3 Model Estimation
Having gathered bond yields and other economic time series data, the ATSM estimation is carried out in three steps. Firstly, one needs to provide the number of country-specific spanned factors to be included in the global ATSM. Secondly, it is decided on the form of the risk factor dynamics under the \(P\)-measure. Finally, the user needs to select general features for the model optimization.
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:
w <- pca_weights_one_country(Yields, Economy = "Uruguay") 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:
data('CM_Yields')
Economies <- c("China", "Brazil", "Mexico", "Uruguay")
N <- 2
SpaFact <- Spanned_Factors(Yields, Economies, N)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 somewhat restricted (as in the GVAR and JLL frameworks). Below, the usage of each of these dimensions are illustrated.
3.2.1 VAR
Using the VAR function of this package requires simply selecting the appropriated set of risk factors for the desired estimated model. For instance, for the models JPS jointP and VAR jointQ, the estimation of the \(P\)-dynamics parameters is obtained as
data("CM_Factors")
PdynPara <- VAR(RiskFactors, VARtype= "unconstrained")whereas the estimation of a JPS model for China is
FactorsChina <- RiskFactors[1:7,]
PdynPara <- VAR(FactorsChina, VARtype= "unconstrained")In both cases, the outputs generated are the vector of intercepts and both the feedback and variance-covariance matrices.
3.2.2 GVAR
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 of two entities of this same economic system. In this package, the illustration of the transition matrix is based on the average of the cross-border trade flow weights for the period spanning the years from 2006 to 2019. 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)Having defined the form of the transition matrix, one can complete the GVARinputs variable along the lines discussed in Section 2.1.1.
data("CM_Factors_GVAR")
GVARinputs <- list()
GVARinputs$Economies <- Economies
GVARinputs$GVARFactors <- FactorsGVAR
GVARinputs$VARXtype <- "unconstrained"
GVARinputs$Wgvar <- W_gvar
N <- 3
GVARpara <- GVAR(GVARinputs, N)A separate routine is provided for computing the foreign-specific factors (also commonly referred as the star variables) used in the estimation of the VARX models.
data('CM_Factors')
StaFac <- StarFactors(RiskFactors, Economies, W_gvar)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 as presented in Section 2.1.1. See, for instance:
data("CM_Factors")
N <- 3
JLLinputs <- list()
ModelType <- "JLL original"
JLLinputs$Economies <- Economies
JLLinputs$DomUnit <- "China"
JLLinputs$WishSigmas <- 1
JLLinputs$SigmaNonOrtho <- NULL
JLLinputs$JLLModelType <- ModelType
JLLpara <- JLL(RiskFactors, N, JLLinputs)3.3 ATSM estimation
The estimation of ATSM involves defining the model inputs from the log-likelihood function and, subsequently, the parameters used in its optimization process. The structure proposed in this part of the code is in a great extent based on the term structure package by Le and Singleton (2018).
3.3.1 The log-likelihood function
For the sake of simplicity, one illustrates the construction of a log-likelihood function based on a VAR jointQ model. For both the JLL and GVAR-based models, one must further specify the JLLinputs or GVARinputs along the lines described in the Section 2.1.1.
# Inputs to be specified by the user
data("CM_Yields")
data("CM_Factors")
ModelType <- "VAR jointQ"
Yields <- Yields
ZZ <- RiskFactors
Economies <- c("China", "Brazil", "Mexico", "Uruguay")
mat <- Maturities(Yields, Economies, UnitYields = "Month")
DataFrequency <- "Monthly"
GlobalVar <- c("GBC", "CPI_OECD") # Global variables
DomVar <- c("Eco_Act", "Inflation") # Domestic variables
N <- 3 # Number of country-specific spanned factors
# Generate the "Factor Labels" list (necessary preliminarily step)
FactorLabels <- LabFac(N, DomVar, GlobalVar, Economies, ModelType)
# Prepare the inputs for log-Likelihood function
ATSMInputs <- InputsForMLEdensity(ModelType, Yields, ZZ, FactorLabels, mat, Economies, DataFrequency)
# Log-Likelihood function
f <- Functionf(ATSMInputs, Economies, mat, DataFrequency, FactorLabels, ModelType)3.3.2 Optimization parameters
JPS (2014) requires the estimation of a set of parameters containing 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. Each of these parameters must be cast in an individual list that should contain the (i) the starting value of the parameter (if any); (ii) the variable label followed by a ‘:’ and a type of constraint; (iii) a lower bound (if any), and (iv) an upper bound (if any). The variable labels of r0, se, K0Z and K1Z should be preceded by the symbol @ as a manner to account for the fact that the solution of these parameters are known in closed-form. For the remaining ones (K1XQ and SSZ), Le and Singleton (2018) provide standardized routines (already incorporated in this package) to set good initial values so that the optimization process runs fast.
The type of constraint to be set for the parameters r0, se, K0Z and K1Z is typically bounded and are used for bounded matrices. For K1XQ and SSZ, the MultiATSM package currently provides the following options:
- K1XQ: ‘Jordan’ (or Jordan MultiCountry’) for a matrix of Jordan type for single (multi) country specifications. These labels can be extended by ‘; stationary’, if one wishes to impose the largest eigenvalue of the risk-neutral feedback matrix to be strictly smaller than 1;
- SSZ: ‘psd’ for a positive semi-definite matrix; ‘diag’ for a diagonal matrix; ‘BlockDiag’ for a block diagonal matrix (typical from the GVAR-based models) and ‘JLLstructure’ for the models containing the restrictions along the lines of JLL (2015).
One example for the list of parameters specification is
K1XQinputs <- list(NULL, "K1XQ: Jordan" , NULL , NULL)
SSZinputs <- list(NULL, "SSZ: psd", NULL, NULL)
r0inputs <- list(NULL, "@r0: bounded", NULL, NULL)
seinputs <- list(NULL, "@se: bounded", 1e-6, NULL)
K0Zinputs<- list(NULL, "@K0Z: bounded", NULL, NULL)
K1Zinputs<- list(NULL, "@K1Z: bounded", NULL, NULL)To complete the optimization setting, the user needs to specify the level of convergence tolerance (usually, 1e-4 is a reasonable value) and the vector OptRun. The first element of this vector must be filled by the word ‘iter off’, if the user wishes to switch off the printouts of the numerical optimization routines, or simply by ‘iter’ otherwise. The second element of the vector concerns the algorithm that is used in the optimization: the option ‘fminunc only’ only uses fminunc, whereas ‘fminsearch only’ only applies fminsearch. If OptRun is formed exclusively by ‘iter off’, then both fminunc and fminsearch are employed in the optimization.