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In the section 15.4 of Hamilton's book Time Series Analysis ( 15.4: The meaning of Test Unit Roots) the author says: Although it might be very interesting to know whether a time series has a unit root, several recent papers have argued that the question is inherently unanswerable on the basis of a finite sample observations. The argument takes the form of two observations:

  1. For any unit root process there exists a stationary process that will be impossible to distinguish from the unit root representation for any given sample size $T$.
  2. For any stationary process and a given sample size, there exists a unit root process that will be impossible to distinguish from the unit root representation.

The author presents an example for each case and we can easily conclude that Unit root and stationary processes differ in their implications (Forecasting, MSE etc) at infinite time horizons, but for any given finite number of observations on the time series, there is a representative from either class of models that could account for all the observed features of the data. We therefore need to be careful with our choice of wording:

  • Testing whether a particular time series ``contains a unit root" or;
  • testing whether innovations ``have a permanent effect on the level of the series"

however interesting, is simply impossible to do.

These observations notwithstanding, there are several closely related and very interesting questions that are answerable. Given enough data, we certainly can ask whether innovations have a significant effect on the level of the series over a specified finite horizon. For a fixed time horizon ($s=3$ years), there exists a sample size such that we can meaningfully inquire whether $\frac{\partial y_{t+s}}{\partial \varepsilon_t }$ is close to zero.

I'm very confused because this last paragraph seems to contradict what the author said earlier. First, he is saying that Given enough data, we certainly can ask whether innovations have a significant effect on the level of the series over a specified finite horizon, but before that he said that we have to be careful about saying testing whether innovations ''have a permanent effect on the level of the series".

I'm really confused how one can be possible and the other not. Could you clarify this for me with an example or a better explanation?

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In my mind, these issues are not particularly wedded to unit roots. What Hamilton (and the papers cited there) seems to be saying is that parameters very close to each other are very hard to distinguish (as are predictions they generate) in finite samples, and that is a proposition I deem correct for many statistical problems.

The answerable question you mention, in turn, tests a rather remote hypothesis from either that of a random walk or a very persistent AR-process, namely if there is any impulse response at horizon three, which would be the case unless we have something like a white noise, an MA(1) or MA(2) etc. Finding that there is such an impulse response would still be compatible with both a unit root and a persistent AR-process (and many other processes).

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  • $\begingroup$ If it is possible to test whether there is any impulse response at horizon three (compatible with a unit root and a persistent AR process), why wouldn't it be possible testing whether innovations ``have a permanent effect on the level of the series" ? $\endgroup$
    – Fam
    Commented Jun 21, 2022 at 13:46
  • $\begingroup$ That is possible too, but it will, in finite samples, be difficult to tell permanent effects apart from long-lasting, but not permanent, effects as implied by a very persistent, but not unit root process (say, an AR(1) with coefficient .99). $\endgroup$ Commented Jun 21, 2022 at 13:54

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