RDP 2013-07: An Empirical BVAR-DSGE Model of the Australian Economy Appendix A: Log-linearised Equations of the Model

This appendix lists the equations in the model. ~ denotes that a variable is a log deviation from the steady-state. Recall that 1 denotes the domestic good sector, 2 the export sector, and * foreign variables. Parameters which are functions of the steady state needed for the log-linearised equations are in the subsequent appendix. We have eliminated the nominal exchange rate from the model; all relationships are expressed in terms of the real exchange rate. Inline Equation, namely consumption of domestically produced goods, has also been substituted out. As the economy-wide technology, Zt, is non-stationary, many of the real variables have been normalised by the level of technology, for example Inline Equation. These are (not exclusively) denoted with lower case letters. f is introduced so as the model can be log-linearised around h = 0, and uses the approximation that ln(1 + x) ≈ x. It effectively linearises terms involving h. vm is similar. See Table A1 for a description of all variables.

Table A1: Variables
Variable Description Variable Description
Variables which have been detrended
c Consumption i Investment
w Wages va Value added
mc Marginal costs, domestic goods h Net foreign assets share of value added
y Output f f ≡ h + 1
m Real money balances tm Real transfers
k Capital vm vm ≡ tm + 1
Other variables
R Nominal interest rate π Domestic good inflation
π Consumer inflation r Gross rental rate on capital
Ψ Imports marginal costs s Consumer terms of trade Inline Equation
L Labour Inline Equation Nominal commodity prices in foreign currency
q Tobin's Q Inline Equation Real commodity prices in foreign currency Inline Equation
g Growth in nominal money Rer Real exchange rate
    Ω Ω = κZ Lagrange multiplier on the budget constraint
uc Preference um Import mark-up
ui Investment a2 Export sector technology
urp Risk premium a Technology growth
ε Monetary policy ud Domestic mark-up
The foreign VAR contains 4 reduced-form shocks
Observed variables
Δyobs Output growth πobs Inflation
Δiobs Investment growth s*obs Terms of trade
y*obs Foreign output gap Δrerobs Change in the real exchange rate
robs Interest rate Δxobs Change in exports

A.1 Consumer

Lagrange multiplier on budget constraint

Euler equation

Labour supply, domestic goods

Labour supply, exports

Tobin's Q, domestic goods

Tobin's Q, exports

where Inline Equation.

Investment, domestic goods

Investment exports

Capital accumulation, domestic goods

Capital accumulation, exports

Money demand

A.2 Domestic Good Firm

Phillips Curve


Marginal costs

Demand for capital

Demand for labour

A.3 Market Clearing

A.4 Importer

where the parameters are defined as per the domestic good.

A.5 Exporter

Labour demand

Capital demand

Production function

A.6 Open Economy

Uncovered interest parity

Net foreign assets

Real exchange rate and consumption terms of trade relationship

Consumption terms of trade growth

A.7 Autoregressive Processes

Technology growth

Risk premium

Preference shock

Export sector technology

Investment technology shock

A.8 Aggregates

CPI inflation

Value added

A.9 Monetary Policy

Taylor rule

Growth in nominal money

Real transfers

A.10 Foreign Sector

The foreign sector is modelled as a BVAR in the foreign output gap, inflation, interest rates and the detrended Australian terms of trade.

A.11 Measurement Equations

As the observed data have been demeaned, constants are omitted from the measurement equations.

Output growth

Investment growth

Foreign output gap

Interest rate

Foreign interest rates


Foreign inflation

Terms of trade

Change in the real exchange rate

Change in exports