Paper Replications"
In MultiATSM: Multicountry Term Structure of Interest Rates Models

This vignette leverages the MultiATSM package to reproduce some empirical findings from three academic papers centered around the unspanned macroeconomic risk framework. The first segment focuses on replicating the outcomes of @JoslinPriebschSingleton2014, which laid the foundation for the unspanned risk ATSM literature through a single-country analysis of the U.S. economy. Subsequently, I present the multi-country model extension proposed by @CandelonMoura2024, employing a GVAR approach to capture risk factor dynamics. Lastly, I introduce the program designed to replicate results detailed in @CandelonMoura2023, an extension of the previous framework to investigate the term structure behavior of major emerging markets during the unprecedented COVID$-19$ pandemic. It is worth highlighting that, due to limited public availability of bond yield data, I employ, in certain instances, simulated data closely matching the original papers' construction method. Consequently, while the outputs generated by the scripts below may not exactly mirror those from the original papers, the main features of these results are preserved.

A thorough explanation on the usage of the additional functionalities of the MultiATSM package is available at the package vignette.

library(MultiATSM)

Joslin, Priebisch and Singleton (2014)

The database used in this replication was constructed by @BauerRudebusch2017 (BR, 2017) and is available at Bauer's website. In that paper, BR (2017) investigate whether macro-finance term structure models better suit the unspanned macro risk framework by JPS (2014) or the earlier traditional spanned settings as the one by @AngPiazzesi2003JME. Accordingly, BR (2017) intend to replicate some of the empirical results reported in JPS (2014). The R-code used by BR (2017) is also available at Bauer's website.

Supported by BR (2017)'s data-set, the code below uses the MultiATSM package to estimate the key model parameters from the ATSM along the lines of JPS (2014). The model estimation is performed using the JPS original modeling setup.

# A) Load database data
LoadData("BR_2017")

# B) GENERAL model inputs
ModelType <- "JPS original"

Economies <- c("US") # Names of the economies from the economic system
GlobalVar <- c() # Global Variables
DomVar <- c("GRO", "INF") # Country-specific variables
N <- 3 # Number of spanned factors per country

t0_sample <- "January-1985"
tF_sample <- "December-2007"

DataFreq <- "Monthly" # Frequency of the data

StatQ <- 0 # Stationary condition

#########################################################################################################
############################### NO NEED TO MAKE CHANGES FROM HERE #######################################
#########################################################################################################
# 2) Minor preliminary work
FactorLabels <- LabFac(N, DomVar, GlobalVar, Economies, ModelType) 

Yields <- t(BR_jps_out$Y)
DomesticMacroVar <- t(BR_jps_out$M.o)
GlobalMacroVar <- c()

# 3) Prepare the inputs of the likelihood function
ATSMInputs <- InputsForOpt(t0_sample, tF_sample, ModelType, Yields, GlobalMacroVar, DomesticMacroVar,
                           FactorLabels, Economies, DataFreq)

# 4) Optimization of the model
ModelPara <- Optimization(ATSMInputs, StatQ, DataFreq, FactorLabels, Economies, ModelType)

The tables below compare the ATSM parameter estimates generated from BR (2017) and the MultiATSM. Overall, one can note that the differences in the estimates are economically modest.

options(scipen = 100) # eliminate the scientific notation
data("BR_jps_gro_R3")
data("JPSrep")

RowsQ <- c("$r0$", "$\\lambda_1$", "$\\lambda_2$", "$\\lambda_3$" )
TableQ <- data.frame(matrix(NA, ncol = 0, nrow =length(RowsQ)))
row.names(TableQ) <- RowsQ

PackageQ<- c(ModelPara$ests$r0, diag(ModelPara$ests$K1XQ))
BRq <- c(BR_jps_out$est.llk$rho0.cP, diag(BR_jps_out$est.llk$KQ.XX))
TableQ$MultiATSM <- PackageQ
TableQ$'BR (2017)' <- BRq

TableQ <- round(TableQ, digits = 5)

suppressWarnings(library(magrittr))

kableExtra::kbl(TableQ, align = "c", caption = "$Q$-dynamics parameters") %>%
  kableExtra::kable_classic("striped", full_width = F)  %>%
  kableExtra::row_spec(0, font_size = 14) %>%
  kableExtra::footnote(general = " $\\lambda$'s are the eigenvalues from the risk-neutral feedback matrix and $r0$ is the long-run mean of the short rate under Q.")

data("BR_jps_gro_R3")
data("JPSrep")

RowsP <- c("PC1", "PC2", "PC3", "GRO", "INF")
ColP <- c(" ", RowsP)

# 1) K0Z and K1Z
# Bauer and Rudebusch coefficients
TablePbr <- data.frame(matrix(NA, ncol = length(ColP), nrow =length(RowsP)))
row.names(TablePbr) <- RowsP
colnames(TablePbr) <- ColP

TablePbr[[ColP[1]]] <- BR_jps_out$est.llk$KP.0Z
for(j in 1:length(RowsP) ){TablePbr[[RowsP[j]]] <- BR_jps_out$est.llk$KP.ZZ[,j]}

TablePbr <- round(TablePbr, digits = 5)

# MultiATSM coefficients
TablePMultiATSM <- data.frame(matrix(NA, ncol = length(ColP), nrow =length(RowsP)))
row.names(TablePMultiATSM) <- RowsP
colnames(TablePMultiATSM) <- ColP

IdxVar <- c(3:5, 1:2) # indexes to flip the order of the spanned and unspanned factors
TablePMultiATSM[[ColP[1]]] <- ModelPara$ests$K0Z[IdxVar]
ModelPara$ests$K1Z <- ModelPara$ests$K1Z[IdxVar,IdxVar]
for(j in 1:length(RowsP) ){ TablePMultiATSM[[RowsP[j]]] <- ModelPara$ests$K1Z[,j]}


TablePMultiATSM <- round(TablePMultiATSM, digits = 5)
TableP <- rbind(TablePbr,TablePMultiATSM)
row.names(TableP) <- c(RowsP,paste(RowsP," ",sep="")) # trick to avoid rows to have the same name

kableExtra::kbl(TableP, align = "c", caption = "$P$-dynamics parameters") %>%
  kableExtra::kable_classic("striped", full_width = F)  %>%
  kableExtra::row_spec(0, font_size = 14) %>%
  kableExtra::add_header_above(c(" "= 1, "K0Z"=1, "K1Z" = 5), bold = T) %>%
  kableExtra::pack_rows("BR (2017)", 1, 5) %>%
  kableExtra::pack_rows("MultiATSM", 6, 10) %>%
  kableExtra::footnote(general = " $K0Z$ is the intercept and $K1Z$ is feedback matrix from the $P$-dynamics.")

data("BR_jps_gro_R3")
data("JPSrep")

se <- data.frame(BR_jps_out$est.llk$sigma.e, ModelPara$ests$se )
rownames(se) <- "se"
colnames(se) <- c("MultiATSM","BR (2017)")
se <- round(se, digits = 7)

kableExtra::kbl(se, align = "c", caption ="Portfolio of yields with errors") %>%
  kableExtra::kable_classic("striped", full_width = F)  %>%
  kableExtra::row_spec(0, font_size = 14) %>%
  kableExtra::footnote(general = " $se$ is the standard deviation of the portfolio of yields observed with errors.")

For replicability, it is important to note that the $P-$dynamics results reported in Table \@ref(tab:PdynTab) for the MultiATSM package make use of the principal component weights provided by BR (2017). Such a matrix is simply a scaled-up version of the one provided by the function pca_weights_one_country() of this package. Accordingly, despite the numerical differences on the weight matrices, both methods generate time series of spanned factors which are perfectly correlated. Another difference between the two approaches relates to the construction form of the log-likelihood function (llk): while in the BR (2017) code the llk is expressed in terms of a portfolio of yields, the MultiATSM package generates this same input directly as a function of observed yields (i.e. both procedures lead to equivalent llk up to the Jacobian term).

Candelon and Moura (2024)

The multi-country framework introduced in @CandelonMoura2024 was designed to enhance the tractability of large-scale ATSMs, providing a deeper understanding of the intricate global economic mechanisms that underlie yield curve fluctuations. This framework also reinforces the statistical validity of inference and enhances the predictive capabilities of these models. This novel setup, embodied by the GVAR multi model class, is benchmarked against the findings of @JotikasthiraLeLundblad2015, which are captured by the JLL original model class. The paper showcases an empirical illustration involving China, Brazil, Mexico, and Uruguay.

# A) Load database data
LoadData("CM_2024")

# B) GENERAL model inputs
ModelType <- "GVAR multi" # Options: "GVAR multi" or "JLL original".

Economies <- c("China", "Brazil", "Mexico", "Uruguay")
GlobalVar <- c("Gl_Eco_Act", "Gl_Inflation")
DomVar <- c("Eco_Act", "Inflation")
N <- 3

t0_sample <- "01-06-2004"
tF_sample <- "01-01-2020"

OutputLabel <- "CM_jfec"
DataFreq <-"Monthly"

StatQ <- 0

# B.1) SPECIFIC model inputs
#################################### GVAR-based models ##################################################
GVARlist <- list( VARXtype = "unconstrained", W_type = "Sample Mean", t_First_Wgvar = "2004",
                  t_Last_Wgvar = "2019", DataConnectedness <- TradeFlows )
#################################### JLL-based models ###################################################
JLLlist <- list(DomUnit =  "China")
###################################### BRW inputs  ######################################################
WishBC <- 1
BRWlist <- within(list(flag_mean = TRUE, gamma = 0.001, N_iter = 200, B = 50, checkBRW = TRUE,
                       B_check = 1000),  N_burn <- round(N_iter * 0.15))

# C) Decide on Settings for numerical outputs
WishFPremia <- 1
FPmatLim <- c(24,36)

Horiz <- 25
DesiredGraphs <- c("GIRF", "GFEVD", "TermPremia")
WishGraphRiskFac <- 0
WishGraphYields <- 1
WishOrthoJLLgraphs <- 1

# D) Bootstrap settings
WishBootstrap <- 0 #  Set it to 1, if bootstrap is desired
BootList <- list(methodBS = 'bs', BlockLength = 4, ndraws = 1000, pctg =  95)

# E) Out-of-sample forecast
WishForecast <- 1
ForecastList <- list(ForHoriz = 12,  t0Sample = 1, t0Forecast = 100, ForType = "Rolling")

#########################################################################################################
############################### NO NEED TO MAKE CHANGES FROM HERE #######################################
#########################################################################################################

# 2) Minor preliminary work: get the sets of factor labels and  a vector of common maturities
FactorLabels <- LabFac(N, DomVar, GlobalVar, Economies, ModelType)

# 3) Prepare the inputs of the likelihood function
ATSMInputs <- InputsForOpt(t0_sample, tF_sample, ModelType, Yields, GlobalMacroVar, DomesticMacroVar,
                           FactorLabels, Economies, DataFreq, GVARlist, JLLlist, WishBC, BRWlist)

# 4) Optimization of the ATSM (Point Estimates)
ModelParaList <- Optimization(ATSMInputs, StatQ, DataFreq, FactorLabels, Economies, ModelType)

# 5) Numerical and graphical outputs
# a) Prepare list of inputs for graphs and numerical outputs
InputsForOutputs <- InputsForOutputs(ModelType, Horiz, DesiredGraphs, OutputLabel, StatQ, DataFreq,
                                    WishGraphYields, WishGraphRiskFac, WishOrthoJLLgraphs, WishFPremia,
                                    FPmatLim, WishBootstrap, BootList, WishForecast, ForecastList)

# b) Fit, IRF, FEVD, GIRF, GFEVD, and Term Premia
NumericalOutputs <- NumOutputs(ModelType, ModelParaList, InputsForOutputs, FactorLabels, Economies)

# c) Confidence intervals (bootstrap analysis)
BootstrapAnalysis <- Bootstrap(ModelType, ModelParaList, NumericalOutputs, Economies, InputsForOutputs,
                               FactorLabels, JLLlist, GVARlist, WishBC, BRWlist)

# 6) Out-of-sample forecasting
Forecasts <- ForecastYields(ModelType, ModelParaList, InputsForOutputs, FactorLabels, Economies,
                            JLLlist, GVARlist, WishBC, BRWlist)

Candelon and Moura (2023)

In this paper, @CandelonMoura2023 delve into an exploration of the underlying factors that shape the sovereign yield curves of Brazil, India, Mexico, and Russia in the midst of the COVID$-19$ pandemic crisis. To comprehensively address the intricate web of global macro-financial and particularly health-related interdependencies, the study employs a modeling approach based on a GVAR multi type framework. By delving into these aspects, the authors aim to provide a nuanced understanding of the dynamics influencing sovereign yield curves in these countries during the challenging circumstances of the pandemic.

# A) Load database data
LoadData("CM_2023")

# B) GENERAL model inputs
ModelType <- "GVAR multi"

Economies <- c("Brazil", "India", "Russia", "Mexico")
GlobalVar <- c("US_Output_growth", "China_Output_growth", "SP500")
DomVar <- c("Inflation","Output_growth", "CDS", "COVID")
N <- 2

t0_sample <- "22-03-2020"
tF_sample <- "26-09-2021"

OutputLabel <- "CM_EM"
DataFreq <-"Weekly"

StatQ <- 0

# B.1) SPECIFIC model inputs
#################################### GVAR-based models ##################################################
GVARlist <- list( VARXtype = "constrained: COVID", W_type = "Sample Mean", t_First_Wgvar = "2015",
                  t_Last_Wgvar = "2020", DataConnectedness = Trade_Flows)
###################################### BRW inputs  ######################################################
WishBC <- 0

# C) Decide on Settings for numerical outputs
WishFPremia <- 1
FPmatLim <- c(47,48)

Horiz <- 12
DesiredGraphs <- c("GIRF", "GFEVD", "TermPremia")
WishGraphRiskFac <- 0
WishGraphYields <- 1
WishOrthoJLLgraphs <- 0

# D) Bootstrap settings
WishBootstrap <- 1 #  YES: 1; No = 0.
BootList <- list(methodBS = 'bs', BlockLength = 4, ndraws = 100, pctg =  95)

#########################################################################################################
############################### NO NEED TO MAKE CHANGES FROM HERE #######################################
#########################################################################################################

# 2) Minor preliminary work: get the sets of factor labels and  a vector of common maturities
FactorLabels <- LabFac(N, DomVar, GlobalVar, Economies, ModelType)

# 3) Prepare the inputs of the likelihood function
ATSMInputs <- InputsForOpt(t0_sample, tF_sample, ModelType, Yields, GlobalMacro, DomMacro, FactorLabels,
                           Economies, DataFreq, GVARlist)

# 4) Optimization of the ATSM (Point Estimates)
ModelParaList <- Optimization(ATSMInputs, StatQ, DataFreq, FactorLabels, Economies, ModelType)

# 5) Numerical and graphical outputs
# a) Prepare list of inputs for graphs and numerical outputs
InputsForOutputs <- InputsForOutputs(ModelType, Horiz, DesiredGraphs, OutputLabel, StatQ, DataFreq,
                                    WishGraphYields, WishGraphRiskFac, WishOrthoJLLgraphs, WishFPremia,
                                    FPmatLim, WishBootstrap, BootList)

# b) Fit, IRF, FEVD, GIRF, GFEVD, and Term Premia
NumericalOutputs <- NumOutputs(ModelType, ModelParaList, InputsForOutputs, FactorLabels, Economies)

# c) Confidence intervals (bootstrap analysis)
BootstrapAnalysis <- Bootstrap(ModelType, ModelParaList, NumericalOutputs, Economies, InputsForOutputs,
                               FactorLabels, JLLlist = NULL, GVARlist)