Copula–Based vMEM Specifications versus Alternatives: The Case of Trading Activity
Abstract
:1. Introduction
2. A Copula Approach to Multiplicative Error Models
3. Maximum Likelihood Inference
3.1. Parameters in the Conditional Mean
3.2. Parameters in the pdf of the Error Term
 Normal copula: $ln{K}^{*}=0$, $ln{g}_{1}(x)=ln{g}_{2}(x)=x/2$;
 StudentT copula: $ln{K}^{*}=ln\left[\frac{\mathsf{\Gamma}((\nu +K)/2)\mathsf{\Gamma}{(\nu /2)}^{K1}}{\mathsf{\Gamma}((\nu +1)/2)}\right]$, $ln{g}_{1}(x)=\frac{\nu +K}{2}ln\left(1+\frac{x}{\nu}\right)$, ${g}_{2}(x)=\frac{\nu +1}{2}ln\left(1+\frac{x}{\nu}\right)$.
3.2.1. Expectation Targeting
3.2.2. Concentrated LogLikelihood
3.3. Asymptotic VarianceCovariance Matrix
4. Alternative Specifications of the Distribution of Errors
4.1. Multivariate Gamma
4.2. Multivariate Lognormal
 We compute logresiduals $ln{\widehat{\mathit{\epsilon}}}_{t}$ ($t=1,\dots ,T$), according to the first two steps of (13), at the current parameter estimates;
 We then estimate their variancecovariance matrix $\mathit{V}$;
 We estimate $\mathit{m}=0.5diag(\mathit{V})$;
 We use Equation (37) to update the estimate of $\mathit{\theta}$,
4.3. Semiparametric
5. Trading Activity and Volatility within a vMEM
 we take the log of the original series;
 we fit on each logseries a spline based regression with additive errors, using time t (a progressive counter from the first to the last observation in the sample) as an independent variable;5
 the residuals of the previous regression are then exponentiated to get the adjusted series.
5.1. Modeling Results
5.2. Forecasting
6. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
Appendix A. Expectation Targeting
Appendix A.1. Framework
Appendix A.2. Auxiliary results
Appendix A.3. The Asymptotic Distribution of the Sample Mean
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1  In some applications, especially involving returns, their elliptical symmetry may constitute a limit (cf., for example, Patton (2006), Okimoto (2008), Cerrato et al. (2015) and reference therein). Copulas in the Archimedean family (Clayton, Gumbel, JoeClayton, etc.) offer a way to bypass such a limitation but suffer from other drawbacks and, in any case, seem to be less relevant for the variables of interest for a vMEM (here, different indicators of trading activity). They will not be pursued in what follows. 
2  This is equivalent to variance targeting in a GARCH context (Engle and Mezrich 1995), where the constant term of the conditional variance model is assumed to be a function of the sample unconditional variance and of the other parameters. In this context, other than a preference for the term expectation targeting since we are modeling a conditional mean, the main argument stays unchanged. 
3  Even when $\mathit{R}$ is a $(2,2)$ matrix, the value of ${\mathit{R}}_{12}$ has to satisfy the cubic equation:
$$\begin{array}{c}\hfill {\mathit{R}}_{12}^{3}{\mathit{R}}_{12}^{2}\frac{{q}_{1}^{\prime}{q}_{2}}{T}+{\mathit{R}}_{12}\left[\frac{{q}_{1}^{\prime}{q}_{1}}{T}+\frac{{q}_{2}^{\prime}{q}_{2}}{T}1\right]\frac{{q}_{1}^{\prime}{q}_{2}}{T}=0.\end{array}$$

4  
5  Alternative methods, such as a moving average of fixed length (centered or uncentered), can be used but in practice they deliver very similar results and will not be discussed in detail here. The spline regression is estimated with the gam() function in the R package mgcv by using default settings. 
6  All models are estimated using Expectation Targeting (Section 3.2.1). The Normal copula based specifications are estimated resorting to the concentrated loglikelihood approach (Section 3.2.2). We omit estimates of the constant term $\omega $. 
7  The estimated degrees of freedom are $8.53$ (s.e. $1.16$) and $8.18$ (s.e. $1.05$), respectively, in the AT and ABT formulations. We also tried full ML estimation of the ABN specification getting a value of the loglikelihood equal to $2046.57$, very close to the concentrated loglikelihood approach (Section 3.2.2) used in Table 3. 
8  The stark improvement in the likelihood functions, coming from the explicit consideration of the correlation structure, does not guarantee similar gains in the forecasting ability of the same variables. This is similar to what happens in modeling returns, when the likelihood function of ARMA models is improved by superimposing a GARCH structure on the conditional variances: no substantive change in the fit of the conditional mean and no better predictive ability. 
9  No attempt at pruning the structure of the model following the automated procedure suggested in Cipollini and Gallo (2010) was performed. 
10  Calculations with our routines written in R were performed on an Intel i75500U 2.4Ghz processor. We did not perform an extensive comparison of estimation times and we did not optimize performance by removing tracing of intermediate results. Moreover, the copulabased and the alternative specifications are optimized resorting to different algorithms: the first ones, more cumbersome to optimize, are estimated toward a combination of NEWUOA (Powell 2006) and NewtonRaphson; for the second ones we used a dogleg algorithm (Nocedal and Wright 2006, ch. 4). 
11  Onestep ahead predictions at time t for the original series are computed multiplying the corresponding forecast of the adjusted indicator by the value of the low frequency component at $t1$. 
Original  Low Freq. Comp.  Adjusted  

$vol$  $nt$  $vol$  $nt$  $vol$  $nt$  
$rkv$  0.646  0.778  0.800  0.847  0.609  0.743 
$vol$  0.890  0.780  0.932 
Conditional Mean (Parameters)  Error Distribution  

I  N  T  $\mathit{LN}$  S  
D: ${\mathit{\alpha}}_{1}$, ${\mathit{\beta}}_{1}$, ${\mathit{\gamma}}_{1}$, , ${\mathit{\alpha}}_{2}$ diagonal  DI  
A: ${\mathit{\alpha}}_{1}$ full; ${\mathit{\beta}}_{1}$, ${\mathit{\gamma}}_{1}$, ${\mathit{\alpha}}_{2}$ diagonal  AI  AN  AT  
B: ${\mathit{\beta}}_{1}$ full; ${\mathit{\alpha}}_{1}$, ${\mathit{\gamma}}_{1}$, ${\mathit{\alpha}}_{2}$ diagonal  BN  BT  
$AB$: ${\mathit{\alpha}}_{1}$, ${\mathit{\beta}}_{1}$ full; ${\mathit{\gamma}}_{1}$, ${\mathit{\alpha}}_{2}$ diagonal  ABN  ABT  ABLN  ABS 
DI  AI  AN  AT  BN  BT  ABN  ABT  ABLN  ABS  

$rk{v}_{t1}$  $0.4967$  $0.4575$  $0.4194$  $0.4165$  $0.4048$  $0.4019$  $0.3687$  $0.3724$  $0.3630$  $0.3777$ 
($\mathit{58.10}$)  ($\mathit{51.94}$)  ($\mathit{48.57}$)  ($\mathit{48.72}$)  ($\mathit{45.25}$)  ($\mathit{47.42}$)  ($\mathit{39.13}$)  ($\mathit{40.77}$)  ($\mathit{50.76}$)  ($\mathit{45.77}$)  
$vo{l}_{t1}$  $0.0300$  $0.0339$  $0.0265$  $0.0230$  $0.0337$  $0.0318$  $0.0298$  
($\mathit{1.11}$)  ($\mathit{0.97}$)  ($\mathit{0.75}$)  ($\mathit{0.54}$)  ($\mathit{0.78}$)  ($\mathit{0.93}$)  ($\mathit{0.75}$)  
$n{t}_{t1}$  $0.0668$  $0.0726$  $0.0606$  $0.1897$  $0.1890$  $0.1691$  $0.1726$  
($\mathit{1.77}$)  ($\mathit{1.57}$)  ($\mathit{1.36}$)  ($\mathit{2.83}$)  ($\mathit{2.82}$)  ($\mathit{3.95}$)  ($\mathit{3.51}$)  
$rk{v}_{t2}$  $0.2672$  $0.2485$  $0.1878$  $0.1923$  $0.1950$  $0.2054$  $0.1778$  $0.1827$  $0.1811$  $0.1790$ 
($\mathit{5.12}$)  ($\mathit{3.82}$)  ($\mathit{2.91}$)  ($\mathit{3.54}$)  ($\mathit{4.64}$)  ($\mathit{5.45}$)  ($\mathit{3.75}$)  ($\mathit{4.19}$)  ($\mathit{5.46}$)  ($\mathit{4.64}$)  
$rk{v}_{t1}^{()}$  $0.0252$  $0.0266$  $0.0265$  $0.0308$  $0.0255$  $0.0297$  $0.0171$  $0.0223$  $0.0211$  $0.0222$ 
($\mathit{0.79}$)  ($\mathit{0.79}$)  ($\mathit{0.83}$)  ($\mathit{1.00}$)  ($\mathit{0.94}$)  ($\mathit{1.13}$)  ($\mathit{0.41}$)  ($\mathit{0.53}$)  ($\mathit{0.73}$)  ($\mathit{0.67}$)  
${\mu}_{t1}^{(rkv)}$  $0.7223$  $0.7067$  $0.6793$  $0.6918$  $0.7235$  $0.7410$  $0.7588$  $0.7675$  $0.7626$  $0.7428$ 
($\mathit{14.19}$)  ($\mathit{9.94}$)  ($\mathit{9.49}$)  ($\mathit{11.41}$)  ($\mathit{17.23}$)  ($\mathit{20.73}$)  ($\mathit{14.13}$)  ($\mathit{16.46}$)  ($\mathit{19.22}$)  ($\mathit{15.50}$)  
${\mu}_{t1}^{(vol)}$  $0.0317$  $0.0139$  $0.1138$  $0.0466$  $0.0909$  $0.0938$  
($\mathit{0.68}$)  ($\mathit{0.32}$)  ($\mathit{0.96}$)  ($\mathit{0.44}$)  ($\mathit{0.95}$)  ($\mathit{0.86}$)  
${\mu}_{t1}^{(nt)}$  $0.0492$  $0.0297$  $0.0984$  $0.1568$  $0.0867$  $0.0839$  
($\mathit{0.99}$)  ($\mathit{0.66}$)  ($\mathit{0.76}$)  ($\mathit{1.26}$)  ($\mathit{0.87}$)  ($\mathit{0.73}$)  
$vol$  0.2658  0.3296  0.4558  0.4970  0.7459  0.1705  0.2717  0.0853  0.1550  
$nt$  0.0766  0.1153  0.1724  0.3202  0.5098  0.0051  0.0077  0.0001  0.0004  
logLik  205.53  239.75  2012.39  2086.31  1967.02  2048.80  2045.81  2125.12  2153.79  
AIC  −375.07  $431.49$  $3970.78$  −4116.61  −3880.05  −4041.60  $4025.62$  $4182.23$  $4241.57$  
BIC  $258.11$  $275.55$  $3795.35$  $3934.69$  $3704.62$  $3859.67$  −3811.20  −3961.32  $4027.16$  
LB(12)  0.0000  0.0000  0.0001  0.0001  0.0000  0.0000  0.0388  0.0141  0.0450  0.0596 
LB(22)  0.0000  0.0000  0.0000  0.0000  0.0000  0.0000  0.0178  0.0109  0.0229  0.0264 
LB(32)  0.0000  0.0000  0.0003  0.0003  0.0000  0.0000  0.0192  0.0132  0.0291  0.0267 
time  0.342  0.753  2.628  15.044  2.485  14.707  2.714  14.859  0.155  0.132 
DI  AI  AN  AT  BN  BT  ABN  ABT  ABLN  ABS  

${\sigma}_{1}$  0.209  0.208  0.207  0.212  0.209  0.214  0.206  0.210  0.204  0.226 
${\sigma}_{2}$  0.254  0.251  0.252  0.256  0.257  0.262  0.252  0.256  0.252  0.272 
${\sigma}_{3}$  0.227  0.227  0.229  0.236  0.232  0.240  0.228  0.235  0.228  0.242 
AN  AT  ABN  ABT  ABLN  ABS  

${\mathit{vol}}_{\mathit{t}}$  ${\mathit{nt}}_{\mathit{t}}$  ${\mathit{vol}}_{\mathit{t}}$  ${\mathit{nt}}_{\mathit{t}}$  ${\mathit{vol}}_{\mathit{t}}$  ${\mathit{nt}}_{\mathit{t}}$  ${\mathit{vol}}_{\mathit{t}}$  ${\mathit{nt}}_{\mathit{t}}$  ${\mathit{vol}}_{\mathit{t}}$  ${\mathit{nt}}_{\mathit{t}}$  ${\mathit{vol}}_{\mathit{t}}$  ${\mathit{nt}}_{\mathit{t}}$  
$rk{v}_{t}$  0.483  0.610  0.505  0.626  0.485  0.612  0.505  0.628  0.469  0.596  0.481  0.605 
$vo{l}_{t}$  0.902  0.917  0.903  0.918  0.902  0.906 
Specification  ${\mathit{e}}_{\mathit{N},\mathit{t}}$  ${\mathit{e}}_{\mathit{G},\mathit{t}}$  

Adjusted  Original  Adjusted  Original  
DI  −2.490  −2.107  −2.428  −2.132 
AI  −1.358  −1.200  −1.277  −1.173 
AN  $0.668$  −0.621  −0.563  −0.591 
AT  $0.510$  −0.449  −0.422  −0.431 
BN  $0.553$  −0.557  $0.449$  −0.560 
BT  $0.474$  −0.479  −0.389  −0.490 
ABN  $0.768$  −0.757  $0.737$  −0.708 
ABLN  0.868  0.720  0.980  0.742 
ABS  $0.277$  $0.230$  $0.180$  $0.164$ 
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Cipollini, F.; Engle, R.F.; Gallo, G.M. Copula–Based vMEM Specifications versus Alternatives: The Case of Trading Activity. Econometrics 2017, 5, 16. https://doi.org/10.3390/econometrics5020016
Cipollini F, Engle RF, Gallo GM. Copula–Based vMEM Specifications versus Alternatives: The Case of Trading Activity. Econometrics. 2017; 5(2):16. https://doi.org/10.3390/econometrics5020016
Chicago/Turabian StyleCipollini, Fabrizio, Robert F. Engle, and Giampiero M. Gallo. 2017. "Copula–Based vMEM Specifications versus Alternatives: The Case of Trading Activity" Econometrics 5, no. 2: 16. https://doi.org/10.3390/econometrics5020016