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That is illegal because an I(1) series wanders ( doesn't have a constant mean ) and an I(0) series doesn't so they can't be set equal. In order to obtain a valid time series regression, the order on the LHS has to be the same as the order on the RHS. Note that if both sides are I(1), then a time series regression is okay but only if what is on the RHS is ...

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Causality between time-series variables does not require the two to be cointegrated. First, cointegration requires that each series be $I(1)$. It is certainly possible for two $I(0)$ series to follow a causal relationship (or two $I(d)$ variables for that matter). Second, cointegration implies a long-run equilibrium among the series, which is not ...

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If your question is the broad: "How do I estimate the differences between these two groups under these problematic contexts?" I recommend taking a look at event study literature, which has a lot of similarities to the DiD literature, but is frequently applied in the context of stocks (which can be non-stationary). MacKinlay, A. Craig. "Event ...

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It is not necessary to detrend the series although the trend has to be dealt with and detrending is one of the options but there are also other options. Other options include explicitly modeling the trend. For example, many tests for cointegration including Johansen cointegration test, it is possible to estimate the test together with a linear or quadratic ...

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I think that the wikipedia steps are not completely correct. When you perform Johansen cointegration test you first have to pretest the data to find if they have the same order of integration. That is all variables have to be either $I(1)$ or $I(2)$ etc. (see Verbeek, 2008). There are some cointegration tests and models that relax this assumption but ...

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Intuitively, you test for cointegration because if two variables are cointegrated, they represent only a "one dimensional" family of data points - even if you have a million data points from that sample, they will all fall close to the same subspace, and in general that will mean that you will have many values of parameters for your regression problem which ...

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