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I am quite new to time series analysis and I am delving for the first time into stationary processes. I don't seem to understand the concepts of non-stationarity and the presence of roots within the unit circle. Particularly, these are my questions:

  1. Where does the characteristic equation come from (i.e. what is its meaning) and why is it important? (I know the way it is built, I just do not understand why it is built that way)

  2. Why can we say that a process is stationary if and only if the roots of the characteristic equation are outside of the unit circle?

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  1. Not restricted to time-series analysis, characteristic equations (CE) are used in many applications or problems, such as differential/difference equation solving, signal processing, control systems etc. And, it is directly related with commonly used transforms, e.g. Z, (Disc./Cont.) Fourier, Laplace transform etc. Using back-shift operator is a type of analysis allowing one to transform the time-series equations into another domain and deduce related properties. Most analyses are not easy to do in time-domain, which is why transforms exist.

    And, CE obtained by the backshift operator is a way of analyses. This is a matter of which transform you use. You could get the CE of $y_t+\alpha y_{t-1}+\beta y_{t-2}=\epsilon_t$ as $1+\alpha B+\beta B^2$ or if you use $\mathcal{Z}$ transform, that'd be $1+\alpha z^{-1}+\beta z^{-2}=0$, which requires your roots, i.e. $z$'s, should be inside the unit circle if you want to be stationary, instead of outside as in $B$, since roots obtained by $B$ and $z$ are reciprocals of each other.

  2. If we go back to what you're accustomed to, i.e. using the back-shift operator, $B$, we can try to find the roots' relation to stationarity. Consider the time series $y_t=2y_{t-1}+\epsilon_t$, we use the previous output, double it, and add it up with $\epsilon_t$ to obtain current output. The process is clearly exploding. The CE of this is $1-2B=0$, which yields $|B|=1/2<1$. But, if it were $y_t=0.5y_{t-1}+\epsilon_t$, the root would be $|B|=2>1$. This is for just an intuition why would you need $|B|>1$ for being stationary.

    Now consider the general case, $y_t=\alpha y_{t-1}+\epsilon_t$. We can obtain the following equation by back-substitution: i.e. $y_t=\alpha^2y_{t-2}+\alpha\epsilon_{t-1}+\epsilon_t$, and so... $$y_t=\sum_{i=0}^\infty{\alpha^i\epsilon_{t-i}}$$ mean and variance of $y_t$ would be $$E[y_t]=\sum_{i=0}^{\infty}{\alpha^iE[\epsilon_{t-i}]}=\mu_\epsilon\sum_{i=0}^\infty\alpha^i=\frac{\mu_\epsilon}{1-\alpha} \ \ \text{iff} \ \ |\alpha|<1 \rightarrow |B|>1$$ since CE is $1-B\alpha=0\rightarrow B=1/\alpha$. Similarly for the variance, we have $$\sigma_Y^2=var(y_t)=\sum_{i=0}^{\infty}\alpha^{2i}\sigma_{\epsilon}^2=\frac{\sigma_{\epsilon}^2}{1-\alpha^2} \ \ \text{iff} \ \ |\alpha^2|<1 \rightarrow |\alpha=1/B|<1$$

    The auto-correlation is $r_Y(k)=E[y_t y_{t-k}]:$ $$E[y_t y_{t-1}]=E[(\alpha y_{t-1}+\epsilon_t)y_{t-1}]=\alpha\sigma_Y^2+\mu_\epsilon\mu_Y$$ $$E[y_t y_{t-2}]=E[(\alpha y_{t-1}+\epsilon_t)y_{t-2}]=\alpha r_Y(1)+\mu_\epsilon\mu_Y=\alpha^2\sigma_Y^2+\mu_\epsilon\mu_Y(1+\alpha)$$ ... $$E[y_t y_{t-k}]=\alpha^k\sigma_Y^2+\mu_\epsilon\mu_Y(1+\alpha+...+\alpha^{k-1})$$ which is irrespective of $t$, and conditioned on $\sigma_Y^2$ and the means don't depend on $t$.

    This is for AR(1), but AR(k) can be reduced down to a series of AR(1)'s: $$(1-\alpha B)(1-\beta B)y_t=\epsilon_t \rightarrow x_t = (1-\beta B)y_t=x_t \ \ \ \& \ \ (1-\alpha B)x_t=\epsilon_t$$ and the analysis can be performed recursively. Here, we're actually referring to weak stationarity but if $\epsilon_t$ is Gaussian as usual (i.e. which is the noise process is assumed to be distributed with typically), weak stationarity corresponds to stationarity.

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