Building Econometric Models

The Dynamic Conditional Correlation Model

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• Several different formulations of the dynamic conditional correlation (DCC) model are available, but a popular specification is due to Engle (2002)
• The model is related to the CCC formulation but where the correlations are allowed to vary over time.
• Define the variance-covariance matrix, Ht, as Ht = DtRtDt
• Dt is a diagonal matrix containing the conditional standard deviations (i.e. the square roots of the conditional variances from univariate GARCH model estimations on each of the N individual series) on the leading diagonal
• Rt is the conditional correlation matrix
• Numerous parameterisations of Rt are possible, including an exponential smoothing approach

The DCC Model – A Possible Specification

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• A possible specification is of the MGARCH form:
• where:
• S is the unconditional correlation matrix of the vector of standardised residuals (from the first stage estimation), ut = Dt−1ϵt
• ι is a vector of ones
• Qt is an N × N symmetric positive definite variance-covariance matrix
• ◦ denotes the Hadamard or element-by-element matrix multiplication procedure
• This specification for the intercept term simplifies estimation and reduces the number of parameters.

The DCC Model – A Possible Specification

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• Engle (2002) proposes a GARCH-esque formulation for dynamically modelling Qt with the conditional correlation matrix, Rt then constructed as
• where diag(·) denotes a matrix comprising the main diagonal elements of (·) and Q∗ is a matrix that takes the square roots of each element in Q
• This operation is effectively taking the covariances in Qt and dividing them by the product of the appropriate standard deviations in Qt∗ to create a matrix of correlations.