ERM Chapter 19

Question 1

What is the method of moments?

Answer 1

• Establish the parameters of a distribution empirically by equating sample moments to population (or true) moments e.g. if we need three parameters we would have to equate three moments and solve simultaneously
• To specify the Gumbel, Clayton and Frank copulas we need to estimate the value of a, which can be done by equating the rank correlation to the underlying correlation
  (r = 1 - 1/a)

A: - more straightforward than the alternatives
D: - parameters are not necessarily the most likely ones
- parameter values may be outside their acceptable ranges

Question 2

What is maximum likelihood estimates (MLE)?

Answer 2

• expresses the joint probability of the actual observations occurring, given the choice of candidate distribution. The method then looks to maximise (with respect to each parameter) the log likelihood function:
  ln(L) = sum[ln(f(Xt))]
  L = product[f(Xt)]
  dlnL/dpi = 0

A: - only generates parameter values that are within the acceptable ranges

• any bias in the parameter estimates reduces as the no. of observations increases
• the distribution of each parameter estimate tends towards the normal distribution

Question 3

Outline ordinary least squares (OLS) regression.

Answer 3

• parameters are selected as to minimise the sum of the squared errors

b = (X’X)^-1X’Y

• Assumptions:
  > linear relationship exists between variables
  > inverse of the data exists
  > explanatory variables should not be correlated with the error terms
  > error terms are not correlated with each other - this would imply that serial correlation exists
  > error terms have a constant and finite variance o^2
  > error terms are normally distributed

Question 4

Outline generalised least squares (GLS) regression.

Answer 4

• the variance of error terms is not necessarily assumed to be constant and not necessarily assumed to be uncorrelated with each other

Question 5

Outline how the fit of GLS regression might be tested.

Answer 5

• test the overall fit using the coefficient of determination. The adjusted coefficient of determination does not automatically increase due to the addition of an extra explanatory variable. Assuming the error terms are normally distributed, an F test can be used to test the overall regression result.
• test the individual regression coefficients by estimating the variance of the error terms, and testing whether the coefficient differs from zero
• test the fit of the regression as a whole by testing the overall regression result
• test the fit of individual regression coefficients by testing whether the inclusion of each individual variable is significant, and hence necessary in our model

Question 6

Outline the testing of modelling based on a likelihood function.

Answer 6

1. Likelihood ratio test: tests whether the addition of variables results in significantly improved explanatory power. Uses nested models, where the second model contains all independent variables of the first model plus some additional variables.
   LR = -2ln(L1/L2)
2. Information Criteria: used to compare alternative models. The lower the value the better the fit. BIC penalises the addition of another independent variable more severely compared to the AIC.
   AIC = 2N - 2lnL
   BIC = NlnT - 2lnL

• IC are not restricted to the comparison of nested models, however IC only enable the ranking of alternative models. They do not quantify the statistical significance of any differences.

Question 7

Testing of the fit for Principal Component Analysis (PCA).

Answer 7

• an advantage is that it readily facilitates stochastic projections
• the model parameters produced do not necessarily have any intuitive interpretation and so its explanatory powers are limited

Question 8

Testing of the fit for Singular value decomposition (SVD).

Answer 8

• unlike PCA, SVD foes not require identification of the covariance matrix, and therefore it operates on original data with no requirement to identify independent variables upon which to base the regression

Question 9

How are models selected?

Answer 9

• Choosing the model with the highest log likelihood will likely choose the model with the largest number of parameters
• More complex models can be justified using AIC and BIC, as they incorporate penalties for additional parameters
• The suitability of specific distributions can be tested using:
  > QQ plots
  > histograms with superimposed fitted density functions
  > empirical cumulative distribution functions (CDFs) with superimposed fitted CDFs
  > autocorrelation functions of time series data (ACFs)

Question 10

Outlines Bayesian networks.

Answer 10

• used to model a network of risks
• relationship between pairs of nodes may be classified as:
  > independent
  > correlated
  > causal
• can be used for:
  > calculating probabilities
  > scenario analysis
  > causal analysis

A: - explicitly model cause and effect

• can incorporate expert judgement where there is insufficient data
• provides a framework for decision making, which can be documented and audited
• can facilitate scenario analysis and causal analysis

Question 11

How are models validated?

Answer 11

• A model should be fitted using one set of data and tested on an independent set of data of comparable size
• Backtesting involves fitting a time series model to data from one period and testing how well it predicts observed values from a subsequent period
• Training sets are used to fit a cross-sectional model on one set of data, with an independent dataset used to test