---
title: "Paper Replications"
author: Rubens Moura
date: "`r Sys.Date()`"
output: 
  bookdown::html_document2:
    base_format: rmarkdown::html_vignette
    number_sections: true
    df_print: paged
    toc: yes
    toc_depth: 2
    fig_caption: yes
vignette: >
  %\VignetteIndexEntry{Paper Replications}
  %\VignetteEngine{knitr::rmarkdown}
  \usepackage[utf8]{inputenc} 
bibliography: references.bib
---

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 @CandelonMoura2021, 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](my-vignette.html).

```{r}
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](https://www.michaeldbauer.com/publication/spanning-puzzle/). 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](https://www.michaeldbauer.com/publication/spanning-puzzle/).

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* modeling setup.

```{r, eval=FALSE} 
#########################################################################################################
#################################### USER INPUTS ########################################################
#########################################################################################################
# A) Load database data
data("BR_jps_gro_R3")

# B) Decide on general model inputs
ModelType <- "JPS"

StationarityUnderQ <- 0
BiasCorrection <- 0
WishForwardPremia <- 1


Economies <- c("U.S.") # Names of the economies from the economic system
GlobalVar <- c()
DomVar <- c("GRO", "INF") # Country-specific variables

N <- 3 # Number of spanned factors per country

DataFrequency <- "Monthly" # Frequency of the data
UnitMatYields <- "Month" # time-unit in which yields are expressed. Available options are "Month" or "Year"

mat <- c(0.25, 0.5, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10) # maturities expressed in years

#########################################################################################################
############################### NO NEED TO MAKE CHANGES FROM HERE #######################################
#########################################################################################################
# 2) Minor preliminary work
C <- length(Economies)
M <- length(DomVar)
FactorLabels <- LabFac(N, DomVar, GlobalVar, Economies, ModelType) # Generate the set of labels
Yields <- t(BR_jps_out$Y)
W <- BR_jps_out$W[1:N,] # Use the Wpca matrix from BR (2017)
SpaFac <- W%*%Yields
rownames(SpaFac) <- FactorLabels$Tables$U.S.[-(1:M)]
ZZ <- rbind(t(BR_jps_out$M.o), SpaFac) # Complete set of risk factors


# 3) Prepare the inputs of the likelihood function
i <- length(Economies)
  # 3.1) Compute the inputs that go directly into the log-likelihood function
  ATSMInputs <- InputsForMLEdensity(ModelType, Yields, ZZ, FactorLabels, mat, Economies, DataFrequency)
  # 3.2) Initial guesses for Variables that will be concentrared out of from the log-likelihood function
  K1XQ <- ATSMInputs$K1XQ
  SSZ <- ATSMInputs$SSZ
  # 3.3) Adjust the inputs which are funtion of the W matrix
  ATSMInputs$Wpca <- W
  ATSMInputs$We <- t(pracma::null(W))
  ATSMInputs$WpcaFull <- rbind(ATSMInputs$Wpca, ATSMInputs$We)
  ATSMInputs$PP <- SpaFac

  # 4) Build the objective function
  f <- Functionf(ATSMInputs, Economies, mat, DataFrequency, FactorLabels, ModelType)

  # 5) Set the optimization settings
  VarLab <- ParaLabels(ModelType, StationarityUnderQ)

  varargin <- list()
  varargin$K1XQ <-list(K1XQ, VarLab[[ModelType]][["K1XQ"]] , NULL , NULL)
  varargin$SSZ <- list(SSZ, VarLab[[ModelType]][["SSZ"]], NULL, NULL)
  varargin$r0 <- list(NULL, VarLab[[ModelType]][["r0"]], NULL, NULL)
  varargin$se <- list(NULL, VarLab[[ModelType]][["se"]], 1e-6, NULL)
  varargin$K0Z <- list(NULL, VarLab[[ModelType]][["K0Z"]], NULL, NULL)
  varargin$K1Z <- list(NULL, VarLab[[ModelType]][["K1Z"]], NULL, NULL)
  varargin$OptRun <-  c("iter off")

  LabelVar<- c('Value', 'Label', 'LB', 'UB') # Elements of each parameter
  for (d in 1:(length(varargin)-1)){ names(varargin[[d]]) <-  LabelVar}

  tol <- 1e-4

  # 6) Optimization of the model
  ModelPara <- Optimization(f, tol, varargin, FactorLabels, Economies, ModelType)$Summary

```



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.



```{r, echo= FALSE}
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.")
```

```{r, echo= FALSE}
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.")
```


```{r, echo= FALSE}
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.")
```



It is relevant to highlight that the script above makes 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 (2021)
The multi-country framework introduced in @CandelonMoura2021 was meticulously 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 jointQ* 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.


```{r, eval=FALSE}
#########################################################################################################
#################################### USER INPUTS ########################################################
#########################################################################################################
# A) Load database data
data("CM_Factors")
data('CM_Factors_GVAR')
data('CM_Trade')
data('CM_Yields')

# B) Decide on general model inputs
ModelType <- "GVAR jointQ"  # Options: "GVAR jointQ", "JLL original" 

StationarityUnderQ <- 0  
BiasCorrection <- 1
WishForwardPremia <- 0 
FPmatLim <- c(24,36) 

Economies <- c("China","Brazil","Mexico", "Uruguay") 
GlobalVar <- c("GBC", "CPI_OECD") 
DomVar <- c("Eco_Act", "Inflation") 

N <- 3 # Number of spanned factors per country

OutputLabel <- "CM_2024"
DataFrequency <- "Monthly"
UnitMatYields <- "Month"

# B.1) Decide on specific model inputs
#################################### GVAR-based models ##################################################
t_First_Wgvar <- "2004"
t_Last_Wgvar <-  "2019"
W_type <- 'Sample Mean'
VARXtype <- "unconstrained"
#################################### JLL-based models ###################################################
DomUnit <- "China"
WishSigmas <- 0
SigmaNonOrtho <- NULL
JLLModelType <- ModelType
###################################### BRW inputs  ######################################################
flag_mean <- TRUE
gamma <- 0.001
N_iter <- 200
N_burn <- N_iter*0.15
B <- 50
checkBRW <- 1
B_check <- 1000
#########################################################################################################

# C) Decide on Settings for numerical outputs
Horiz <- 25
DesiredGraphs <- c("Fit", "GIRF", "GFEVD")
WishGraphRiskFac <- 0
WishGraphYields <- 1
WishOrthoJLLgraphs <- 1

# D) Bootstrap settings
WishBootstrap <- 0 
Bootlist <- list()
Bootlist$methodBS <- 'bs' 
Bootlist$BlockLength <- 4 
Bootlist$ndraws <-  1000
Bootlist$pctg   <-  95 

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

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

# 2) Minor preliminary work
C <- length(Economies)
FactorLabels <- LabFac(N, DomVar,GlobalVar, Economies, ModelType)
ZZ <- RiskFactors

Data <- list()
Data$GVARFactors <- FactorsGVAR
mat <- Maturities(Yields, Economies, UnitYields = UnitMatYields)

# 2.1) Generate GVARinputs, JLLinputs and BRWinputs
ModInputs <- ListModelInputs(ModelType, Data, Economies, VARXtype, t_First_Wgvar, t_Last_Wgvar, W_type,
                             DomUnit, WishSigmas, SigmaNonOrtho, BiasCorrection, flag_mean, gamma, 
                             N_iter, N_burn, B, checkBRW, B_check)

GVARinputs <- ModInputs$GVARinputs
JLLinputs <- ModInputs$JLLinputs
BRWinputs <- ModInputs$BRWinputs

# 3) Prepare the inputs of the likelihood function
ModelParaList <- list()

# 3.1) Compute the inputs that go directly into the log-likelihood function
ATSMInputs <- InputsForMLEdensity(ModelType, Yields, ZZ, FactorLabels, mat, Economies, DataFrequency, 
                                  JLLinputs, GVARinputs, BRWinputs)


# 3.2) Initial guesses for Variables that will be concentrared out of from the log-likelihood function
K1XQ <- ATSMInputs$K1XQ
if (ModelType == "JLL original"){ SSZ <- NULL}else{SSZ <- ATSMInputs$SSZ}

# 4) Build the objective function
f <- Functionf(ATSMInputs, Economies, mat, DataFrequency, FactorLabels, ModelType)

# 5) Set the optimization settings
VarLab <- ParaLabels(ModelType, StationarityUnderQ)

varargin <- list()
varargin$K1XQ <-list(K1XQ, VarLab[[ModelType]][["K1XQ"]] , NULL , NULL)
varargin$SSZ <- list(SSZ, VarLab[[ModelType]][["SSZ"]], NULL, NULL)
varargin$r0 <- list(NULL, VarLab[[ModelType]][["r0"]], NULL, NULL)
varargin$se <- list(NULL, VarLab[[ModelType]][["se"]], 1e-6, NULL)
varargin$K0Z <- list(NULL, VarLab[[ModelType]][["K0Z"]], NULL, NULL)
varargin$K1Z <- list(NULL, VarLab[[ModelType]][["K1Z"]], NULL, NULL)
varargin$OptRun <-  c("iter off")

LabelVar<- c('Value', 'Label', 'LB', 'UB') 
for (d in 1:(length(varargin)-1)){ names(varargin[[d]]) <-  LabelVar}

tol <- 1e-4
# 6) Optimization of the model
ModelParaList[[ModelType]] <- Optimization(f, tol, varargin, FactorLabels, Economies, ModelType,
                                           JLLinputs, GVARinputs)$Summary

# 7) Numerical Outputs
InputsForOutputs <- InputsForOutputs(ModelType, Horiz, DesiredGraphs, OutputLabel, StationarityUnderQ,
                                     UnitMatYields, WishGraphYields, WishGraphRiskFac, WishOrthoJLLgraphs, 
                                     WishForwardPremia, FPmatLim, WishBootstrap, Bootlist, WishForecast,
                                     ForecastList)
# A) Fit, IRF, FEVD, GIRF, GFEVD and Risk premia
NumericalOutputs <- NumOutputs(ModelType, ModelParaList, InputsForOutputs, FactorLabels, Economies)

# B) Bootstrap
Bootstrap <- Bootstrap(ModelType, ModelParaList, NumericalOutputs, mat, Economies, InputsForOutputs,
                       FactorLabels, DataFrequency, varargin, JLLinputs, GVARinputs, BRWinputs)

# C) Out-of-sample forecasting
Forecasts <- ForecastYields(ModelType, ModelParaList, InputsForOutputs, FactorLabels, Economies,
                            DataFrequency, JLLinputs, GVARinputs, BRWinputs)



```


# 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 jointQ* 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.


```{r, eval=FALSE}
# A) Load database data
data("CM_Factors_2023")
data('CM_Factors_GVAR_2023')
data('CM_Trade_2023')
data('CM_Yields_2023')

# B) Decide on general model inputs
ModelType <- "GVAR jointQ" 

StationarityUnderQ <- 0 
BiasCorrection <- 0

WishForwardPremia <- 1 
FPmatLim <- c(47,48) 

Economies <- c("Brazil", "India", "Russia", "Mexico") # Names of the economies from the economic system
GlobalVar <- c("US_Output_growth", "China_Output_growth", "SP500") # Global Variables
DomVar <- c("Inflation","Output_growth", "CDS", "COVID") # Country-specific variables

N <- 2 # Number of spanned factors per country

OutputLabel <- "CM_2023" 
DataFrequency <- "Weekly" 
UnitMatYields <- "Month" 

# C.1) Decide on specific model inputs
#################################### GVAR-based models ##################################################
t_First_Wgvar <- "2015" 
t_Last_Wgvar <-  "2020" 
W_type <- 'Sample Mean' 
VARXtype <- "constrained: COVID" 
###################################### BRW inputs  ######################################################

# D) Decide on Settings for numerical outputs
Horiz <- 12
DesiredGraphs <- c("GIRF", "GFEVD", "TermPremia") 
WishGraphRiskFac <- 0
WishGraphYields <- 1
WishOrthoJLLgraphs <- 0

# E) Bootstrap settings
WishBootstrap <- 0 #  YES: 1; No = 0.
Bootlist <- list()
Bootlist$methodBS <- 'bs' 
Bootlist$BlockLength <- 4 # necessary input if one chooses the block bootstrap
Bootlist$ndraws <-  100
Bootlist$pctg   <-  95 # confidence level

# F) Out-of-sample forecast
WishForecast <- 0 
ForecastList <- list()
ForecastList$ForHoriz <- 12 # forecast horizon
ForecastList$t0Sample <- 1 # initial sample date
ForecastList$t0Forecast <- 50 # last sample date for the first forecast
ForecastList$ForType <- c()

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

Data <- list()
Data$GVARFactors <- GVARFactors
Data$Wgvar <-  Trade_Flows
mat <- Maturities(Yields, Economies, UnitYields = UnitMatYields)

# 2.1) Generate GVARinputs, JLLinputs and BRWinputs
ModInputs <- ListModelInputs(ModelType, Data, Economies, VARXtype, t_First_Wgvar, t_Last_Wgvar, W_type)
                            
GVARinputs <- ModInputs$GVARinputs
JLLinputs <- NULL

# 3) Prepare the inputs of the likelihood function
# 3.1) Compute the inputs that go directly into the log-likelihood function
ATSMInputs <- InputsForMLEdensity(ModelType, Yields, ZZ, FactorLabels, mat, Economies, DataFrequency, 
                                  JLLinputs, GVARinputs)

# 3.2) Initial guesses for Variables that will be concentrared out of from the log-likelihood function
K1XQ <- ATSMInputs$K1XQ
SSZ <- ATSMInputs$SSZ

# 4) Build the objective function
f <- Functionf(ATSMInputs, Economies, mat, DataFrequency, FactorLabels, ModelType)

# 5) Set the optimization settings
VarLab <- ParaLabels(ModelType, StationarityUnderQ)

varargin <- list()
varargin$K1XQ <-list(K1XQ, VarLab[[ModelType]][["K1XQ"]] , NULL , NULL)
varargin$SSZ <- list(SSZ, VarLab[[ModelType]][["SSZ"]], NULL, NULL)
varargin$r0 <- list(NULL, VarLab[[ModelType]][["r0"]], NULL, NULL)
varargin$se <- list(NULL, VarLab[[ModelType]][["se"]], 1e-6, NULL)
varargin$K0Z <- list(NULL, VarLab[[ModelType]][["K0Z"]], NULL, NULL)
varargin$K1Z <- list(NULL, VarLab[[ModelType]][["K1Z"]], NULL, NULL)
varargin$OptRun <-  c("iter off")

LabelVar<- c('Value', 'Label', 'LB', 'UB') # Elements of each parameter
for (d in 1:(length(varargin)-1)){ names(varargin[[d]]) <-  LabelVar}

tol <- 1e-4
# 6) Optimization of the model
ModelParaList <- list()
ModelParaList[[ModelType]] <- Optimization(f, tol, varargin, FactorLabels, Economies, ModelType, 
                                           JLLinputs, GVARinputs)$Summary

# 7) Numerical Outputs
InputsForOutputs <- InputsForOutputs(ModelType, Horiz, DesiredGraphs, OutputLabel, StationarityUnderQ,
                                     UnitMatYields, WishGraphYields, WishGraphRiskFac, WishOrthoJLLgraphs,
                                     WishForwardPremia, FPmatLim, WishBootstrap, Bootlist, WishForecast,
                                     ForecastList)
# A) Fit, IRF, FEVD, GIRF, GFEVD, and Term Premia
NumericalOutputs <- NumOutputs(ModelType, ModelParaList, InputsForOutputs, FactorLabels, Economies)

# B) Bootstrap
Bootstrap <- Bootstrap(ModelType, ModelParaList, NumericalOutputs, mat, Economies, InputsForOutputs,
                       FactorLabels, DataFrequency, varargin, JLLinputs, GVARinputs)
```

# References

