index.qmd

---
title: ""
author: "Tomasz Woźniak"
email: "tomasz.wozniak@unimelb.edu.au"
title-slide-attributes:
  data-background-color: "#0056B9"
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 = "#0056B9"
bsyell = "#FFD800"
bsvars_grad = grDevices::colorRampPalette(c(bspink, bsyell))(5)
```




##  {background-color="#0056B9"}

![](bsvars_ukr.png){.absolute top=40 right=275 width="500"}


##  {background-color="#0056B9"}
<!-- FFD800 -->

$$ $$

$$ $$

### [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} package features {style="color:white;"}

### [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} models and identification{style="color:white;"}

### [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy{style="color:white;"}


## {background-color="#0056B9"}

$$ $$

### Slides [as a Website](https://bsvars.github.io/2024-08-bsvars-w4UKR/){style="color:#FFD800;"} {style="color:white;"}

### GitHub [repo](https://github.com/bsvars/2024-08-bsvars-w4UKR){style="color:#FFD800;"} to reproduce the slides and results {style="color:white;"}

### [**R** script](https://github.com/bsvars/2024-08-bsvars-w4UKR/blob/main/w4UKR_empirical.R){style="color:#FFD800;"} to reproduce empirical results {style="color:white;"}

$$ $$

### [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} on CRAN{style="color:white;"}

### [bsvars.github.io/bsvars/](https://bsvars.github.io/bsvars/){style="color:#FFD800;"} {style="color:white;"}

### [bsvars.github.io](https://bsvars.github.io){style="color:#FFD800;"} {style="color:white;"}



## {background-color="#0056B9"}

![](social@w4UKR.png){.absolute top=0 right=200 width="735"}



## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} package features {background-color="#0056B9"}




##

![](cran.png){.absolute top=10 right=30 width="1000"}


## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} package features

$$ $$

### - Bayesian estimation of Structural VARs

### - 5 volatility & 3 non-normal models for shocks

### - identification using 

####   - exclusion restrictions
####   - heteroskedasticity
####   - and non-normality

### - efficient and fast Gibbs samplers



## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} package features

$$ $$

### - excellent computational speed

### - frontier econometric techniques

### - compiled code using [cpp]{style="color:#FFD800;"} via [Rcpp](https://cran.r-project.org/package=Rcpp){style="color:#FFD800;"} and [RcppArmadillo](https://cran.r-project.org/package=RcppArmadillo){style="color:#FFD800;"}

### - data analysis in [R](https://cran.r-project.org/){style="color:#FFD800;"}





## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} package features

$$ $$

### - 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)
```






## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} package features

$$ $$

### - 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)
```



## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} package features

$$ $$

### - 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()
```




## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} package features

$$ $$

### - specify different models

```{r}
#| eval: false
spec1 = specify_bsvar$new(us_fiscal_lsuw, p = 4)                  # SVAR(4)
spec2 = specify_bsvar_sv$new(us_fiscal_lsuw)                      # SVAR(1)-SV
spec3 = specify_bsvar_t$new(us_fiscal_lsuw, p = 2)                # SVAR(2)-t
spec4 = specify_bsvar_msh$new(us_fiscal_lsuw, M = 2)              # SVAR(1)-MSH(2)
spec5 = specify_bsvar_msh$new(us_fiscal_lsuw, finiteM = FALSE)    # SVAR(1)-MSH(Inf)
spec6 = specify_bsvar_mix$new(us_fiscal_lsuw, M = 2)              # SVAR(1)-MIX(2)
spec7 = specify_bsvar_mix$new(us_fiscal_lsuw, finiteM = FALSE)    # SVAR(1)-MIX(Inf)
```

### - from now on it's the same, e.g.


```{r}
#| eval: false
spec1 |> 
  estimate(S = 1000) |> 
  estimate(S = 10000) -> post

spec2 |> 
  estimate(S = 1000) |> 
  estimate(S = 10000) -> post
```








## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} package features

### - progress bar

![](bsvars_progress.png){.absolute top=150 right=100 width="900"}




## My first steps with [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"}

### Exercise.

1. Download **R** script [w4UKR_bsvars.R](https://github.com/bsvars/2024-08-bsvars-w4UKR/blob/main/w4UKR_bsvars.R)
2. Open the file in your **RStudio**
3. Execute the code line by line

#### You have just estimated your first model using the **bsvars** package! YAY!

4. Modify the line specifying the model to 
```
specify_bsvar_t$new(p = 4) |>
```
5. Run the code line by line



## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} models and identification {background-color="#0056B9"}




## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} models

$$ $$

### Structural VAR
\begin{align}
\text{reduced form:}&&\mathbf{y}_t &= \mathbf{A}_1\mathbf{y}_{t-1} + \dots + \mathbf{A}_p\mathbf{y}_{t-p} + \mathbf{A}_d\mathbf{d}_{t} + \boldsymbol{\varepsilon}_t \\
\text{structural form:}&&\mathbf{B}_0\boldsymbol{\varepsilon}_t &= \mathbf{u}_t
\end{align}

- system modelling of dependent variables $\mathbf{y}_t$
- system dynamics captured by modeling effects of lags $\mathbf{y}_{t-1},\dots,\mathbf{y}_{t-p}$
- deterministic terms and exogenous variables $\mathbf{d}_{t}$
- economic structure introduced by structural matrix $\mathbf{B}_0$
- well-isolated structural shocks $\mathbf{u}_t$





## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} models

$$ $$

### 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


## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} models

### Reduced form hierarchical prior
\begin{align}
\text{autoregressive slopes:}&& [\mathbf{A}]_{n\cdot}'\mid\gamma_{A.n} &\sim N_{Np+1}\left( \underline{\mathbf{m}}_{n.A}, \gamma_{A.n}\underline{\Omega}_A \right) \\
\text{autoregressive shrinkage:}&&\gamma_{A.n} | s_{A.n}  &\sim IG2\left(s_{A.n}, \underline{\nu}_A\right)\\
\text{local scale:}&&s_{A.n} | s_{A} &\sim G\left(s_{A}, \underline{a}_A\right)\\
\text{global scale:}&&s_{A} &\sim IG2\left(\underline{s}_{s_A}, \underline{\nu}_{s_A}\right)
\end{align}

- Minnesota prior mean and shrinkage decay with increasing lags
- Flexibility in shrinkage and scale hyper-parameters
- 3-level equation-specific local-global hierarchical prior




## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} models

### Structural form hierarchical prior
\begin{align}
\text{exclusion restrictions:}&& [\mathbf{B}_0]_{n\cdot} &= \mathbf{b}_n\mathbf{V}_n\\
\text{structural relations:}&& \mathbf{B}_0\mid\gamma_{B}&\sim |\det(\mathbf{B}_0)|^{\underline{\nu}_B - N}\exp\left\{-\frac{1}{2} \sum_{n=1}^{N} \gamma_{B.n}^{-1} \mathbf{b}_n\mathbf{b}_n' \right\} \\
\text{structural shrinkage:}&&\gamma_{B.n} | s_{B.n}  &\sim IG2\left(s_{B.n}, \underline{\nu}_b\right)\\
\text{local scale:}&&s_{B.n} | s_{B} &\sim G\left(s_{B}, \underline{a}_B\right)\\
\text{global scale:}&&s_{B} &\sim IG2\left(\underline{s}_{s_B}, \underline{\nu}_{s_B}\right)
\end{align}

- Highly adaptive equation-by-equation exclusion restrictions
- Likelihood-shape preserving prior
- Flexibility in shrinkage and scale hyper-parameters
- 3-level equation-specific local-global hierarchical prior



## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} models



### Volatility models

- Homoskedastic $\sigma_{n.t}^2 = 1$
- Stochastic Volatility: non-centred and centred
- Stationary Markov-switching heteroskedastisity
- Sparse Markov-switching heteroskedastisity

### Non-normal models

- Student-t shocks
- Finite mixture of normal components
- Sparse mixture of normal components


## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} models

$$ $$

### 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 $\sigma_{n.t}^2 = 1$
- homoskedasticity verification by testing $\omega_n = 0$


## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} models

$$ $$

### Centred Stochastic Volatility

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

- excellent volatility forecasting performance
- weak standardisation
- no homoskedasticity verification available


## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} models

### Stochastic Volatility: conditional variance prior


```{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](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} identification (simplified)

\begin{align}
&\\
\mathbf\Sigma &= \mathbf{B}_0^{-1}\mathbf{B}_0^{-1\prime}\\[1ex]
\end{align}

- $\mathbf\Sigma$ can be estimated using data easily
- The relationship presents a system of equations to be solved for $\mathbf{B}_0$
- $\mathbf\Sigma$ is a *symmetric* $N\times N$ matrix
- $\mathbf\Sigma$ has $N(N+1)/2$ unique elements (equations)
- $\mathbf{B}_0$ is an $N\times N$ matrix with $N^2$ unique elements to estimate
- We cannot estimate all elements of $\mathbf{B}_0$ using $N(N+1)/2$ equations
- $\mathbf{B}_0$ is <text style="color:#0056B9;">**not identified**</text>



## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} identification (simplified)

\begin{align}
&\\
\mathbf\Sigma &= \mathbf{B}_0^{-1}\mathbf{B}_0^{-1\prime}\\[1ex]
\end{align}

### Identification.

- Only $N(N+1)/2$ elements in $\mathbf{B}_0$ can be estimated
- Impose $N(N-1)/2$ restrictions on $\mathbf{B}_0$ to solve the equation
- This identifies the rows of $\mathbf{B}_0$ up to a sign
- Change the sign of any number of $\mathbf{B}_0$ rows and $\mathbf\Sigma$ will not change
- Often $\mathbf{B}_0$ is made lower-triangular



## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} identification (simplified)

Let $N=2$ 

\begin{align}
\begin{bmatrix}\sigma_1^2&\sigma_{12}\\ \sigma_{12}&\sigma_2^2\end{bmatrix} &\qquad
\begin{bmatrix}B_{0.11}&B_{0.12}\\ B_{0.21}&B_{0.22}\end{bmatrix}\\[1ex]
\end{align}

- 3 unique elements in $\mathbf\Sigma$ - 3 equations in the system
- 4 elements in $\mathbf{B}_0$ cannot be estimated

### Identification.

\begin{align}
\begin{bmatrix}\sigma_1^2&\sigma_{12}\\ \sigma_{12}&\sigma_2^2\end{bmatrix} &\qquad
\begin{bmatrix}B_{0.11}& 0\\ B_{0.21}&B_{0.22}\end{bmatrix}\\[1ex]
\end{align}

- 3 equations identify 3 elements in $\mathbf{B}_0$



## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} identification (simplified)

### Identification through Heteroskedasticity.

Suppose that: 

- there are two covariances, $\mathbf\Sigma_1$ and $\mathbf\Sigma_2$, associated with the sample
- matrix $\mathbf{B}_0$ does not change over time
- structural shocks are heteroskedastic with covariances $\text{diag}\left(\boldsymbol\sigma_1^2\right)$ and $\text{diag}\left(\boldsymbol\sigma_2^2\right)$

\begin{align}
\mathbf\Sigma_1 &= \mathbf{B}_0^{-1}\text{diag}\left(\boldsymbol\sigma_1^2\right)\mathbf{B}_0^{-1\prime}\\[1ex]
\mathbf\Sigma_2 &= \mathbf{B}_0^{-1}\text{diag}\left(\boldsymbol\sigma_2^2\right)\mathbf{B}_0^{-1\prime}
\end{align}



## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} identification (simplified)

### Identification through Heteroskedasticity.
\begin{align}
\mathbf\Sigma_1 &= \mathbf{B}_0^{-1}\text{diag}\left(\boldsymbol\sigma_1^2\right)\mathbf{B}_0^{-1\prime}\\[1ex]
\mathbf\Sigma_2 &= \mathbf{B}_0^{-1}\text{diag}\left(\boldsymbol\sigma_2^2\right)\mathbf{B}_0^{-1\prime}
\end{align}


### Identification.

- $\mathbf\Sigma_1$ and $\mathbf\Sigma_2$ contain $N^2+N$ unique elements
- All $N^2$ elements of $\mathbf{B}_0$ can be estimated
- Both $N$-vectors $\boldsymbol\sigma_1^2$ and $\boldsymbol\sigma_2^2$ can be estimated due to additional restriction: $\boldsymbol\sigma_i^2 \approx \boldsymbol\imath_N$


## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} identification (simplified)

### Identification through Heteroskedasticity.

The setup can be generalised to conditional heteroskedasticity of structural shocks

\begin{align}
u_t |Y_{t-1} &\sim N_N\left(\mathbf{0}_N, \text{diag}\left(\boldsymbol\sigma_t^2\right)\right)\\[1ex]
\mathbf\Sigma_t &= \mathbf{B}_0^{-1}\text{diag}\left(\boldsymbol\sigma_t^2\right)\mathbf{B}_0^{-1\prime}
\end{align}

### Identification.

- Matrix $\mathbf{B}_0$ is identified up to its rows' sign change and equations' reordering
- shocks are identified if changes in their conditional variances are non-proportional 
- Structural shocks' conditional variances $\boldsymbol\sigma_t^2$ can be estimated
- Choose any (conditional) variance model for $\boldsymbol\sigma_t^2$ that fits the data well.






## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} identification (simplified)

### Identification through Non-normality.

\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 $\nu \rightarrow\infty$

## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} identification (simplified) {background-color="#0056B9"}


### Identification through Non-normality.{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 = "#0056B9",
  mfrow = c(1,2),
  col = scales::alpha("#FFD800", .5),
  col.main = "#FFD800",
  col.lab = "#FFD800"
)
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 = "#FFD800")
abline(a = 0, b = -1, col = "#FFD800")

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 = "Student-t"
)
abline(a = 0, b = 1, col = "#FFD800")
abline(a = 0, b = -1, col = "#FFD800")

```




## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} identification verification

$$ $$

### Savage-Dickey Density Ratio.

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 favor of the restriction
- simple to compute given the unrestricted model estimation output




## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} identification verification

### ...in non-centred Stochastic Volatility.

A structural shock is homoskedastic if
$$
H_0:\quad\omega_n = 0
$$

$$ $$

* if $\omega_n = 0$ the shock is homoskedastic with $\sigma_{nt}^2 = 1$
* the only homoskedastic shock in the system is identified
* two or more homoskedastic shocks have not identified through heteroskedasticity
* heteroskedastic shocks are identified with probability 1
* use the `verify_identification()` function





## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} identification verification

### ...in t-distributed shocks model.

A structural shock is homoskedastic if
$$
H_0:\quad\nu \rightarrow\infty
$$

$$ $$

* if $\nu \rightarrow\infty$ all the shocks are normal
* when $\nu$ is finite and less than 30, the shocks are leptokurtic
* then all the shocks are identified through non-normality
* SDDR derivations and computation require some fancy stats
* use the `verify_identification()` function





## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} hypothesis verification

### ...for autoregressive parameters.

A structural shock is homoskedastic if
$$
H_0:\quad\mathbf{S}\text{vec}(\mathbf{A})  = \mathbf{r}
$$

$$ $$

* verify restrictions on the autoregressive parameters
* can be used, e.g., to verify Granger causality
* use the `verify_autoregression()` function




## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} hypothesis verification

### ...for autoregressive parameters.

### Exercise.

1. Download the **R** script [w4UKR_bsvars_Granger.R](https://github.com/bsvars/2024-08-bsvars-w4UKR/blob/main/w4UKR_bsvars_Granger.R)
2. Verify the hypothesis of no Granger causality from ttr to gdp


## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy   {background-color="#0056B9"}






## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy

### Domestic economy.

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://cran.r-project.org/package=bsvars){style="color:#FFD800;"} 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](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy

### Lower-triangular 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.

- $u_t^{mps}$ - monetary policy shock identified via Taylor's Rule
- $u_t^{ex}$ - currency shock




<!-- ## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy -->

<!-- ### Extended 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}&B_{0.34}&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 -->
<!-- - identification through heteroskedasticity helps identifying the shocks -->



## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy

### Specify and estimate a lower-triangular 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
load("empiRical/bsvars01.rda")
```


## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy

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

```{r bs00}
#| cache: true
#| fig-align: center
#| fig-width: 8
post |> compute_impulse_responses(horizon = 60) |> plot(probability = 0.68, col = "#0056B9")
```


## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy

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

```{r bs00fevd}
#| cache: true
#| fig-align: center
#| fig-width: 8
post |> compute_variance_decompositions(horizon = 60) |> plot(col = bsvars_grad)
```


## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy

### SVAR-SV: Structural shocks.

```{r bs00sss}
#| cache: true
#| fig-align: center
#| fig-width: 8
post |> compute_structural_shocks() |> plot(col = "#0056B9")
```



## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy

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

```{r bs00csd}
#| cache: true
#| fig-align: center
#| fig-width: 8
post |> compute_conditional_sd() |> plot(col = "#0056B9")
```


## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy

### SVAR-SV: Homoskedasticity verification.

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



## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy

### SVAR-SV: Fitted Values.

```{r bs00fit}
#| cache: true
#| fig-align: center
#| fig-width: 8
post |> compute_fitted_values() |> plot(col = "#0056B9")
```



## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy

### 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()
```



<!-- ## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy -->

<!-- ### Forecasting. -->

<!-- ```{r bs00for} -->
<!-- #| cache: true -->

<!-- post |> forecast(horizon = 24) |> plot(probability = 0.68, col = "#0056B9") -->
<!-- ``` -->









## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy

### Specify and estimate a lower-triangular SVAR-t.

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

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

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

```{r load01}
#| echo: false
#| cache: true
load("empiRical/bsvars06.rda" )
```

## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy

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

```{r bs01}
#| cache: true
#| fig-align: center
#| fig-width: 8
post |> compute_impulse_responses(horizon = 60) |> plot(probability = 0.68, col = "#0056B9" )
```


## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy

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

```{r bs01fevd}
#| cache: true
#| fig-align: center
#| fig-width: 8
post |> compute_variance_decompositions(horizon = 60) |> plot(col = bsvars_grad )
```


## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy

### SVAR-t: Structural shocks.

```{r bs01sss}
#| cache: true
#| fig-align: center
#| fig-width: 8
post |> compute_structural_shocks() |> plot(col = "#0056B9" )
```




## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy

### SVAR-t: Normality verification.

```{r bs01vv}
#| cache: true
#| fig-align: center
#| fig-width: 8
post |> verify_identification() |>  summary()
```




## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy

### SVAR-t: Degrees-of-freedom posterior density.

```{r bs01df}
#| cache: true
#| fig-align: center
#| fig-width: 8
post$posterior$df |> hist(breaks = 100, col = "#0056B9", border = "#0056B9", bty = "n" )
```





## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy

### SVAR-t: Fitted Values.

```{r bs01fit}
#| cache: true
#| fig-align: center
#| fig-width: 8
post |> compute_fitted_values() |> plot(col = "#0056B9")
```





##  {background-color="#0056B9"}

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