This function fits the multivariate models within the chain ladder framework. Corresponding to the model argument, there are two major models that are incorporated into this function. One is the Multivariate Chain Ladder (MCL) model proposed by Prohl and Schmidt (2005), which is characterized by a diagonal development matrix, allowing multiple lines to be developed together while reflecting the correlations among lines. The other is a natural generalization of the MCL model, the General Multivariate Chain Ladder (GMCL) model proposed by Zhang (2010), which has a non-diagonal development matrix and intercepts, and can be used to develop structurally related triangles, such as paid and incurred or paid and case reserve, as well as contemporaneously related ones. The MCL model is a sub-model of GMCL, but it is programmed separately because: a) its stand-alone importance; b) different MSE methods are only available for the MCL model; c) extrapolation is not allowed for GMCL. Some technical details about the GMCL model. Assume N triangles are available. Denote $Y_{i,k}=(Y^{(1)}_{i,k}, \ldots ,Y^{(N)}_{i,k})$ as an $N \times 1$ vector of cumulative losses at accident year i and development year k where (n) refers to the n-th triangle. The GMCL model in development period k is:
$$Y_{i,k+1}=A_k + B_k \cdot Y_{i,k}+\epsilon_{i,k},$$
where $A_k$ is a column of intercepts and $B_k$ is the usual development matrix. By default, MultiChainLadder sets $A_k$ to be zero, but one can specify a model with intercepts using the int argument.
Assumptions for this model are:
$$E(\epsilon_{i,k}|Y_{i,1},\ldots,Y_{i,I+1-k})=0.$$
$$cov(\epsilon_{i,k}|Y_{i,1},\ldots,Y_{i,I+1-k})=\Sigma_{\epsilon_{i,k}}=D(Y_{i,k}^{-\delta/2})\Sigma_k D(Y_{i,k}^{-\delta/2}).$$
$$\mbox{losses of different accident years are independent}.$$
$$\epsilon_{i,k} \, \mbox{are symmetrically distributed}.$$
The GMCL model is very flexible since different parameter restrictions can be specified. It will be equivalent to the MCL model if the model does not have intercepts and the development matrix is restricted to be diagonal. When applied to paid and incurred triangles, it can reflect the development relationship between the two triangles, as described by Quarg and Mack (2004). The full bivariate model is identical to the "double regression" model described by Mack (2003), which is argued by him to be equivalent to the Munich Chain Ladder (MuCL) model. GMCL with intercepts can also help improve model adequacy as described by Barnett and Zehnwirth (2000).
Currently the model GMCL can only work for trapezoid data, and it only allows for estimation method mse.method="Mack", while the model MCL allows extrapolation and the mse method that assumes independence among estimated parameters. The model MCL under estimation method "OLS" will be equivalent to separate chain ladders (SCL). When one triangle is specified (as a list), MCL is equivalent to MackChainLadder.
GMCL allows different model structures to be specified across the development periods. This is usually achieved through the combination of the int and restrict.regMat arguments. int indicates which periods will have intercepts, and restrict.regMat allows different parameter restrictions to be specified in a list.
In using the multivariate method, one often specifies separate chain ladder for later periods to stabilize the estimation. In this case, one can use "[", defined for class triangles to split the input data, and use the MultiChainLadder to fit two models, either MCL or GMCL, and join them together using Join2Fits, which creates an object of class MCLFit or GMCLFit. Then methods of predict and Mse can be called to produce predictions and mean square errors. The function JoinFitMse is written to make it easy to construct an object of class MultiChainLadder, for which a couple of methods are defined to produce statistical results and diagonostic plots.