Cointegration of two time series

#1
Hello - this is more of a conceptual question. I'm whether a linear relationship exists between two time series: short-term interest rate differential of U.S. and U.K. and the GBP/USD exchange rate where observations are taken on a daily frequency since 2015.

I ran Dickey-Fuller test on both and find both series to be non-stationary, which indicates that as a framework a linear regression model may not be useful (here short-term rates are the independent variable and GBP/USD is the dependent variable) . However, if the two series are non-stationary and cointegrated then the residuals from a linear regression model will be stationary and hence the linear regression model could be useful. However, when I test for cointegration using Johansen's test I find no cointegration equation between the two series (the output results are below using XLSTAT), which undermines the case for a linear regression I presume.

I'm not sure what to make of this result as it's hard to believe that differentials in short-term interest rates between U.S. and U.K. and the respective exchange rate aren't cointegrated. What am I missing here?

Results from Johansen's test at 5% significance level (the variable 'Actual' is the GBP/USD exchange rate and SHORT_RATES is the variable capturing the differential in short-term interest rates):
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noetsi

Fortran must die
#2
As I noted in the other thread my understanding is that the correct way of doing time series regression with cointegration is actually vector error correction models - although I suspect relatively few do.

I don't think you can use linear regression with non-stationary data period unless they are cointegrated. The slopes are influenced by the drift over time of both variables.