06Class05

1. Review of last class: Model Risk and Evaluation

1.1. Uncertainty

Particularly important when highly leveraged, for OTC derivative products…

Liquidity, transaction costs…

Mote Carlo: missing the peak

Ultimate goal: decision

Common sense (Nick Leeson, to make USD 10M profit in one week in Jan 1995 would have had to trade more than 4 times the volume of Nikkei futures contract in both S’pore and Osaka”

Reference: Crouhy, Galai, Mark: Model Risk, J of Fin Engineering 1998 pp. 267-288.

1.1.1.

1.2. Kinds of models

1.2.1. Statistical models

· Data-driven / Observations

Neural Network

Assumptions of statistical nature

1.2.2. Structural models

· First Principles / Assumptions

Black Scholes

Financial theory that can be wrong

1.3. Limitations

1.3.1. Analytical

1.3.2. Computational

1.3.3. Data

Effective number of data points

1.4. Goals

1.4.1. Prediction

1.4.2. Description

1.4.3. Causal inference

Variables we can vary, vs variables we can’t

1.5. Evaluation

Diebold Chapter 12

1.6. Combination

Ibid.

2. Linear Regression

2.1. Goal?

Why do we do linear regression?

What really is a linear model?

2.2. Data Generating Process (DGP)

2.2.1. General concept

2.2.2. Specific for linear regression

3. Notation

3.1. x: Input

3.1.1. Explanatory variables

3.1.2. p of them

Curse of Dimensionality

3.2. y: Output

3.2.1. Response

3.2.2. Outcome

4. Matlab

4.1.

4.2. X\y

4.3. regress(y,X)

y = X b

X is n x p matrix

y is n x 1 vector

Returns statistics

5. Interpretation

5.1.1. y is conditional expectation

5.2. Densities

5.2.1. Normal, constant variance

5.2.2. Noninformative prior

Uniform on slope

Uniform on log(variance), i.e., p(sigma^2) ~ sigma ^{-2}

5.3. Priors

5.3.1. Pseuda data

5.3.2. Hints

6. Sanity checks

6.1. Residuals

6.1.1. Plot vs X

6.1.2. Standardize by dividing by predicted variance

7. Surprises

7.1. Regress x on y vs y on x

7.1.1. Don’t obtain inverse slopes

7.2. Collinearity

Determinant

Cov of data

Numerical problems

8. Analysis

8.1. Influence of the individual data points

8.1.1. Hat matrix

9. Tricks

9.1. Continuous variables

9.1.1. Ensure positivity

Log

9.1.2. Ensure range between 0 and 1

“sigmoid”, i.e., ½ (tanh (.) +1)

9.2. Discrete variables

9.2.1. Do not impose metric if there is none

9.3. Polynomial

polyfit

9.4. Interactions

“Bilinear model”