Large dimensional portfolio allocation based on a mixed frequency dynamic factor model

Abstract

In this paper, we propose a mixed-frequency dynamic factor model (MFDFM) taking into account the high-frequency variation and low-frequency variation at the same time. The factor loadings in our model are affected by the past quadratic variation of factor returns, while the process of the factor quadratic variation is under a mixed-frequency framework (DCC-RV). By combing the variations from the high-frequency and low-frequency domain, our approach exhibits a better estimation and forecast of the assets covariance matrix. Our empirical study compares our MFDFM model with the sample realized covariance matrix and the traditional factor model with intraday returns or daily returns. The results of the empirical study indicate that our proposed model indeed outperforms other models in the sense that the Markowitz’s portfolios based on the MFDFM have a better performance.

Keywords:

• Dynamic structure
• high frequency regression
• Markowitz’s portfolio allocation
• mixed-frequency factor models
• volatility forecasting

Notes

1 We exclude jumps from the price dynamics and volatility process in order not to complicate the model and technicality. We believe that applying the truncation technique in Jing et al. (2014) and Pelger (2019) on our returns could help remove jumps. In our empirical study below we use the bipower variation to construct the threshold, see Subsection 4.1.

2 This condition is an assumption to ensure the central limit theorem for quadratic variation, see Jacod and Protter (2011, Section 5.4). There is another condition to derive the asymptotic distribution of quadratic variation, see Barndorff-Nielsen and Shephard, (2004). Here we use the former assumption since it is weaker than the latter and could include more processes.

3 A quadratic covariation for semimartingale X_t and Y_t is defined as $${\left[X,Y\right]}_t=\underset{\left|\right|P\left|\right|\to 0}{\lim }{\sum }_{k=1}^M(X_{t_k}-X_{t_{k-1}})(Y_{t_k}-Y_{t_{k-1}}),$$ where P ranges over partitions of the interval $[0,t]$ and the norm of the partition P is mesh, i.e. sup$|t_k-t_{k-1}|\to 0.$ Thus the quadratic (co)variation can be thought of as the sum of outer products of return vectors computed over infinitesimal time intervals calculated during the period from time 0 up to time t. $〈·,·〉\mathrm{}$ is the predictable compensator of quadratic variation process (so the difference of both is local martingale), and in the case when X_t and Y_t are both continuous, we have ${〈X,Y〉}_t$ = ${\left[X,Y\right]}_t.$

4 The invertibility in the formula of $\mathit{b}$ is discussed as below. Traditionally in the high-frequecy factor model, only the invertibility of ${\mathit{c}}_{F,t}$ is required. However even if ${\mathit{c}}_{F,t}$ is invertible, ${\tilde{\mathit{c}}}_{F,t}={\mathit{\gamma }}_{\lfloor t/T\rfloor }{\mathit{c}}_{F,t}{\mathit{\gamma }}_{\lfloor t/T\rfloor }^{\prime }$ would still be singular. Actually, if we only consider one day process, ${\int }_{lT}^{(l+1)T}{\tilde{\mathit{c}}}_{F,s}ds={\mathit{\gamma }}_l{\int }_{lT}^{(l+1)T}{\mathit{c}}_{F,s}ds{\mathit{\gamma }}_l^{\prime }$ is also singular since ${\mathit{\gamma }}_{\lfloor t/T\rfloor }$ is the same when $t\in [lT,(l+1)T].$ Therefore to ensure the invertibility we here consider the whole process across all trading days so that the assumption of the invertibility of ${\int }_T^{LT}{\tilde{\mathit{c}}}_{F,s}ds$ is reasonable.

5 Here we assume the observable factors have captured the most of co-movements of assets, and thus ${\mathbf{\Sigma }}_{\epsilon }$ is assumed to be diagonal. Actually ${\widehat{\mathbf{\Sigma }}}_{\epsilon }$ is estimated by averaging the estimated realized variance of residuals for each asset.

6 Here, T is fixed since it denotes the intraday trading interval length, which is obvious constant. And since $\delta =T/M,\delta \to 0$ indicates $M\to \infty .$

7 The existence of ${\mathbf{\Lambda }}_l$ is assured by the existence of the asymptotic covariance of ${\widehat{\mathbf{\Sigma }}}_{F,l}$ since ${\widehat{\mathit{\gamma }}}_{l-1}={\mathit{I}}_q\otimes \text{vech}({\widehat{\mathbf{\Sigma }}}_{F,l-1}).$