A note on the co-moments in the IFACD model

The Independent Factor Autoregressive Conditional Density (IFACD) model of Ghalanos, Rossi and Urga (2014) uniquely, in its class of parametric models, generates time varying higher co-moment forecasts, as a consequence of the ACD specification of the conditional density of the standardized innovations. In this short note I discuss in more detail the properties of the conditional co-moments of this model, certain interesting properties as relates to the higher moment CAPM and a fast algorithm for populating these very large flattened tensors.

Conditional Co-Moments

The conditional co-moments of \( \mathbf{r}_t \) of order 3 and 4 are represented as tensor matrices
\[ \label{eq1}
\mathbf{M}_{t}^3 = \mathbf{A} \mathbf{M}_{f,t}^3(\mathbf{A} \otimes \mathbf{A})’, \quad \\
\mathbf{M}_{t}^4 = \mathbf{A} \mathbf{M}_{f,t}^4(\mathbf{A} \otimes \mathbf{A} \otimes \mathbf{A})’
\]
where \( \mathbf{M}_{f,t}^3 \) and \( \mathbf{M}_{f,t}^4 \) are the \( N \times N^2 \) conditional third comoment matrix and the \( N \times N^3 \) conditional fourth comoment matrix of the factors, respectively. \( \mathbf{M}_{f,t}^3 \) and \( \mathbf{M}_{f,t}^4 \), are defined as
\[ \begin{eqnarray}
\mathbf{M}_{f,t}^3 & =&
\begin{bmatrix}
\mathbf{M}_{1,f,t}^3,\mathbf{M}_{2,f,t}^3,\ldots,\mathbf{M}_{N,f,t}^3
\end{bmatrix}\\
\mathbf{M}_{f,t}^4 & = &
\begin{bmatrix}
\mathbf{M}_{11,f,t}^4,\mathbf{M}_{12,f,t}^4,\ldots,\mathbf{M}_{1N,f,t}^4|\ldots|\mathbf{M}_{N1,f,t}^4,\mathbf{M}_{N2,f,t}^4,\ldots,\mathbf{M}_{NN,f,t}^4
\end{bmatrix}\\
\end{eqnarray}
\]
where \( \mathbf{M}_{k,f,t}^3, k=1,\ldots,N \) and \( \mathbf{M}_{kl,f,t}^4, k,l=1,\ldots,N \) are the \( N\times N \) submatrices of \( \mathbf{M}_{f,t}^3 \) and \( \mathbf{M}_{f,t}^4 \), respectively, with elements
\[
\begin{eqnarray*}
m_{ijk,f,t}^3 & = & E[f_{i,t}f_{j,t}f_{k,t}|\mathfrak{F}_{t-1}] \\
m_{ijkl,f,t}^4 & = & E[f_{i,t}f_{j,t}f_{k,t}f_{l,t}|\mathfrak{F}_{t-1}].
\end{eqnarray*}
\]
Since the factors \( f_{it} \) can be decomposed as \( z_{it}\sqrt{h_{it}} \), and given the assumptions on \( z_{it} \), then \( E[f_{i,t}f_{j,t}f_{k,t}|\mathfrak{F}_{t-1}] = 0\ \). It is also true that for \( i \neq j\neq k \neq l \), then \( E[f_{i,t}f_{j,t}f_{k,t}f_{l,t}|\mathfrak{F}_{t-1}] = 0 \) and when \( i=j \) and \( k=l \), then \( E[f_{i,t}f_{j,t}f_{k,t}f_{l,t}|\mathfrak{F}_{t-1}] = h_{it}h_{kt}. \)
Thus, under the assumption of mutual independence, all elements in the conditional co-moments matrices with at least 3 different indices are zero. Finally, we standardize the conditional co-moments to obtain conditional coskewness and cokurtosis of \( \mathbf{r}_t \)
\[
\mathbf{S}_{ijk,t} = \frac{m_{ijk,t}^3}{({\sigma_{i,t}}{\sigma _{j,t}}{\sigma _{k,t}})}, \quad \\
\mathbf{K}_{ijkl,t} = \frac{m_{ijkl,t}^4}{({\sigma_{i,t}}{\sigma _{j,t}}{\sigma _{k,t}}{\sigma _{l,t}})},
\]
where \( \mathbf{S}_{ijk,t} \) represents the coskewness between elements \( i,j,k \) of \( \mathbf{r}_t \), \( \sigma_{i,t} \) the standard deviation of \( \mathbf{r}_{i,t} \), and in the case of \( i=j=k \) represents the skewness of asset \( i \) at time \( t \), and similarly for the cokurtosis tensor \( \mathbf{K}_{ijkl,t} \)

Location of non-zero entries

The flattened tensors grow quite quickly in size as n (factors) and m (moment) become larger. Populating the factor matrices with the values from the ACD dynamics in a fast and efficient manner is key if this model is to be called ‘feasible’ and estimation ‘large-scale’. For the third co-moment matrix, this is very simply since the column based vectorized index of the location of non-zero entries is found to be \( ((1:n)-1)n^2 + ((1:n)-1)n + (1:n) \).
These represent the (column based vectorized) location in the third co-moment matrix of the factor unstandardized skewness estimated from the ACD model (and determined jointly by the skew and shape dynamics). In the case of the fourth co-moment matrix of the factors, the situation is slightly more involved since in addition to the entries \( \{i=j=k=l\} \) for which the column based vectorized location is found to be \( ((1:n)-1)n^3 + ((1:n)-1)n^2 + ((1:n)-1)n + (1:n) \), there are also the entries \( \{i=j,k=l\} \) to consider as discussed in the previous section. To this end, consider the \( n\times n^3 \) matrix of the flattened tensor \( M_{f,ijkl}^4 \), with the indices illustrated as in the \( 4\times4^3 \) matrix below:

\[ \small
\begin{array}{*{14}{c}}
{j:} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & {…} & 4 & 4 \\
{k:} & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & {…} & 4 & 4 \\
{l:} & 1 & 2 & 3 & 4 & 1 & 2 & 3 & 4 & {…} & 3 & 4 \\
{i:1} & {\{ 1111\} } & {\{ 1112\} } & {\{ 1113\} } & {\{ 1114\} } & {\{ 1121\} } & {\{ 1122\} } & {\{ 1123\} } & {\{ 1124\} } & {…} & {\{ 1443\} } & {\{ 1444\} } \\
{i:2} & {\{ 2111\} } & {\{ 2112\} } & {\{ 2113\} } & {\{ 2114\} } & {\{ 2121\} } & {\{ 2122\} } & {\{ 2123\} } & {\{ 2124\} } & {…} & {\{ 2443\} } & {\{ 2444\} } \\
{i:3} & {\{ 3111\} } & {\{ 3112\} } & {\{ 3113\} } & {\{ 3114\} } & {\{ 3121\} } & {\{ 3122\} } & {\{ 3123\} } & {\{ 3124\} } & {…} & {\{ 3443\} } & {\{ 3444\} } \\
{i:4} & {\{ 3111\} } & {\{ 4112\} } & {\{ 4113\} } & {\{ 4114\} } & {\{ 3121\} } & {\{ 4122\} } & {\{ 4123\} } & {\{ 4124\} } & {…} & {\{ 4443\} } & {\{ 4444\} } \\
\end{array}
\normalsize
\]
We are interested in finding the index location of all the pairs where \( \{i=j,k=l\} \) and their permutations, e.g. {1122},{1212},{2211},{2112},{1221},{2121}.
To do this, we first note the following:

• There are \( \frac{n!}{2(n-2)!} \) unique pairs.
• Each pair has 6 permutations.
• The number of unique pairs whose difference is \( d \) is \( n-d \) e.g. for \( d=1 \),\( n=4 \), we have 3 unique pairs {2,1},{3,2} and {4,3}.
• The first pair has columnwise vector based index \( v=n+2 \) e.g. for \( n=4 \), the first pair is {2112} in the example matrix above with columnwise vector based index of 6.
• The \( p^{th} \) unique pair, representing the first in the set of pairs with \( n-d \) differences, has vector based index \( (p-1)+(n+1) \), which given the starting pair with index \( n+2 \) means that we can calculate the indices of each unique pair in the matrix given the previous unique pair’s position.
• Within each \( p^{th} \) unique pair, there are 6 permutations. Starting from the index of the first pair in the series of 6, denoted \( v_1 \), the second permutation has index \( v_1+((n-1)n)d \), the third \( v_2+(n-1)d \), the fourth \( v_3+d(n+1)(n-1)^2 \), the fifth \( v_4+(n-1)d \) and the sixth \( v_5+((n-1)n)d \).
• The first pair of the next set of pairs whose difference is \( n-d \) has and index which is equal to \( n^3+n^2+n+1 \) more than the previous pair.

This fast method for calculating the location of each entry and populating it accordingly is already implemented in the rmgarch package for the GO-GARCH (NIG/GH) model. In addition, in order to avoid memory problems when it comes time to perform the kronecker multiplications using the mixing matrix \( \mathbf{A} \) on the factor higher co-moments to arrive at the asset higher co-moments, the method of Buis amd Dyksen (1996) is used which is also implemented in the klin package.

The Higher Moment Time Varying Statistical Factor CAPM

A very interesting application made possible as a result of the model’s properties is the estimation of the (higher moment) time varying betas from the CAPM model. Consider a universe of \( n \) assets with returns \( \mathbf{r}_t \), with benchmark \( b \) whose return is \( r_{b,t} \) and determined by a linear combination based on a pre-specified weighting vector \( \mathbf{w}_t \):
\[
r_{b,t} = \mathbf{w}_t’ \mathbf{r}_t
\]
such that \( \mathbf{w}_t’\mathbf{1}=1 \). It could be the case that \( \mathbf{w}_t \) represents the weights of the benchmark index or some other pre-deteremined weighting scheme (e.g. equal weight). To calculate the CAPM betas with respect to the benchmark, we need not model the pairwise combination of assets-benchmark but instead only the assets which comprise the benchmark. The following formulae outline the steps:

• Define the conditional covariance: \( M_t^2=AH_tA’ \), where \( {{\mathbf{H}}_{\mathbf{t}}} = E\left[ {{f_t}{{f’}_t}{{\left| \Im \right.}_{t – 1}}} \right], \in {\mathbb{R}^{N \times N}} \)
• Define the conditional 3rd co-moment: \( {\mathbf{M}}_{\mathbf{t}}^3{\mathbf{ = AM}}_{{\mathbf{f,t}}}^{\mathbf{3}}\left( {{\mathbf{A’}} \otimes {\mathbf{A’}}} \right), \in {\mathbb{R}^{N \times {N^2}}} \)
• Define the conditional 4th co-moment: \( {\mathbf{M}}_{\mathbf{t}}^4{\mathbf{ = AM}}_{{\mathbf{f,t}}}^4\left( {{\mathbf{A’}} \otimes {\mathbf{A’}} \otimes {\mathbf{A’}}} \right), \in {\mathbb{R}^{N \times {N^3}}} \)

The conditional beta Covariance:
\[
{\beta _{i,t}} = \frac{{E\left[ {\left( {{R_{i,t}} – {{\bar R}_{i,t}}} \right)\left( {{R_{m,t}} – {{\bar R}_{m,t}}} \right)} \right]}}
{{E\left[ {{{\left( {{R_{m,t}} – {{\bar R}_{m,t}}} \right)}^2}} \right]}} = \frac{{{\mathbf{m}}_{i,t}^2{\mathbf{w}}}}
{{{\mathbf{w’M}}_{\mathbf{t}}^{\mathbf{2}}{\mathbf{w}}}}
\]
where \( {\mathbf{m}}_{i,t}^2: = ro{w_i}{\mathbf{M}}_{\mathbf{t}}^{\mathbf{2}}\left( {i = 1,…,N} \right) \).

The conditional beta Coskewness:
\[
{s_{i,t}} = \frac{{E\left[ {\left( {{R_{i,t}} – {{\bar R}_{i,t}}} \right){{\left( {{R_{m,t}} – {{\bar R}_{m,t}}} \right)}^2}} \right]}}
{{E\left[ {{{\left( {{R_{m,t}} – {{\bar R}_{m,t}}} \right)}^3}} \right]}} = \frac{{{\mathbf{m}}_{i,t}^3\left( {{\mathbf{w}} \otimes {\mathbf{w}}} \right)}}
{{{\mathbf{w’M}}_{\mathbf{t}}^{\mathbf{3}}\left( {{\mathbf{w}} \otimes {\mathbf{w}}} \right)}}
\]
where \( {\mathbf{m}}_{i,t}^3: = ro{w_i}{\mathbf{M}}_{\mathbf{t}}^3\left( {i = 1,…,N} \right) \).

The conditional beta Cokurtosis:
\[
{{\text{k}}_{i,t}} = \frac{{E\left[ {\left( {{R_{i,t}} – {{\bar R}_{i,t}}} \right){{\left( {{R_{m,t}} – {{\bar R}_{m,t}}} \right)}^3}} \right]}}
{{E\left[ {{{\left( {{R_{m,t}} – {{\bar R}_{m,t}}} \right)}^4}} \right]}} = \frac{{{\mathbf{m}}_{i,t}^4\left( {{\mathbf{w}} \otimes {\mathbf{w}} \otimes {\mathbf{w}}} \right)}}
{{{\mathbf{w’M}}_{\mathbf{t}}^4\left( {{\mathbf{w}} \otimes {\mathbf{w}} \otimes {\mathbf{w}}} \right)}}
\]
where \( {\mathbf{m}}_{i,t}^4: = ro{w_i}{\mathbf{M}}_{\mathbf{t}}^4\left( {i = 1,…,N} \right) \)

Thus, considering that we can include dimensionality reduction at the PCA whitening stage (the n.comp argument in the rmgarch::fastica algorithm), the IFACD (and the non-time varying higher moments restricted GO-GARCH sub-model) provides a clear avenue for estimating a large scale statistical factor based time varying higher moment CAPM model. A demonstration is available here.

Fast weighted VaR calculation using the Cornish-Fisher Expansion

In the IFACD model, the conditional weighted density can be calculated using the inversion of the NIG/GH characteristic function via FFT as discussed in Ghalanos, Rossi and Urga (2014) and the rmgarch vignette (for the GO-GARCH model). Methods for working with the weighted density are already available in the rmgarch package including methods for the density, distribution and quantile (dfft, pfft and qfft) on estimated, forecast and simulated objects. An alternative avenue for the calculation of the weighted quantile is to use the Cornish-Fisher expansion which makes use of the conditional higher moments:
\[
Va{R_{t,\alpha }} = {\mu _t} + {\sigma _t}\left( {\phi + (1 – {\phi ^2})\frac{{{S_t}}}
{6} + \left( {{\phi ^3} – 3\phi } \right)\frac{{{K_t}}}
{{24}} + \left( {5\phi – 2{\phi ^3}} \right)\frac{{S_t^2}}
{{36}}} \right)
\]
where \( \phi={\Phi ^{ – 1}}\left( \alpha \right) \), represents the quantile of the standard normal distribution evaluated at the coverage level \( \alpha \), \( S_t \) the skewness at time \( t \) and \( K_t \) the excess kurtosis at time \( t \). The weighted moments \( (\mu_t,\sigma_t,S_t,K_t) \) can be calculated as follows:
\[ \begin{gathered}
\mu_t = \mathbf{w}_t’ \mathbf{M_{t}^1},\\
\sigma_{{t}}^2 = \mathbf{w}_t'{\mathbf{\Sigma}_{t}}\mathbf{w}_t, \\
{S_{t}} = \frac{{\mathbf{w}_t’\mathbf{M}_{_t}^3(\mathbf{w}_t \otimes \mathbf{w}_t)}}{{{{(\mathbf{w}_t'{\mathbf{\Sigma}_{t}}\mathbf{w}_t)}^{3/2}}}}, \\
{K_{t}} = \frac{{\mathbf{w}_t’\mathbf{M}_{_t}^4(\mathbf{w}_t \otimes \mathbf{w}_t \otimes \mathbf{w}_t)}}{{{{(\mathbf{w}_t'{\mathbf{\Sigma}_{t}}\mathbf{w}_t)}^2}}}, \\
\end{gathered}
\]
where \( \mathbf{M}_{t}^1 \) is the conditional mean vector, \( \mathbf{M}_{t}^3 \) and \( \mathbf{M}_{t}^4 \) are the third and fourth co-moment matrices described in the previous section, and \( \mathbf{\Sigma}_{t} \) the conditional covariance (the notation \( \mathbf{M}_{t}^2 \) has also been used).
A demonstration is available here.

References

Buis, P. E., & Dyksen, W. R. (1996). Efficient vector and parallel manipulation of tensor products. ACM Transactions on Mathematical Software (TOMS), 22(1), 18-23.

Ghalanos, A., Rossi, E. & Urga G. (2014). Independent Factor Autoregressive Conditional Density model. Econometric Reviews (forthcoming).