I am trying to model interest rate data using ARIMA in order to estimate the effect of an external shock. However, when I run the ADF and KPSS tests, both conclude that stationarity is rejected. My understanding is that this means that there is a unit root in the series, and that any external shock will be persistent forever. However, many posts suggest that differencing a non-stationary series will allow forecasting through ARIMA. Can a persistent shock be predicted through an ARIMA model?
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Suppose you have a random walk $y_t=y_{t-1}+\epsilon_t$, i.e., an ARIMA(0,1,0). Differencing gives $\Delta y_t=\epsilon_t$. When $E(\epsilon_t)=0$, that means our forecast given our information set at time $T$ for the change will be $$ E_T(\Delta y_{T+h})=0 $$ for any horizon $h=1,2,\ldots$.
You may now (always, not only for a random walk) write $$ y_{T+h}=y_T+\Delta y_{T+1}+\ldots+\Delta y_{T+h} $$ so that we can forecast the ARIMA process $y_t$ at $T+h$ as $$ E_T(y_{T+h})=y_T+0 $$ Some terminology: ADF cannot reject stationarity, as the null is nonstationarity, so at best you may not reject the null of nonstationarity.
answered Jun 2, 2017 at 11:17
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$\begingroup$ Now suppose that the ADF does not reject the null of nonstationarity, do we automatically assume that there is a unit root? What is the purpose to modeling a nonstationary series with ARIMA if all we are getting in the future is the previous value + a random error term? $\endgroup$– rback20Jun 2, 2017 at 17:28
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$\begingroup$ Also, if the ADF test fails to reject the null hypothesis, and we are still able to fit an AR(1) model with a significant coefficient of 0.99, can we be confident in this coefficient or no? $\endgroup$– rback20Jun 2, 2017 at 18:08
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$\begingroup$ Q1: Like with any other hypothesis test, not rejecting is not proof that the null is true, but common practice is to proceed as if it is, because the data does not provide evidence against the null. $\endgroup$ Jun 2, 2017 at 18:12
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$\begingroup$ Q2: first, a random walk is of course the simplest possible ARIMA process with $d>0$. But in essence, yes, that is the forecast, and it is the optimal forecast if the underlying process is a random walk, despite the fact that it is not an "exciting" one. Still, it may well be a plausible one. Think of stock prices, where the efficient market hypothesis strongly suggests that the best we can say about future stock prices (at least in the short run, abstracting from drift) is that they are just as likely to go up as down, so to stay where they are on average. $\endgroup$ Jun 2, 2017 at 18:14
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$\begingroup$ Q3: yes, the coefficient estimate in a regression of $y$ on its lag is consistent for the true value no matter if the true process is a stationary AR(1) or a random walk. $\endgroup$ Jun 2, 2017 at 18:15