Autoregressive spectrum

BROCKWELL, P.J., and DAVIS, R.A. (2006) point out that for any real-valued stationary process $\left(x_{t}\right)$ with continuous spectral density $f\left(\omega\right)$ it is possible to find both $AR(p)$ and $MA(q)$ processes which spectral densities are arbitrarily close to $f\left(\omega\right)$. For this reason, in some sense, $\left(x_{t}\right)$ can be approximated by either $AR(p)$ or $MA(q)$ process. This fact is a basis of one of the methods of achieving a consistent estimator of the spectrum, which is called an autoregressive spectrum estimation. It is based on the approximation of the stochastic process $\left(x_{t}\right)$ by an autoregressive process of sufficiently high order $p$:

\(x_{t} = \mu + \left( \phi_{1}B + \ldots + \phi_{p}B^{p} \right)x_{t} + \varepsilon_{t}\) [1]

where \(\varepsilon_{t}\) is a white-noise variable with mean zero and a constant variance.

The autoregressive spectrum estimator for the series $x_{t}$ is defined as: ¹

\(\widehat{s}\left( \omega \right) = 10\operatorname{\times}{\log_{10}\frac{\sigma_{x}^{2}}{2\pi{|1 - \sum_{k = 1}^{p}{\widehat{\phi}}_{k}e^{- ik\omega}|}^{2}}}\) [2]

where:

$\omega$– frequency, $0 \leq \omega \leq \pi$;

$\sigma_{x}^{2}$ – the innovation variance of the sample residuals;

\({\widehat{\phi}}_{k}\) – $\text{AR}\left(k\right)$ coefficient estimates of the linear regression of $x_{t} - \overline{x}$ on $x_{t - k} - \overline{x}$, $1 \leq k \leq p$.

The autoregressive spectrum estimator is used in the visual spectral analysis tool for detecting significant peaks in the spectrum. The criterion of visual significance, implemented in JDemetra+, is based on the range ${\widehat{s}}^{\max} - {\widehat{s}}^{\min}$ of the $\widehat{s}\left( \omega \right)$ values, where ${\widehat{s}}^{\max} = \max_{k}\widehat{s}\left( \omega_{k} \right)$; ${\widehat{s}}^{\min} = \min_{k}\widehat{s}\left( \omega_{k} \right);$ and $\widehat{s}\left( \omega_{k} \right)\ $is $k^{\text{th}}$ value of autoregressive spectrum estimator.

The particular value is considered to be visually significant if, at a trading day or at a seasonal frequency $\omega_{k}$ (other than the seasonal frequency $\omega_{60} = \pi$), $\widehat{s}\left( \omega_{k} \right)\ $is above the median of the plotted values of $\widehat{s}\left( \omega_{k} \right)$ and is larger than both neighbouring values $\widehat{s}\left( \omega_{k - 1} \right)$ and $\widehat{s}\left( \omega_{k + 1} \right)$ by at least $\frac{6}{52}$ times the range ${\widehat{s}}^{\max} - {\widehat{s}}^{\min}$.

Following the suggestion of SOUKUP, R.J., and FINDLEY, D.F. (1999), JDemetra+ uses an autoregressive model spectral estimator of model order 30. This order yields high resolution of strong components, meaning peaks that are sharply defined in the plot of $\widehat{s}\left( \omega \right)$ with 61 frequencies. The minimum number of observations needed to compute the spectrum is set to $n =$ 80 for monthly data and to $n =$ 60 for quarterly series while the maximum number of observations considered for the estimation is 121. Consequently, with these settings it is possible to identify up to 30 peaks in the plot of 61 frequencies. By choosing $\omega_{k} = \frac{\text{πk}}{60}$ for $k = $0,1,…,60 the density estimates are calculated at exact seasonal frequencies (1, 2, 3, 4, 5 and 6 cycles per year).

The model order can also be selected based on the AIC criterion (in practice it is much lower than 30). A lower order produces the smoother spectrum, but the contrast between the spectral amplitudes at the trading day frequencies and neighbouring frequencies is weaker, and therefore not as suitable for automatic detection.

SOUKUP, R.J., and FINDLEY, D.F. (1999) also explain that the periodogram can be used in the visual significance test as it has as good as those of the AR(30) spectrum abilities to detect trading day effect, but also has a greater false alarm rate².