Let us take this $\operatorname{AR}(p)$ equation

$$\left(1 - \sum_{i=1}^{p} \phi_i L^i \right)X_t = \mu + \epsilon_t$$

as an example.

When $p=0$ I read this to mean

\begin{align*} \mu + \epsilon_t &= \left(1 - \sum_{i=1}^{0} \phi_i L^i \right)X_t \\ & = \left(1 - \phi_1 L^1 - \phi_0L^0 \right)X_t \\ & = X_t - \phi_1 L^1X_t - \phi_0L^0X_t \\ &= X_t - \phi_1 X_{t-1} - \phi_0 X_t \\ &= (1-\phi_0)X_t - \phi_1 X_{t-1} \\ \end{align*}

which could then be solved for $X_t$ to give

$$X_t = \frac{\mu + \epsilon_t + \phi_1 X_{t-1}}{1-\phi_0}.$$

If we take

\begin{align*} \delta &:= \frac{\mu}{1 - \phi_0} \\ \gamma_t &:= \frac{\epsilon_t}{1 - \phi_0} \\ \theta_{t} &:= \frac{\phi_1}{1 - \phi_0} \end{align*}

then we get the form of an $\operatorname{AR}(1)$ model as follows

$$X_t = \delta + \theta_t X_{t-1} + \gamma_t.$$

Prima facie this tells us $\operatorname{AR}(0) = \operatorname{AR}(1)$, but I thought that $\operatorname{AR}(0)$ should reduce to $$X_t = \mu + \epsilon_t.$$ Although I think many people take $\operatorname{AR}(0)$ to mean $$X_t = \epsilon_t$$ but I included $\mu$ in my definition of $\operatorname{AR}(p)$ at the start in a way where it won't fall out due any choice of $p$.

Some possibilities I have considered:

  • There exists a convention that $p=0 \implies \sum_{i=1}^{p} \phi_i L^i = 0$.
  • I've made a set of mistakes doing the basic algebra (it happens).
  • The sources I have found are mistaken.

What is going on here?

  • 1
    $\begingroup$ That's not the usual meaning of the sum: when $p=0,$ the sum is empty. $\endgroup$
    – whuber
    Jun 24, 2022 at 15:43
  • $\begingroup$ For future readers, see: math.stackexchange.com/questions/924203/… $\endgroup$
    – Galen
    Jun 24, 2022 at 16:03
  • $\begingroup$ For future readers, see: math.stackexchange.com/questions/1714491/… $\endgroup$
    – Galen
    Jun 24, 2022 at 16:03
  • 2
    $\begingroup$ Note the comments attached to the latter, such as math.stackexchange.com/questions/1714491/…. IMHO, that math thread is skating on thin ice, because it's really discussing programming conventions. The formal mathematical definition of summation is recursive (relying on the Peano postulates), beginning with defining the empty sum as zero. Many of the remarks in that Math thread implicitly point out there's no reason to define a summation with descending values of the index. $\endgroup$
    – whuber
    Jun 24, 2022 at 16:05


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.