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When dealing with non-stationary processes, can deterministic trends be converted into stochastic trends? If yes, why?

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No (to my knowledge). A deterministic trend is something like $$ y_t=\delta t+u_t $$ while a stochastic trend is something like $$ y_t=y_{t-1}+u_t=y_0+\sum_{s=1}^tu_s, $$ and these behave fundamentally differently, as you can verify by running something like

t <- 1:100
u <- rnorm(100)
y_st <- cumsum(u)
y_dt <- .1*t + u
plot(t, y_st, type="l", col="red", lwd=2)
lines(t, y_dt, col="green", lwd=2)
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