ECE478 Financial Signal Processing
Problem Set V: Financial Time-Series Analysis
Here, (m) denotes the covariance at lag m for a block of data. There are di§erent ways
to compute (m), speciÖcally the scaling factor, so let us use the following convention:
assume the data is 0 mean; if not subtract the sample mean Örst. Then (with data indexed
1 n N):
(m) = 1
N
X
N
n=m+1
xnxnm
The (auto-)correlation coe¢ cients are:
(m) = (m) = (0)
For example, the MATLAB function autocorr(x) will compute and plot (m) for 0 m 20,
with signiÖcance bounds (in the graph, if a displayed value is below the bounds, then it can be
considered as 0). For our purposes, it is Öne if we use j (m)j > 0:2 as a test of ìsigniÖcanceî.
Our formula for (m) is technically biased, and can be interpreted as a Bartlett (i.e., similar
to triangular) windowed form of unbiased estimates. In any case, for our purposes, the lags
of interest are M, where M N so this distinction is not of interest for us, here.
For pairs of signals, we can deÖne:
xy (m) = 1
N
X
N
n=m+1
xnynm
for m 0, and similarly for m < 0. Then the cross-correlation coe¢ cient is:
xy (m) =
xy (m)
q
xx (0) yy (0)
and in MATLAB, crosscorr(x; y) will compute and graph this, by default for 20 m 20.
1. Heavy Tail Distributions
First we are going to explore some distributions. Generate N = 1e6 iid samples of
N (0; 1), Cauchy with = 1, and Studentsí t-distribution with = 5 and = 10
degrees of freedom. The variance of Studentsít-distribution is Önite for 3 (well,
technically > 2 but here we consider only integer values for ) and is given by:
2
For comparison, you should normalize your Studentsít-distribution data sets so they
have variance 1. Obviously donít normalize Cauchy since it has no variance! However,
= 0:544 is a reasonable "match" to N (0; 1) in the sense that both yield approximatel
