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FINANCIAL ECONOMETRICS

FINANCIAL ECONOMETRICS. SPRING 2013 WEEK VII MULTIVARIATE MODELLING OF VOLATILITY Prof. Dr. Burç ÜLENGİN. MULTIVARIATE VOLATILITY. There may be interactions among the conditional variance of the return series. Also covariance of the return series may change over the time.

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FINANCIAL ECONOMETRICS

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  1. FINANCIAL ECONOMETRICS SPRING 2013 WEEK VII MULTIVARIATE MODELLING OF VOLATILITY Prof. Dr. Burç ÜLENGİN

  2. MULTIVARIATE VOLATILITY • There may be interactions among the conditional variance of the return series. • Also covariance of the return series may change over the time. • Therefore the full perspective of volatility modelling requires the treatment of variances and covariances together- simultaneously. • When the variances and covariances are modelled it means that correlations are modelled too.

  3. MOVING CORRELATION OF THE RETURNS OF TWO FINANCIAL ASSETS

  4. MULTIVARIATE GARCH • In multivariate GARCH models, yt is a vector of the conditional means (Nx1), the conditional variance of yt is an matrix H (NxN). • The diagonal elements of H are the variance terms hii, and the off-diagonal elements are the covariance terms hij.

  5. MULTIVARIATE GARCH • There are numerous different representations of the multivariate GARCH model. • The main representations are: • VECH • Diagonal • BEKK- Baba, Engle, Kraft, Kroner • Constant correlation representation • Principle component representation

  6. VECH REPRESANTATION • Full treatment of the matrix H • In the VECH model, the number of parameters can be exteremely large. • Estimating a large number of parameters is not in theory a problem as long as there is large enough sample size. • The parameters of VECH are estimated by maximum likelihood and the obtaining convergence of the typical optimization algorithm employed in practice be very difficult when a large number of parameters are involved. • Also estimated variances must be positive and it requires the additional restrictions on parameters

  7. VECH REPRESANTATION 2 Variable Case A and B are {Nx(N+1)/2 , Nx(N+1)/2} matrices . In the case of 2 variables, 3 equations and 21 parameters. 5 variables, 20 equations and 820 parameters. 10 variables, 55 equations and 4025 parameters.

  8. DIAGONAL REPRESENTATION • The diagonal representation is based on the assumption that the individual conditional variances and conditional covariances are functions of only lagged values of themselves and lagged squared residuals. • Bollerslev, Engle and Woodridge (1988) proposed • In the case of 2 variables, this representation reduces the number of parameters to be estimated from 21 to 9. • At the expense of losing information on certain interrelationships, such as the relationship between the individual conditional variances and the conditional covariances. • Also estimated variances must be positive and it requires the additional restrictions on parameters

  9. DIAGONAL REPRESENTATION2 Variable Case

  10. DIAGONAL REPRESENTATIONOIL & NATURAL GAS PRICES

  11. DIAGONAL REPRESENTATION ESTIMATIONOIL & NATURAL GAS PRICES

  12. DIAGONAL REPRESENTATION ESTIMATIONOIL & NATURAL GAS PRICES

  13. DIAGONAL REPRESENTATION VOLATILITY FORECAST OF OIL & NATURAL GAS PRICES

  14. DIAGONAL REPRESENTATION REVISED MODEL ESTIMATION OIL & NATURAL GAS PRICES

  15. DIAGONAL REPRESENTATION REVISED MODEL ESTIMATION OIL & NATURAL GAS PRICES

  16. DIAGONAL REPRESENTATION REVISED MODEL ESTIMATION OIL & NATURAL GAS PRICES

  17. DIAGONAL REPRESENTATION VOLATILITY FORECAST OF OIL & NATURAL GAS PRICES

  18. DIAGONAL REPRESENTATION TARCH MODEL ESTIMATION OIL & NATURAL GAS PRICES I1=1 if e1t-1<0 =0 otherwise I2=1 if e2t-1<0 =0 otherwise

  19. DIAGONAL REPRESENTATION TARCH MODEL ESTIMATION OIL & NATURAL GAS PRICES

  20. DIAGONAL REPRESENTATION TARCH MODEL ESTIMATION OIL & NATURAL GAS PRICES

  21. DIAGONAL REPRESENTATION TARCH MODEL FORECAST OIL & NATURAL GAS PRICES VOLATILITY

  22. BEKK REPRESENTATION • Engle and Kroner(1995) developed the Baba(1990) approach. • BEKK representation of multivariate GARCH improves on both the VECH and diagonal representation, since H is almost guaranteed to be positive definite. • BEKK representation require more parameters than Diagonal rep. but less parameters than VECH. • It is more general than diagonal rep. as it allows for interaction effects that diagonal rep. does not.

  23. BEKK REPRESENTATION OF OIL & NATURAL GAS PRICES

  24. BEKK ESTIMATION OF OIL & NATURAL GAS PRICES

  25. BEKK ESTIMATION OF OIL & NATURAL GAS PRICES

  26. REVISED BEKK REPRESENTATION OF OIL & NATURAL GAS PRICES

  27. REVISED BEKK REPRESENTATION OF OIL & NATURAL GAS PRICES

  28. REVISED BEKK REPRESENTATION OF OIL & NATURAL GAS PRICES

  29. REVISED BEKK FORECASTING OF OIL & NATURAL GAS PRICES VOLATILITY

  30. CONSTANT CORRELATION REPRESENTATION • Bollerslev(1990) employes the conditional corelation matrix R to derive a representation of the multivariate GARCH model. • In his R matrix, Bollerslev restricts the conditional correlations to be equal to the correlation coefficients between variables, which are simply constants. Thus R is constant over time. • This representation has the advantage that H will be positive definite.

  31. CONSTANT CORRELATION REPRESENTATION The individual variance terms hiit are taken to be individual GARCH processes

  32. CONSTANT CORRELATION REPRESENTATION OF OIL & NATURAL GAS PRICES

  33. CONSTANT CORRELATION REPRESENTATION OF OIL & NATURAL GAS PRICES

  34. CONSTANT CORRELATION FORECAST OF OIL & NATURAL GAS PRICES VOLATILITY

  35. REVISED CONSTANT CORRELATION REPRESENTATION OF OIL & NATURAL GAS PRICES

  36. REVISED CONSTANT CORRELATION REPRESENTATION OF OIL & NATURAL GAS PRICES

  37. REVISED CONSTANT CORRELATION FORECAST OF OIL & NATURAL GAS PRICES VOLATILITY

  38. FACTOR VOLATILITY MODELS

  39. Principal Component Analysis

  40. Principal Component Analysis • We collect 120 financial ratios in order to asses financial health of the firms. How can we reduce these ratios a few indices? • The production control department collect several measures in order to control process. Can we develop some indices in order to summarize the process outcomes? • In order to carry out efficient regression analysis we have to reduce multicollinearity among the explanatory variables if it exists. Can we generate some new indices in order to get orthogonal explanatory series that also contain most of the information of the original variables?

  41. Principal Component Analysis in Finance • To reduce number of risk factors to a manageable dimension. For example, instead of 60 yields of different maturities as risk factors, we might use just 3 principal component. • To identy the key sources of risk. Typically the most important risk factors are parallel shifts, changes in slope and changes in convexity of the curves. • To facilitate the measurement of portfolio risk, for instance by introducing scenarios on the movements in the major risk factors.

  42. Basics & Background • A is square matrix • X is a column vector • is a scalar quantity-eigenvalue u normalized eigenvector Basic properties

  43. Basics & Background IF matrix A composes of some observed x values Pricipal Component Scores

  44. MATHEMATICAL BACKGROUND A nxn square matrix

  45. A BASIC EXAMPLE OF EIGENVALUES AND EIGENVECTORS Normalization

  46. MATHEMATICAL EXAMPLE U1 U2

  47. Basics & Background • Eigenvalue and Eigenvector: • Eigen originates in the German language and can be loosely translated as “of itself” • Thus an Eigenvalue of a matrix could be conceptualized as a “value of itself” • Eigenvalues and Eigenvectors are utilized in a wide range of applications (PCA, calculating a power of a matrix, finding solutions for a system of differential equations, and growth models)

  48. GEOMETRICAL APPROACH TO PRINCIPAL COMPONENT ANALYSIS x2      x1      Mean corrected data

  49. AXIS ROTATION x2      x1     

  50. AXIS ROTATION DIMEMSION REDUCTION x2’ x2 x1’      x1     

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