Module 19: Fitting models

Question 1

Method of moments:

univariate distribution

Answer 1

The parameters of a univariate distribution are established by equating population moments to respective sample moments.

Question 2

Method of moments:

copula

Answer 2

The parameters of a copula might be established by equating the true, underlying correlation of the copula to estimated correlations from the sample data.

Question 3

Method of Maximum Likelihood:

univariate distribution

Answer 3

The parameters of a univariate distribution are established as those that maximise the log-likelihood function, where the likelihood function is the joint probability of the actual observations occurring.

Question 4

Ordinary Lease Squares regression (OLS)

Answer 4

Ordinary least squares regression explains a dependent variable, Y, (t = 1, …, T) in terms of a linear combination of response variables: X₁, … Xₙ, (n = 1, …, N) plus an error term ε ~ N(0 σ²).

In matrix form, this gives:
Y = Xβ + ε

Under OLS, the parameters are selected to mimise the sum of the squared error terms (ε’ε). A closed form solution for the vector of parameters is:
Y^ = (X’X)⁻¹ X’ Y

Question 5

Generalised least squares regression (GLS)

Answer 5

Under GLS, the variance of the error terms is transformed by a matrix to allow for non-constant variance and/or serial correlation in the error terms.

Question 6

Singular Value Decomposition

Answer 6

SVD is a type of least squares optimisation that does not require identification of the covariance matrix. It operates on the original data with no requirement to (pre)specify the explanatory variables upon which to base a regression. It may result in a smaller number of explanatory variables although these may not have any intuitive meaning.

Question 7

F-test

Answer 7

(based on the coefficients of determination)

can be used to test the fit of the regression as a whole, assuming that the error terms are normally distributed.

Question 8

t-test

Answer 8

To test the fit of the individual parameters, a t-test can be used to test whether each β is significantly different to zero.

Question 9

Likelihood-ratio (LR) test

Answer 9

Can be used for nested models, to test whether the addition of extra parameters results in significantly improved explanatory power.

Question 10

Information criteria

Answer 10

Can be used to compare alternative models, although they do not quantify the statistical significance of any differences. A lower value implies a better fit.

Question 11

2 Examples of information criteria

Answer 11

• Akaike’s information criterion (AIC)
• Bayesian information criterion (BIC).

The BIC penalises the introduction of additional independent variables more severely than the AIC.

Question 12

4 graphical diagnostic tests of the suitability of specific models

Answer 12

• QQ plots
• histograms with superimposed fitted density functions
• empirical CDFs with superimposed fitted CDFs
• autocorrelation functions of time series data (ACFs)

Question 13

Bayesian Networks

Answer 13

Can be used to model a network of risks using directed acyclic graphs (DAGs)

Each node in a DAG represents an event or state. The relationship between pairs of nodes may be classified (visualised) as:

• independent (no line)
• correlated (line)
• casual (arrow, from parent to child)

Question 14

4 Benefits of modelling using Bayesian networks

Answer 14

• explicitly model cause and effect
• can incorporate expert judgement where there is insufficient data (hence useful for operational risk modelling)
• provides a framework for decision making, which can be documented and audited
• can facilitate scenario analysis and causal analysis