Many observed time series exhibit serial autocorrelation; that is, linear association between lagged observations. This suggests past observations might predict current observations. The autoregressive (AR) process models the conditional mean of y_{t} as a function of past observations, $${y}_{t-1},{y}_{t-2},\dots ,{y}_{t-p}$$. An AR process that depends on p past observations is called an AR model of degree p, denoted by AR(p).
The form of the AR(p) model in Econometrics Toolbox™ is
$${y}_{t}=c+{\varphi}_{1}{y}_{t-1}+\dots +{\varphi}_{p}{y}_{t-p}+{\epsilon}_{t},$$ | (5-6) |
where $${\epsilon}_{t}$$ is an uncorrelated innovation process with mean zero.
In lag operator polynomial notation, $${L}^{i}{y}_{t}={y}_{t-i}$$. Define the degree p AR lag operator polynomial $$\varphi (L)=(1-{\varphi}_{1}L-\dots -{\varphi}_{p}{L}^{p})$$ . You can write the AR(p) model as
$$\varphi (L){y}_{t}=c+{\epsilon}_{t}.$$ | (5-7) |
The signs of the coefficients in the AR lag operator polynomial, $$\varphi (L)$$, are opposite to the right side of Equation 5-6. When specifying and interpreting AR coefficients in Econometrics Toolbox, use the form in Equation 5-6.
Consider the AR(p) model in lag operator notation,
$$\varphi (L){y}_{t}=c+{\epsilon}_{t}.$$
From this expression, you can see that
$${y}_{t}=\mu +{\varphi}^{-1}(L){\epsilon}_{t}=\mu +\psi (L){\epsilon}_{t},$$ | (5-8) |
where
$$\mu =\frac{c}{\left(1-{\varphi}_{1}-\dots -{\varphi}_{p}\right)}$$
is the unconditional mean of the process, and $$\psi (L)$$ is an infinite-degree lag operator polynomial, $$(1+{\psi}_{1}L+{\psi}_{2}{L}^{2}+\dots )$$.
Note:
The |
By Wold's decomposition [1], Equation 5-8 corresponds to a stationary stochastic process provided the coefficients $${\psi}_{i}$$ are absolutely summable. This is the case when the AR polynomial, $$\varphi (L)$$, is stable, meaning all its roots lie outside the unit circle.
Econometrics Toolbox enforces stability of the AR polynomial.
When you specify an AR model using arima
, you get
an error if you enter coefficients that do not correspond to a stable
polynomial. Similarly, estimate
imposes stationarity
constraints during estimation.
[1] Wold, H. A Study in the Analysis of Stationary Time Series. Uppsala, Sweden: Almqvist & Wiksell, 1938.