Is it appropriate to run the Error Correction Model on data which are not I(1)? I have intraday data (frequency = 1 min.) for 6 stocks and 1950 observations per each time series.
I checked stationarity for the level data and first difference and it appears that:


*

*5 stocks's level data are non-stationary and 1 is stationary,

*all 6 stocks's first differences are stationary,
meaning that that 1 stock is not I(1) and that 5 stocks are I(1).


My question is: Is it appropriate to run Engle-Granger methodology, ending with Error Correction Model, on the data pair which includes 1 time series which is I(1) and the other one which is not I(1)?
 A: Running a VECM requires testing of cointegration as a prerequisite. Only if the series are cointegrated , can we proceed with a VECM.  Also prior to running cointegration, the mandatory stationarity testing requires that the variables are non stationary at level and stationary on first difference ( Integrated of the same order).  
A: The short answer is "no." It's actually OK to use ECMs if ALL variables are stationary. And it's obviously OK if they are ALL I(1).
The real requirement is that the equation is "balanced" which means that all the variables are at the SAME order of integration (whatever that order happens to be). If one is stationary and the other is I(1) that's bad, and will potentially lead to spurious results.
Of course in reality you may not have enough power to actually determine the order of integration of variables...and they may actually be "fractionally integrated" which causes other issues.
See this paper  (which is part of a longer back and forth about the proper use of ECMs) for more details
https://www.cambridge.org/core/journals/political-analysis/article/abs/treating-time-with-all-due-seriousness/6819BB223EB54EF830FCE7065FBFF965
