Testing for cointegration using Johansen’s Full Information Maximum Likelihood (FIML) estimator:
o Estimating the cointegrating rank, long-run parameters and equilibrium adjustment parameters.
• Testing PPP and weak exogeneity restrictions in the Johansen model.
The learning outcomes from this analysis will be to develop your understanding of:
• Applying and interpreting the Engle-Granger 2-Step estimator in Eviews.
• The contexts in which the Engle-Granger 2-Step estimator may be a poor estimator of long-run relationships:
o When regressors are not weakly exogenous for the long run parameters
o In the presence of multiple cointegrating relationships.
• The utility of the Johansen estimator in the above contexts.
• Applying and interpreting the Johansen estimator in Eviews.
• The pitfalls of Johansen. Principally the importance of basing your analysis soundly in economic/finance theory to help you to:
o Determine the number of cointegrating relationships.
o Identify the long-run parameters.
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1. Background and overview
/The application of cointegration tests is conditional on the individual series following I(1) processes.
Be sure to go through the unit root tests for the (logs of the) nominal exchange rate, Indian WPI and US WPI (as outlined in the handout for Seminars 6/7). Testing for unit roots in the individual series is a necessary preliminary step in a cointegration analysis.
The evidence of the ADF, P-P and KPSS tests indicated conclusively that the (logs of the) nominal exchange rate and Indian WPI are I(1) processes. The evidence for the US WPI was less conclusive with there being some evidence that the series is a trend stationary I(0) process. However this is an unusual result and we will take this series as being I(1) for the purposes of the cointegration analysis.
Firstly we will carry out a single equation cointegration test: the Engle-Granger 2-step estimator. Then we will look at evidence for cointegration based on Johansen’s systems estimator. A summary of the techniques is given below (see also the notes for Lectures 8 and 9).
Engle-Granger 2-Step Estimator
Step 1 is centred on the long-run equation.
ttttPPSεβββ+++=*321lnlnln
The residual term is tested for a unit root using an ADF test: a unit root (non-stationarity) implies that there is no cointegrating relationship. If we reject the null there is evidence of a cointegrating relationship. In this framework, cointegration is a necessary condition for long-run PPP.
PPP also implies the following restrictions on the long-run parameters:
1:320=−=ββH.
These restrictions imply which is the form we used when we tested the (log) real exchange rate for non-stationarity and applied the GPH (long memory) test. The existence of a cointegrating relationship which satisfies the above parameter restrictions provides the ttttPPSε=+−*lnlnlnnecessary and sufficient conditions for long-run PPP.
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However we cannot test these restrictions in the OLS equation because classical t and F tests are invalid for inferences.
The main problem in this regard is that the variables are I(1) (which gives rise to non-standard test distributions – see Lecture 8). Also there is autocorrelation in the error term, due to omitted short run dynamics, which invalidates OLS standard errors. Our analysis at this stage is therefore restricted to a test of the necessary condition for PPP (i.e., cointegration). Tests of the necessary and sufficient conditions (i.e., cointegration along with the parameter restrictions) will be deferred until the Johansen analysis.
The Granger Representation Theorem says that cointegration is both a necessary and sufficient condition for the existence of an Error Correction Model (ECM). Accordingly, if we reject the null of a unit root in the long-run residuals (i.e., if we find evidence for cointegration) then we can go on to estimate an ECM for the nominal exchange rate (step 2 of Engle-Granger).
Step 2 therefore involves estimation of the following model
ΣΣΣ=−−=−=−++Δ+Δ+Δ+=ΔmjttjtjmjjtjmjjtjtvSPPS111*1ˆlnlnlnlnεαφγδμ
Short run dynamics
Error correction term (lagged equilibrium error). The coefficientαmeasures the speed of adjustment of the exchange rate back to the equilibrium
This equation allows us to estimate the short run dynamics and the speed of adjustment to dis-equilibrium.
Classical inferences (t and F tests) are valid in the ECM since all the variables in this model are I(0) (stationary).
The Engle-Granger test assumes that the domestic and foreign wholesale price series (the regressors) are weakly exogenous for the long-run parameters. This means that the estimator assumes that there is no information about the long-run parameters contained in the ECMs for the wholesale prices. If weak exogeneity does not hold then the Engle-Granger test is inefficient because it ignores this additional information (see Lecture 9). 3
Johansen’s Full Information Maximum Likelihood Estimator
In this context the efficient estimator is Johansen’s Full Information Maximum Likelihood estimator (FIML):
• FULL INFORMATION:
⇒ the estimator uses information from all the ECM equations in the system.
• MAXIMUM LIKELIHOOD:
⇒ the parameters are chosen to maximize the likelihood of generating the observed sample of data (see lecture 6). The likelihood function is based on the assumption that the data are jointly normally distributed.
This approach is efficient because it estimates the cointegrating relationships using the entire system of equations for the exchange rate and wholesale price series. Therefore, if there is information about the long run in the wholesale price equations, it will be incorporated into the Johansen estimator.
Another potential benefit of Johansen’s estimator is that it can estimate the number of cointegrating relationships:
In a system of I(1) variables there can, in principle, be up to n1−n cointegrating relationships (see Lecture 9):
1−≤nr
Engle-Granger 2-step is unable to do identify multiple cointegrating relationships because it is a single equation estimator.
Note however, in the context of PPP, economic theory predicts there is only one long-run relationship. So if we found two cointegrating vectors in a PPP analysis it would be hard to interpret and identify the second relationship. Therefore our reason for using Johansen in this context is not because it can estimate multiple cointegrating vectors. Rather, we are using it because it is an efficient estimator of a single cointegrating vector (if the wholesale price indices are not weakly exogenous).
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Implementing Johansen
To put Johansen into practice we first need to estimate an unrestricted VAR in the levels of the data; this will determine the lag length for the short-run dynamics and provide an estimate of the long-run matrix Π
()⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=++++=−−*211lnlnln,0~ ,...ttttttptpttPPSyNIIDvvyAyAyσμ
We need to choose the lag length p so that the error terms are normally and independently distributed. The normality requirement is because Johansen is a maximum likelihood estimator based on a Gaussian distribution. However, the large sample properties of the estimator depend only on an IID assumption for the errors – practically speaking this means that models with non-normal errors are OK so long as there is no autocorrelation in the error terms.
The next stage is to estimate the long-run dynamics conditional on the lag length chosen for the VAR. This involves estimating the following factorization of the long run coefficient matrix
()IAAp−++=Π′=Π...1βα
rn×matrix of equilibrium
adjustment parameters
nr×matrix of
cointegrating vectors
The first step in this factorization is to estimate the number of cointegrating vectors - . The basic intuition here is that the cointegrating relationships are the linear combinations of the I(1) levels of y which are ()Π=rankrcorrelated with the I(0) differences of y; conversely I(1) linear combinations of the levels are uncorrelated with the differences.1 These correlations are estimated by the eigenvalues ()λof the matrix of ‘squared’ correlations between the levels and differences (see Lecture 9 notes). The number of non-zero eigenvalues/squared correlations therefore gives the number of cointegrating vectors.
1 Note that an I(1) variable will have a zero correlation with an I(0) variable. Suppose x~I(1) and y~I(0) then()()yxxyyxσσρ,cov=. This correlation will be asymptotically zero because the standard deviation of the I(1) process, xσ, is asymptotically infinite.
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Johansen proposed two tests of cointegrating rank based on these eigenvalues:
Maximum eigenvalue statistic
()()()1::1,...,1,0 ,ˆ1log101max+=Π≤Π−=−−=+rrankHrrankHnrTrλλ
This is a test of significance of the marginal eigenvalue. If there are at most r cointegrating vectors this marginal, , eigenvalue should be zero.
Trace statistic
()()().::1,...,1,0 ,ˆ1log101rrankHrrankHnrTnriitrace>Π≤Π−=−−=Σ+=λλ
This is a test of the joint significance of the remaining n−r eigenvalues. If there are at most r cointegrating vectors then the remaining n−r eigenvalues should be zero.
Estimates of the cointegrating vectors ()βˆ are given by the r eigenvectors corresponding to the r non-zero eigenvalues.2 The linear combinations of the levels of y, implied by these eigenvectors, are correlated with the differences of y and are therefore stationary:. The adjustment parameters can then be estimated given the estimated cointegrating vectors. Specifically )0(~ˆIyβ′αˆcan be estimated from a regression of on (the vector error correction term) given the short run dynamics tyΔpty−′βˆ
11,...,+−−ΔΔpttyy
2 These eigenvectors Βdiagonalize the correlation matrix Π~ into its matrix of eigenvalues:
Λ=ΒΠΒ′~
where: is a matrix with the eigenvalues/correlations on the lead diagonal and zeroes everywhere else; and the cointegrating vectors Λβ′are the first r rows of Β′ (corresponding to the non-zero eigenvalues in ). Λ
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Having estimated the long-run features of the system (the cointegrating rank, cointegrating vectors and adjustment parameters) the short-run dynamics are estimated in the VECM
tptptpttvyyyy+′+ΔΓ++ΔΓ+=Δ−+−−−βαμ1111...
Short run dynamics
The lag length of p in the levels VAR translates into a lag length of p-1 in the VECM.
Error correction term
In practice the ECT can be entered in the equation at any lag (often at the first lag).
As with any econometric modelling apply the rule: ‘test, test, test’. That is, subject the estimated VECM to rigorous misspecification testing. Any evidence of misspecification could be due to a poor selection of the lag length or possibly there are exogenous variables which need to be included in the model. The misspecification testing should take place before testing the restrictions on the model parameters implied by the theory.
Testing economic hypotheses about the cointegrating vectors and adjustment parameters
One of the bonuses of the Johansen approach is that it gives a general framework for testing hypotheses about the cointegrating vectors and adjustment parameters. In the context of testing PPP we can use this framework to:
• Test the long-run PPP restrictions.
• Test the weak exogeneity of domestic and foreign prices (which was assumed in the Engle-Granger test).
To illustrate these restrictions, in the context of PPP, look at the VECM (ignoring short-run dynamics and assuming there is one cointegrating vector as implied by theory)
()⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛+⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛+⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛ΔΔΔ−−−tttttttttvvvPPSPPS321*111131211312111321*lnlnlnlnlnlnβββαααμμμ
PPP implies the following restrictions on the βvector
()111−=′β
Weak exogeneity of domestic and foreign prices implies
⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=0011αα
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Re-writing the VECM with these restrictions we get
()()ttttttttttttttttttvPvPvPPSSvvvPPSPPS33*221*111111321*11111321*lnlnlnlnlnlnlnlnln11100lnlnln+=Δ+=Δ++−+=Δ⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛+⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛+⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛ΔΔΔ−−−−−−μμαμαμμμ
We can see that, if the weak exogeneity restrictions hold, there is no information about the long run parameters ()β in the domestic and foreign price equations. Therefore, for purposes of estimating the long-run parameters, it would be sufficient (and efficient) to estimate the model using a single equation (i.e., Engle-Granger) estimator of the log nominal exchange rate:
()tttttvPPSS1*111111lnlnlnln++−+=Δ−−−αμ
In summary our empirical objectives are to:
1. Test for cointegration using the Engle-Granger 2-step estimator.
2. Estimate the cointegrating rank, long-run parameters and equilibrium adjustment parameters using Johansen Full Information Maximum Likelihood.
3. Test PPP and weak exogeneity restrictions in the Johansen model.
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2. Engle-Granger 2-Step
E-G Step 1: Estimate the cointegrating relationship using OLS
On the main toolbar click Quick/Estimate equation and enter
lns c lnp lnpstar
Dependent Variable: LNSMethod: Least SquaresDate: 03/03/06 Time: 17:56Sample: 1973M01 2005M10Included observations: 394VariableCoefficientStd. Errort-StatisticProb. C2.4832680.1071923.1670LNP1.4435920.01642487.89270LNPSTAR-1.164620.037385-31.15240R-squared0.984767 Mean dependent va2.91605Adjusted R0.984689 S.D. dependent var0.690374S.E. of reg0.085425 Akaike info criterion-2.07477Sum squar2.853298 Schwarz criterion-2.04449Log likeliho411.7297 F-statistic12638.48Durbin-Wa0.082395 Prob(F-statistic)0
The coefficients appear to be far from the values implied by PPP – but, as mentioned above, we can’t test the PPP restrictions in this equation. But we can test whether the exchange rate and relative prices cointegrate…
Accordingly, obtain the residuals from the OLS equation to test whether or not they are non-stationary
On the equation toolbar click Proc/Make Residual Series
Name for resid series: coint_resid
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Now open the series coint_resid and
View/Unit Root Test
Test type: ADF
Test for unit root: in level
Include in test equation: none
(the alternative hypothesis is stationarity around a zero mean since, given that the OLS equation included a constant, the OLS residuals have a zero mean by construction)
Null Hypothesis: COINT_RESID has a unit rootExogenous: NoneLag Length: 1 (Automatic based on SIC, MAXLAG=16)t-Statistic Prob.*Augmented Dickey-Fuller test stat-3.202270.0014Test critica1% level-2.570885% level-1.9416310% level-1.61616*MacKinnon (1996) one-sided p-values.
This p-value is incorrect since the series coint_resid was estimated from an OLS regression. Since OLS minimises the residual variance, residuals which are in fact non-stationary will tend to appear more stationary than they actually are. The correct critical values take into account that the residuals were estimated from an n-variable regression (see the MacKinnon Critical Values given in a separate handout).
Using MacKinnon’s Critical Values for n=3, constant no trend (the constant appeared in the OLS equation used to obtain the residual series) we can calculate the correct critical value for a 5% test
()7641844.339441.13394352.87429.3%522211−=−−−=++=−−∞TTCφφφ
This critical value implies the residuals are non-stationary (-3.202>-3.764). The inference from this is that there is:
• No cointegration between the exchange rate and relative prices
• No long-run PPP relationship.
The Granger Representation Theorem says that cointegration between a set of variables is a necessary and sufficient condition for the existence of an error correction representation for those variables.
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The above result therefore implies that there is no error correction representation for the exchange rate. However, purely out of interest, we will attempt to estimate the ECM
E-G Step 2: Estimate the ECM
On the main toolbar click Quick/Estimate Equation and enter
d(lns) c coint_resid(-1) d(lns(-1)) d(lnp(-1)) d(lnpstar(-1))
(assuming a first-order ECM i.e., only one lag of the differenced series)
Dependent Variable: D(LNS)Method: Least SquaresDate: 03/03/06 Time: 18:31Sample (adjusted): 1973M03 2005M10Included observations: 392 after adjustmentsVariableCoefficientStd. Errort-StatisticProb. C0.0044380.0012233.6288660.0003COINT_RESID(-1)-0.01460.012069-1.209540.2272D(LNS(-1))0.021280.0509110.4179740.6762D(LNP(-1))0.0346690.1013360.3421230.7324D(LNPSTAR(-1))-0.050640.128434-0.394250.6936R-squared0.004537 Mean dependent va0.004569Adjusted R-squared-0.00575 S.D. dependent var0.019865S.E. of regression0.019922 Akaike info criterion-4.9813Sum squared resid0.153596 Schwarz criterion-4.93065Log likelihood981.3353 F-statistic0.440956Durbin-Watson stat2.00463 Prob(F-statistic)0.778997
This is a poor model (just as the Granger Representation Theorem would have predicted). None of the coefficients are significant. In particular, the coefficient on the error correction term (lagged long-run residuals) is insignificant – this is exactly what we would expect given that we found no cointegration at step 1. Also the short-run coefficients (coefficients on the differenced variables) are highly insignificant.
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3. Johansen FIML estimator
Estimate the VAR
On the main toolbar click Quick/ Estimate VAR…
VAR Type: Unrestricted VAR
Endogenous variables: lns lnp lnpstar
Exogenous variables: c
The aim here is to determine the VAR lag order p. We will use information criteria and tests for residual normality and autocorrelation to arrive at this lag order. Start with the information criteria
On the VAR toolbar…
View/Lag Structure/Lag Length Criteria
Lags to include: 8
VAR Lag Order Selection CriteriaEndogenous variables: LNS LNP LNPSTAR Exogenous variables: C Date: 03/13/07 Time: 23:45Sample: 1973M01 2005M10Included observations: 386 LagLogLLRFPEAICSCHQ0323.0781NA 3.82E-05-1.658436-1.627691-1.6462413563.5756413.8342.05E-12-18.40194-18.27896-18.353223616.604104.1351.63E-12-18.63007 -18.41486* -18.54473*33625.18416.714731.63E-12-18.6279-18.32045-18.50643634.57218.144351.63E-12-18.62991-18.23023-18.471453643.53517.183161.63E-12-18.62972-18.1378-18.434663653.174 18.32896* 1.62e-12* -18.63303*-18.04888-18.401473658.69610.41381.65E-12-18.615-17.93862-18.346883663.9549.8345651.69E-12-18.59562-17.82699-18.2908 * indicates lag order selected by the criterion LR: 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
The Schwarz and Hannan-Quinn criteria are minimized for a VAR(2) model. The Akaike criterion is minimized for a VAR(6).
We will therefore try estimating a VAR(2) (on grounds of parsimony). However make sure there is no autocorrelation in the residuals before proceeding. If there is autocorrelation you need to estimate a model with more lags.
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To test for residual autocorrelation in the VAR we can look at a multivariate test for autocorrelation:
On the VAR toolbar click View/Residual Tests/Autocorrelation LM test
VAR Residual Serial Correlation LH0: no serial correlation at lag ordeDate: 03/13/07 Time: 23:56Sample: 1973M01 2005M10Included observations: 392LagsLM-StatProb115.405440.0804215.060890.0893327.951470.001415.394470.0807519.320390.022666.1323870.7266717.211480.0455812.543650.18446.6901360.66931013.627660.13621119.194470.02361225.553960.0024Probs from chi-square with 9 df.
The residuals appear to be linearly independent at most lag orders. This suggests that a VAR(2) is sufficient to mop-up all the dynamics in the system.
Conditional on p=2 we can now proceed to estimating the cointegrating rank and the long-run parameters.
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Estimate the cointegrating rank, cointegrating vectors and adjustment parameters
On the VAR toolbar…
View/Cointegration Test
Deterministic trend assumption of test:
Allow for linear deterministic trend in data
3. Intercept (no trend) in CE and test VAR
Lag Intervals: 1 1
Date: 03/14/07 Time: 00:07Sample (adjusted): 1973M03 2005M10Included observations: 392 after adjustmentsTrend assumption: Linear deterministic trendSeries: LNS LNP LNPSTAR Lags interval (in first differences): 1 to 1Unrestricted Cointegration Rank Test (Trace)HypothesizedTrace0.05No. of CE(s)EigenvalueStatisticCritical ValueProb.**None *0.08790747.2041729.797070.0002At most 10.02182611.1349315.494710.2033At most 20.0063182.4843953.8414660.115 Trace test indicates 1 cointegrating eqn(s) at the 0.05 level * denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-valuesUnrestricted Cointegration Rank Test (Maximum Eigenvalue)HypothesizedMax-Eigen0.05No. of CE(s)EigenvalueStatisticCritical ValueProb.**None *0.08790736.0692421.131620.0002At most 10.0218268.65053314.26460.3164At most 20.0063182.4843953.8414660.115 Max-eigenvalue test indicates 1 cointegrating eqn(s) at the 0.05 lev * denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values
Note that a VAR(2) implies a VECM(1) model – so only specify 1 lag at this stage.
Allow for a trend in the data (recall from seminar 6 that the line graphs of the individual series showed trends).
However don’t include a trend in the cointegrating relationship – long-run PPP is an equilibrium relationship: it does not have a trend!
Both the trace and maximum eigenvalue tests indicate there is one cointegrating vector. This is the number of cointegrating relationships predicted by the theory (i.e., one equilibrium equation relating to long-run PPP).
Note that in principle we could have found two cointegrating vectors in a system of n=3 I(1) variables. However a second cointegrating vector has no economic interpretation in the context of testing PPP.
When using Johansen your analysis should be firmly grounded in theory. Otherwise it’s very easy to get confused!
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The estimated cointegrating vector and adjustment parameters are reported at the same stage of the analysis.
1 Cointegrating Equation(s): Log likelihood3633.076Normalized cointegrating coefficients (standard error in parentheses)LNSLNPLNPSTAR1-1.6341981.928470.087090.20224Adjustment coefficients (standard error in parentheses)D(LNS)0.0038950.00729D(LNP)0.0015810.00338D(LNPSTAR)-0.0161580.00277
Based on PPP, we were expecting a cointegrating vector of the form ()111−=′β for PPP. Our estimate is which appears to be some way from the hypothetical values – we will test if the PPP restrictions hold shortly. (946.1641.11ˆ−=′β
Note that the cointegrating vector has been normalised on the first coefficient in the vector (111=β: this is the coefficient on ln). Since r=1, this normalisation is sufficient to identify the cointegrating vector.
However, in general, if there are r cointegrating vectors at least r restrictions per cointegrating vector are required to achieve identification (see the example given for identifying the cointegrating vectors in testing the Expectations Hypothesis of the term structure in Lecture 9).
Also note that the estimated adjustment parameters in the exchange rate equation is positive (the wrong sign), small and statistically insignificant. This suggests that the exchange rate does not adjust to previous disequilibrium. In fact only the adjustment coefficient in the foreign (US) price equation (D(LNPSTAR)) is statistically significant – the adjustment back to equilibrium appears to take place through this equation.
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Based on r=1 we can now estimate the VECM.
On the VAR toolbar click on the Estimate button
Change the VAR Type to Vector Error Correction
Lag intervals for Endogenous: 1 1
Click on the Cointegration tab to check the long run settings
Number of cointegrating equations: 1
Deterministic trend assumption of test:
Allow for linear deterministic trend in data
3. Intercept (no trend) in CE and test VAR
Then click OK
(Output omitted to save space)
Before proceeding to test the PPP restrictions in the model carry out a basic misspecification test on the estimated VECM.
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On the VAR toolbar click View/Residual Tests/Autocorrelation LM test
VEC Residual Serial Correlation LH0: no serial correlation at lag ordeDate: 03/14/07 Time: 01:07Sample: 1973M01 2005M10Included observations: 392LagsLM-StatProb116.030280.0663215.766060.0719326.823690.0015415.873320.0696519.935530.018366.7714220.6609716.944230.0496812.434310.18997.2130170.6151013.152760.15581120.274280.01631224.649440.0034Probs from chi-square with 9 df.
Overall the LM test suggests the VECM(1) model with r=1 (one cointegrating vector) is an adequate model for the data.
Now proceed to testing the PPP restrictions in the cointegrating vector
Test the PPP and weak exogeneity restrictions
On the VAR toolbar click Estimate
VAR Type: Vector Error Correction
Lag intervals: 1 1
Click on the ‘Cointegration’ tab…
Rank/number of cointegrating vectors: 1
Deterministic Trend Specification: Option 3 (see above)
Click on the ‘VEC Restrictions’ tab…
Check: ‘Impose Restrictions’
b(1,1)=1, b(1,2)=-1, b(1,3)=1
Click on OK…
Vector Error Correction Estimates Date: 03/14/07 Time: 01:23 Sample (adjusted): 1973M03 2005M10 Included observations: 392 after adjustments Standard errors in ( ) & t-statistics in [ ]Cointegration Restrictions: B(1,1)=1, B(1,2)=-1, B(1,3)=1 Convergence achieved after 1 iterations.Restrictions identify all cointegrating vectorsLR test for binding restrictions (rank = 1): Chi-square(2)20.11246Probability0.000043Cointegrating Eq: CointEq1LNS(-1)1LNP(-1)-1LNPSTAR(-1)1C-3.45498Error Correction:D(LNS)D(LNP)D(LNPSTAR)CointEq1-0.001329-0.001379-0.00591-0.00382-0.00177-0.00148
Note: b(i,j) is the long-run coefficient for variable j in the ith cointegrating vector
The cointegration restriction test (‘LR – likelihood ratio – test for binding restrictions’) is significant which indicates that the PPP restrictions do not hold.
As noted before, the adjustment parameters are very small in magnitude and only the adjustment in the foreign (US) price equation is significant.
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We can test the weak exogeneity restrictions in much the same way as above. To test the weak exogeneity of the domestic and foreign price indices, alongside the PPP restrictions, type in the VEC restrictions box:
b(1,1)=1, b(1,2)=-1, b(1,3)=1,
a(2,1)=0, a(3,1)=0
Additional weak exogeneity restrictions
Vector Error Correction Estimates Date: 03/14/07 Time: 01:36 Sample (adjusted): 1973M03 2005M10 Included observations: 392 after adjustments Standard errors in ( ) & t-statistics in [ ]Cointegration Restrictions: B(1,1)=1, B(1,2)=-1, B(1,3)=1, A(2,1)=0, A(3,1)=0 Convergence achieved after 2 iterations.Restrictions identify all cointegrating vectorsLR test for binding restrictions (rank = 1): Chi-square(4)35.87545Probability0Cointegrating EqCointEq1LNS(-1)1LNP(-1)-1LNPSTAR(-1)1C-3.45498Error Correction:D(LNS)D(LNP)D(LNPSTAR)CointEq1-0.00170200-0.0038100[-0.44628][ NA][ NA]
The joint test of the weak exogeneity and PPP restrictions is strongly rejected. This highlights that the weak exogeneity assumption made by the Engle-Granger test is invalid for our sample.
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4. Conclusions
• The Engle-Granger 2 Step test found no cointegrating relationship between the nominal exchange rate and wholesale price series for India-US.
• In contrast, the Johansen system estimator found the existence of a single cointegrating relationship (a necessary condition for long-run PPP).
• However tests of the PPP restrictions on the long-run parameters were rejected.
• Accordingly, there is only partial evidence for long-run PPP between India and the US.
• /In fact, rejection of the long-run parameter restrictions is quite a common finding in empirical tests of PPP (even if there is evidence for cointegration). The presence of transportation costs and/or differences in the bundles of goods used to form the domestic and foreign price indices (measurement errors) could explain the deviation of the long-run parameters from their theoretical values.
Reading
IB9Y6 Empirical Finance, Notes for lectures 8 and 9.
代写留学生论文Brooks (2002), Introductory econometrics for finance, Chapter 7: See in particular Section 7.13 for another worked Eviews application of Engle-Granger 2-step and Johansen.
Stuart Fraser, March 2007.
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