Elsevier

Journal of Empirical Finance

Volume 48, September 2018, Pages 162-180
Journal of Empirical Finance

Multivariate models with long memory dependence in conditional correlation and volatility

https://doi.org/10.1016/j.jempfin.2018.06.011Get rights and content

Highlights

  • Multivariate models are proposed that allow for long memory in conditional correlation and volatility.

  • The models are the first multivariate long memory conditional correlation processes in the literature.

  • The models are applied to a data set consisting of 10 US stocks.

  • Relative to a number of benchmarks, the QLIKE loss function indicates that the models provide superior forecasts over 20–80 day horizons. When forecasting minimum variance portfolio weights, the benefits diminish completely.

Abstract

Multivariate models with long memory (LM) in conditional correlation and volatility are proposed. The models employ a fractionally integrated version of the dynamic conditional correlation GARCH (DCC-GARCH) process (Engle, 2002), and can be used to forecast conditional covariance matrices of high dimension. The models are applied to a data set consisting of ten US stocks and out of sample forecasts over 1–80 days evaluated using statistical and economic loss functions. If intraday data is unavailable, the statistical loss function reveals that LM correlation models provide superior return covariance matrix forecasts over 20–80 days. When intraday data is available, LM correlation models provide superior forecasts of the realised covariance matrix over the same horizons, however the gains when forecasting the return covariance matrix are small. Finally, when forecasting minimum variance portfolio weights, even though the benefits from LM correlation models diminish completely, they are not consistently outperformed by any of the benchmarks.

Introduction

Long memory models for the first and second moments have been well studied over the last two to three decades Baillie (1996), Robinson (2003), Kirman and Teyssiere (2007), with more recent attention focused on the realised covariance matrix Bauer and Vorkink (2011), Chiriac and Voev (2011). Long memory in conditional correlations has been documented as far back as Andersen et al. (2003), yet no multivariate long memory conditional correlation models have been developed. This paper seeks to fill this gap by proposing fractionally integrated versions of the dynamic conditional correlation (DCC) process (Engle, 2002). Analytical conditions for positive definiteness (PD) are derived, and out of sample forecast performance is evaluated against a number of benchmarks.

Long memory is commonly used to describe persistent dependence between time series observations as the lag increases. This is typically characterised by hyperbolic decay of the autocovariance function which is not absolutely summable. In contrast, weakly dependent or short memory (SM) processes like DCC-GARCH have exponential decay of the auto-covariance function which is absolutely summable.

A voluminous literature has documented long memory in the volatility of equities, currencies and commodities. Long memory in volatility may arise from aggregation of multiple volatility components caused by heterogeneous information flows (Andersen and Bollerslev, 1997) or heterogeneous traders (Müller et al., 1997), it may also arise from a heavy tailed regime switching process (Liu, 2000). Multivariate extensions have examined spectral density estimators of the fractional differencing parameter d Lobato (1999), Lobato and Velasco (2000), common long memory factors in large systems (Morana, 2007), fractional cointegration (Brunetti and Gilbert, 2000), long memory stochastic volatility models (So and Kwok, 2006) and vector fractionally integrated ARMA (VARFIMA) models on Cholesky factors from realised covariance matrices (Chiriac and Voev, 2011).

This paper proposes two long memory correlation models. Both employ fractionally integrated versions of the DCC process, with the key difference being the assumed information set. The first model extends the multivariate HEAVY (M-HEAVY) specification of Noureldin et al. (2012), which conditions on intraday data and uses realised covariance matrices as innovations. Noureldin et al. (2012) use a SM model via a BEKK GARCH structure, to model conditional covariance matrix dynamics. Instead the proposed Long Memory Multivariate HEAVY (LM-M-HEAVY) model allows for LM in the conditional correlations and volatilities via fractional integration. The second model has similar correlation dynamics to the first, but conditions on a lower frequency data set, employing innovations that are a function of daily returns. The proposed LM-DCC model therefore represents a fractionally integrated version of the more conventional DCC process (Engle, 2002). This model is considered given its popularity in the literature, and that high frequency intraday data may not always be available.

The proposed long memory correlation models are motivated by the following. First, the HEAVY specification jointly models the dynamics of the realised covariance matrix as well as the daily return covariance matrix. The vast majority of the literature that employs realised volatility or covariance matrices assumes that investors open their position at the commencement of trade and close out at the end of the day Corsi (2009), Chiriac and Voev (2011), Bauer and Vorkink (2011), Golosnoy et al. (2012). This is inadequate for the majority of investors who hold positions over night, because they require close to close forecasts. The HEAVY model meets this need as it can be fit to close to close or open to close returns. HEAVY model forecasts may also exhibit short-run momentum and respond rapidly to sudden changes in volatility or correlation, outperforming univariate (Shephard and Sheppard, 2010) and multivariate (Noureldin et al., 2012) GARCH forecasts over short term horizons.

Second, dynamic correlation models fit to daily returns and realised correlation proxies support the presence of long memory. Short memory dynamic conditional correlation (SM-DCC) models often imply near unit root behaviour Engle (2002), Tse and Tsui (2002), Janus et al. (2014), Hafner and Manner (2010). It is now well understood that near unit roots may occur if the data generating process (DGP) has long memory or is weakly dependent with occasional breaks. These two processes may be easily confused: in the presence of long memory, a weakly dependent model will spuriously identify occasional breaks; while long memory will be spuriously identified if the DGP is weakly dependent with occasional breaks. See Banerjee and Urga (2005) and Perron and Qu (2010) for comprehensive reviews.

Both approaches have been adopted when examining dynamic correlations. SM-DCC models have captured occasional breaks via dummy variables (Cappiello et al., 2006) and threshold effects (Kwan et al., 2009). Constant correlation models with regime switching (Pelletier, 2006) and smooth transitions (Silvennoinen and Teräsvirta, 2009) have also been used. In contrast, Audrino and Corsi (2010) and Asai (2013) approximate long memory in correlation via the heterogeneous autoregressive (HAR) structure of Corsi (2009), Andersen et al. (2003) fit ARFIMA models to realised correlations, and Janus et al. (2014) estimate a LM-DCC model via an ARFIMA specification that employs a Student t copula. These LM correlation models are only applied in a bivariate setting (i.e a single pairwise correlation), and have not been generalised to a multivariate setting given that they cannot ensure positive definiteness.

This paper remains agnostic on whether correlations have long memory dependence or are short memory with occasional breaks. A long memory specification is employed because even if data has occasional breaks and is weakly dependent, a fractional model may be useful for forecasting (Diebold and Inoue, 2001). Long memory processes may be viewed as a convenient forecasting tool because they do not require forecasts of break points out of sample. This is important because ex-ante break point identification is difficult and failure to do so may be costly. Dacco and Satchell (1999) for example show that failure to forecast the regime may result in regime switching model forecasts having a higher mean square error than forecasts from a random walk.1

The third motivation is the shortcomings in the limited literature that extends fractionally integrated GARCH models to a multivariate setting. Teyssiere (1998) and Pafka and Matyas (2001) estimate a multivariate FIGARCH model via a diagonal specification similar to Bollerslev et al. (1988). There are no analytical PD conditions and the diagonal specification is generally only suitable for several assets. Chiriac and Voev (2011) combine long memory volatility models (FIGARCH) with the SM-DCC model of Engle (2002). Whilst the SM-DCC model can be used for a large number of assets and analytical conditions for PD easily imposed, the correlation dynamics exhibit near unit root behaviour and so the use of long memory in volatility but not in the correlations may be sub-optimal. Niguez and Rubia (2006) propose an orthogonal HYGARCH (Davidson, 2004) model which may be used for a large number of assets, however principal components are difficult to interpret.

The final motivation is the limitations in existing LM (or near LM) models when applied to realised covariance matrices Chiriac and Voev (2011), Bauer and Vorkink (2011), Golosnoy et al. (2012), Lucas and Opschoor (2017). These models lack the flexibility of the DCC structure and are less amenable to large scale system estimation. The VARFIMA model of Chiriac and Voev (2011) imposes an ARFIMA structure on all the Cholesky factors from the realised covariance matrix. Identification requires the complicated Eschelon form (Lütkepohl and Poskitt, 1996) or the final equations form, which imposes a scalar autoregressive polynomial. Bauer and Vorkink (2011) fit a HAR model with a latent factor structure to the matrix log transformation of the realised covariance matrix. Estimation requires a GMM procedure which may be problematic when the number of assets is large. The conditional autoregressive wishart (CAW) model (Golosnoy et al., 2012), approximates long memory via a high order VARMA process and is therefore heavily parameterised. Their five asset model for example has 116 parameters, and even the most restricted version has 41 parameters. Estimation is computationally burdensome requiring a bottom up approach.2  Lucas and Opschoor (2017) model the conditional covariance matrix via a FIGARCH specification with score driven dynamics. Their model is tightly parameterised and therefore lacks the flexibility of the DCC process. Further, the non linear transformations of the data in Chiriac and Voev (2011) and Bauer and Vorkink (2011) (which are used to ensure PD), induce a bias in the forecast Cholesky factors or log matrix transformations, because they must be inverted. Bauer and Vorkink (2011) derive an expression for the theoretical bias under heavily restrictive assumptions, and their data driven alternative is not feasible when forecasting out of sample. Chiriac and Voev (2011) acknowledge that modelling Cholesky factors induces an even larger bias, but do not provide any theoretical results.

This paper overcomes these limitations by proposing models with fractionally integrated volatility and a fractionally integrated DCC type structure. Following Engle (2002) the covariance matrix is disentangled into separate models for the conditional volatilities and correlation. This is in contrast to Chiriac and Voev (2011) and Bauer and Vorkink (2011) who fit their models to the covariance matrix (or some transformation of it) directly. The quasi maximum likelihood estimator is therefore decomposed into the sum of the variance and correlation components. This simplifies estimation and allows for flexibility across asset variance dynamics. The LM-M-HEAVY model extends the univariate HEAVY models in Shephard and Sheppard (2010) to allow for fractional integration and volatility asymmetries. The LM-DCC model fit to daily returns also allows for asymmetric fractionally integrated volatility via the FIEGARCH process (Bollerslev and Mikkelsen, 1996). The flexibility in the variance and correlation equations allows the data to determine whether a long or short memory process is required. This enables an examination into whether LM in covariance is driven by LM in conditional volatility and correlation, or just LM in volatility (with correlation subject to SM). Finally, both models are applied directly to the data, avoiding any bias that comes from modelling non linear transformations of data.

The models are applied to a data set consisting of ten US stocks. Out of sample forecasts are evaluated relative to a large number of benchmarks using a statistical and economic loss function. If intraday data is available, the statistical loss function reveals that a HEAVY-DCC model with fractional integration in realised correlations provides superior out of sample forecasts of the realised covariance matrix over 20–80 day horizons. Over 1–10 day horizons, the model provides comparable forecasts to a HEAVY-DCC specification with SM in correlations. When using the model to forecast the return covariance matrix, the gains over the longer term horizons are small. In contrast, if only daily data is available, the proposed LM-DCC process clearly provides superior forecasts of the return covariance matrix over 20–80 days relative to a SM-DCC process. Over 1–10 day forecast horizons, the LM-DCC and SM-DCC processes are comparable.

The difference in the return covariance matrix results can be explained by the innovation matrix used in the respective models. When intraday data is available, the lagged realised correlation matrix is used as the innovation matrix in the HEAVY-DCC model. This matrix has much higher explanatory power than the innovation matrix used in the DCC process (which is a function of daily standardised residual cross products). The higher explanatory power of the most recent innovation matrix, reduces the importance of the more distant innovations and therefore the gains from a LM process.

Results using the statistical loss function also support HEAVY-DCC models over a number of benchmarks, which highlights the benefits from conditioning on high frequency data, the flexibility of the DCC process, and the ability to avoid modelling non linear data transformations. Finally, results are mixed when forecasting weights for the minimum variance portfolio with no model clearly dominant. Even though the benefits from LM correlation models diminish completely, their performance is generally comparable to the best performing benchmarks.

Section 2 presents the proposed models. Section 3 applies the models to the data set consisting of ten large cap US stocks. Concluding remarks follow in Section 4.

Section snippets

Multivariate LM correlation models

This section presents two LM correlation models. Rather than model the covariance matrix (or some transformation of it) directly Chiriac and Voev (2011), Bauer and Vorkink (2011), both models disentangle the covariance matrix into its variance and correlation components. Following Engle (2002), the quasi maximum likelihood estimator is decomposed into the sum of the likelihoods from the univariate conditional variance equations and the correlation equation. This greatly simplifies estimation

Data and preliminary results

The proposed models are applied to a data set consisting of 10 large cap US equities: Apple Inc (AAPL), AT&T Inc (T), Pfizer Inc (PFE), Bank of America Corp (BAC), Walmart Stores Inc (WMT), Microsoft Corp (MSFT), Wells Fargo & Co (WFC), JPMorgan Chase & Co (JPM), Procter & Gamble (PG) and General Electric (GE). Intraday five minute returns from 9.30 am to 4 pm, as well as daily close to close and daily open to close returns are obtained from Thomson Reuters Tick History commencing January 2,

Conclusion

Multivariate models with long memory dependence in conditional correlations are proposed that modify the DCC process of Engle (2002) to allow for fractional integration. The models were applied to a data set consisting of ten US stocks and out of sample forecasts evaluated relative to a large number of benchmarks.

If intraday data is available, the QLIKE loss function found that the HEAVY-DCC model with LM in realised correlation provided superior forecasts of the realised covariance matrix over

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