## Alpha Spectral Analysis

One of the questions of interest is the optimal sampling frequency to use for extracting the alpha signal from an alpha generation function.  We can use Fourier transforms to help identify the cyclical behavior of the strategy alpha and hence determine the best time-frames for sampling and trading.  Typically, these spectral analysis techniques will highlight several different cycle lengths where the alpha signal is strongest.

The spectral density of the combined alpha signals across twelve pairs of stocks is shown in Fig. 1 below.  It is clear that the strongest signals occur in the shorter frequencies with cycles of up to several hundred seconds. Focusing on the density within
this time frame, we can identify in Fig. 2 several frequency cycles where the alpha signal appears strongest. These are around 50, 80, 160, 190, and 230 seconds.  The cycle with the strongest signal appears to be around 228 secs, as illustrated in Fig. 3.  The signals at cycles of 54 & 80 (Fig. 4), and 158 & 185/195 (Fig. 5) secs appear to be of approximately equal strength.
There is some variation in the individual pattern for of the power spectra for each pair, but the findings are broadly comparable, and indicate that strategies should be designed for sampling frequencies at around these time intervals.

Fig. 1 Alpha Power Spectrum

Fig.2

Fig. 3

Fig. 4

Fig. 5

PRINCIPAL COMPONENTS ANALYSIS OF ALPHA POWER SPECTRUM
If we look at the correlation surface of the power spectra of the twelve pairs some clear patterns emerge (see Fig 6):

Fig. 6

Focusing on the off-diagonal elements, it is clear that the power spectrum of each pair is perfectly correlated with the power spectrum of its conjugate.   So, for instance the power spectrum of the Stock1-Stock3 pair is exactly correlated with the spectrum for its converse, Stock3-Stock1.

But it is also clear that there are many other significant correlations between non-conjugate pairs.  For example, the correlation between the power spectra for Stock1-Stock2 vs Stock2-Stock3 is 0.72, while the correlation of the power spectra of Stock1-Stock2 and Stock2-Stock4 is 0.69.

We can further analyze the alpha power spectrum using PCA to expose the underlying factor structure.  As shown in Fig. 7, the first two principal components account for around 87% of the variance in the alpha power spectrum, and the first four components account for over 98% of the total variation.

PCA Analysis of Power Spectra

Fig. 7

Stock3 dominates PC-1 with loadings of 0.52 for Stock3-Stock4, 0.64 for Stock3-Stock2, 0.29 for Stock1-Stock3 and 0.26 for Stock4-Stock3.  Stock3 is also highly influential in PC-2 with loadings of -0.64 for Stock3-Stock4 and 0.67 for Stock3-Stock2 and again in PC-3 with a loading of -0.60 for Stock3-Stock1.  Stock4 plays a major role in the makeup of PC-3, with the highest loading of 0.74 for Stock4-Stock2.

Fig. 8  PCA Analysis of Power Spectra

## A Practical Application of Regime Switching Models to Pairs Trading

In the previous post I outlined some of the available techniques used for modeling market states.  The following is an illustration of how these techniques can be applied in practice.    You can download this post in pdf format here.

The chart below shows the daily compounded returns for a single pair in an ETF statistical arbitrage strategy, back-tested over a 1-year period from April 2010 to March 2011.

The idea is to examine the characteristics of the returns process and assess its predictability.

The initial impression given by the analytics plots of daily returns, shown in Fig 2 below, is that the process may be somewhat predictable, given what appears to be a significant 1-order lag in the autocorrelation spectrum.  We also see evidence of the
customary non-Gaussian “fat-tailed” distribution in the error process.

An initial attempt to fit a standard Auto-Regressive Moving Average ARMA(1,0,1) model  yields disappointing results, with an unadjusted  model R-squared of only 7% (see model output in Appendix 1)

However, by fitting a 2-state Markov model we are able to explain as much as 65% in the variation in the returns process (see Appendix II).
The model estimates Markov Transition Probabilities as follows.

P(.|1)       P(.|2)

P(1|.)       0.93920      0.69781

P(2|.)     0.060802      0.30219

In other words, the process spends most of the time in State 1, switching to State 2 around once a month, as illustrated in Fig 3 below.

In the first state, the  pairs model produces an expected daily return of around 65bp, with a standard deviation of similar magnitude.  In this state, the process also exhibits very significant auto-regressive and moving average features.

Regime 1:

Intercept                   0.00648     0.0009       7.2          0

AR1                            0.92569    0.01897   48.797        0

MA1                         -0.96264    0.02111   -45.601        0

Error Variance^(1/2)           0.00666     0.0007

In the second state, the pairs model  produces lower average returns, and with much greater variability, while the autoregressive and moving average terms are poorly determined.

Regime 2:

Intercept                    0.03554    0.04778    0.744    0.459

AR1                            0.79349    0.06418   12.364        0

MA1                         -0.76904    0.51601     -1.49   0.139

Error Variance^(1/2)           0.01819     0.0031

CONCLUSION
The analysis in Appendix II suggests that the residual process is stable and Gaussian.  In other words, the two-state Markov model is able to account for the non-Normality of the returns process and extract the salient autoregressive and moving average features in a way that makes economic sense.

How is this information useful?  Potentially in two ways:

(i)     If the market state can be forecast successfully, we can use that information to increase our capital allocation during periods when the process is predicted to be in State 1, and reduce the allocation at times when it is in State 2.

(ii)    By examining the timing of the Markov states and considering different features of the market during the contrasting periods, we might be able to identify additional explanatory factors that could be used to further enhance the trading model.