Introduction to Empirical Macroeconomics with R

Workshops for Ukraine

Xiaolei (Adam) Wang

University of Melbourne

About me

  • PhD student at the University of Melbourne

  • Interested in Bayesian econometrics

  • Author of the R package bsvarSIGNs

Plan for today

  • Vector Autoregression (VAR)

  • Structural Vector Autoregression (SVAR)

  • Applications in Macroeconomics

  • Many fun exercises in R!

Prerequisites

  • Install the bsvarSIGNs package
install.packages("bsvarSIGNs")
  • Install the readxl package
install.packages("readxl")

Vector Autoregression

Vector Autoregression (VAR)

  • A VAR model is a system of equations where each variable is regressed on its own lagged values and the lagged values of all other variables

  • This allows us to model the linear dynamic interdependencies

  • e.g. a bivariate VAR(1):

    \[ \begin{align*} \begin{bmatrix} y_{1,t} \\ y_{2,t} \\ \end{bmatrix} & = \begin{bmatrix} A_{1,11} & A_{1,12} \\ A_{1,21} & A_{1,22} \\ \end{bmatrix} \begin{bmatrix} y_{1,t-1} \\ y_{2,t-1} \\ \end{bmatrix}+ \begin{bmatrix} \varepsilon_{1,t} \\ \varepsilon_{2,t} \\ \end{bmatrix}, & \begin{bmatrix} \varepsilon_{1,t} \\ \varepsilon_{2,t} \\ \end{bmatrix} ~ & \sim N\left( \begin{bmatrix} 0 \\ 0 \\ \end{bmatrix}, \begin{bmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{12} & \Sigma_{22} \\ \end{bmatrix} \right) \\ y_t & = A_1 y_{t-1} + \varepsilon_t , & \varepsilon_t & \sim N(0, \Sigma) \end{align*} \]

  • Useful for forecasting, Granger causality test etc.

Matrix representation

  • Generally, a VAR(p) model:

    \[ \begin{align*} y_t & = A_1 y_{t-1} + A_2 y_{t-2} + \dots + A_p y_{t-p} + A_dd_t + \varepsilon_t \\ y_t & = \begin{bmatrix} A_1 & \dots & A_p & A_d \end{bmatrix} \begin{bmatrix} y_{t-1} \\ \vdots \\ y_{t-p} \\ d_t \end{bmatrix} + \varepsilon_t \\ \end{align*} \]

  • Where \(d_t\) contains exogenous variables e.g. the constant term

  • More compatly,

    \[ \begin{align*} y_t & = Ax_t + \varepsilon_t, & \varepsilon_t & \sim N(0, \Sigma) \\ \end{align*} \]

Bayesian estimation

  • Want to estimate \(A\) and \(\Sigma\), there is a conjugate prior

    \[ A|\Sigma\sim MN(\underline{A}, \Sigma, \underline{V}),\quad \Sigma\sim IW(\underline{S}, \underline{\nu}) \]

  • This generates closed-form posterior distributions

    \[ A|\Sigma,\text{data}\sim MN(\bar{A}, \Sigma, \bar{V}),\quad \Sigma|\text{data}\sim IW(\bar{S}, \bar{\nu}) \]

  • \(MN\) is the matrix normal distribution

    \[ A\sim MN(M,U,V)\Leftrightarrow \text{vec}(A)\sim N(\text{vec}(M),V\otimes U) \]

  • \(IW\) is the inverse Wishart distribution

    • Distribution for positive-definite matrices (covariance)
    • A generalization of the inverse gamma distribution

Minnesota prior

  • Still need to decide \(\underline{A}\), \(\underline{V}\), \(\underline{S}\), \(\underline{\nu}\)

    • for non-stationary variables, set corresponding elements in \(\underline{A}\) to 1
    • Shrink distant lagged coefficients to zero with small values in \(\underline{V}\)
    • \(\underline{S}=I\), \(\underline{\nu}=n+2\)

  • Improves forecast by reducing overfitting

An example

library(bsvarSIGNs)
head(optimism)
     productivity stock_prices consumption real_interest_rate hours_worked
[1,]    0.2172072    -11.28895   -4.331866        0.008022252    -7.599184
[2,]    0.2129482    -11.17647   -4.324255        0.021877748    -7.592445
[3,]    0.2069894    -11.11805   -4.318455        0.018811208    -7.580577
[4,]    0.2003908    -11.08317   -4.301382        0.012867114    -7.572133
[5,]    0.1945243    -11.02211   -4.293948        0.024503568    -7.577512
[6,]    0.2002138    -11.06306   -4.291474        0.000876887    -7.577491

  • The optimism dataset is available in the bsvarSIGNs package

  • Contains 5 quarterly US data from 1955Q1 to 2010Q41

    • productivity: total factor productivity on John Fernald’s website
    • stock prices: S&P 500 index divided by CPI, logged
    • consumption: real consumption expenditures, logged
    • real interest rate: federal funds rate minus inflation
    • hours worked: hours of all persons in the non-farm sector, logged


Estimation in R

set.seed(2025)
spec <- specify_bsvarSIGN$new(optimism, p = 4)
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
**************************************************|
  • Two lines of code!

Fitted values

fit <- compute_fitted_values(post)
plot(fit)

Forecast

forecast <- forecast(post, horizon = 4)
plot(forecast, data_in_plot = .2)

Exercise 1

  • Open the ex1.R file

  • Run the whole script to plot the fitted values and forecasts

Structural Vector Autoregression

Structural Vector Autoregression (SVAR)

  • Recall the VAR equation

    \[ y_t = Ax_t + \varepsilon_t \]

  • SVAR defines a linear relation between \(\varepsilon_t\) and strutural shocks \(u_t\)

    \[ u_t = B\varepsilon_t,\quad u_t\sim N(0, I) \]

  • \(u_{1,t},\dots,u_{n,t}\) are independent with each other and across time

    • they are \(n\) distinct causal drivers

  • While \(\varepsilon_t\) is a mixture of the structural shocks, causal relation is not clear \[ \varepsilon_t = B^{-1}u_t = B_0u_t \]

Simultaneous equation

  • SVAR is a system of simultaneous equations

    \[ \begin{align*} y_t & = Ax_t + \varepsilon_t \\ y_t & = Ax_t + B_0u_t \\ By_t & = BAx_t + u_t \end{align*} \]

  • e.g. suppose supply and demand are determined simultaneously

    \[ \begin{align*} p_{t} & = \frac{-B_{12}}{B_{11}}q_t + \dots + \frac{1}{B_{11}}u_{t}^D, & B_{11}>0,B_{12}>0 \\ q_{t} & = \frac{-B_{21}}{B_{22}}p_t + \dots + \frac{1}{B_{22}}u_{t}^S, & B_{22}>0,B_{21}<0 \end{align*} \]

  • In the SVAR notation, \[ \begin{align*} \begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \\ \end{bmatrix} \begin{bmatrix} p_{t} \\ q_{t} \\ \end{bmatrix} & = \dots+ \begin{bmatrix} u_{t}^D \\ u_{t}^S \\ \end{bmatrix} \\ By_t & = \dots+u_t \end{align*} \]

Identification problem

  • After fitting a VAR,

    \[ y_t = Ax_t + \varepsilon_t,\quad \varepsilon_t\sim N(0, \Sigma) \]

  • We have \(A\) and \(\Sigma\), but without restrictions cannot recover \(B\)

    \[ y_t = Ax_t + B^{-1}u_t,\quad u_t\sim N(0, I) \]

  • Since \(B\) has \(n^2\) elements, and \(\Sigma\) is symmetric and has only \(n(n+1)/2\) elements

  • Need \(n(n-1)/2\) restrictions to identify \(B\)

    • e.g. fix \(B\) to be lower triangular,

    \[ \begin{align*} \begin{bmatrix} \ast & 0 \\ \ast & \ast \\ \end{bmatrix} \begin{bmatrix} p_{t} \\ q_{t} \\ \end{bmatrix} & =\dots+ \begin{bmatrix} u_{t}^D \\ u_{t}^S \\ \end{bmatrix} \end{align*} \]

  • Can be controversial, price does not depend on quantity in demand equation?

Sign restriction 1: \(B\)

  • A less controversial way is to impose sign restrictions

  • e.g. demand (supply) curve should have negative (positive) slope

    • higher price \(\Rightarrow\) lower (higher) quantity demanded (supplied)
    • normalize \(B_{11}\) and \(B_{22}\) positive
    • restrict \(B_{12}\) positive and \(B_{21}\) negative

    \[ \begin{align*} B_{11}p_{t} & = -B_{12}q_t + \dots + u_{t}^D \\ B_{22}q_{t} & = -B_{21}p_t + \dots + u_{t}^S \end{align*} \]

  • In the SVAR notation, \[ \begin{align*} \begin{bmatrix} + & + \\ - & + \\ \end{bmatrix} \begin{bmatrix} p_{t} \\ q_{t} \\ \end{bmatrix} & =\dots + \begin{bmatrix} u_{t}^D \\ u_{t}^S \\ \end{bmatrix} \end{align*} \]

Exercise 2

  • Open the ex2.R file

  • This exercise simulates (log) price and (log) quantity data from

    \[ \begin{align*} \begin{bmatrix} 1 & 1 \\ -1 & 1 \\ \end{bmatrix} \begin{bmatrix} p_{t} \\ q_{t} \\ \end{bmatrix} & = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \begin{bmatrix} p_{t-1} \\ q_{t-1} \\ \end{bmatrix}+ \begin{bmatrix} u_{t}^D \\ u_{t}^S \\ \end{bmatrix} \end{align*} \]

  • Goal: estimate matrix \(B\) by imposing restrictions on

    • structural matrix \(B\)
    • 1 for positive, -1 for negative, NA for unrestricted

  • Hint: slide 20

Sign restriction 2: IRFs

  • There is nothing stopping us from imposing restrictions on \(B_0\)

  • \(B_0\) is the impulse response function (IRF) at horizon 0, why?

    \[ y_t = Ax_t + B_0u_t \]

  • e.g. consider the optimism dataset, can define optimistic shock as

    \[ \begin{align*} \begin{bmatrix} \text{productivity}_t \\ \text{stock_prices}_t \\ \text{consumption}_t \\ \text{real_interest_rate}_t \\ \text{hours_worked}_t \\ \end{bmatrix} =\dots+ \begin{bmatrix} \ast & \ast & \ast & \ast & \ast \\ + & \ast & \ast & \ast & \ast \\ \ast & \ast & \ast & \ast & \ast \\ \ast & \ast & \ast & \ast & \ast \\ \ast & \ast & \ast & \ast & \ast \\ \end{bmatrix} \begin{bmatrix} u_{t}^\text{optimism} \\ u_{2,t} \\ u_{3,t} \\ u_{4,t} \\ u_{5,t} \\ \end{bmatrix} \end{align*} \]

  • Can impose restrictions on IRFs at further horizons

    • e.g. \(A_1B_0\) at horizon 1, etc. \[ \begin{align*} y_{t+1} & = A_1 y_{t} + \dots \\ & =A_1(Ax_t+B_0u_t)+\dots \\ & =A_1B_0u_t+\dots \end{align*} \]

Sign restriction 3: narrative

  • Suppose there is strong evidence for the history of the shocks

  • e.g. a negative (unfavorable) supply shock at the begining of COVID-19

    \[ u_{2020Q1}^S < 0 \]

  • e.g. a negative optimism shock at the begining of Global Financial Crisis

    • a pessimistic shock! \[ u_{2007Q3}^{\text{optimism}} < 0 \]

Sign and zero restriction

  • Placing \(n(n-1)/2\) zero restrictions can be controversial

  • Might desire to impose just few zeros and signs on the IRFs

  • e.g. again the optimism dataset

    \[ \begin{align*} \begin{bmatrix} \text{productivity}_t \\ \text{stock_prices}_t \\ \text{consumption}_t \\ \text{real_interest_rate}_t \\ \text{hours_worked}_t \\ \end{bmatrix} =\dots+ \begin{bmatrix} 0 & \ast & \ast & \ast & \ast \\ + & \ast & \ast & \ast & \ast \\ \ast & \ast & \ast & \ast & \ast \\ \ast & \ast & \ast & \ast & \ast \\ \ast & \ast & \ast & \ast & \ast \\ \end{bmatrix} \begin{bmatrix} u_{t}^\text{optimism} \\ u_{2,t} \\ u_{3,t} \\ u_{4,t} \\ u_{5,t} \\ \end{bmatrix} \end{align*} \]

  • Can interpret first shock as an optimistic shock

    • No effect on productivity
    • Positive effect on stock prices

Dynamic causal effects: IRFs

Visualize structural shocks

Exercise 3

  • Open the ex3.R file

  • This exercise analyzes the US optimism dataset

  • Goal: replicate the IRF plot by imposing restrictions on

    • IRFs
    • 1 for positive, -1 for negative, NA for unrestricted, 0 for zero

  • Hint: slide 24

Exercise 4

  • Open the ex4.R file

  • This exercise extends the analysis on the US optimism dataset

  • Goal: compare IRF plots by additionally imposing

    • optimism shock is negative in 2007Q3, the start of the GFC

  • Hint: slide 23, row 211 of the dataset corresponds to 2007Q3, and

?specify_narrative

Appliactions in Macroeconomics

Monetary policies

  • According to Wikipedia…

Monetary policy is the policy adopted by the monetary authority of a nation to affect monetary and other financial conditions to accomplish broader objectives like high employment and price stability.

  • Central banks can

    • raise interest rate \(\Rightarrow\) save more \(\Rightarrow\) less money in market, contractionary
    • lower interest rate \(\Rightarrow\) save less \(\Rightarrow\) more money in market, expansionary

  • Think of economy like a machine

    • high inflation/output means overheating, need to cool down, contractionary monetary policy!
    • low inflation/output means underperforming, need to heat up, expansionary monetary policy!

Taylor rule

  • A simple rule to describe monetary policy

    \[ i_t = r_t + \pi_t + a_\pi(\pi_t-\pi^*) + a_y(y_t-y^*), \quad a_\pi, a_y > 0 \]

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

  • Raise interest rate (cool down) when inflation/output is too high

  • Lower interest rate (heat up) when inflation/output is too low

  • Usually, there is some form of the Taylor rule in a SVAR model

An Australian example

  • Is raising interest rate really contractionary?

  • Let’s analyze the effect of a positive monetary policy shock

  • 4 variables, from 1982Q1 to 2019Q41

    • CASH: cash rate, the policy interest rate in Australia
    • GDP: real gross domestic product
    • CPI: trimmed mean consumer price index, a measure of inflation
    • TWI: trade-weighted exchange rate, higher TWI means stronger AUD
# A tibble: 6 × 5
  Date                 CASH    GDP   CPI   TWI
  <dttm>              <dbl>  <dbl> <dbl> <dbl>
1 1982-03-31 00:00:00  15.4 156752  29.8  88.8
2 1982-06-30 00:00:00  18.4 158171  30.6  88.2
3 1982-09-30 00:00:00  17.8 157114  31.5  83.8
4 1982-12-31 00:00:00  13.7 154627  32.3  83.4
5 1983-03-31 00:00:00  13.9 153074  33.0  76.1
6 1983-06-30 00:00:00  12.1 152783  33.6  77.7


Contemporaneous relations \(B\)

  • Treat first shock as a monetary policy shock, using the SVAR notation,

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

  • The first row is similar to the Taylor rule

    \[ B_{11}\text{CASH}_t = -B_{12}\text{GDP}_t - B_{13}\text{CPI}_t - B_{14}\text{TWI}_t + \dots + u_{t}^{MP} \]

  • Base on previous discussion, we can restrict

    • \(B_{11}>0\), normalization
    • \(B_{12}<0\), cool down when high GDP
    • \(B_{13}<0\), cool down when high inflation
    • \(B_{14}>0\), discourage investing in AUD when high exchange rate

IRFs

  • Restricting \(B\) is not enough, let’s consider the IRFs

  • What happends to the variables when there is a positive monetary policy shock?

    • CASH: increases, by definition
    • GDP: unrestricted, since we are interested in its response
    • CPI: decreases, expect negative inflation
    • TWI: increases, encourage investing in AUD

    \[ \begin{align*} \begin{bmatrix} \text{CASH}_t \\ \text{GDP}_t \\ \text{CPI}_t \\ \text{TWI}_t \\ \end{bmatrix} =\dots+ \begin{bmatrix} + & \ast & \ast & \ast \\ \ast & \ast & \ast & \ast \\ - & \ast & \ast & \ast \\ + & \ast & \ast & \ast \\ \end{bmatrix} \begin{bmatrix} u_{t}^{MP} \\ u_{2,t} \\ u_{3,t} \\ u_{4,t} \\ \end{bmatrix} \end{align*} \]

  • Can extend these restrictions to horizons \(1,2,3\)

Result

Exercise 5

  • Open the ex5.R file

  • This exercise analyzes the Australian monetary policy dataset

  • Goal: replicate the IRF plot by imposing restrictions on

    • structural matrix \(B\)
    • IRFs
    • 1 for positive, -1 for negative, NA for unrestricted

  • Hint: slide 33 and 34

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