I would think that over the 53 years of your analysis, the composition of your sources of energy and their CO2 content has changed over the years. I do not see how it could be possible to have a stable relationship involving CO2 and energy consumption.

You should make inquiries about how CO2 omissions are calculated in your country. You will find that it is a deterministic function of various economic variables including possibly consumption of carboniferous fuels, some types of consumption, and some types of production. In this way, you should be able to formulate a better model before you start your econometrics.

Ideally, u model your ardl according to the break in your dependent variable. So the focus will be on the break you observed in your CO2 emission. In addition, in modelling this, You need to understand what kind of break you want to model. That is, based on your findings on the nature of the trend and your understanding of the pattern of co2 in your country, is the break deterministic( additive break) or stochastic (innovative break), this will enable you generate the appropriate dummy to be incorporated into your ardl as an exogenous variable...

To deal with structural breaks in a multivariate ARDL (Autoregressive Distributed Lag) model, you can:

This allows you to control for breaks without losing the benefits of the ARDL model's flexibility.

This is a question about statistical inference in Banach spaces.

Some less-than-educated economists claimed they could test for structural changes. Either the ar wrong Godel was brain p the wrong tree

To handle structural breaks in a multivariate ARDL model, you can include dummy variables representing break periods, use the Bai-Perron test to detect multiple breaks, and incorporate interaction terms or split the sample accordingly to maintain model robustness and validity.