1. Tutorial Financial Econometrics/Statistics
2005 SAMSI program on Financial Mathematics, Statistics, and Econometrics

2. Goal

3. At the index level

4. Part I: Modeling
... in which we see what basic properties of stock prices/indices we want to capture

5. Contents Returns and their (static) properties Pricing models
Time series properties of returns

6. Why returns? Prices are generally found to be non-stationary
Makes life difficult (or simpler...)
Traditional statistics prefers stationary data
Returns are found to be stationary

7. Which returns? Two type of returns can be defined Discrete compounding
Continuous compounding

8. Discrete compounding
If you make 10% on half of your money and 5% on the other half, you have in total 7.5%
Discrete compounding is additive over portfolio formation

9. Continuous compounding
If you made 3% during the first half year and 2% during the second part of the year, you made (exactly) 5% in total
Continuous compounding is additive over time

10. Empirical properties of returns
Mean
St.dev.
Annualized volatility
Skewness
Kurtosis
Min
Max
IBM
-0.0%
2.46%
39.03%
-23.51
-138%
12.4%
(corr)
0.0%
1.64%
26.02%
-0.28
15.56
-26.1%
S&P
0.95%
15.01%
-1.4
39.86
-22.9%
8.7%
Data period: July December 2004; daily frequency

11. Stylized facts Expected returns difficult to assess
What’s the ‘equity premium’?
Index volatility < individual stock volatility
Negative skewness
Crash risk
Large kurtosis
Fat tails (thus EVT analysis?)

12. Pricing models
Finance considers the final value of an asset to be ‘known’
as a random variable , that is
In such a setting, finding the price of an asset is equivalent to finding its expected return:

13. Pricing models 2
As a result, pricing models model expected returns ...
... in terms of known quantities or a few ‘almost known’ quantities

14. Capital Asset Pricing Model
One of the best known pricing models
The theorem/model states

15. Black-Scholes Also Black-Scholes is a pricing model
(Exact) contemporaneous relation between asset prices/returns

16. Time series properties of returns
Traditionally model fitting exercise without much finance
mostly univariate time series and, thus, less scope for tor the ‘traditional’ cross-sectional pricing models
lately more finance theory is integrated
Focuses on the dynamics/dependence in returns

17. Random walk hypothesis
Standard paradigm in the
Prices follow a random walk
Returns are i.i.d.
Normality often imposed as well
Compare Black-Scholes assumptions

18. Box-Jenkins analysis

19. Linear time series analysis
Box-Jenkins analysis generally identifies a white noise
This has been taken long as support for the random walk hypothesis
Recent developments
Some autocorrelation effects in ‘momentum’
Some (linear) predictability
Largely academic discussion

20. Higher moments and risk

21. Risk predictability
There is strong evidence for autocorrelation in squared returns
also holds for other powers
‘volatility clustering’
While direction of change is difficult to predict, (absolute) size of change is
risk is predictable

22. The ARCH model First model to capture this effect
No mean effects for simplicity
ARCH in mean

23. ARCH properties Uncorrelated returns Correlated squared returns
martingale difference returns
Correlated squared returns
with limited set of possible patterns
Symmetric distribution if innovations are symmetric
Fat tailed distribution, even if innovations are not

24. The GARCH model
Generalized ARCH
Beware of time indices ...

25. GARCH model Parsimonious way to describe various correlation patterns
for squared returns
Higher-order extension trivial
Math-stat analysis not that trivial
See inference section later

26. Stochastic volatility models
Use latent volatility process

27. Stochastic volatility models
Also SV models lead to volatility clustering
Leverage
Negative innovation correlation means that volatility increases and price decreases go together
Negative return/volatility correlation
(One) structural story: default risk

28. Continuous time modeling
Mathematical finance uses continuous time, mainly for ‘simplicity’
Compare asymptotic statistics as approximation theory
Empirical finance (at least originally) focused on discrete time models

29. Consistency
The volatility clustering and other empirical evidence is consistent with appropriate continuous time models
A simple continuous time stochastic volatility model

30. Approximation theory
There is a large literature that deals with the approximation of continuous time stochastic volatility models with discrete time models
Important applications
Inference
Simulation
Pricing

31. Other asset classes So far we only discussed stock(indices)
Stock derivatives can be studied using a derivative pricing models
Financial econometrics also deals with many other asset classes
Term structure (including credit risk)
Commodities
Mutual funds
Energy markets
...

32. Term structure modeling
Model a complete curve at a single point in time
There exist models
in discrete/continuous time
descriptive/pricing
for standard interest rates/derivatives
...

33. Part 2: Inference

34. Contents Parametric inference for ARCH-type models
Rank based inference

35. Analogy principle
The classical approach to estimation is based on the analogy principle
if you want to estimate an expectation, take an average
if you want to estimate a probability, take a frequency
...

36. Moment estimation (GMM)
Consider an ARCH-type model
We suppose that can be calculated on the basis of observations if is known
Moment condition

37. Moment estimation - 2 The estimator now is taken to solve
In case of “underidentification”: use instruments
In case of “overidentification”: minimize distance-to-zero

38. Likelihood estimation
In case the density of the innovations is known, say it is , one can write down the density/likelihood of observed returns
Estimator: maximize this

39. Doing the math ... Maximizing the log-likelihood boils down to solving
with

40. Efficiency consideration
Which of the above estimators is “better”?
Analysis using Hájek-Le Cam theory of asymptotic statistics
Approximate complicated statistical experiment with very simple ones
Something which works well in the approximating experiment, will also do well in the original one

41. Quasi MLE
In order for maximum likelihood to work, one needs the density of the innovations
If this is not know, one can guess a density (e.g., the normal)
This is known as
ML under non-standard conditions (Huber)
Quasi maximum likelihood
Pseudo maximum likelihood

42. Will it work?
For ARCH-type models, postulating the Gaussian density can be shown to lead to consistent estimates
There is a large theory on when this works or not
We say “for ARCH-type models the Gaussian distribution has the QMLE property”

43. The QMLE pitfall One often sees people referring to Gaussian MLE
Then, they remark that we know financial innovations are fat-tailed ...
... and they switch to t-distributions
The t-distribution does not possess the QMLE property (but, see later)

44. How to deal with SV-models?
The SV models look the same
But now, is a latent process and hence not observed
Likelihood estimation still works “in principle”, but unobserved variances have to be integrated out

45. Inference for continuous time models
Continuous time inference can, in theory, be based on
continuous record observations
discretely sampled observations
Essentially all known approaches are based on approximating discrete time models

46. ... in which we discuss the main ideas of rank based inference

47. The statistical model Consider a model where ‘somewhere’ there
exist i.i.d. random errors
The observations are
The parameter of interest is some
We denote the density of the errors by

48. Formal model
We have an outcome space , with the number of observations and the dimension of
Take standard Borel sigma-fields
Model for sample size :
Asymptotics refer to

49. Example: Linear regression
Linear regression model (with observations )
Innovation density and cdf

50. Example ARCH(1) Consider the standard ARCH(1) model
Innovation density and cdf

51. Maintained hypothesis
For given and sample size , the
innovations can be calculated from the
observations
For cross-sectional models one may even often write
Latent variable (e.g., SV) models ...

52. Innovation ranks The ranks are the ranks of the innovations
We also write for the ranks
of the innovations based on
a value for the parameter of interest
Ranks of observations are generally not very useful

53. Basic properties The distribution does not depend on nor on
permutation of
This is (fortunately) not true for
at least ‘essentially’

54. Invariance
Suppose we generate the innovations as transformation with i.i.d. standard uniform
Now, the ranks are even invariant with respect to

55. Reconstruction
For large sample size we have
and, thus,

56. Rank based statistics
The idea is to apply whatever procedure you have that uses innovations on the innovations reconstructed from the ranks
This makes the procedure robust to distributional changes
Efficiency loss due to ‘ ’?

57. Rank based autocorrelations
Time-series properties can be studied using rank based autocorrelations
These can be interpreted as ‘standard’ autocorrelations
rank based
for given reference density and distribution free

58. Robustness
An important property of rank based statistics is the distributional invariance
As a result: a rank based estimator is consistent for any reference density
All densities satisfy the QMLE property when using rank based inference

59. Limiting distribution
The limiting distribution of depends on both the chosen reference density and the actual underlying density
The optimal choice for the reference density is the actual density
How ‘efficient’ is this estimator?
Semiparametrically efficient

60. Remark
All procedures are distribution free with respect to the innovation density
They are, clearly, not distribution free with respect to the parameter of interest

61. Signs and ranks

62. Why ranks?
So far, we have been considering ‘completely’ unrestricted sets of innovation densities
For this class of densities ranks are ‘maximal invariant’
This is crucial for proving semiparametric efficiency

63. Alternatives Alternative specifications may impose
zero-median innovations
symmetric innovations
zero-mean innovations
This is generally a bad idea ...

64. Zero-median innovations
The maximal invariant now becomes the ranks and signs of the innovations
The ideas remain the same, but for a more precise reconstruction
Split sample of innovations in positive and negative part and treat those separately

65. But ranks are still ... Yes, the ranks are still invariant
... and the previous results go through
But the efficiency bound has now changed and rank based procedures are no longer semiparametrically efficient
... but sign-and-rank based procedures are

66. Symmetric innovations
In the symmetric case, the signed-ranks become maximal invariant
signs of the innovations
ranks of the absolute values
The reconstruction now becomes still more precise (and efficient)

67. Semiparametric efficiency

68. General result
Using the maximal invariant to reconstitute the central sequence leads to semiparametrically efficient inference
in the model for which this maximal invariant is derived
In general use

69. Proof
The proof is non-trivial, but some intuition can be given using tangent spaces