smfd12(13a).tex
Day 12. 21.11.2013.
General case: M A(q ). As above,
q
Xt = t +
j tj = (B )t ,
where () = 1 +
j =1
q
j =1
j j .
So formally, if we invert this we obtain
t = (B )1 Xt ,
and as () = 1 +
smfd1(13a).tex
Day 1. 11.11.2013
I. ESTIMATION OF PARAMETERS
1. PARAMETERS; LIKELIHOOD
To do Statistics to handle the mathematics and data analysis of situations involving randomness we need to model
smfd3(13a).tex
Day 3. 18.11.2013
6. Complements
1. CAPM.
All of this is highly relevant to Mathematical Finance. Finance was an
art rather than a science before the 1952 PhD thesis of Harry MARKOWITZ
smfd4.tex
Day 4. 24.10.2013.
3. Likelihood Ratio Tests
We turn now to the general case: composite H0 v. composite H1 . We may
not be able to nd UMP (best) tests. Instead, we seek a general procedure
f
smfd5.tex
Day 5. 25.10.2013.
5. Multiplication Theorem for Determinants.
If A, B are n n (so AB , and BA, are dened),
|AB | = |A|.|B |.
Proof. We can display a matrix A as a row of its columns, A = [a
smfd6.tex
Day 6. 31.10.2013.
As always, n may be large the larger the better, as large samples are
more informative than small ones. The size of p varies with the problem.
But typically p might be of
smfd7(13a).tex
Day 7. 1.11.2013.
IV. REGRESSION
1. Least Squares
The idea of regression is to take some sample of size n from some unknown
population (typically n is large the larger the better), and
smfd8(13a).tex
Day 8. 7.11.2013.
We turn now to a technical result, which is important in reducing ndimensional problems to one-dimensional ones.
Theorem (Cramr-Wold device). The distribution of a ran
smfd9(13a).tex
Day 9. 8.11.2013.
The MLE for is x, as this reduces the last term (the only one involving
) to its minimum value, 0. For a square matrix A = (aij ), its determinant
is
aij Aij
|A| =
j
f
smfd10(13a).tex
Day 10. 14.11.2013.
V. TIME SERIES (TS).
1. Stationary processes and autocorrelation
A TS - a sequence of observations indexed by time - may well exhibit, on
visual inspection after pl
smfd11(13a).tex
Day 11 15.11.2013.
4. General autoregressive processes, AR(p).
Again working with the zero-mean case for simplicity, the extension of
the above to p parameters is the model
Xt = 1 Xt1
smfd13(13a).tex
Day 13. 22.11.2013.
10. ARCH and GARCH; Econometrics ([BF, 9.4.1, 220-222)
There are a number of stylised facts in mathematical nance. E.g.:
(i). Financial data show skewness. This is
smfd2(13a).tex
Day 2. 17.11.2013
3. Large-sample properties of maximum-likelihood estimators
We assume the following regularity conditions:
(i) dierentiability under the integral sign twice (as before