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Surely, you could run a logit, probit, or a cloglog model; in fact any appropriate binary model works well. The logit model (and to some extent the cloglog) model tend to be used more often, but this is only because of the familiarity of the users with these models. See the following link for more details: https://files.nyu.edu/mrg217/public/btscs.pdf

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I assume you refer to "weak stationarity", namely, constant unconditional mean, constant unconditional variance, and unconditional covariance that if it is not zero, it depends on the distance between two elements of the time series, and not on the time index. More over, who said that dummy variables are not-stationary? The binary dummy variable ...

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In connection to what jmbejara is saying, you can convert many models to a markov model by simply increasing the dimension of the state space and then precomputing the nonlinearities to make it a linear model. As far as I know this technique cannot be applied under conditions when number of parameters is not deterministic but stochastic. Or the process' ...

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You've got to distinguish the condition "linear in parameters" from "linear in variables." Often it is the case that a nonlinear relationship between variables can be transformed into a linear relationship in transformed variables. For example, $$y = a + bx + cx^2 + \epsilon$$ can be transformed into $$y = a + bx + cz + \epsilon,$$ with $z=x^2$ as a new ...

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You can take any two vectors of numbers, and, given that their elements form meaningful pairs (i.e. magnitudes of two variables in the same point in time), estimate their covariance. In your case, it should be negative. This would give you the strength of any linear stochastic dependence. To try to model CPI as a function of e-commerce alone, say a linear ...

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K nearest neighbors is a simple solution you could use (although it might be too slow for your needs). There are several implementations in R, although for your purposes it would probably be easier to just roll your own (it's a very simple algorithm to code). This would also give you the flexibility of defining your own distance function, since you may ...

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If you don't have a very large sample, the jarque.bera test will almost always reject normality of the residuals for a VAR. I suggests you plot your residuals in a histogram and a qqplot to see for yourself if it is reasonable to believe that the residuals are behaving like white noise.

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