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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 u