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I am examing if there is a cointegrating relationhip based on the Engle-Granger approach between EUR/USD and GBP/USD currency series. Applying the ADF test in the log-price and log-return series I found out that the level series are both I(1). Then I regressed the logeuro levels on the loggbp levels. Applying the ADF test (with constant but not trend) on the residuals estimated from the regression, I get the following output:

Null Hypothesis: RESID01 has a unit root
Exogenous: Constant
Lag Length: 0 (Automatic - based on SIC, maxlag=21)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -3.025600 0.0329
Test critical values: 1% level -3.435957
5% level -2.863904
10% level -2.568079
*MacKinnon (1996) one-sided p-values.

1st question: Are the above critical values the ones I must use for the rejection of the null hypothesis? I believe there is another set of critical values when we test for the stationarity on the residuals of an estimated LR model. Where can I found this set?

2nd question: Should I regress the loggbp levels on the logeur levels? Doing that and applying the ADF test on the estimated residuals I get:

Null Hypothesis: RESID02 has a unit root
Exogenous: Constant
Lag Length: 0 (Automatic - based on SIC, maxlag=21)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -3.079654 0.0284
Test critical values: 1% level -3.435957
5% level -2.863904
10% level -2.568079

*MacKinnon (1996) one-sided p-values.

3rd question: From an economical point of view is it logical the two time series to cointegrate (ie have a long run relationship)?

4th question: If the two series are cointegrated how can we estimate the ECM (error correction model)? I believe that the ECM model is: dlogeur=b0+b1*dloggbp+b2*RESID(-1). Does this model captures the long-run relationship between the series as well as the short-run relationship? What is the interpretation of the coefficients b1 and b2?

To answer just the first question, yes there is a different set of critical values. The built-in Engle-Granger test from a group object performs the relevant regression and test computation.

To answer just the first question, yes there is a different set of critical values. The built-in Engle-Granger test from a group object performs the relevant regression and test computation.

I believe this set of critical values is the one tabulated by MacKinnon: Critical Values for Cointegration Tests

Using these critical values the null hypothesis cannot be rejected so the residuals are not stationary and thus the two series are not cointegrated.