Explain what is meant by a deterministic and stochastic trend in relation to the following time series process?
$y_t = c + y_{t-1} + \varepsilon_t$ where $\varepsilon_t\sim iid(0, \sigma^2)$
this should be a random walk with drift, how should it relate to deterministic and stochastic trend?
The deterministic trend is one that you can determine from the equation directly, for example for the time series process $y_t = ct + \varepsilon$ has a deterministic trend with an expected value of $E[y_t] = ct$ and a constant variance of $Var(y_t) = \sigma^2$ (with $\varepsilon - iid(0,\sigma^2)$. This will produce basically a straight line in time, with some tiny fluctuations at each point.
The stochastic trend is one that can change in each run due to the random component of the process, as is the case in $y_t = c + y_{t-1} + \varepsilon_t$; this produces the same expected value of $y_t$ but has a non-constant variance of $Var(y_t) = t\sigma^2$, since the random component generated by $\varepsilon _t$ becomes accumulated in time by summation of the $y_{t-1}$ terms. This can produce wildly different runs in each iteration, which is done in random walks. The 'average' run over many iterations will still follow the general trend but with a lot more noise, and the trend for any given iteration is stochastic in nature.
For further clarification I recommend watching these videos in order, they clear things up rather nicely (he does a better job explaining than I do). https://www.youtube.com/watch?v=ouahL4HbwBE
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