index.qmd

---
title: "<span style='color: #ffd700;'>Bayesian Structural Vector Autoregressions</span>" 
author: "<span style='color: #ffd700;'>by Tomasz Woźniak</span>"
email: "tomasz.wozniak@unimelb.edu.au"
title-slide-attributes:
  data-background-color: "#ff69b4"
number-sections: false
format: 
  revealjs: 
    theme: [simple, theme.scss]
    slide-number: c
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---

##  {background-color="#ff69b4"}

$$ $$

### Structural Vector Autoregressions {style="color:#ffd700;"}

### Identification of Structural VARs {style="color:#ffd700;"}

### Dynamic Causal Effects {style="color:#ffd700;"}

### Bayesian Estimation {style="color:#ffd700;"}

### Monetary Policy Analysis Using the  [bsvars](https://cran.r-project.org/package=bsvars) Package {style="color:#ffd700;"}





##  {background-color="#ff69b4"}

![](bsvars.png){.absolute top=30 right=275 width="500"}



## <span style="color: #ffd700;">Materials</span> {background-color="#ff69b4"} 

$$ $$

### Lecture Slides [as a Website](https://bsvars.github.io/2024-10-be24-bsvars) {style="color:#ffd700;"}

### [R script](https://github.com/bsvars/2024-10-be24-bsvars/blob/main/be24-bsvars.R) for your own Australian monetary policy analysis{style="color:#ffd700;"}

### GitHub [repo](https://github.com/bsvars/2024-10-be24-bsvars) to reproduce the slides and results{style="color:#ffd700;"}

### Tasks{style="color:#ffd700;"}


## <span style="color: #ffd700;">Structural Vector Autoregressions</span> {background-color="#ff69b4"}

## Structural Vector Autoregressions

- go-to models for the analysis of policy effects

::: incremental
- facilitate the analysis of **dynamic causal effects** of a well-isolated cause
- extensively used for: *monetary* and *fiscal* policy, *financial* markets, ...
- relatively simple to work with data and provide *empirical evidence on the propagation of shocks* through economies and markets
- provide data-driven stylised facts to be incorporated in theoretical model
- require identification of the cause of the dynamic effects
- extendible: *featuring many variations in specification*
    -   non-normality
    -   heteroskedasticity
    -   time-varying parameters
    -   Bayesian
-   Proposed by [Sims (1980)](https://doi.org/10.2307/1912017)
:::

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::



## Structural Vector Autoregressions

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

### The model.
\begin{align}
\text{VAR equation: }&& y_t &= \mathbf{A}_1 y_{t-1} + \dots + \mathbf{A}_p y_{t-p} + \boldsymbol\mu_0 + \epsilon_t\\[1ex]
\text{structural equation: }&& \mathbf{B}\epsilon_t &= u_t\\[1ex]
\text{structural shocks: }&& u_t |Y_{t-1} &\sim N_N\left(\mathbf{0}_N,\mathbf{I}_N\right)
\end{align}

::: {.fragment}
### Notation.
- $\mathbf{B}$ - $N\times N$ structural matrix of contemporaneous relationships
- $u_t$ - $N$-vector of structural shocks at time $t$
  
  Isolating these shocks allows us to *identify dynamic effects of
uncorrelated shocks* on variables $y_t$

- $\epsilon_t$ - $N$-vector with error terms at time $t$
- the rest as in the lecture on [Bayesian VARs](https://bsvars.github.io/2024-10-be24-bsvarSIGNs/#/varp-model)
:::



## Structural Vector Autoregressions

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

### The VAR errors.
\begin{align}
&&&\\
\text{structural equation: }&& \epsilon_t &= \mathbf{B}^{-1}u_t = \mathbf{\Theta}_0 u_t\\[1ex]
\text{error terms: }&& \epsilon_t |Y_{t-1} &\sim N_N\left(\mathbf{0}_N,\Sigma\right)\\[1ex]
\text{covariance: }&& \mathbf\Sigma &= \mathbf{B}^{-1}\mathbf{B}^{-1\prime} = \Theta_0\Theta_0'
\end{align}

### Notation.
- $\mathbf\Sigma$ - $N\times N$ covariance matrix of error terms 
- $\Theta_0 = \mathbf{B}^{-1}$ - $N\times N$ matrix of **contemporaneous effects**


## Structural Vector Autoregressions

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

Plug the *VAR equation* into the *structural equation* to obtain:

\begin{align}
\mathbf{B}y_t &= \mathbf{B}\mathbf{A}_1 y_{t-1} + \dots + \mathbf{B}\mathbf{A}_p y_{t-p} + \mathbf{B}\boldsymbol\mu_0 + u_t\\[1ex]
&\\
\end{align}

### Contemporaneous relationships.

Let $N=2$

\begin{align}
\mathbf{B}y_t &= \begin{bmatrix}B_{11}&B_{12}\\B_{21}&B_{22}\end{bmatrix}\begin{bmatrix}y_{1t}\\y_{2t}\end{bmatrix}
\end{align}





## Structural Vector Autoregressions

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

Plug the *structural equation* for $\epsilon_t$ into the *VAR equation* to obtain:

\begin{align}
y_t &= \mathbf{A}_1 y_{t-1} + \dots + \mathbf{A}_p y_{t-p} + \boldsymbol\mu_0 + \mathbf{B}^{-1}u_t\\[1ex]
y_t &= \mathbf{A}_1 y_{t-1} + \dots + \mathbf{A}_p y_{t-p} + \boldsymbol\mu_0 + \mathbf{\Theta}_0 u_t
\end{align}

### Contemporaneous effects.

Let $N=2$

\begin{align}
\begin{bmatrix}y_{1t}\\y_{2t}\end{bmatrix} &= \dots +
\begin{bmatrix}\Theta_{11}&\Theta_{12}\\\Theta_{21}&\Theta_{22}\end{bmatrix}\begin{bmatrix}u_{1t}\\ u_{2t}\end{bmatrix}
\end{align}

### Task.

What is the contemporaneous effect of the first shock on the second variable?





## <span style="color: #ffd700;">Identification of Structural VARs </span>  {background-color="#ff69b4"}

## Identification of SVARs (Simplified)

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

### Covariance and structural relationships.

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

- $\mathbf\Sigma$ can be estimated using data easily

::: incremental
- The relationship presents a system of equations to be solved for $\mathbf{B}$
- $\mathbf\Sigma$ is a *symmetric* $N\times N$ matrix
- $\mathbf\Sigma$ has $N(N+1)/2$ unique elements given equations
- $\mathbf{B}$ is an $N\times N$ matrix with $N^2$ unique elements to estimate
- We cannot estimate all elements of $\mathbf{B}$ using $N(N+1)/2$ equations
- $\mathbf{B}$ is <text style="color:#ff69b4;">**not identified**</text>
:::



## Identification of SVARs (Simplified)

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

### Covariance and structural relationships.

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

### Identification.

- Only $N(N+1)/2$ elements in $\mathbf{B}$ can be estimated
- Impose $N(N-1)/2$ restrictions on $\mathbf{B}$ to solve the equation
- This identifies the rows of $\mathbf{B}$ (and the columns of $\mathbf\Theta_0$) up to a sign
- Change the sign of any number of $\mathbf{B}$ rows and $\mathbf\Sigma$ will not change
- Often $\mathbf{B}$ is made lower-triangular



## Identification of SVARs (Simplified)

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

### Covariance and structural relationships.

Let $N=2$ 

\begin{align}
\begin{bmatrix}\sigma_1^2&\sigma_{12}\\ \sigma_{12}&\sigma_2^2\end{bmatrix} &\qquad
\begin{bmatrix}B_{11}&B_{12}\\ B_{21}&B_{22}\end{bmatrix}\\[1ex]
\end{align}

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

### Identification.

\begin{align}
\begin{bmatrix}\sigma_1^2&\sigma_{12}\\ \sigma_{12}&\sigma_2^2\end{bmatrix} &\qquad
\begin{bmatrix}B_{11}& 0\\ B_{21}&B_{22}\end{bmatrix}\\[1ex]
\end{align}

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




## Identification of Monetary Policy Shock

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

Consider a system of four variables:

\begin{align}
y_t = \begin{bmatrix} \Delta rgdp_t & \pi_t & cr_t & \Delta rtwi_t \end{bmatrix}'
\end{align}

- $\Delta rgdp_t$ - real Gross Domestic Product growth
- $\pi_t$ - Consumer Price Index inflation
- $cr_t$ - Cash Rate Target - Australian nominal interest rate
- $\Delta rtwi_t$ - real Trade-Weighted Index rate of return (exchange rate)

### Identified system.

A lower-triangular matrix identifies: 

- contemporaneous relationships $\mathbf{B}$
- contemporaneous effects $\mathbf\Theta_0$
- structural shocks $u_t$


## Identification of Monetary Policy Shock

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

### Identified system.

\begin{align}
\begin{bmatrix}
B_{11}&0&0&0\\
B_{21}&B_{22}&0&0\\
B_{31}&B_{32}&B_{33}&0\\
B_{41}&B_{42}&B_{43}&B_{44}
\end{bmatrix}
\begin{bmatrix} \Delta rgdp_t \\ \pi_t \\ cr_t \\ \Delta rtwi_t \end{bmatrix} &= \dots + 
\begin{bmatrix} u_t^{ad} \\ u_t^{as} \\ u_t^{mps} \\ u_t^{ex} \end{bmatrix}
\end{align}

### Identified shocks.

$u_t^{ad}$ - aggregate demand shock is exogenous to the rest of the system

$u_t^{as}$ - aggregate supply shock

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

$u_t^{ex}$ - currency shock




## Identification of Monetary Policy Shock

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

### Identified system.

\begin{align}
\begin{bmatrix}
B_{11}&0&0&0\\
B_{21}&B_{22}&0&0\\
B_{31}&B_{32}&B_{33}&0\\
B_{41}&B_{42}&B_{43}&B_{44}
\end{bmatrix}
\begin{bmatrix} \Delta rgdp_t \\ \pi_t \\ cr_t \\ \Delta rtwi_t \end{bmatrix} &= \dots + 
\begin{bmatrix} u_t^{ad} \\ u_t^{as} \\ u_t^{mps} \\ u_t^{ex} \end{bmatrix}
\end{align}

### Tasks.

- Write out the third equation for the cash rate.
- Let $B_{33}>0$. What values of $B_{31}$ and $B_{32}$ does theory imply?




## Identification of Monetary Policy Shock

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

### Identified system.

\begin{align}
\begin{bmatrix}
B_{11}&0&0&0\\
B_{21}&B_{22}&0&0\\
B_{31}&B_{32}&B_{33}&0\\
B_{41}&B_{42}&B_{43}&B_{44}
\end{bmatrix}
\begin{bmatrix} \Delta rgdp_t \\ \pi_t \\ cr_t \\ \Delta rtwi_t \end{bmatrix} &= \dots + 
\begin{bmatrix} u_t^{ad} \\ u_t^{as} \\ u_t^{mps} \\ u_t^{ex} \end{bmatrix}
\end{align}

### Monetary policy shock.

- is uncorrelated with any other shocks
- consists of the unanticipated part of the *monetary policy instrument*, interest rate
- In this model, the systematic part of the monetary policy consists of:
  - contemporaneous relationships with GDP and inflation
  - lagged relationships with all variables



## Alternative Identification

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

### Identified system.

\begin{align}
\begin{bmatrix}
B_{11}&B_{12}&B_{13}&B_{14}\\
B_{21}&B_{22}&B_{23}&B_{24}\\
B_{31}&B_{32}&B_{33}&B_{34}\\
B_{41}&B_{42}&B_{43}&B_{44}
\end{bmatrix}
\begin{bmatrix} \Delta rgdp_t \\ \pi_t \\ cr_t \\ \Delta rtwi_t \end{bmatrix} &= \dots + 
\begin{bmatrix} u_t^{ad} \\ u_t^{as} \\ u_t^{mps} \\ u_t^{ex} \end{bmatrix}
\end{align}

### Identification.

Assume the structural shocks are:
\begin{align}
\text{heteroskedastic: }&& u_t^{(i)}\mid\mathbf{Y}_{t-1} &\sim N(0,\sigma_{i.t}^2) \\
\text{non-normal: }&& u_t^{(i)}\mid\mathbf{Y}_{t-1} &\sim t(0,1, \nu_i) \\
\end{align}




## <span style="color: #ffd700;">Alternative Identification</span> {background-color="#ff69b4"}

```{r Alternative_Identification}
#| echo: false
#| cache: true

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 = "#ff69b4",
  mfrow = c(1,2),
  col = scales::alpha("#ffd700", .5),
  col.main = "#ffd700",
  col.lab = "#ffd700"
)
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 = "homoskedastic and normal"
)
abline(a = 0, b = 1, col = "#ffd700")
abline(a = 0, b = -1, col = "#ffd700")

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 = "heteroskedastic or non-normal"
)
abline(a = 0, b = 1, col = "#ffd700")
abline(a = 0, b = -1, col = "#ffd700")

```















## <span style="color: #ffd700;">Dynamic Causal Effects</span> {background-color="#ff69b4"}


## Dynamic Causal Effects

### Impulse response functions

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

**Impulse response functions** to *orthogonal shocks* computed for an empirically relevant SVAR model are considered the <text style="color:#ff69b4;">**dynamic causal effects**</text> of the underlying shocks $u_t$ on economic measurements $y_{t+i}$ $i$ periods ahead.

\begin{align*}
\frac{\partial y_{n.t+i}}{\partial u_{j.t}}&=\theta_{nj.i}
\end{align*}

- $\theta_{nj.i}$ - response of $n$th variable to $j$th shock $i$ periods after shock's occurrence

  for $i=0,1,\dots,h$ and $n,j=1,\dots,N$




## Dynamic Causal Effects

### Impulse response functions

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

**Impulse response functions** to *orthogonal shocks* computed for an empirically relevant SVAR model are considered the <text style="color:#ff69b4;">**dynamic causal effects**</text> of the underlying shocks $u_t$ on economic measurements $y_{t+i}$ $i$ periods ahead.

\begin{align*}
\frac{\partial y_{t+i}}{\partial u_t}&=\underset{N\times N}{\mathbf\Theta_i}
\end{align*}

- $\mathbf\Theta_i$ - responses of all of the variables to all of the shocks $i$ periods after shocks' occurrence

  for $i=0,1,\dots,h$ and $n,j=1,\dots,N$





## Dynamic Causal Effects

### Impulse response functions at finite horizon.

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

Define matrices

$$
\underset{(pN\times pN)}{\mathbb{A}} = \begin{bmatrix}\mathbf{A}_1 & \mathbf{A}_2 &\dots& \mathbf{A}_p\\ &\mathbf{I}_{N(p-1)}&&\mathbf{0}_{N(p-1)\times N} \end{bmatrix}\quad\text{and}\quad
\underset{(N\times pN)}{\mathbf{J}} = \begin{bmatrix} \mathbf{I}_{N} & \mathbf{0}_{N\times N(p-1)} \end{bmatrix}
$$
Impulse response at horizon $i=0,1,\dots,h$ are equal to:

\begin{align}
\mathbf\Theta_i &= \mathbf{J}\mathbb{A}^i\mathbf{J}'\mathbf{\Theta}_0
\end{align}
where $\mathbb{A}^0=\mathbf{I}_N$, $\mathbb{A}^1=\mathbb{A}$, $\mathbb{A}^2=\mathbb{A}\mathbb{A}$, ...

### At infinite horizon.

Inform about the value of the effect in the long run.

\begin{align}
\mathbf\Theta_{\infty} &= \left( \mathbf{I}_N - \mathbf{A}_1 - \dots - \mathbf{A}_p \right)^{-1}\mathbf{\Theta}_0
\end{align}




## Dynamic Causal Effects

### Tasks.

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

Consider a bivariate ($N=2$) Structural VAR model with $p=1$ lag and no constant term:
\begin{align}
y_t &= \mathbf{A}y_{t-1} + \epsilon_t\\
\mathbf{B}\epsilon_t &= u_t
\end{align}

Derive:

- on impact impulse response, $\mathbf\Theta_0$
- impulse response at horizon 1, $\mathbf\Theta_1$



## Dynamic Causal Effects

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

### Bayesian Estimation.

#### Step 1. Estimate the model

Obtain a sample from the posterior distribution 
$$\left\{ \mathbf{A}^{(s)},\mathbf{B}^{(s)} \right\}_{s=1}^{S}$$

#### Step 2. Compute impulse responses

For each of the $S$ draws, compute impulse responses $\mathbf\Theta_0^{(s)}, \dots, \mathbf\Theta_h^{(s)}$ as a function of $\mathbf{A}^{(s)}$ and $\mathbf{B}^{(s)}$ and return
$$\left\{\mathbf\Theta_0^{(s)}, \dots, \mathbf\Theta_h^{(s)}\right\}_{s=1}^{S}$$ 
as a sample drew from the posterior distribution the impulse responses.














## <span style="color: #ffd700;">Bayesian Estimation</span> {background-color="#ff69b4"}

## Bayesian Estimation

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

$\left.\right.$

#### **Gibbs sampler** by [Waggoner & Zha (2003)](https://doi.org/10.1016/S0165-1889(02)00168-9) 

facilitates estimation of Bayesian SVARs for

- lower-triangular and non-recursive identification patterns of exclusion restrictions
- over-identifying (more than $N(N − 1)/2)$ exclusion restrictions
- models identified via heteroskedasticity and non-normality

$\left.\right.$

#### Other approaches facilitate estimation of SVARs

- identified through non-normal residuals
- identified by zero and sign restrictions
- identified using instrumental variables (Proxy SVARs)





## Bayesian Estimation

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

### Exclusion restrictions on the rows of $\mathbf{B}$

$$
\underset{(1\times N)}{\mathbf{B}_{[n\cdot]}} = \underset{(1\times r_n)}{\mathbf{b}_n} \underset{(r_n\times N)}{V_n} \qquad\text{such that}\qquad
\mathbf{B} = \begin{bmatrix} \mathbf{b}_1V_1\\ \vdots \\ \mathbf{b}_NV_N \end{bmatrix}
$$

- $\mathbf{b}_n$ - a $1\times r_n$ vector of unrestricted elements of $n$ row of $\mathbf{B}$
- $V_n$ - an $r_n\times N$ *fixed* matrix of ones and zeros

### Example.

$$\mathbf{b}_n = \begin{bmatrix} b_1 & b_2\end{bmatrix}\quad V_n = \begin{bmatrix} 1&0&0\\0&0&1\end{bmatrix} \quad\rightarrow\quad \mathbf{B}_{[n\cdot]} = \begin{bmatrix} b_1&0 & b_2\end{bmatrix}
$$




## Bayesian Estimation

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

### The $n$th Structural Equation.
\begin{align*}
\mathbf{b}_nV_n\epsilon_t &= u_{n.t}\\
u_{n.t} &\sim N(0,1)
\end{align*}

### Matrix Notation.
\begin{align*}
E V_n' \mathbf{b}_n' &= U_n\\
U_n &\sim N_T\left(\mathbf{0}_T,I_T\right)\\[2ex]
\underset{(T\times1)}{U_n}  &= \begin{bmatrix} u_{n.1} & \dots & u_{n.T} \end{bmatrix}'\\
\underset{(T\times N)}{E} &\text{ - defined as before}
\end{align*}



## Bayesian Estimation

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

### Likelihood function.
\begin{align*}
L(\mathbf{A},\mathbf{B}|Y,X) &\propto 
|\text{det}\left( \mathbf{B} \right)|^{T}\exp\left\{ -\frac{1}{2}\sum_{n=1}^N \mathbf{b}_nV_nE'EV_n'\mathbf{b}_n' \right\}\\[1ex]
E &= Y - X\mathbf{A}
\end{align*}

### Hierarchical prior for $\mathbf{B}$
\begin{align*}
\mathbf{b}_n | \gamma_B &\sim N_{r_n}\left(\mathbf{0}_{r_n}, \gamma_B V_n\underline{S}^{-1}V_n'\right)\\[1ex]
\gamma_B &\sim IG2(\underline{s},\underline{\nu})
\end{align*}

- $\underline{S}$ - $N\times N$ prior scale matrix
- $\underline{s}$ and $\underline{\nu}$ positive scalars of scale and shape




## Bayesian Estimation

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

### Kernel of the full conditional posterior for $\mathbf{B}$

\begin{align*}
p(\mathbf{B}|Y,X,\mathbf{A}, \gamma_B)&\propto
|\text{det}\left( \mathbf{B} \right)|^{T}\exp\left\{ -\frac{1}{2}\sum_{n=1}^N \mathbf{b}_n \overline{S}_n^{-1}\mathbf{b}_n' \right\}\\[1ex]
\overline{S}_n^{-1} &= V_n\left[ \gamma_B^{-1}\underline{S}^{-1} + (Y-X\mathbf{A})'(Y-X\mathbf{A}) \right]V_n'\\[2ex]
\end{align*}

- This is a kernel of a *Generalised-Normal* distribution
- A feasible Gibbs sampler was proposed by by [Waggoner & Zha (2003)](https://doi.org/10.1016/S0165-1889(02)00168-9)
- The Gibbs sampler draws from the full conditional posterior for $n = 1,\dots,N$:
$$ p(\mathbf{b}_n | \mathbf{b}_1,\dots, \mathbf{b}_{n-1},\mathbf{b}_{n+1}, \mathbf{b}_N, \mathbf{A}, \gamma_B, Y, X) $$







## <span style="color: #ffd700;">Monetary Policy Analysis Using R Package</span> [bsvars](https://cran.r-project.org/package=bsvars) {background-color="#ff69b4"}


## [bsvars](https://cran.r-project.org/package=bsvars) an R Package

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

![](cran.png)




## [bsvars](https://cran.r-project.org/package=bsvars) an R Package: Features

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

- Bayesian estimation of Structural VARs
- identification via:
  - exclusion restrictions 
  - heteroskedasticity
  - non-normality
- six heteroskedastic processes
- efficient and fast Gibbs sampler
- excellent computational speed
  - frontier econometric techniques
  - compiled code using **C++**
- data analysis in **R**



## [bsvars](https://cran.r-project.org/package=bsvars) an R Package: Features

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

![](progress.png)



## [bsvars](https://cran.r-project.org/package=bsvars) an R Package: Features

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

- simple model setup using `specify_*()` 
- flexibility in setting priors, restrictions, etc.
- one function for estimation `estimate()` 
- posterior processing utility functions

### The simplest workflow using pipe.

```{r} 
#| cache: true
#| eval: false

library(bsvars)

data(us_fiscal_lsuw)
set.seed(1)

us_fiscal_lsuw |>
  specify_bsvar_sv$new(p = 2) |>
  estimate(S = 1000) |>
  estimate(S = 5000) |>
  compute_impulse_responses(horizon = 8) -> irfs
```





## Australian Monetary Policy Analysis

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

Based on [Turnip (2017)]( https://doi.org/10.1111/1475-4932.12345)

### System of four variables.

\begin{align}
y_t = \begin{bmatrix} \Delta rgdp_t & \pi_t & cr_t & \Delta rtwi_t \end{bmatrix}'
\end{align}

### Alternative identification patterns.

\begin{align}
\textbf{lower-triangular} && \textbf{extended}\\
\begin{bmatrix}
B_{11}&0&0&0\\
B_{21}&B_{22}&0&0\\
B_{31}&B_{32}&B_{33}&0\\
B_{41}&B_{42}&B_{43}&B_{44}
\end{bmatrix}
\begin{bmatrix} \Delta rgdp_t \\ \pi_t \\ cr_t \\ \Delta rtwi_t \end{bmatrix} &&
\begin{bmatrix}
B_{11}&0&0&0\\
B_{21}&B_{22}&0&0\\
B_{31}&B_{32}&B_{33}&B_{34}\\
B_{41}&B_{42}&B_{43}&B_{44}
\end{bmatrix}
\begin{bmatrix} \Delta rgdp_t \\ \pi_t \\ cr_t \\ \Delta rtwi_t \end{bmatrix}
\end{align}

- In the **extended** model, the monetary policy shock is not identified
- Use identification via heteroskedasticity to identify it


## Four-Variable Monetary System

```{r} 
#| label: data
#| cache: true
#| warning: false
#| fig-align: "center"
#| fig-height: 6
#| output-location: slide

# Gross domestic product (GDP); Chain volume
rgdp_dwnld      = readrba::read_rba(series_id = "GGDPCVGDP")
rgdp_tmp        = xts::xts(rgdp_dwnld$value, rgdp_dwnld$date, tclass = 'yearqtr')
drgdp           = na.omit(400 * diff(log(rgdp_tmp)))
drgdp           = xts::to.quarterly(drgdp, OHLC = FALSE)

# Consumer price index; All groups; Quarterly change (in per cent)
picpi_dwnld     = readrba::read_rba(series_id = "GCPIAGSAQP")
pi              = 4 * xts::xts(picpi_dwnld$value, picpi_dwnld$date, tclass = 'yearqtr')
pi              = xts::to.quarterly(pi, OHLC = FALSE)

# Interbank Overnight Cash Rate
cr_dwnld        = readrba::read_rba(series_id = "FIRMMCRID")   # Cash Rate Target
cr_tmp          = xts::xts(cr_dwnld$value, cr_dwnld$date)
cr              = xts::to.quarterly(cr_tmp, OHLC = FALSE)

# Real Trade-Weighted Index
rtwi_dwnld      = readrba::read_rba(series_id = "FRERTWI")
rtwi_tmp        = xts::xts(rtwi_dwnld$value, rtwi_dwnld$date, tclass = 'yearqtr')
rtwi            = 100 * na.omit(diff(log(rtwi_tmp)))
drtwi            = xts::to.quarterly(rtwi, OHLC = FALSE)

y               = na.omit(merge(drgdp, pi, cr, drtwi))
plot(y, main = "Australian monetary system",
     legend.loc = "bottomleft", col = c("#FF00FF","#990099","#ff69b4","#330033"))
```

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::


## Estimation Setup

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

```{r} 
#| cache: true

# setup
############################################################
library(bsvars)
set.seed(123)

N       = ncol(y)
p       = 9
S_burn  = 1e4
S       = 2e4
thin    = 2
```



## Extended Model Identification Setup

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

```{r} 
#| cache: true
# structural matrix - extended model
############################################################
B_LR = matrix(TRUE, N, N)
B_LR[upper.tri(B_LR)] = FALSE
B_LR[3,4] = TRUE
B_LR
```



## Alternative Models Estimation

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

### Lower-triangular model with zero prior mean for $\mathbf{A}$.

```{r}
#| cache: true
#| eval: true
# estimation - lower-triangular model
############################################################
spec_bsvar0     = specify_bsvar$new(as.matrix(y), p = p, stationary = rep(TRUE, N))
spec_bsvar0 |>
  estimate(S = S_burn) |>
  estimate(S = S, thin = thin) -> soe_bsvar0
```




## Alternative Models Estimation

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

### Lower-triangular model with MLE prior mean for $\mathbf{A}$.

```{r}
#| cache: true
#| code-line-numbers: "5-9"
#| eval: true

# estimation - lower-triangular model - MLE prior for A
############################################################
spec_bsvar      = specify_bsvar$new(as.matrix(y), p = p, stationary = rep(TRUE, N))

A_mle           = t(solve(
  tcrossprod(spec_bsvar$data_matrices$X),
  tcrossprod(spec_bsvar$data_matrices$X, spec_bsvar$data_matrices$Y)
))
spec_bsvar$prior$A = A_mle

spec_bsvar |>
  estimate(S = S_burn) |>
  estimate(S = S, thin = thin) -> soe_bsvar
```






## Alternative Models Estimation

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

### Extended model with MLE prior mean for $\mathbf{A}$.

```{r}
#| cache: true
#| code-line-numbers: "3-4"
#| eval: true

# estimation - extended model - MLE prior for A
############################################################
spec_bsvar_lr   = specify_bsvar$new(as.matrix(y), p = p, B = B_LR, stationary = rep(TRUE, N))
spec_bsvar_lr$prior$A = A_mle

spec_bsvar_lr |>
  estimate(S = S_burn) |>
  estimate(S = S, thin = thin) -> soe_bsvar_lr
```





## Alternative Models Estimation

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

### Lower-triangular model with Stochastic Volatility heteroskedastisity.

```{r}
#| cache: true
#| code-line-numbers: "3"
#| eval: true

# estimation - lower-triangular SV heteroskedastic model
############################################################
spec_bsvar_sv = specify_bsvar_sv$new(as.matrix(y), p = p, stationary = rep(TRUE, N))
spec_bsvar_sv$prior$A = A_mle

spec_bsvar_sv |>
  estimate(S = S_burn) |>
  estimate(S = S, thin = thin) -> soe_bsvar_sv
```





## Alternative Models Estimation

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

### Extended model with Stochastic Volatility heteroskedastisity.

```{r}
#| cache: true
#| code-line-numbers: "3"
#| eval: true

# estimation - extended SV heteroskedastic model
############################################################
spec_bsvar_lr_sv = specify_bsvar_sv$new(as.matrix(y), p = p, B = B_LR, stationary = rep(TRUE, N))
spec_bsvar_lr_sv$prior$A = A_mle

spec_bsvar_lr_sv |>
  estimate(S = S_burn) |>
  estimate(S = S, thin = thin) -> soe_bsvar_lr_sv

```





## IRFs: Compute impulse responses

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

```{r}
#| cache: true

ir_bsvar0       = compute_impulse_responses(soe_bsvar0, horizon = 20)
ir_bsvar0[,3,,] = 0.25 * ir_bsvar0[,3,,] / mean(ir_bsvar0[3,3,1,])

ir_bsvar        = compute_impulse_responses(soe_bsvar, horizon = 20)
ir_bsvar[,3,,]  = 0.25 * ir_bsvar[,3,,] / mean(ir_bsvar[3,3,1,])

ir_bsvar_lr        = compute_impulse_responses(soe_bsvar_lr, horizon = 20)
ir_bsvar_lr[,3,,]  = 0.25 * ir_bsvar_lr[,3,,] / mean(ir_bsvar_lr[3,3,1,])

ir_bsvar_sv        = compute_impulse_responses(soe_bsvar_sv, horizon = 20)
ir_bsvar_sv[,3,,]  = 0.25 * ir_bsvar_sv[,3,,] / mean(ir_bsvar_sv[3,3,1,])

ir_bsvar_lr_sv        = compute_impulse_responses(soe_bsvar_lr_sv, horizon = 20)
ir_bsvar_lr_sv[,3,,]  = 0.25 * ir_bsvar_lr_sv[,3,,] / mean(ir_bsvar_lr_sv[3,3,1,])

```



## IRFs: Zero vs. MLE prior mean

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

```{r}
#| cache: true
#| echo: false
#| fig-align: "center"
#| fig-height: 8

par(mfrow = c(4,2))
for (n in 1:4) {
  bsvarTVPs::ribbon_plot(
    ir_bsvar0[n,3,,],
    main = paste("Response of",colnames(y)[n],"to the mps"),
    ylab = colnames(y)[n],
    xlab = "horizon [quarters]",
    bty  = "n",
    col = "#ff69b4"
  )
  abline(h = 0)
  bsvarTVPs::ribbon_plot(
    ir_bsvar[n,3,,],
    main = paste("Response of",colnames(y)[n],"to the mps"),
    ylab = colnames(y)[n],
    xlab = "horizon [quarters]",
    bty  = "n",
    col = "#ff69b4"
  )
  abline(h = 0)
}
```



## IRFs: Recursive vs. Extended (homosk)

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

```{r}
#| cache: true
#| echo: false
#| fig-align: "center"
#| fig-height: 8

par(mfrow = c(4,2))
for (n in 1:4) {
  bsvarTVPs::ribbon_plot(
    ir_bsvar[n,3,,],
    main = paste("Response of",colnames(y)[n],"to the mps"),
    ylab = colnames(y)[n],
    xlab = "horizon [quarters]",
    bty  = "n",
    col = "#ff69b4"
  )
  abline(h = 0)
  bsvarTVPs::ribbon_plot(
    ir_bsvar_lr[n,3,,],
    main = paste("Response of",colnames(y)[n],"to the mps"),
    ylab = colnames(y)[n],
    xlab = "horizon [quarters]",
    bty  = "n",
    col = "#ff69b4"
  )
  abline(h = 0)
}
```


## IRFs: Homosk vs. SV (Recursive)

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

```{r}
#| cache: true
#| echo: false
#| fig-align: "center"
#| fig-height: 7

par(mfrow = c(4,2))
for (n in 1:4) {
  bsvarTVPs::ribbon_plot(
    ir_bsvar[n,3,,],
    main = paste("Response of",colnames(y)[n],"to the mps"),
    ylab = colnames(y)[n],
    xlab = "horizon [quarters]",
    bty  = "n",
    col = "#ff69b4"
  )
  abline(h = 0)
  bsvarTVPs::ribbon_plot(
    ir_bsvar_sv[n,3,,],
    main = paste("Response of",colnames(y)[n],"to the mps"),
    ylab = colnames(y)[n],
    xlab = "horizon [quarters]",
    bty  = "n",
    col = "#ff69b4"
  )
  abline(h = 0)
}
```


## IRFs: Homosk vs. SV (Extended)

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

```{r}
#| cache: true
#| echo: false
#| fig-align: "center"
#| fig-height: 7

par(mfrow = c(4,2))
for (n in 1:4) {
  bsvarTVPs::ribbon_plot(
    ir_bsvar_lr[n,3,,],
    main = paste("Response of",colnames(y)[n],"to the mps"),
    ylab = colnames(y)[n],
    xlab = "horizon [quarters]",
    bty  = "n",
    col = "#ff69b4"
  )
  abline(h = 0)
  bsvarTVPs::ribbon_plot(
    ir_bsvar_lr_sv[n,3,,],
    main = paste("Response of",colnames(y)[n],"to the mps"),
    ylab = colnames(y)[n],
    xlab = "horizon [quarters]",
    bty  = "n",
    col = "#ff69b4"
  )
  abline(h = 0)
}
```




## SDs: homoskedastic vs. SV (Recursive)

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

```{r}
#| cache: true
#| echo: false
#| fig-align: "center"
#| fig-height: 7

sd      = compute_conditional_sd(soe_bsvar)
sd_sv   = compute_conditional_sd(soe_bsvar_sv)

par(mfrow = c(4,2))
for (n in 1:4) {
  bsvarTVPs::ribbon_plot(
    sd[n,,],
    main = paste("SD of",colnames(y)[n]),
    ylab = colnames(y)[n],
    xlab = "time",
    bty  = "n",
    col = "#ff69b4"
  )
  abline(h = 1)
  bsvarTVPs::ribbon_plot(
    sd_sv[n,,],
    main = paste("SD of",colnames(y)[n]),
    ylab = colnames(y)[n],
    xlab = "time",
    bty  = "n",
    col = "#ff69b4"
  )
  abline(h = 1)
}
```




## SDs: homoskedastic vs. SV (Extended)

::: footer
[Bayesian Structural VARs](https://bsvars.github.io/2024-10-be24-bsvars)
:::

```{r}
#| cache: true
#| echo: false
#| fig-align: "center"
#| fig-height: 7

sd      = compute_conditional_sd(soe_bsvar_lr)
sd_sv   = compute_conditional_sd(soe_bsvar_lr_sv)

par(mfrow = c(4,2))
for (n in 1:4) {
  bsvarTVPs::ribbon_plot(
    sd[n,,],
    main = paste("SD of",colnames(y)[n]),
    ylab = colnames(y)[n],
    xlab = "time",
    bty  = "n",
    col = "#ff69b4"
  )
  abline(h = 1)
  bsvarTVPs::ribbon_plot(
    sd_sv[n,,],
    main = paste("SD of",colnames(y)[n]),
    ylab = colnames(y)[n],
    xlab = "time",
    bty  = "n",
    col = "#ff69b4"
  )
  abline(h = 1)
}
```




##  {background-color="#ff69b4"}

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