Skip to content
All library documents

ARMA Polynomial Normalization and Rescaling the Series

Article Quant Q&A · Author: Alfons Ingomar
Read the original

Summary

The document considers how to express an ARMA equation as lag polynomials when the coefficient on the current observation is not one. In the example, the observation equation has a leading coefficient of negative four and a lag-two term, while the noise side includes a moving-average term. The question is whether the autoregressive polynomial must begin with one.

The answer explains that the polynomials can retain the coefficients implied by the original equation. If a unit leading coefficient is preferred, the observed series can be rescaled by multiplying it by negative one quarter; this changes the representation while preserving the underlying relation. Normalization can make the model easier to interpret as explaining the original series' value, but the document offers no discussion of stationarity, invertibility, or parameter estimation, so its focus is limited to polynomial form and scaling.

Key ideas

  • A lag polynomial's leading coefficient need not be one in every valid representation of an ARMA equation.
  • Rescaling the observed series can produce an equivalent model with a normalized leading coefficient.
  • Normalization is often convenient when interpreting the modeled variable in its original scale.
  • The discussion addresses notation and scaling rather than ARMA diagnostics or estimation.

Tags

Statistics
Full text
# Define polynomials of an ARMA process


# Define polynomials of an ARMA process












I just started out with financial time series and I'm a bit stuck with ARMA models. I have the following ARMA process:

$-4X_t + X_{t-2} = Z_t + 0.2 Z_{t-1}$

Now I am being asked for the polynomials of $\Phi$ and $\Theta$ so we can write the model as: $\Phi (B) X_t = \Theta (B) Z_t$.

This is how I am deriving my solution:

$ \Phi(B) = 1-\phi_1 B - \phi_2 B^2 - ... - \phi_p B^p$

$=-4 +0B --1B^2 $

$= -4 +1B^2$

However, I'm not convinced that this answer is legit. Shouldn't this polynomial always start with 1?

## Answer by Louis. B (score 2, accepted)

https://quant.stackexchange.com/a/21474

There is no particular issue with your polynomials. However if you really want them to both start with a 1, you can apply a change of variable by defining : \begin{equation}Y_t = -\frac{1}{4}X_t\end{equation} Then your polynomials $\Phi_y(B)$ and $\Theta(B)$ such that : \begin{equation}\Phi_y(B)Y_t=\Theta(B)Z_t\end{equation} will both start with a $1$.

It is indeed often more convenient for the economic intuition to have both of them starting with $1$ with the idea that you want to explain the value of $X_t$ and not the value of $3X_t$ or $\lambda\cdot X_t$ with $\lambda\in\mathbf{R}$.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.