Seasonal Dependence and Spectral Peaks

Seasonal Dependence and Frequency

A time series can have a constant mean and variance while still depending strongly on the calendar. For example, monthly observations may be more closely related to observations twelve months apart than to observations one month apart. This is seasonal dependence: dependence that recurs at lags associated with a seasonal period. In the frequency domain, such dependence can appear as a peak at a seasonal frequency.

A peak is not automatically evidence that the observed series follows a fixed seasonal path. It describes how variation is distributed over frequencies. A stationary process can have a pronounced seasonal peak while its mean remains constant and its realizations remain random.

Definition: A process \(X_t\) is covariance stationary if its mean is constant over time, its variance is finite and constant, and its autocovariance \(\gamma(h)=\operatorname{Cov}(X_t,X_{t+h})\) depends only on the lag \(h\). A process has seasonal dependence with period \(s\) when its dependence pattern has recurring features at lags related to \(s\), such as elevated autocovariance at \(s,2s,\ldots\), or a periodic autocovariance sequence.

For observations taken at equally spaced times, angular frequency \(\omega\) is measured in radians per observation. A cycle with period \(s\) observations has fundamental frequency \(\omega_0=2\pi/s\). Integer multiples \(2\pi k/s\) are its seasonal harmonics, interpreted modulo \(2\pi\). For monthly data with an annual cycle, \(s=12\), so the fundamental frequency is \(2\pi/12\) radians per month.

Under the convention used here, a spectral density \(f(\omega)\), when it exists, satisfies

\gamma(h)=\int_{-\pi}^{\pi} e^{ih\omega} f(\omega)\,d\omega, \qquad f(\omega)=\frac{1}{2\pi}\sum_{h=-\infty}^{\infty}\gamma(h)e^{-ih\omega}.

Thus the height of a spectral density describes variance per unit frequency, and its integral over \([-\pi,\pi]\) equals the process variance. Some stationary processes have spectral mass concentrated at isolated frequencies rather than a spectral density. The distinction matters: a sharp continuous peak and a spectral line are not the same mathematical object.

A Stationary Seasonal Harmonic

A simple way to create exactly periodic autocovariance is to combine sine and cosine waves with random coefficients. The random coefficients make the level and phase of the oscillation uncertain, rather than fixing the process to one deterministic seasonal path.

Worked Example: A Random Seasonal Oscillation

Let \(A\) and \(B\) be independent \(N(0,4)\) random variables, and define, for integer \(t\),

X_t=A\cos(\omega_0t)+B\sin(\omega_0t), \qquad \omega_0=\frac{2\pi}{12}.

Mean and variance. Since \(E(A)=E(B)=0\), \(E(X_t)=0\). Independence and the variance of each coefficient give

\operatorname{Var}(X_t) =4\cos^2(\omega_0t)+4\sin^2(\omega_0t)=4.

Autocovariance. For any lag \(h\), independence makes the cross terms vanish, so

\begin{aligned} \gamma(h) &=E(X_tX_{t+h})\\ &=4\cos(\omega_0t)\cos(\omega_0(t+h)) +4\sin(\omega_0t)\sin(\omega_0(t+h))\\ &=4\cos(\omega_0h). \end{aligned}

The covariance depends on \(h\), not on \(t\), so the process is covariance stationary. It is also Gaussian, and a Gaussian process is strictly stationary when its mean and covariance are invariant under time shifts. The autocovariance repeats every 12 lags: \(\gamma(12)=4\), while \(\gamma(6)=-4\).

Spectrum. This process has spectral lines, with mass 2 at each of \(\omega_0\) and \(-\omega_0\), rather than an ordinary continuous spectral density. The two masses sum to its variance, 4. Each realization is sinusoidal, but its amplitude and phase are random. The spectral lines express persistent random oscillation at a frequency; they do not imply a deterministic trend or a changing mean.

Seasonal Filters and Continuous Spectral Peaks

A different construction produces seasonal dependence with an ordinary spectral density. Let \(\varepsilon_t\) be a zero-mean white-noise sequence with variance \(\sigma^2\), and filter it using current and seasonally lagged innovations:

X_t=\varepsilon_t+\theta\varepsilon_{t-s}.

The process is covariance stationary because it is a finite linear filter of white noise. Its variance is \(\sigma^2(1+\theta^2)\), and its only nonzero autocovariances apart from lag zero are \(\gamma(s)=\gamma(-s)=\theta\sigma^2\). Applying the spectral formula gives

\begin{aligned} f(\omega) &=\frac{\sigma^2}{2\pi} \left|1+\theta e^{-is\omega}\right|^2\\ &=\frac{\sigma^2}{2\pi} \left(1+\theta^2+2\theta\cos(s\omega)\right). \end{aligned}

When \(\theta>0\), the spectrum is largest where \(s\omega\) is an integer multiple of \(2\pi\), or at the seasonal harmonics. It is smallest halfway between those frequencies. For \(-1<\theta<0\), the alignment reverses: minima occur at the seasonal harmonics and maxima halfway between them. These are continuous spectral peaks, not spectral lines.

Worked Example: Peaks at Annual Frequencies

Suppose monthly innovations have variance \(\sigma^2=4\), and let \(X_t=\varepsilon_t+0.6\varepsilon_{t-12}\). The model is a finite filter of white noise, so it is covariance stationary. Its variance is \(4(1+0.6^2)=5.44\), and its autocovariance at lag 12 is \(0.6(4)=2.4\); at other nonzero lags it is zero.

The spectrum is

f(\omega)=\frac{4}{2\pi} \left(1+0.6^2+1.2\cos(12\omega)\right).

At an annual harmonic, such as \(\omega=2\pi/12\), \(\cos(12\omega)=1\). Therefore

f_{\max} =\frac{4}{2\pi}(1+0.6)^2 =\frac{10.24}{2\pi} \approx 1.630.

Halfway between adjacent annual harmonics, \(\omega=\pi/12\), the cosine is \(-1\), giving

f_{\min} =\frac{4}{2\pi}(1-0.6)^2 =\frac{0.64}{2\pi} \approx 0.1019.

The peak-to-trough ratio calculated from the unrounded values is \(\big((1+0.6)/(1-0.6)\big)^2=16.00\). The displayed values are rounded, so their quotient may differ slightly from the unrounded ratio. The spectrum indicates more variance around annual frequencies than halfway between them; it does not establish that the series repeats the same values each year.

Concentrating Variance Near Seasonal Frequencies

A filter that combines several innovations separated by one seasonal period can make the peaks more concentrated. For an integer \(m\geq 1\), define

Y_t=\sum_{j=0}^{m-1}\varepsilon_{t-js},

where \(\varepsilon_t\) is zero-mean white noise with variance \(\sigma^2\). The autocovariance counts how many innovations the two sums share. At lag \(qs\), the sums share \(m-|q|\) innovations when \(|q|<m\), and share none when the lag is not a multiple of \(s\). Hence

\gamma(qs)= \begin{cases} \sigma^2(m-|q|), & |q|<m,\\ 0, & |q|\geq m, \end{cases} \qquad \gamma(h)=0\quad\text{if }h\text{ is not a multiple of }s.

The filter's frequency response is a geometric sum. Using the finite-sum identity gives the density

f_Y(\omega) =\frac{\sigma^2}{2\pi} \left|\sum_{j=0}^{m-1}e^{-ijs\omega}\right|^2 =\frac{\sigma^2}{2\pi} \left(\frac{\sin(ms\omega/2)}{\sin(s\omega/2)}\right)^2.

At a seasonal harmonic, numerator and denominator both approach zero, but the geometric sum itself has \(m\) terms that align. Taking the limit therefore gives a peak height \(\sigma^2m^2/(2\pi)\). Increasing \(m\) makes the filter combine more seasonally spaced innovations and raises the peak relative to frequencies where those terms cancel.

Worked Example: A Three-Seasonal-Lag Filter

Take \(s=12\), \(m=3\), and \(\sigma^2=1\). Then \(Y_t=\varepsilon_t+\varepsilon_{t-12}+\varepsilon_{t-24}\). Directly counting shared innovations gives \(\gamma(0)=3\), \(\gamma(12)=\gamma(-12)=2\), and \(\gamma(24)=\gamma(-24)=1\); all other autocovariances are zero.

At the annual frequency \(\omega=2\pi/12\), all three terms in the filter response align. Thus

f_Y(2\pi/12)=\frac{1}{2\pi}(1+1+1)^2 =\frac{9}{2\pi}\approx 1.432.

At the midpoint \(\omega=\pi/12\), the geometric-sum formula has numerator \(\sin(3\pi/2)=-1\) and denominator \(\sin(\pi/2)=1\). Therefore

f_Y(\pi/12)=\frac{1}{2\pi} \left(\frac{-1}{1}\right)^2 =\frac{1}{2\pi}\approx 0.1592.

The peak is nine times the midpoint value. This is a continuous spectrum: the filter concentrates variance around annual frequencies without placing all variance at isolated frequencies.

How to Interpret a Seasonal Peak

A spectral peak answers a frequency-domain question: which oscillation rates account for a relatively large share of the process variance? Its location suggests a characteristic period, and its width gives a qualitative indication of how tightly variation is concentrated around that frequency. It does not by itself identify the cause of the dependence, establish a deterministic seasonal pattern, or show that the mean changes over time.

A deterministic seasonal mean, such as a fixed sequence of month-specific averages, is a different feature from stationary seasonal dependence. A stationary process can instead have a constant mean and seasonal autocovariance. The distinction is important in practice: a regular-looking realization may be generated by random fluctuations with strong dependence, not by a fixed calendar path.

Check stationarity.
State whether the mean is constant and whether covariance depends only on lag before interpreting a stationary spectrum.

Translate frequency to period.
For a peak at \(\omega\), the corresponding period in observations is \(2\pi/\omega\), when \(\omega\neq 0\).

Identify the type of spectral feature.
A continuous peak represents concentrated variance over a frequency range; a spectral line represents point mass at a frequency.

Keep the conclusion limited.
Describe seasonal dependence or oscillatory variation; do not infer a deterministic trend or a causal mechanism from the peak alone.

Common Mistake

A common error is to see a strong annual-frequency peak and conclude that the series has a deterministic annual trend. This confuses variation around a frequency with the time path of the mean. In the seasonal moving-average example, the mean remains zero at every date even though the spectrum has distinct annual peaks. The appropriate conclusion is that variation is more concentrated near annual frequencies under the stated stationary model. A changing mean or deterministic seasonal component would require separate evidence and modeling.

Key takeaway: Seasonal dependence can be created by stationary harmonic components or by filters that link innovations across seasons. Its autocovariance may repeat periodically, and its spectrum may show seasonal peaks or spectral lines. These features describe dependence and variance by frequency, not deterministic trends.

Check Your Understanding

Answer each question using the definitions and results above.

For monthly data with seasonal period \(s=12\), what is the fundamental angular frequency in radians per month?
In the random harmonic example, why does the autocovariance depend on lag \(h\) but not on time \(t\)?
For \(X_t=\varepsilon_t+\theta\varepsilon_{t-s}\) with \(\theta>0\), where are the spectral maxima and minima?
In the three-seasonal-lag filter, why is the spectral peak at an annual harmonic larger than the value halfway between adjacent harmonics?
Why is it incorrect to interpret a stationary seasonal spectral peak as proof of a deterministic seasonal trend?