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\title{EC821:\ Time Series Econometrics \\
Spring 2003\\
Notes Section 10 Part 1}
\author{Christopher F\ Baum \and Department of Economics \and Boston College \and %
baum@bc.edu}
\maketitle

\section{Spectral analysis\protect\footnote{%
This presentation draws heavily from Ch. 6 of Hamilton (1994).}}

Our focus thus far has been on considering how the timeseries $Y$ reacts to
a sequence of innovations $\epsilon _{t},$ and the implications of the
relationship between $Y$ and $\epsilon $ for the covariance (or correlation)
between $Y_{t}$ and $Y_{\tau }$ at distinct dates $t$ and $\tau .$ We speak
of this as analysing the properties of $Y$ in the time domain. We might also
consider the timeseries behavior in the frequency domain, where $Y_{t}$ is
described as a weighted sum of periodic (sine and cosine) functions: 
\begin{equation}
Y_{t}=\mu +\int_{0}^{\pi }\alpha \left( \omega \right) \cos \left( \omega
t\right) d\omega +\int_{0}^{\pi }\delta \left( \omega \right) \sin \left(
\omega t\right) d\omega  \label{spectral}
\end{equation}

This is known as the spectral representation of the timeseries, and its
study is known as spectral analysis. The goal will be to determine how
important cycles of different frequencies are in accounting for the behavior
of $Y.$ The two types of analysis (time-domain and frequency-domain
modelling) are not mutually exclusive. Any CS\ process has both a
time-domain representation and a frequency-domain representation, and any
feature of the data that can be described by one representation can be
equally well described by the other representation. For some features, the
time-domain description may be simpler or more convenient to work with,
while for other features the frequency-domain representation may be simpler.

\subsection{The population spectrum}

Let $Y$ be a CS\ process with mean $E(Y_{t})=\mu $ and $j^{th}$
autocovariance 
\[
E\left( Y_{t}-\mu \right) \left( Y_{t-j}-\mu \right) =\gamma _{j}. 
\]
If the autocovariances are absolutely summable, as previously defined, then
we may define the autocovariance generating function as 
\begin{equation}
g_{Y}\left( z\right) =\sum_{j=-\infty }^{\infty }\gamma _{j}z^{j}
\label{acgen}
\end{equation}
where $j$ denotes a complex scalar. If (\ref{acgen}) is divided by $2\pi $, $%
z$ is represented by $z=e^{-i\omega }$ for $i=\sqrt{-1}$ and $\omega $ a
real scalar, then we define the population spectrum of $Y:$%
\begin{equation}
s_{Y}=\frac{1}{2\pi }g_{Y}\left( e^{-i\omega }\right) =\frac{1}{2\pi }
\sum_{j=-\infty }^{\infty }\gamma _{j}e^{-i\omega j}  \label{spectrum}
\end{equation}

Note that the spectrum is a function of the frequency $\omega :$ given any
particular value of $\omega $ and a sequence of autocovariances $\left\{
\gamma _{j}\right\} $ we could in principle calculate the value of $%
s_{Y}\left( \omega \right) .$ By De Moivre's theorem, 
\[
e^{-i\omega j}=\cos \left( \omega j\right) -i\sin \left( \omega j\right) 
\]

so 
\[
s_{Y}=\frac{1}{2\pi }\sum_{j=-\infty }^{\infty }\left[ \cos \left( \omega
j\right) -i\sin \left( \omega j\right) \right] 
\]

For a CS\ process, $\gamma _{j}=\gamma _{-j},$ so that we can consider only
the positive half line. Using some elementary results from trigonometry, we
may simplify the expression to 
\[
s_{Y}\left( \omega \right) =\frac{1}{2\pi }\left\{ \gamma
_{0}+2\sum_{j=1}^{\infty }\gamma _{j}\cos \left( \omega j\right) \right\} . 
\]

This implies that the population spectrum $s_{Y}\left( \omega \right) $
exists, and is a continuous, real-valued function of $\omega .$ If the $%
\gamma _{j}$ are the autocovariances of a CS\ process, then $s_{Y}\left(
\omega \right) $ will be nonnegative for all $\omega ,$ and given symmetry
of the cosine function, the spectrum is symmetric around $\omega =0.$ Since
cos$\left[ (\omega +2\pi k)j\right] =\cos \left[ \omega j\right] $ for any
integers $k$ and $j,$ it follows that $s_{Y}\left( \omega +2\pi k\right)
=s_{Y}\left( \omega \right) $ for any integer $k.$ Thus,. the spectrum is a
periodic function of $\omega .$ If we know the value of $s_{Y}\left( \omega
\right) $ for all $\omega $ between 0 and $\pi ,$ we can infer the value of $%
s_{Y}\left( \omega \right) $ for any $\omega .$ We routinely express the
empirical spectrum as a function defined over $\left( 0,\pi \right) ,$ with
the understanding that it is symmetric around 0.

We may define the spectrum, or spectral density function, for a number of $%
ARMA$ processes analytically. For instance, a white noise process has a flat
spectrum, with $s_{Y}\left( \omega \right) =\sigma ^{2}/2\pi ,$ and the
meaning of white noise reflects the equiprobability of all frequencies in
the representation of the process. A $MA(1)$ process $Y_{t}=\epsilon
_{t}+\theta \epsilon _{t-1}$ will have a monotonic spectrum, decreasing over 
$\left( 0,\pi \right) $ for $\theta >0,$ increasing for $\theta <0.$ An $%
AR(1)$ process $Y_{t}=c+\phi Y_{t-1}+\epsilon _{t}$ with $\phi >0$ will have
a monotonic spectrum, decreasing over $\left( 0,\pi \right) $, while for $%
\phi <0,$ $s_{Y}\left( \omega \right) $ will be a monotone increasing
function of $\omega .$

How may we interpret the spectrum? We may demonstrate that 
\begin{eqnarray*}
\int_{-\pi }^{\pi }s_{Y}\left( \omega \right) e^{iwk}d\omega &=&\gamma
_{k},or \\
\int_{-\pi }^{\pi }s_{Y}\left( \omega \right) \cos \left( wk\right) d\omega
&=&\gamma _{k}.
\end{eqnarray*}
This implies, for $k=0,$ that the integral of the spectrum over its range
equals $\gamma _{0}:$ the variance of the series. Thus the spectrum is an
alternative representation of the variance, or variability, of a series,
with the interpretation that the total variance arises from those components
defined over the various frequencies contained in the range $\left( 0,\pi
\right) .$ Since $s_{Y}\left( \omega \right) $ is nonnegative, if we
calculate 
\[
\int_{-\omega _{1}}^{\omega _{1}}s_{Y}\left( \omega \right) d\omega 
\]
for any $\omega _{1}\in \left( 0,\pi \right) ,$ the result would be a
positive number that we could interpret as the portion of the variance of $Y$
associated with frequencies $\omega $ that are less than $\omega _{1}$ in
absolute value. Since $s_{Y}\left( \omega \right) $ is symmetric, we may
express this quantity as 
\[
2\int_{0}^{\omega _{1}}s_{Y}\left( \omega \right) d\omega , 
\]
the portion of the variance of $Y$ attributed to periodic random components
with frequencies less than or equal to $\omega _{1}.$ Thus, the variance of
a timeseries may be decomposed into those portions due to low-frequency
variation (or long waves); medium-frequency variations (e.g. business cycle
frequencies); and high-frequency variations (such as seasonal variations in
quarterly or monthly data).

\subsection{The sample periodogram}

Since the population spectrum is defined in terms of the sequence $\left\{
\gamma _{j}\right\} ,$ the population second moments of the timeseries, we
may compute a sample counterpart from the empirical autocorrelations of the
series measured over $T$ observations, 
\begin{equation}
\hat{\gamma}_{j}=T^{-1}\sum_{t=j+1}^{T}\left( y_{t}-\bar{y}\right) \left(
y_{t-j}-\bar{y}\right) ,j=0,1,2,\ldots ,T-1  \label{sampac}
\end{equation}
where $\bar{y}$ is the sample mean of the timeseries. For any given $\omega $
we may then construct the sample periodogram, 
\begin{eqnarray}
\hat{s}_{Y}\left( \omega \right) &=&\frac{1}{2\pi }\sum_{j=-T+1}^{T-1}\hat{
\gamma}_{j}e^{-i\omega j},or  \label{samper} \\
&=&\frac{1}{2\pi }\left\{ \hat{\gamma}_{0}+2\sum_{j=1}^{T-1}\hat{\gamma}%
_{j}\cos \left( \omega j\right) \right\}  \nonumber
\end{eqnarray}
which may constructed from any timeseries of sufficient length, using the
estimates of the sample autocovariance function. As we discussed above, the
sample periodogram allows the timeseries to be represented by components of
different frequencies, and the empirical variance of the timeseries to be
decomposed into that emanating from certain frequency bands. The highest
frequency that may be considered in this representation is $\omega =\pi ,$
known as the Nyquist frequency. Since a frequency $\omega $ is associated
with a period of $2\pi /\omega $ periods, the Nyquist frequency is
associated with a cycle of 2 periods' length (two months for monthly data,
two days for daily data...) and our analysis cannot detect a cycle of higher
frequency (for instance, a set of daily stock prices cannot be used to
analyse intra-daily movements in stock prices). The lowest frequency that
can be represented in a finite sample of $T$ observations is $\omega
_{1}=2\pi /T,$ corresponding to a period of $T.$ We cannot infer anything
about cycles that last longer than that (e.g. 50-year swings in economic
activity cannot be analysed using 40 years of postwar data).

\subsection{Estimating the population spectrum}

Although it might seem natural to estimate the population spectrum $%
s_{Y}\left( \omega \right) $ with the sample periodogram $\hat{s}_{Y}\left(
\omega \right) ,$ this approach has its limitations. Unless the sample is
very large, estimates from the periodogram may be very imprecise. Just as
the sample autocorrelation function may be not very smooth, the sample
periodogram is likely to be something other than the smooth function that
theory would suggest for the population spectrum. Most approaches to
estimation of the spectrum involve the use of a window, or kernel estimator:
a technique that averages adjacent estimates of the periodogram to create a
smooth function. Such an estimator reduces the variance of the resulting
spectrum at the cost of introducing some bias. A certain amount of
subjective judgment is often employed to consider the appropriate bandwidth
for a spectral window, or kernel estimator.

\begin{thebibliography}{9}
\bibitem{ham94}  Hamilton, James D., 1994. Time series analysis. Princeton:\
Princeton University Press.
\end{thebibliography}

\end{document}
