Dummy interaction variables are always non-stationary? I want to know why we can include dummy interaction terms into time series models if they're always non-stationary?
For example let $X_t$ be $I(0)$, $X_t \sim N(\mu,\sigma^2)$ and $D_t \in \{0,1\}$. Suppose the estimation window is $200$ and $D_t =0$ for $t=1,...,100$ and $D_t =1$ for $t=101,...,200$.  The interaction is $D_tX_t$. 
We know that 


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*$E[D_tX_t|t\in\{1,...,100\}]=0$

*$E[D_tX_t|t\in\{101,...,200\}]=\mu$


Therefore it's non-stationary? The variance also changes for this variable at $t=101$. 
This of course holds for intercept dummy variables ($D_t$ by themselves); however in this case the variance is stationary (but the mean changes). 
 A: 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 (Bernoulli r.v.) you refer too will be non-stationary only if you specify a priori that its probability of occurring in each time period is not constant (its specific realization to which you refer, does not affect this a priori assumption).
If, therefore, the dummy variable is (or is assumed) also $I(0)$ then the variable $Y_t = X_tD_t$ is a continuous transformation of two stationary r.v's - and stationarity is preserved under continuous transformations - so $Y_t$ is also stationary.
Now, say you are faced with a specific "time-window" and a specific realization of the dummy variable, that makes you suspect that this dummy variable does "not look like" a stationary process. Then test it for stationarity if a test is available, or at least discuss your visual suspicions and work from there. If the dummy represents something seasonal, use the tools available to deal with seasonal time series.
A: I think it depends on the specific form of the time series model. de Boef and Keele (2008) give a nice overview of dynamic time-series models and how they relate to one another.
I tend to prefer the single-equation generalized error correction model (1; Banerjee, 1993) because it is agnostic as to the stationarity of predictors, and seems to have decent estimation properties for near-integrated data as per de Boef (2001).


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*$\Delta y_{t} = \beta_{0} + \beta_{c}(y_{t-1} - \Sigma \mathbf{X}_{t-1}) + \mathbf{B}_{\Delta}\Delta \mathbf{X} + \mathbf{BX}$



References
Banerjee, A., Dolado, J. J., Galbraith, J. W., and Hendry, D. F. (1993). Co-integration, error correction, and the econometric analysis of non-stationary data. Oxford University Press, USA.
De Boef, S. (2001). Modeling equilibrium relationships: Error correction models with strongly autoregressive data. Political Analysis, 9(1):78–94.
De Boef, S. and Keele, L. (2008). Taking time seriously. American Journal of Political Science, 52(1):184–200.
