Does it matter which is the dependent variable in regression of time series data? I am testing for cointegration between the Real GDP per capita of England and France. I use a Dickey-Fuller test to test for stationarity and concluded that both of my series are non-stationary. So I then ran the regression and did the second Dickey-Fuller test for co-integration on the residual. Here is what I am confused about: 


If I ran the regression and had rgdpe of England as a dependent variable I concluded that my regression was spurious. (My residuals were non-stationary).
If I ran the regression and had rgdpe of France as a dependent variable I concluded that the rgdpe are cointegrated (My residuals were stationary).


Can anyone explain the reason for this discrepancy?
Asymptotically, you should be able to interchange the role of $Y_t$ and $X_t$ and estimate lets say $$X_t =\alpha^* + \beta^* Y_t+u^*_t$$ where $\alpha^* = -\alpha/\beta$ and $\beta^*= 1/\beta$ and still get cointegrated relationship.

However, crucial caveat here is that this holds only asymptotically and only for $R^2$ close to unity. In finite samples you can find discrepancy between the two. Moreover, if $R^2$ is not high you will also find discrepancy and in that case you should not even model cointegrated relationship with standard OLS but rather with something like Fully Modified OLS (FMLS) or Canonical OLS. Also, which variable is exogenous and which endogenous will matter if the variables are not integrated of the same order or one of them is stationary. Maybe you just got false negative in one of the previous tests.

All this aside if you believe that there could be a cointegrated relationship that could go both ways it is not really appropriate to use Engle-Granger approach to test for it by two Engle-Granger tests. Rather you should use Johansen cointegration test that tests for it by estimating simultaneous system of these all in one model.