# RDP 2013-06: Estimating and Identifying Empirical BVAR-DSGE Models for Small Open Economies Appendix B: Log-linearised Equations of the DSGE Model

These equations are based on code provided by Bruce Preston. I make two changes, namely eliminating the nominal exchange rate from the Taylor rule (and hence the model), which removes the need to include the real exchange rate identity, and making the steady-state mark-up of the small economy importers depend on the small economy elasticity of substitution, rather than the large economy's elasticity (although these were calibrated to be the same). The notation follows Justiniano and Preston (2010a), except that πt denotes log-linearised inflation. I denote US variables with a *, log-linearised variables in lower case, the jth shock by εj,t, and the innovation to it by vj,t.

## B.1 Variables

The mnemonics for the variables are given in Table B1.

Table B1: Mnemonics
Mnemonic Variable Mnemonic Variable
y Output s Terms of trade
i Nominal interest rate q Real exchange rate
π Inflation z Net foreign assets
w Real wages πF Imports inflation
πw Nominal wage inflation Shocks
n Hours worked Foreign cost-push
c Consumption εn Labour disutility
CH Consumption of home goods εa Technology
Y Level of output εi Monetary policy
Rp of domestic goods øt Risk premium
Rp of imports εg Preference
ψF Law on one price gap
Note: Rp denotes relative price to consumption

## B.2 Large Economy

IS curve

Price Phillips curve

where

and , where f* denotes the production function. Following

Justiniano and Preston (2006) this is calibrated at 0.33.

Wage Phillips curve

Where

and

Real wages

Labour

Taylor rule

Preference shock

Technology shock

Labour disutility shock

## B.3 Small Economy

Euler equation

Market clearing

where

and

Change in the terms of trade

Law of one price

Net foreign assets

Uncovered interest parity

Change in the real exchange rate

Wage inflation

where

and

Real wage

Labour

Taylor rule

Domestic good inflation

where

and where f denotes the production function. Following Justiniano and Preston (2006) this is calibrated at 0.33.

Foreign good inflation

where

CPI

Preference shock

Technology shock

Labour disutility shock