3 Model Estimation
Once bond yields and economic time series data are gathered and model features are selected, ATSM estimation proceeds in three steps. First, estimate the country-specific spanned factors for inclusion in the global ATSM. Next, estimate the parameters governing risk factor dynamics under the \(P-\)measure. Finally, optimize the entire selected ATSM, including the \(Q-\)measure parameters.
3.1 Spanned Factors
The spanned factors are yield-related variables that are used to fit the cross-section dimensions of the term structures. Typically, the spanned factors of country \(i\), \(P_{i,t}\), are computed as the N first principal components (PCs) of the set of observed bond yields. Formally, \(P_{i,t}\) is constructed as \(P_{i,t} = w_i Y_{i,t}\) where \(w_i\) is the PC weight matrix and \(Y_{i,t}\) is a country-specific column-vector of yields with increasing maturities.
For N=3, the spanned factors are traditionally interpreted as level, slope, and curvature. The nature of such interpretability results from the features of the PC weight matrix as illustrated below:
In matrix w, each row stores the weights used for constructing each spanned factor. The entries of the first row are linked to the composition of the level factor in that they load rather equally on all yields. Accordingly, high (low) values of the level factor indicate an overall high (low) value of yields across all maturities. In the second row, the weights monotonically increase with the maturities and, therefore, they capture the degree of steepness (slope) of the term structure. High slope factor values imply a steep yield curve, whereas low ones entail flat (or, possibly, downward) curves. In the third row, the weights of the curvature factor are presented. The name of this factor follows from the fact that the weights have a more pronounced effect on the middle range maturities of the curve. These concepts are graphically illustrated below.
Figure 3.1: Yield loadings on the spanned factors
To directly obtain the time series of the country-specific spanned
factors, the user can simply use the Spanned_Factors() function as
follows:
3.2 The P-dynamics estimation
As presented in Table 0.1, the dynamics of the risk factors under the \(P\)-measure evolve according to a VAR(1) model, which may be fully unrestricted (as in the case of the JPS-related models) or partially restricted (as in the GVAR and JLL frameworks). While the focus here is their integration into an ATSM setup, each specification can also be estimated independently to study dynamic systems. The usage of these models is illustrated below.
3.2.1 VAR
Using the VAR() function of this package requires simply selecting the
set of risk factors for the desired estimated model. For instance, for
the models JPS global and JPS multi, the estimation of the
\(P\)-dynamics parameters is obtained as
whereas the estimation of a JPS original model for China is
In both cases, the outputs generated are the vector of intercepts and both the feedback and variance-covariance matrices of an unconstrained VAR model.
3.2.2 GVAR
The GVAR() function is designed to estimate a first-order GVAR which
is built from country-specific VARX(1,1,1) models. For a comprehensive
discussion of GVAR models, see Chudik and Pesaran (2016).
The GVAR() function requires two key inputs: the number of
country-specific spanned factors (N) and a set of supplementary inputs
stored in the GVARinputs list. The former is composed by four
elements:
Economies: A character vector containing the names of the economies included in the system;
GVAR list of risk factors: a list of risk factors sorted by country in addition to the global variables. See the outputs of the
DatabasePrep()function;VARX type: a string-vector containing the desired estimated form of the VARX(1,1,1). Three possibilities are available. The “unconstrained” form estimates the model without any constrains, using standard OLS regressions for each model equation. The “constrained: Spanned Factors” form prevents foreign-spanned-factors to impact any domestic factor in the feedback matrix, whereas “constrained: ’ followed by the name of the risk factor restricts this same factor to be influenced only by its own lagged values and the lagged values of its own star variables. In the last two cases, the VARX(1,1,1) is estimated by restricted least squares.
GVARinputs <- list(Economies = Economies, GVARFactors = FactorsGVAR, VARXtype ="constrained: Inflation")- Transition matrix: The estimation of a GVAR model requires defining a measure of interdependence among the countries of the economic system. This information is reported in the transition matrix, the entries of which reflect the degree of interconnection across two entities of this same economic system. The GVAR setups can be estimated using fixed or, starting from package’s version \(0.3.3\), time-varying interdependence weights.
Time-varying interdependence implies that the star factors are constructed by applying trade flow weights from the specific year to adjust the risk factors of the corresponding year. To employ this feature, users need to specify the transition matrix type as Time-varying and select the same specific date (year) for both the initial and final years within the transition matrix. This implies that the trade weights of this year are used for solving (link matrices) the GVAR system.
Below the illustration of the transition matrix is computed as the average of the cross-border trade flow weights for the period spanning the years from 2006 to 2019. Therefore, the degree of dependence across countries is assumed to be fixed over the entire sample span. Note that each row sums up to \(1\).
data("CM_Trade")
t_First <- "2006"
t_Last <- "2019"
Economies <- c("China", "Brazil", "Mexico", "Uruguay")
type <- "Sample Mean"
W_gvar <- Transition_Matrix(t_First, t_Last, Economies, type, DataPath = NULL, TradeFlows)
print(W_gvar)## China Brazil Mexico Uruguay
## China 0.0000000 0.6549066 0.31550095 0.029592464
## Brazil 0.8268799 0.0000000 0.12340661 0.049713473
## Mexico 0.8596078 0.1326284 0.00000000 0.007763753
## Uruguay 0.3811195 0.5497525 0.06912795 0.000000000Having defined the form of the transition matrix, one can complete the GVARinputs list to estimate a GVAR(1) model. Note that the CheckInputs parameter is set as TRUE to perform a prior consistency check on the inputs provided in GVARinputs.
data("CM_Factors_GVAR")
GVARinputs <- list(Economies = Economies, GVARFactors = FactorsGVAR, VARXtype = "unconstrained",
Wgvar = W_gvar)
N <- 3
GVARpara <- GVAR(GVARinputs, N, CheckInputs = TRUE)A separate routine is provided for computing the foreign-specific factors (also commonly referred to as the star variables) used in the estimation of the VARX models.
3.2.3 JLL
Calculating the \(P\)-dynamics parameters in the form proposed by JLL (2015) requires the following inputs: (i) the time series of the risk factors in non-orthogonalized form; (ii) the number of country-specific spanned factors and (iii) the specification of the JLLinputs list, which includes the following elements:
Economies: A character vector containing the names of the economies included in the system;
Dominant Country: a string-vector containing the name of the economy which is assigned as the dominant country (applicable for the JLL original and JLL joint Sigma models) or None (applicable for the JLL No DomUnit);
Wish the estimation of the sigmas matrices: this is a binary variable which assumes value equal to \(1\) if the user wishes the estimation of all JLL sigma matrices (i.e. variance-covariance and the Cholesky factorization matrices) and, \(0\) otherwise. Since this estimation is done numerically, its estimation can take several minutes;
Sigma of the non-orthogonalized variance-covariance matrix: to save time, the user may provide the variance-covariance matrix from the non-orthogonalized dynamics. Otherwise, this input should be set as NULL.
JLL type: a string-vector containing the label of the JLL model to be estimated according to the features described in Table 0.1.
One possible list of JLLinputs is as follows:
ModelType <- "JLL original"
JLLinputs <- list(Economies = Economies, DomUnit = "China", WishSigmas = 1, SigmaNonOrtho = NULL,
JLLModelType = ModelType)As such, the \(P-\)dynamics dimension of a JLL-based setup can be computed as:
3.3 ATSM estimation
ATSM estimation involves optimizing the log-likelihood function by
optimally selecting the model parameters. The optimization structure in
this package is based on routines from the term structure package by
Le and Singleton (2018). The ATSM otimization is carried out using the
Optimization() function.
Broadly, the unspanned risk framework from JPS (2014) (and, therefore, all its multi-country extensions) requires the estimation parameter set consisting of 6 elements, namely: the risk-neutral long-run mean of the short rate (r0), the risk-neutral feedback matrix (K1XQ), the variance-covariance matrix (SSZ) from the VAR processes, the standard deviation of the errors from the portfolios of yields observed with error (se), in addition to the intercept (K0Z) and the feedback (K1Z) matrices of the physical dynamics.
The parameters K0Z and K1Z are estimated as part of the model’s \(P-\)dynamics and are known in closed form. Likewise, r0 and se have analytical solutions and can be factored out of the log-likelihood function. However, the remaining parameters (K1XQ and SSZ) require numerical estimation. Le and Singleton (2018) provides standardized routines (integrated into this package) to set effective initial values, facilitating the optimization process.
The MultiATSM package offers two optimization algorithms: fminunc and fminsearch. By default, both algorithms are run sequentially twice to ensure the model reaches the global optimum. Specifically in the default bootstrap setup, only fminunc is used to save time.