Why Not Fourier Methods?

1. Motivation

There are many ways that you might measure the typical horizon of a stock’s demand shocks. For instance, Fourier methods might at first appear to be a promising approach, but first impressions can be deceiving. Here’s why: spikes in trading volume tend to be asynchronous. For example, you might see a 1-hour burst of trading activity starting at 9:37am, then another starting at 11:03am, and a third at 2:42pm, but you’d never see bursts of trading activity arriving and subsiding every hour like clockwork. Wavelet methods, as described in my earlier post, can handle these kinds of asynchronous shocks. Fourier methods can’t.

2. Spectral Analysis

Suppose there’s a minute-by-minute trading-volume time series that realizes hour-long shocks. To recover the 1-hour horizon from this time series, Fourier analysis tells us to estimate a collection of regressions at frequencies ranging from 1 cycle per month to 1 cycle per minute:

(1)   \begin{align*}   \mathit{vlm}_t - \langle \mathit{vlm}_t\rangle &= \alpha_f \cdot \sin(\pi \cdot f \cdot t) + \beta_f \cdot \cos(\pi \cdot f \cdot t) + \varepsilon_t,   \qquad \text{with } f \in [\sfrac{1}{22},390] \end{align*}

A frequency of f = 390 cycles per day, for example, denotes the 1-minute time horizon since there are 6.5 \, \sfrac{\mathrm{hr}}{\mathrm{day}} \times 60 \, \sfrac{\mathrm{min}}{\mathrm{hr}} = 390 minutes in a trading day. The amount of variation at a particular frequency is then proportional to the power of that frequency, S_f, defined as:

(2)   \begin{align*}   S_f &=  \frac{\alpha_f^2 + \beta_f^2}{2} \end{align*}

If the time series realizes hour-long shocks, then shouldn’t we find a peak in the power of the series at the hourly horizon, f = 6.5? Isn’t this what computing the power of a time series at a particular frequency is designed to capture?

3. Asynchronous Shocks

Yes, but only if the shocks come at regular intervals. For instance, if the first 60 minutes realized a positive shock, minutes 61 through 120 realized a negative shock, minutes 121 through 180 realized a positive shock again, and so on… then Fourier analysis would be the right approach. However, trading-volume shocks have irregular arrival times and random signs. Fourier analysis can’t handle this sort of asynchronous structure.

4. Simulation-Based Example

Let’s consider a short example to solidify this point. We simulate a month-long time series of minute-by-minute trading-volume data with 60-minute shocks by first randomly selecting J = 1000 minutes during which hour-long jumps begin, \{ \tau_1, \tau_2, \tau_3, \cdots, \tau_J\}, and then adding white noise, \varepsilon_t \overset{\scriptscriptstyle \mathrm{iid}}{\sim} \mathrm{N}(0,1):

(3)   \begin{align*}   \mathit{vlm}_t - \langle\mathit{vlm}_t\rangle &= \sigma \cdot \varepsilon_t + \sum_{j=1}^{1000} \lambda \cdot 1_{\{ t \in [\tau_j,\tau_j + 59] \}}   \cdot    \begin{cases}      +1 &\text{w/ prob. } 50{\scriptstyle \%}     \\     -1 &\text{w/ prob. } 50{\scriptstyle \%}   \end{cases} \end{align*}

Each of the jumps has a magnitude of \lambda = 0.05 \times 10^4 \, \sfrac{\mathrm{sh}}{\mathrm{min}} and is equally likely to be positive or negative. The white noise has a standard deviation of \sigma = 0.06 \times 10^4 \, \sfrac{\mathrm{sh}}{\sqrt{\mathrm{min}}}. The resulting series is shown in the left-most panel of the figure below.

plot--wavelet-variance-illustration--17dec2014

By construction, this process only has white noise and 60-minute-long shocks. That’s it. There are no other time scales to worry about. None. If Fourier analysis were the correct tool for identifying horizon-specific trading-volume fluctuations, then you’d expect there to be a spike in the power of the time series at the 60-minute horizon. But, what happens if we look for evidence of this 60-minute timescale by estimating the power spectrum shown in the middle panel of the figure above? Do we see any evidence of a 60-minute shock? No. There is nothing at the 60-minute horizon. Asynchronous shocks of a fixed length don’t show up in the Fourier power spectrum. They do, however, show up in the Wavelet-variance plot as shown in the right-most panel of the figure above where there is a clear spike at the 1-hour horizon.

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