index.qmd

---
title: ""
author: "Tomasz Woźniak"
email: "tomasz.wozniak@unimelb.edu.au"
title-slide-attributes:
  data-background-color: "#00bf63"
number-sections: false
format: 
  revealjs: 
    footer: "<a href='https://bsvars.github.io'>bsvars.github.io</a>"
    theme: [simple, theme.scss]
    transition: concave
    smaller: true
    multiplex: true
    code-line-numbers: false
execute: 
  echo: true
---

```{r}
#| echo: false
bspink = "#00bf63"
bsyell = "#8c52ff"
bsvars_grad = grDevices::colorRampPalette(c(bspink, bsyell))(5)
```





##  {background-color="#00bf63"}

![](bsvars.png){.absolute top=40 right=540 width="500"}
![](bsvarSIGNs.png){.absolute top=40 right=10 width="500"}





## {background-color="#00bf63"}

$$ $$

### This presentation{style="color:#8c52ff;"}

#### Slides as a [Website](https://bsvars.github.io/2024-08-bsvars-monash/){style="color:#8c52ff;"} {style="color:white;"}

#### GitHub [repo](https://github.com/bsvars/2024-08-bsvars-monash){style="color:#8c52ff;"} to reproduce the slides and results {style="color:white;"}

$$ $$

### Resources{style="color:#8c52ff;"}

#### [bsvars.github.io](https://bsvars.github.io){style="color:#8c52ff;"} {style="color:white;"}

#### [bsvars](https://cran.r-project.org/package=bsvars){style="color:#8c52ff;"} on CRAN {style="color:white;"}

#### [bsvarSIGNs](https://cran.r-project.org/package=bsvarSIGNs){style="color:#8c52ff;"} on CRAN {style="color:white;"}





## {background-color="#00bf63"}

![](social@monash.png){.absolute width="735"}


##  {background-color="#00bf63"}

$$ $$

$$ $$

### [bsvars](https://bsvars.github.io){style="color:#8c52ff;"} features

### [bsvars](https://bsvars.github.io){style="color:#8c52ff;"} models and identification{style="color:white;"}

### [bsvars](https://bsvars.github.io){style="color:#8c52ff;"} modeling of  monetary policy{style="color:white;"}






## [bsvars](https://bsvars.github.io){style="color:#8c52ff;"} features {background-color="#00bf63"}




##

:::: {.columns}

::: {.column width="50%"}
![](bsvars-cran.png){.absolute top=10 width="530"}
:::

::: {.column width="50%"}
![](bsvarSIGNs-cran.png){.absolute top=10 width="530"}
:::

::::



## [bsvars](https://bsvars.github.io){style="color:#8c52ff;"} features {background-color="#00bf63"}

$$ $$

- Bayesian estimation of Structural VARs
- coherent code structure, workflows, and objects
- excellent computational speed
- frontier econometric techniques
- efficient and fast Gibbs samplers
- compiled code using [cpp]{style="color:#8c52ff;"} via [Rcpp](https://cran.r-project.org/package=Rcpp){style="color:#8c52ff;"} and [RcppArmadillo](https://cran.r-project.org/package=RcppArmadillo){style="color:#8c52ff;"}
- data analysis in [R](https://cran.r-project.org/){style="color:#8c52ff;"}





## {background-color="#00bf63"}

:::: {.columns}

::: {.column width="50%"}
![](bsvars.png){width="150"}

- 5 volatility & 3 non-normal models
- identification using 
  - exclusion restrictions
  - heteroskedasticity, and
  - non-normality
- Priors: 3-level eq-specific local-global shrinkage
:::

::: {.column width="50%"}
![](bsvarSIGNs.png){width="150"}

- flexible Bayesian VAR
- identification using 
  - sign restrictions
  - sign & zero restrictions
  - narrative restrictions
- Priors: Minnesota with dummy observation and estimated shrinkage

:::

::::







## {background-color="#00bf63"}

:::: {.columns}

::: {.column width="50%"}
![](bsvars.png){width="150"}

- package and data loading

```{r}
#| eval: false
library(bsvars)
data(us_fiscal_lsuw)
```

- simple model setup

```{r}
#| eval: false
spec = specify_bsvar$new(us_fiscal_lsuw)
```

- simple estimation

```{r}
#| eval: false
burn = estimate(spec, S = 1000)
post = estimate(burn, S = 10000)
```

:::

::: {.column width="50%"}
![](bsvarSIGNs.png){width="150"}

- package and data loading

```{r}
#| eval: false
library(bsvarSIGNs)
data(optimism)
```

- simple model setup

```{r}
#| eval: false
spec = specify_bsvarSIGN$new(optimism)
```

- simple estimation

```{r}
#| eval: false
post = estimate(spec, S = 10000)
```

:::

::::






## {background-color="#00bf63"}

:::: {.columns}

::: {.column width="50%"}
![](bsvars.png){width="150"}

- structural analyses
```{r}
#| eval: false
irfs = compute_impulse_responses(post , horizon = 12)
fevd = compute_variance_decompositions(post, horizon = 12)
hds  = compute_historical_decompositions(post)
ss   = compute_structural_shocks(post)
csds = compute_conditional_sd(post)
sddr = verify_identification(post)
```

- predictive analyses

```{r}
#| eval: false
fvs  = compute_fitted_values(post)
fore = forecast(post, horizon = 12)
```


- plots and summaries

```{r}
#| eval: false
plot(irfs)
summary(irfs)
```

:::
  
::: {.column width="50%"}
![](bsvarSIGNs.png){width="150"}


- structural analyses
```{r}
#| eval: false
irfs = compute_impulse_responses(post , horizon = 12)
fevd = compute_variance_decompositions(post, horizon = 12)
hds  = compute_historical_decompositions(post)
ss   = compute_structural_shocks(post)
csds = compute_conditional_sd(post)
```

- predictive analyses

```{r}
#| eval: false
fvs  = compute_fitted_values(post)
fore = forecast(post, horizon = 12)
```

- plots and summaries

```{r}
#| eval: false
plot(irfs)
summary(irfs)
```

:::
  
::::



## {background-color="#00bf63"}

:::: {.columns}

::: {.column width="50%"}
![](bsvars.png){width="150"}

- workflow with the pipe

```{r}
#| eval: false
library(bsvars)
data(us_fiscal_lsuw)

us_fiscal_lsuw |> 
  specify_bsvar$new() |> 
  estimate(S = 1000) |> 
  estimate(S = 10000) -> post

post |> compute_impulse_responses(horizon = 12) |> plot()
post |> compute_variance_decompositions(horizon = 12) |> plot()
post |> compute_historical_decompositions() |> plot()
post |> compute_structural_shocks() |> plot()
post |> compute_conditional_sd() |> plot()
post |> forecast(horizon = 12) |> plot()
post |> verify_identification() |> summary()
```
:::
  
::: {.column width="50%"}
![](bsvarSIGNs.png){width="150"}

- workflow with the pipe

```{r}
#| eval: false
library(bsvarSIGNs)
data(optimism)

optimism |> 
  specify_bsvarSIGN$new() |> 
  estimate(S = 10000) -> post

post |> compute_impulse_responses(horizon = 12) |> plot()
post |> compute_variance_decompositions(horizon = 12) |> plot()
post |> compute_historical_decompositions() |> plot()
post |> compute_structural_shocks() |> plot()
post |> compute_conditional_sd() |> plot()
post |> forecast(horizon = 12) |> plot()
```
:::
  
::::


## {background-color="#00bf63"}

:::: {.columns}

::: {.column width="50%"}
![](bsvars.png){width="150"}

- progress bar

![](bsvars_progress.png){width="500"}
:::
  
::: {.column width="50%"}
![](bsvarSIGNs.png){width="150"}

- progress bar

![](bsvarSIGNs_progress.png){width="500"}
:::
  
::::








## [bsvars](https://bsvars.github.io){style="color:#8c52ff;"} models and identification {background-color="#00bf63"}






## 

![](bsvars.png){.absolute right=10 width="120"}

$$ $$

### Structural VAR
\begin{align}
\text{reduced form:}&&\mathbf{y}_t &= \mathbf{A}\mathbf{x}_t + \boldsymbol{\varepsilon}_t \\
\text{structural form:}&&\mathbf{B}_0\boldsymbol{\varepsilon}_t &= \mathbf{u}_t \\
\text{structural shocks:}&&\mathbf{u}_t\mid\mathbf{x}_t &\sim N\left( \mathbf{0}_N, \text{diag}\left(\boldsymbol{\sigma}_t^2\right) \right)
\end{align}

- interpretable structural specification
- identification through 
  - exclusion restrictions
  - heteroskedasticity
  - non-normality
- facilitates application of frontier numerical techniques
- features 3-level equation-specific local-global prior shrinkage






## 

![](bsvarSIGNs.png){.absolute right=10 width="120"}

$$ $$

### Structural VAR
\begin{align}
\text{reduced form:}&&\mathbf{y}_t &= \mathbf{A}\mathbf{x}_t + \boldsymbol{\varepsilon}_t \\
\text{error term:}&&\boldsymbol{\varepsilon}_t\mid\mathbf{x}_t &\sim N\left( \mathbf{0}_N, \boldsymbol{\Sigma} \right)\\
\text{rotation:}&&\mathbf{Q} &\sim Haar \\
\text{structural form:}&&\left(\mathbf{A}, \boldsymbol{\Sigma}, \mathbf{Q}\right) &\rightarrow\left(\mathbf{A}, \mathbf{B}_0\right)
\end{align}

- identification through 
  - sign & zero restrictions on structural matrix or impulse responses
  - narrative restrictions on shocks
  - all at once!
- facilitates independent sampling & application of frontier numerical techniques
- features Minnesota and dummy observation priors with estimated shrinkage




## 

![](bsvars.png){.absolute right=10 width="120"}

$$ $$

### Non-centred Stochastic Volatility

\begin{align}
\text{conditional variance:}&&\sigma_{n.t}^2 &= \exp\left\{\omega_n h_{n.t}\right\}\\
\text{log-volatility:}&&h_{n.t} &= \rho_n h_{n.t-1} + v_{n.t}\\ 
\text{volatility innovation:}&&v_{n.t}&\sim N\left(0,1\right)\\
\end{align}

- excellent volatility forecasting performance
- standardization around $\sigma_{n.t}^2 = 1$
- `verify_identification()` by checking $H_0:\quad\omega_n = 0$




## 

![](bsvars.png){.absolute right=10 width="120"}

### Stochastic Volatility: conditional variances


```{r plot_cv_prior}
#| echo: false
p_svnc_log      = function(x,t){
  # log-variances non-centered SV
  (pi*sqrt(t*sigma.omega.sq))^(-1)*besselK(abs(x)/sqrt(t*sigma.omega.sq),0)
}
p_sv_log        = function(x,t){
  # log-variances SV
  gamma((sigma.nu+1)/2)/gamma((sigma.nu)/2)*pi^(-1/2)*((t+1)*sigma.s)^(-1/2)*(1+((t+1)*sigma.s)^(-1)*x^(2))^(-(sigma.nu+1)/2)
}
p_svnc          = function(x,t){
  # variances non-centered SV
  (pi*sqrt(t*sigma.omega.sq)*x)^(-1)*besselK(abs(log(x))/sqrt(t*sigma.omega.sq),0)
}
p_sv            = function(x,t){
  # variances SV
  gamma((sigma.nu+1)/2)/gamma((sigma.nu)/2)*pi^(-1/2)*sigma.s^(sigma.nu/2)*(t+1)^(-1/2)*(x)^(-1)*(sigma.s+(t+1)^(-1)*(log(x))^(2))^(-(sigma.nu+1)/2)
}

T               = 5
zlimabrar       = 2.5
grid            = seq(from=-2.5, to=2.5, by=0.00001)
grid_var        = seq(from=0.00000001, to=3.2, by=0.00001)

s               = 0.1
sigma.omega.sq  = s   # conditional variance hyper-parameter
sigma.s         = s   # log-conditional variance hyper-parameter
sigma.nu        = 3   # log-conditional variance hyper-parameter

plot(
  x = grid_var, 
  y = p_svnc(grid_var,T), 
  type = "l", 
  main = "", 
  col = bspink, 
  lwd = 2,
  xlim = c(0,3), 
  ylim = c(0,zlimabrar), 
  xlab = "conditional variance", 
  ylab = "density", 
  frame.plot = FALSE, 
  axes = FALSE
)
lines(
  x = grid_var, 
  y = p_sv(grid_var, T), 
  col = bsyell,
  lwd = 2
)
legend(
  "topright",
  legend = c("non-centered", "centered"),
  col = c(bspink, bsyell),
  lwd = 2,
  bty = "n"
)
axis(2, c(0,1,2), c(0,1,2))
axis(1, c(0,1,2,3), c(0,1,2,3))
abline(h = 0, lwd = 0.5)
```







## 

![](bsvars.png){.absolute right=10 width="120"}

$$ $$

### Markov-switching and mixture models.

\begin{align}
\text{structural shocks:}&&\mathbf{u}_t\mid s_t &\sim N\left( \mathbf{0}_N, \text{diag}\left(\boldsymbol{\sigma}_{s_t}^2\right) \right)\\
\text{prior & standardisation:}&& M^{-1}\left(\boldsymbol{\sigma}_{1}^2, \dots, \boldsymbol{\sigma}_{M}^2\right) &\sim Dirichlet(\underline{a}\boldsymbol\imath')\\
\text{latent process:}&& s_t&\sim \text{Markov or independent}
\end{align}

- modeling Markov-switching or mixture models
- non-normality provides identification information
- potential gains in forecasting precision
- `verify_identification()`  by checking $H_0:\quad\boldsymbol{\sigma}_{1}^2, \dots, \boldsymbol{\sigma}_{M}^2 = 1$





## 

![](bsvars.png){.absolute right=10 width="120"}

$$ $$

### Student-t shocks.

\begin{align}
\text{structural shocks:}&&\mathbf{u}_t\mid\mathbf{x}_t &\sim t\left( \mathbf{0}_N, \mathbf{I}_N, \nu \right)
\end{align}

- $\nu$ - the degrees of freedom parameter is estimated
- fat tails provide identification information
- potential gains in forecasting precision
- robustness to outliers
- `verify_identification()`  by checking $H_0:\quad\nu \rightarrow\infty$











## {background-color="#00bf63"}

![](bsvars.png){.absolute right=10 width="120"}

### Student-t shocks.{style="color:#FFFFFF;"}
```{r}
#| echo: false
set.seed(1)

ax_lim = 3.6
T = 500
df = 3
B = matrix(c(1,-1,1,1), 2, 2)
Bit = t(solve(B))
en = matrix(rnorm(2 * T), T, 2)
et = sqrt((df - 2) / df) * matrix(rt(2 * T, df = df), T, 2)

yn = en %*% Bit
yt = et %*% Bit

par(
  bg = "#00bf63",
  mfrow = c(1,2),
  col = scales::alpha("#8c52ff", .5),
  col.main = "#8c52ff",
  col.lab = "#8c52ff"
)
plot(
  x = yn[,1], y = yn[,2], 
  ylim = c(-ax_lim, ax_lim), 
  xlim = c(-ax_lim, ax_lim),
  bty="n", pch = 16,
  ylab = "y", xlab = "x",
  axes = FALSE,
  main = "normal"
)
abline(a = 0, b = 1, col = "#8c52ff")
abline(a = 0, b = -1, col = "#8c52ff")

plot(
  x = yt[,1], y = yt[,2], 
  ylim = c(-ax_lim, ax_lim), 
  xlim = c(-ax_lim, ax_lim),
  bty="n", pch = 16,
  ylab = "", xlab = "x",
  axes = FALSE,
  main = "non-normal"
)
abline(a = 0, b = 1, col = "#8c52ff")
abline(a = 0, b = -1, col = "#8c52ff")

```




## 

![](bsvars.png){.absolute right=10 width="120"}

$$ $$

### Identification verification via SDDR.

Verify the restriction through the posterior odds ratio using the SDDR:
$$
SDDR = \frac{\Pr[H_0 | data]}{\Pr[H_1 | data]}= \frac{p(H_0 | data)}{p(H_0 )}
$$

- suitable to verify sharp restrictions on parameters
- is interpreted as posterior odds ratio
- values greater than 1 provide evidence in favour of the restriction
- simple to compute given the unrestricted model estimation output
- see methods `verify_identification()` and `verify_autoregression()`






## [bsvars](https://bsvars.github.io){style="color:#8c52ff;"} modeling of  monetary policy{background-color="#00bf63"}



## [bsvars](https://bsvars.github.io){style="color:#8c52ff;"} modeling of  monetary policy

### Australian domestic sector.

Consider a system of five domestic variables:

\begin{align}
y_t = \begin{bmatrix} rgdp_t & cpi_t & CR_t & EX_t & aord_t \end{bmatrix}'
\end{align}

- $rgdp_t$ - log real Gross Domestic Product
- $cpi_t$ - log Consumer Price Index
- $CR_t$ - Cash Rate Target - Australian nominal interest rate
- $EX_t$ - USD/AUD exchange rate
- $aord_t$ - log All Ordinaries Index

- monthly data from August 1990 to March 2024
- quarterly variables interpolated to monthly frequency




## [bsvars](https://bsvars.github.io){style="color:#8c52ff;"} modeling of  monetary policy

### Foreign sector.

The foreign sector includes three US variables:

- $rgdp_t^{(US)}$ - log real Gross Domestic Product
- $cpi_t^{(US)}$ - log Consumer Price Index
- $FFR_t$ - Cash Rate Target - Australian nominal interest rate
- monthly data from August 1990 to March 2024
- quarterly variables interpolated to monthly frequency
- contemporaneous and four lagged values are included in the model as exogenous variables



## 

![](bsvars.png){.absolute right=10 width="120"}

$$ $$

### Zero-restriction-identified system.

\begin{align}
\begin{bmatrix}
B_{0.11}&0&0&0&0\\
B_{0.21}&B_{0.22}&0&0&0\\
B_{0.31}&B_{0.32}&B_{0.33}&0&0\\
B_{0.41}&B_{0.42}&B_{0.43}&B_{0.44}&0\\
B_{0.51}&B_{0.52}&B_{0.53}&B_{0.54}&B_{0.55}
\end{bmatrix}
\begin{bmatrix}rgdp_t \\ cpi_t \\ CR_t \\ EX_t\\ aord_t \end{bmatrix} &= \dots +
\begin{bmatrix} u_t^{ad} \\ u_t^{as} \\ u_t^{mps} \\ u_t^{ex} \\ u_t^{aord} \end{bmatrix}
\end{align}

### Identified shocks.

- $u_t^{mps}$ - identified via Taylor's Rule extended by exchange rate
- $u_t^{ex}$ - currency shock indistinguishable from the monetary policy shock






## 

![](bsvars.png){.absolute right=10 width="120"}

$$ $$

### Specify and estimate the SVAR-SV.

```{r spec_lt}
#| eval: false

library(bsvars)
load("soe.rda")
soe           = as.matrix(soe)

TT            = nrow(soe)
lag_order     = 8
lag_exogenous = 4
T             = TT - max(lag_order, lag_exogenous)

exogenous     = matrix(NA, TT - lag_exogenous, 0)
for (i in 0:lag_exogenous) {
  exogenous   = cbind(exogenous, as.matrix(soe[(lag_exogenous - i + 1):(TT - i), 6:8]))
}

set.seed(1234)
spec          = specify_bsvar_sv$new(
  data        = tail(soe[,1:5], T),
  p           = lag_order,
  exogenous   = tail(exogenous, T)
)

burn          = estimate(spec, 1e4)
post          = estimate(burn, 1e4)
```


```{r load00}
#| echo: false
#| cache: true
library(bsvars)
load("bsvars01.rda")
```


## 

![](bsvars.png){.absolute right=10 width="120"}

$$ $$

### SVAR-SV: Compute and plot impulse responses.

```{r bs00}
#| cache: true
post |> compute_impulse_responses(horizon = 60) |> plot(probability = 0.68, col = "#00bf63")

```


## 

![](bsvars.png){.absolute right=10 width="120"}

$$ $$

### SVAR-SV: Forecast error variance decompositions.

```{r bs00fevd}
#| cache: true
post |> compute_variance_decompositions(horizon = 60) |> plot(col = bsvars_grad)

```


## 

![](bsvars.png){.absolute right=10 width="120"}

$$ $$

### SVAR-SV: Structural shocks.

```{r bs00sss}
#| cache: true
post |> compute_structural_shocks() |> plot(col = "#00bf63")

```



## 

![](bsvars.png){.absolute right=10 width="120"}

$$ $$

### SVAR-SV: Shocks' conditional standard deviations.

```{r bs00csd}
#| cache: true
post |> compute_conditional_sd() |> plot(col = "#00bf63")

```


## 

![](bsvars.png){.absolute right=10 width="120"}

$$ $$

### SVAR-SV: identification verification.

```{r bs00vv}
#| cache: true
post |> verify_identification() |> summary()

```



## 

![](bsvars.png){.absolute right=10 width="120"}

$$ $$

### SVAR-SV: Fitted Values.

```{r bs00fit}
#| cache: true
post |> compute_fitted_values() |> plot(col = "#00bf63")

```



## 

![](bsvars.png){.absolute right=10 width="120"}

$$ $$

### SVAR-SV: Does foreign sector matter?

```{r bs00va}
#| cache: true

A0 = matrix(NA, 5, 56)
A0[,45:56] = 0
post |> verify_autoregression(hypothesis = A0) |> summary()

```





## 

![](bsvarSIGNs.png){.absolute right=10 width="120"}

$$ $$

### Sign-restricted system.

\begin{align}
\begin{bmatrix}rgdp_t \\ cpi_t \\ CR_t \\ EX_t\\ aord_t \end{bmatrix} &= \dots +
\begin{bmatrix}
-&*&*&*&*\\
-&*&*&*&*\\
+&*&*&*&*\\
*&*&*&*&*\\
*&*&*&*&*\\
\end{bmatrix}\begin{bmatrix} u_t^{ad} \\ u_t^{as} \\ u_t^{mps} \\ u_t^{ex} \\ u_t^{aord} \end{bmatrix}
\end{align}

### Identified shocks.

- $u_t^{mps}$ - identified via Taylor's Rule extended by exchange rate
- $u_t^{ex}$ - currency shock indistinguishable from the monetary policy shock
- identification through heteroskedasticity helps identifying the shocks





## 

![](bsvarSIGNs.png){.absolute right=10 width="120"}

$$ $$

### Specify and estimate a sign-restricted SVAR.

```{r spec01}
#| eval: false

sign_irf0     = matrix(c(-1, -1, 1, rep(NA, 22)), 5, 5)
sign_irf      = array(NA, c(5, 5, 6))
for (i in 1:6) sign_irf[,,i] = sign_irf0

set.seed(1234)
spec          = specify_bsvarSIGN$new(
  data        = tail(soe[,1:5], T),
  p           = lag_order,
  exogenous   = tail(exogenous, T),
  sign_irf    = sign_irf
)

post          = estimate(spec, 1e4)

sign_irf0
```


```{r spec011}
#| echo: false

sign_irf0     = matrix(c(-1, -1, 1, rep(NA, 22)), 5, 5)
sign_irf0
```



```{r load01}
#| echo: false
#| cache: true
library(bsvarSIGNs)
load("bsvars07.rda")
```





## 

![](bsvarSIGNs.png){.absolute right=10 width="120"}

$$ $$

### SVAR: Compute and plot impulse responses.

```{r bs01}
#| cache: true
post |> compute_impulse_responses(horizon = 60) |> plot(probability = 0.68, col = "#00bf63" )
```


## 

![](bsvarSIGNs.png){.absolute right=10 width="120"}

$$ $$

### SVAR: Forecast error variance decompositions.

```{r bs01fevd}
#| cache: true
post |> compute_variance_decompositions(horizon = 60) |> plot(col = bsvars_grad )
```


## 

![](bsvarSIGNs.png){.absolute right=10 width="120"}

$$ $$

### SVAR: Structural shocks.

```{r bs01sss}
#| cache: true
post |> compute_structural_shocks() |> plot(col = "#00bf63" )
```



## 

![](bsvarSIGNs.png){.absolute right=10 width="120"}

$$ $$

### SVAR: Fitted values.

```{r bs01ff}
#| cache: true
post |> compute_fitted_values() |> plot(col = "#00bf63" )
```





##  {background-color="#00bf63"}

![](bsvars.png){.absolute top=40 right=540 width="500"}
![](bsvarSIGNs.png){.absolute top=40 right=10 width="500"}