DCC GARCH

The aim of this section is to explain how the DCC-GARCH(1,1) model is fitted in order to examine the evolution of pairwise dynamic conditional correlations between sets of markets. More specifically, this step of the study is performed to obtain estimates of dynamic correlations between the decomposed return series obtained in the MODWT stage in a bivariate setup.

5.2.1 Background of the DCC-GARCH model

The dynamic conditional correlation GARCH model proposed by Engle (2002) generalizes a framework previously introduced in Bollerslev (1990), namely the constant conditional correlation GARCH model (CCC-GARCH). A distinct difference between the two is that instead of keeping correlation constant, the DCC analysis uses GARCH time series analysis to incorporate conditional correlations in a dynamic context. Correlation is often assumed not to be constant for financial assets, so while the CCC-GARCH allows for time-varying conditional variances and covariances it has a clear disadvantage to the DCC-GARCH when it comes to examining time-varying conditional correlations.

Furthermore, there are computational advantages to using the DCC-GARCH model over other multivariate GARCH models. First, it combines the flexibility of univariate GARCH models and parsimonious parametric models for correlations while avoiding the complexity that comes with conventional multivariate GARCH models. One benefit of the DCC-GARCH model is that the number of parameters that are estimated in the correlation process is independent of the number of series to be correlated. This allows for the estimation of potentially very large matrices. In comparison, models such as the VECH model introduced in Bollerslev et al (1988) or the BEKK model by Engle & Kroner (1995) are good examples of set-ups where the complexity is a serious issue, because the number of estimated parameters rapidly increases when adding more dependent variables. (Lehkonen & Heimonen, 2014) For a comprehensive discussion on the benefits and drawbacks of various multivariate GARCH models, see Bauwens et al (2006).

5.2.2 The DCC-GARCH(1,1) model

The process for estimating dynamic conditional variances and correlations is divided into two stages, which in practice separates the procedure’s univariate and multivariate dynamics. The first stage involves fitting univariate GARCH models to each univariate series to obtain time-varying conditional standard deviations, √ℎ𝑖,𝑡 (defined below). The estimates from the univariate model stage are then used as inputs in the second stage in which we can subsequently estimate the DCC parameters which describe the conditional correlation dynamics. A detailed description of how this whole process is implemented is provided in the paragraphs that follow.

First, we specify the return equation as an AR(1) model with GARCH(1,1) error term

𝑟_𝑖,𝑡 = 𝛾_𝑖,𝑜+ 𝛾_𝑖,1𝑟_{𝑖,𝑡−1}+ 𝜀_𝑖,𝑡, 𝜀_𝑖,𝑡~𝐺𝐴𝑅𝐶𝐻(1,1), (1)

where 𝛾_𝑖,𝑜 is a constant and 𝛾_𝑖,1is the lag-1 coefficient of the AR(1) model for the 𝑖th series, 𝑖 = 1,2, … , 𝑛 = 11, and 𝑟_𝑖,𝑡 is the return for the 𝑖th series at time 𝑡 = 1,2, . . , 𝑇. The equation includes an AR(1) term to account for autocorrelation which is often present when dealing with financial asset returns. Then the error from the AR(1) model, 𝑟_𝑖,𝑡− (𝛾_𝑖,𝑜+ 𝛾_𝑖,1𝑟_{𝑖,𝑡−1})=𝜀_𝑖,𝑡, is assumed to be a GARCH(1,1) process, namely, 𝜀_𝑖,𝑡=

√ℎ𝑖,𝑡𝑧_𝑖,𝑡 ,where {𝑧_𝑖,𝑡} is an IID normally distributed sequence (mean 0 and variance 1) and the conditional variance is ℎ_𝑖,𝑡 = 𝜔_𝑖+ 𝛼_𝑖𝜀_{𝑖,𝑡−1}² + 𝛽_𝑖ℎ_{𝑖,𝑡−1}.

Before we move to the multivariate part, we mention here that since we have decomposed our initial return series using MODWT over various timescales, the return series 𝑟_𝑖 = (𝑟_𝑖,1, 𝑟_𝑖,2, … , 𝑟_𝑖,𝑇) in the above equation will be the sequence of wavelet coefficients of series 𝑖, at a given level (d1 or d2 or... or d6 - six sets of analyses per one country or country pair) and not the returns themselves (one analysis per one country or country pair). In addition, because the sequences d1, d2, ..., d6 have zero-mean (by construction), one could opt for removing the constant term in the above equation, but we will keep it there, in line with Lehkonen & Heimonen (2014).

In the multivariate stage, we put the error-related terms, at time 𝑡, in vectors ε_t = (ε_1,t,

ε-2,t, … , ε_n,t)′ and z_t = (z_1,t, z_2,t, … , z_n,t)′, and assume that 𝐸[𝑧_𝑡𝑧_𝑡^′] = 𝐼_𝑛 , 𝐸[ε_𝑡|Ω_𝑡−1] = 0_𝑛 and 𝐸[ε_𝑡ε_𝑡^′|Ω_𝑡−1] = 𝐻_𝑡, where 𝐻_𝑡 is the conditional covariance matrix of ε_t, given the information available up to time 𝑡 − 1; 𝐼_𝑛 is the identity matrix of size 𝑛 and 0_𝑛 is a column vector of zeros of length 𝑛. Choosing a specific form of the conditional covariance matrix 𝐻_𝑡 leads to a specific multivariate model.

In the DCC-GARCH model the conditional covariance matrix is expressed as

𝐻_𝑡 = 𝐷_𝑡𝑅_𝑡𝐷_𝑡,

where the time-varying standard deviations obtained from fitting the series with univariate GARCH models are included in the diagonal of matrix 𝐷_𝑡 which at time 𝑡 is expressed as

𝐷_𝑡 =

The second unique element of the conditional covariance matrix, 𝐻_𝑡, is the time-varying conditional correlation matrix of ε_t, 𝑅_𝑡. This symmetric matrix is expressed as

𝑅_𝑡=

There are two necessary criteria for defining the DCC-GARCH process:

1. The conditional variance matrix (𝐻_𝑡) must be positive-definite. This means in practice that the correlation matrix 𝑅_𝑡 needs to be positive-definite since 𝐷_𝑡 is always positive-definite due to its structure with only positive diagonal elements.

2. The elements of the 𝑅_𝑡 matrix should all be equal or less than 1.

Assuming these criteria are met, 𝑅_𝑡can additionally be defined in the following way which includes the dynamic conditional correlation specification, 𝑅_𝑡=

𝑑𝑖𝑎𝑔(𝑄_𝑡)^−1/2𝑄_𝑡𝑑𝑖𝑎𝑔(𝑄_𝑡)^−1/2 , where

𝑄_𝑡= (1 − 𝛼 − 𝛽)𝑄̅ + 𝛼(ε_𝑡−1ε_𝑡−1′) + 𝛽𝑄_𝑡−1. (2)

The matrix 𝑄̅ (𝑛𝑥𝑛) is the unconditional covariance matrix of vector ε_𝑡 and 𝑄_𝑡 = [𝑞_{𝑖𝑗,𝑡}], 𝑖, 𝑗 = 1,2 … . , 𝑛 is the time-varying covariance matrix of vector ε_𝑡 from which the dynamical conditional correlation 𝜌_{𝑖𝑗,𝑡} between series 𝑖 and 𝑗, 𝑖 ≠ 𝑗, 𝑖, 𝑗 = 1,2, … , 𝑛 for 𝑡 = 1, … , 𝑇 can be extracted.

Maximum likelihood is used to estimate the model and in the case of the DCC-GARCH(1,1) model we obtain estimates of the two non-negative scalar parameters α and β. These parameters are required to satisfy the condition α + β < 1.

This process thereby gives us a time-varying correlation estimate between two series - in our case two series will be AR(1)-filtered wavelet coefficients from a given level for two countries. When the series are moving in the same direction, the correlation between the series increases. Conversely, we observe a low correlation value when the series move in

opposite directions. The restriction that the scalar parameters should always have a combined value of less than one (α + β < 1) implies that after a shock, the time-dependent correlation between the two series eventually returns to a long-run (average) level. The length of time it takes for the correlation to revert to that level depends on the parameters α and β, where α and β depict short- and long-run persistence, respectively. In the next section, the results from fitting the DCC-GARCH (1,1) model in a pairwise setup will be presented, and the focus will be on the obtained estimates of dynamic conditional correlation and estimates of the parameters α and β. How these estimates differ when taking into account timescales is of special interest.