Paper ID: | 2122 |
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Title: | Cross-sectional Learning of Extremal Dependence among Financial Assets |

-Did you run any experiments exploring the efficiency of the proposed fitting procedure? For a known model, how many samples are needed to reliably estimate the up/down tail dependence parameters? How does this vary with the strength of the dependence parameters (both rho and u/v)? -Can you provide more detail on the proposed fitting procedure. How many probability levels do you consider in the quantile regression? After estimating the parameters of y1, you invert g() to draw samples of z1, which are then used to estimate the parameters of y2, etc... It seems that the estimation errors will compound, making the estimate of the parameters of yn incredible noisy. Is it possible to remove some of the estimation error by proceeding along a different variable ordering? -Please define/address the constant A in eqn 2/3. -Why are u and v required to be greater than 1 in eqn 3? -How are the ideal number of violations in table 1 calculated?

The authors proposed a new statistical model for heavy tailed random vectors. They model the heavy tailed random vector as a known random vector transformed by a new quantile function. The model not only have parameters controlling the marginal tail heaviness, but also capture the tail dependence by separate parameters from the correlation parameters. To fit the model, the authors proposed an algorithm that recursively fits moment estimators and quantile regressions. The authors also combined the proposed model with a GARCH filter to describe the conditional distribution of financial asset returns, and demonstrated its superior performance by coverage tests in modeling historical stock return data. Quality: The technical content of the paper looks sound to me. The proposed model and fitting algorithm make intuitive sense, and the characterization of tail dependence is well demonstrated by simulation (Figure 1). Clarity: I find the methodology of the paper well presented with intuitive explanations. The experimental results are easy to understand as well in general, though a few implementation details could be revealed to improve clarity (see #3 and #4 below). Originality: The main contribution of the paper seems to be the generative model based on transformation by quantile functions, which captures correlations, tail heaviness, and tail dependence by separate parameters. The new quantile function seems novel, but supporting evidence justifying its advantage over existing ones seems to be missing (see #1 below). Coupling the GARCH-type models with heavy tailed models for the innovation terms is not new, though the experiment in Section 5.1 seems to show superior performance of the proposed model over common heavy tailed models. Significance: The paper makes relevant contribution on the methodology for modeling tail dependence. Based on the good simulation results, it looks promising for the methodology to be applied to modeling financial asset returns. On the other hand, one weakness of the paper is a lack of theoretical results on the proposed methodology. Most of the benefits of the new model have been demonstrated by simulations. It would be very helpful if the authors could provide some theoretical insights on the relation between the model parameters and the tail dependence measures, and on the performance (consistency, efficiency etc) of the parameter estimators. Itemized comments: 1. The advantage of the new quantile function (3) compared to the existing function (2) seems unjustified. Compared with (2), (3) changes the multiplicative factors containing the up and down tail parameters into an additive term. While this makes the function less sensitive to the tail parameters when they are large, the paper does not present supporting data on why the reduced sensitivity is desired. 2. On Line 132, the authors concluded that v_{ij} determines mainly the down-tail dependence of y_i and y_j. For any 1 <= k < j, does v_{ik} also have similar interpretation as v_{ij}? For example, in Equation (4), by symmetry, v_{31} and v_{32} seems to have similar effect on the tail dependence between y_3 and y_2. 3. In Algorithm 1 on Page 5, \Psi (the set of \tau's in Equation (7)) should also be an input parameter of the algorithm. Moreover, since it determines which quantiles are estimated in the loss function, I'd expect it to have notable effect on the results. I think it would be helpful to discuss how \Psi was chosen in the experiments, and provide some guidance on its choice in general. 4. Equation (13) doesn't seem to have closed form solution in general. Some details about how it's solved in the experiments and on the computational complexity would be helpful. 5. In addition to the up and down tail dependences, how could we also model negative tail dependence, e.g., P(X < Q_X(t), Y > Q_Y(1 - t)) / t? This is the counterpart of negative correlations, and is also notably common in financial asset returns (e.g., when money flow from one asset class (e.g., stocks) another (e.g., bonds)). Minor comments: 1. In Figures 2 and 3, it may be clearer to see the fitting errors if we overlay the oracle and the fitted lines in the same plot. Update: Thanks to the authors for the feedback. I believe Items 2 and 5 above are well addressed. On the other hand, as pointed out by another reviewer as well, a lack of theoretical results still seems to be the main weakness of the paper, though I agree that due to the complexity of the learning procedure, an extensive theoretical analysis would be a luxury at this stage.

The paper proposes a novel way to model correlations separately from tail dependence. The model comes in two flavors: lower-triangular model (an extension of the Gaussian vector factorization case) and one-factor tail dependence model. The authors demonstrate an algorithm to learn model parameters, and show through simulations its flexibility to model tail dependence and, combined with GARCH-type models, to forecast the conditional distribution of multiple asset returns. Originality I believe this is a novel approach to model tail dependency. Existing approaches such as the copula approach and elliptical distributions are compared. Quality The proposed model lacks theoretical analysis, e.g. its consistency properties but simulations are provided to ground the method. Note also that the model comes with great flexibility to model tail dependency but at the same time allows more degrees of freedom to overfit. I would be interested in a way to cope with this problem. Clarity The paper is quite well-written, symbols are clearly defined, and the algorithm is clearly specified. Significance This paper proposes a new way to address an important problem of tail dependence model with some evidence of method soundness