modelling the effects of monetary policy in Australia

bsvars.org

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Australia and its monetary policy

structural vector autoregressions

effects estimation using bsvarSIGNs

resources

bsvars.org

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slides as a website

GitHub repo with quarto presentation

R scripts and data for results reproduction

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bsvars.org official website

R package bsvars on CRAN

R package bsvarSIGNs on CRAN

Australia and its monetary policy

Australia and its monetary policy

bsvars.org

  • Australia is a small open economy
    • foreign sector determines its macroeconomic outcomes
    • domestic sector does not affect the foreign sector
  • its macroeconomic outcomes are jointly determined in a dynamic system
  • most of its international exchange in underwritten in US dollars
  • its currency floats

Australia and its monetary policy

bsvars.org

  • the Reserve Bank of Australia conducts monetary policy
    • they set the cash rate target
    • they target inflation as per price stability mandate
    • they target full employment as per dual mandate
    • they coordinate the policy with central banks of other inflation-targeting countries

Australia and its monetary policy

bsvars.org

Taylor’s rule (Taylor, 1993)

  • A simple rule to describe monetary policy of inflation-targetting countries \[\begin{align} i_t = r_t + \pi_t + a_\pi(\pi_t-\pi^*) + a_y(y_t-y^*), \quad a_\pi, a_y > 0 \end{align}\]

    • \(i_t\): nominal policy interest rate
    • \(r_t\): natural interest rate
    • \(\pi_t\): inflation rate
    • \(\pi^*\): inflation target
    • \(y_t\): gross domestic product
    • \(y^*\): potential gross domestic product

structural vector autoregressions

structural vector autoregressions

bsvars.org

  • go-to models for the analysis of policy effects

  • facilitate the analysis of dynamic causal effects of a well-isolated cause

  • 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 models

  • extensively used for: monetary and fiscal policy, financial markets, …

  • extendible: featuring many variations in specification

    • non-normality
    • non-linearity
    • hierarchical modelling

  • Proposed by Sims (1980)

structural vector autoregressions

bsvars.org

data: domestic sector.

  • \(\text{CASH}_t\) - target cash rate
  • \(\text{gdp}_t\) - logarithm of gross domestic product
  • \(\text{cpi}_t\) - logarithm of consumer price index
  • \(\text{TWI}_t\) - trade-weighted index of the Australian dollar

data: foreign sector.

  • \(\text{TOT}_t\) - terms of trade
  • \(\text{gdp}_{US.t}\) - logarithm of US gross domestic product
  • \(\text{FFR}_t\) - Federal Funds Rate - US nominal interest rate

structural vector autoregressions

bsvars.org

data.

  • based on the system by Read (2023)
  • quarterly time series from 1982 to 2020
  • rates are not transformed
  • macroeconomic aggregates are in logarithms
  • data are unit-root non-stationary
  • time series exhibit long memory
  • data source: RBA, ABS, FRED

structural vector autoregressions

bsvars.org

data <- readxl::read_excel("VARData.xlsx")
data[, 3:7] <- 100 * log(data[, 3:7])

domestic <- ts(
  data[, 2:5],
  start = c(1982, 1),
  frequency = 4
)
plot.ts(domestic, col = "#ff68b4", lwd = 2, bty = "l")

structural vector autoregressions

bsvars.org

foreign <- ts(
  data[, 6:8],
  start = c(1982, 1),
  frequency = 4
)
plot(foreign, col = "#ff68b4", lwd = 2, bty = "l")

structural vector autoregressions

bsvars.org

the model.

\[\begin{align} \text{dynamic equation: }&& y_t &= \mathbf{A}_1 y_{t-1} + \dots + \mathbf{A}_p y_{t-4} + \mathbf{A}_d x_{t} + \epsilon_t\\[1ex] \text{structural equation: }&& \mathbf{B}\epsilon_t &= u_t\\[1ex] \text{structural shocks: }&& u_t |Y_{t-1} &\sim N_N\left(\mathbf{0}_N,\text{diag}\left(\boldsymbol\sigma^2\right)\right) \end{align}\]

notation.

  • \(y_t\) - \(N\)-vector of variables at time \(t\)
  • \(\mathbf{A}_i\) and \(\mathbf{B}\) - autoregressive and structural coefficients
  • \(\epsilon_t\) and \(u_t\) - error terms and structural shocks at time \(t\)
  • \(\boldsymbol\sigma^2\) - structural shock variances

structural vector autoregressions

bsvars.org

dynamic equation.

\[\begin{align} \text{dynamic equation: }&& y_t &= \mathbf{A}_1 y_{t-1} + \dots + \mathbf{A}_4 y_{t-4} + \mathbf{A}_d x_{t} + \epsilon_t\\[1ex] \text{domestic sector: }&& y_t &= \begin{bmatrix}\text{CASH}_t &\text{gdp}_t &\text{cpi}_t &\text{TWI}_t \end{bmatrix}'\\[1ex] \text{foreign sector: }&& x_t &= \begin{bmatrix}\text{TOT}_{t-i} &\text{gdp}_{US.t-i} &\text{FFR}_{t-i} \end{bmatrix}' \end{align}\]

  • captures dynamic relationships between variables in domestic sector
  • jointly models the dynamic system
  • includes foreign sector’s impact on domestic sector

structural vector autoregressions

bsvars.org

structural equation.

\[ \begin{align*} \begin{bmatrix} B_{11} & B_{12} & B_{13} & B_{14} \\ B_{21} & B_{22} & B_{23} & B_{24} \\ B_{31} & B_{32} & B_{33} & B_{34} \\ B_{41} & B_{42} & B_{43} & B_{44} \\ \end{bmatrix} \begin{bmatrix} \text{CASH}_t \\ \text{GDP}_t \\ \text{CPI}_t \\ \text{TWI}_t \\ \end{bmatrix} = \dots+ \begin{bmatrix} u_{t}^{MP} \\ u_{2,t} \\ u_{3,t} \\ u_{4,t} \\ \end{bmatrix} \end{align*} \]

  • captures contemporaneous relationships between variables
  • imposes theoretical structure on the system
  • identifies \(u_{t}^{MP}\) as the monetary policy shock
  • restrictions on parameters \(B\) are essential to identify the monetary policy shock

structural vector autoregressions

bsvars.org

structural shocks.

\[\begin{align} \text{jointly normal: }&& u_t |Y_{t-1} &\sim N_4\left(\mathbf{0}_4,\text{diag}\left(\boldsymbol\sigma^2\right)\right)\\[1ex] \text{structural shocks: }&& u_t&=\begin{bmatrix}u_{t}^{MP} & u_{2,t} & u_{3,t} & u_{4,t}\end{bmatrix}' \end{align}\]

  • shocks are contemporaneously and temporarily independent
  • shocks are jointly normal
  • the monetary policy shock, \(u_{t}^{MP}\), is identified as independent source of fluctuations of the cash rate, \(\text{CASH}_t\)
  • identification of the monetary policy shock is essential in estimating the dynamic causal effects of monetary policy on the economy

effects estimation using bsvarSIGNs

bsvars.org

effects estimation using bsvarSIGNs

bsvars.org

# restrictions on impulse response functions
sign_irf <- matrix(NA, 4, 4)
sign_irf[1, 1] <- sign_irf[4, 1] <- 1     # impact on cash rate and exchange rate
sign_irf[3, 1] <- -1                      # impact on consumer price index
sign_irf <- array(sign_irf, c(4, 4, 4))   # last for 4 periods
sign_irf[, , 1]
     [,1] [,2] [,3] [,4]
[1,]    1   NA   NA   NA
[2,]   NA   NA   NA   NA
[3,]   -1   NA   NA   NA
[4,]    1   NA   NA   NA
sign_structural <- matrix(NA, 4, 4)
sign_structural[1, ] <- c(1, -1, -1, 1)
sign_structural
     [,1] [,2] [,3] [,4]
[1,]    1   -1   -1    1
[2,]   NA   NA   NA   NA
[3,]   NA   NA   NA   NA
[4,]   NA   NA   NA   NA

effects estimation using bsvarSIGNs

bsvars.org

library(bsvarSIGNs)

spec <- specify_bsvarSIGN$new(
  domestic,
  p = 4,
  exogenous = foreign,
  sign_irf = sign_irf,
  sign_structural = sign_structural
)
post <- estimate(spec, S = 5000)
**************************************************|
 bsvarSIGNs: Bayesian Structural VAR with sign,   |
             zero and narrative restrictions      |
**************************************************|
 Progress of simulation for 5000 independent draws
 Press Esc to interrupt the computations
**************************************************|

effects estimation using bsvarSIGNs

bsvars.org

irf <- compute_impulse_responses(post, horizon = 20)
plot(irf, probability = 0.68)