A Structural Model of Australia as a Small Open Economy

The Australian Economic Review, vol. 42, no. 1, pp. 24–41

A Structural Model of Australia as a Small Open Economy

Kristoffer P. Nimark∗Economic Research Department, Reserve Bank of Australia

Abstract

This paper sets up and estimates a structuralmodel of Australia as a small open economyusing Bayesian techniques. Unlike other recentstudies, the paper shows that a small micro-founded model can capture the open economydimensions quite well. Specifically, the modelattributes a substantial fraction of the volatil-ity of domestic output and inflation to foreigndisturbances, close to what is suggested by un-restricted VAR studies. The paper also investi-gates the effects of various exogenous shockson the Australian economy.

∗ The author thanks Jarkko Jaaskela, Christopher Kent,Mariano Kulish, Philip Liu, Adrian Pagan and Bruce Pre-ston for valuable comments and discussions. The viewsexpressed in this paper are those of the author and notnecessarily those of the Reserve Bank of Australia.

1. Introduction

This paper presents and estimates a small struc-tural model of the Australian economy with theaim of providing both a theoretically rigorousframework as well as rich enough dynamicsto make the model empirically plausible. Theeconomics of the model are simple. House-holds choose how much to consume and howmuch labour to supply. Firms choose pricesand then produce enough goods to meet de-mand. A fraction of the domestically producedgoods are exported and a fraction of the domes-tically consumed goods are imported, with thesize of the fractions determined by the relativeprice of goods produced at home and abroad.This is the minimal structure needed to cap-ture the open economy dimension of the Aus-tralian economy and it is similar to that usedin many other studies, for example Lubik andSchorfheide (2005), Gali and Monacelli (2005)and Justiano and Preston (2005). In addition tothis basic structure, the model is amended toaccount for the importance of the commoditiessector for Australian exports by adding exoge-nous export demand and income shocks.

Estimated models derived from micro foun-dations have become popular tools at centralbanks around the world. One reason often citedfor this is that structural models can be usedto produce counterfactual scenarios, as wellas to make predictions about how macroeco-nomic outcomes would change if alternativepolicies were implemented. Nessen (2006) pro-vides a useful perspective on how small struc-tural models can be used in the policy process.She argues that a model is not a tool that pro-vides answers to questions, but rather a frame-work of principles in which a structured andtransparent analysis can be conducted.

For any model to be a useful analytical tool,however, one first needs to establish whether or

C©2009 The University of Melbourne, Melbourne Institute of Applied Economic and Social ResearchPublished by Blackwell Publishing Asia Pty Ltd

Nimark: Structural Model of Australia as a Small Open Economy 25

not it provides a reasonable description of thedata. In a series of papers, Smets and Wouters(2003, 2004) show that medium scale mod-els can fit the dynamics of a large (closed)economy well. Some recent papers have askedwhether structural open economy models canprovide a similarly good fit (see, for exam-ple, Justiano and Preston 2005; Fukac, Paganand Pavlov 2006). Particularly, Justiano andPreston (2005) question whether these mod-els can account for the influence of foreignshocks on the domestic economy. This papershows that the influence of foreign shocks canindeed be captured by the dynamics of a smallstructural model and we argue that the model’ssuccess along this dimension is due to the in-clusion of trade quantities in the set of timeseries that are used to estimate the model.

The model is estimated using Bayesianmethods that exploit information from outsidethe data sample to generate posterior estimatesof the structural parameters. The number oftime series used is larger than in most otherstudies to ensure that the data spans the openeconomy dimension of the model. The mag-nitude of measurement errors in some of theobservable time series used is also estimated.This not only allows for errors in the data intro-duced through the data collection process, butalso recognises the fact that some of the theoret-ical variables of the model do not have clear-cutobservable counterparts. This approach also al-lows something to be said about how well thesetime series fit the cross-equation and dynamicimplications of the model.

2. A Small Scale Model of Australia

The structural model is in most respects astandard New Keynesian small open economymodel. But the model has a number of ad-justments to account for some features of theAustralian economy that are peculiar comparedwith many other developed countries. In partic-ular, while international trade for most devel-oped countries appears to be driven by benefitsthat come from specialisation, Australia’s ex-ternal trade appears to be driven more by classi-cal comparative advantage, with exports dom-inated by primary products, while more than

half of imports are manufactured goods (seeComposition of Trade 2005). In the standardmodel, the demand for a country’s exports aredetermined by the level of world output and thedomestic relative cost of production. Australiacan be considered to be a price taker in manyof its export markets and has little influenceover the price of its exports. Exogenous shocksare therefore added to both the volume of ex-port demand as well as the price that exportersreceive for their goods.

Australia is also considered a small economyin the model in the sense that macroeconomicoutcomes and policy in Australia are assumedto have no discernible impact on world output,inflation and interest rates. These foreign vari-ables are thus modelled as being exogenous toAustralia.

2.1 Household Preferences

A continuum of households populate the econ-omy, consume goods and supply labour tofirms. Consider a representative household in-dexed by i ∈ (0, 1) that wishes to maximise thediscounted sum of its expected utility,

Et

{ ∞∑s=0

βsU (Ct+s(i), Nt+s(i))

}(1)

where β ∈ (0, 1) is the household’s subjectivediscount factor. The period utility function inconsumption Ct and labour Nt is given by

U (Ct (i), Nt (i)) = exp(εct )

(Ct (i)H

−ηt

)1−γ

1 − γ

− Nt (i)1+ϕ

1 + ϕ(2)

and reflects the fact that households like to con-sume but dislike work. εc

t is a white noise pro-cess with variance σ 2

c . The variable Ht

Ht =∫

Ct−1(i)di (3)

is a reference level of consumption capturingthe notion that households not only care abouttheir own consumption, but also care about the

C©2009 The University of Melbourne, Melbourne Institute of Applied Economic and Social Research

26 The Australian Economic Review March 2009

lagged consumption of others. This feature—often referred to as ‘external habits’ or a prefer-ence for ‘catching up with the Joneses’—helpsto explain the inertia of aggregate output, sincepast levels of aggregate consumption are posi-tively related to the marginal utility of currentconsumption under this set up.

2.2 The Consumption Bundle

Households’ preferences are specified over acontinuum of differentiated goods that enterthe households’ utility function with decreas-ing marginal weight. Households thus preferto consume a mixture of differentiated goodsrather than consuming just one variety. Theconsumption bundle Ct is a constant elasticityof substitution (CES) aggregated index of do-mestically produced and imported sub-bundlesCd

t and Cmt

Ct ≡[

(1 − α)1δ C

d δ−1δ

t + α1δ C

m δ−1δ

t

] δδ−1

(4)

Cdt ≡

[∫Cd

t (j )υ−1υ

] υυ−1

(5)

Cmt ≡

[∫Cm

t (j )υ−1υ

] υυ−1

(6)

The domestic price index (CPI) that is consis-tent with the specification of the utility functionis then given by

Pt ≡ [(1 − α) P d1−δ

t + αP m1−δt

] 11−δ (7)

This specification implies that in steadystate, domestic households spend a fraction(1 − α) of their income on domestically pro-duced goods.

2.3 Import Demand

The domestic demand for imported goods Cmt

can be shown to be

Cmt = Ct exp

(vm

t

)exp (τt )

−δ (8)

which depends on the relative price of importsτ t as perceived by the domestic consumer

τt = log

(P m

t

Pt

)(9)

Thus, the cheaper are imported goods rela-tive to domestic goods, the larger will be theshare of imported goods in the consumptionbundle. The exogenous shock to the domesticconsumers demand for imported goods vm

t canbe interpreted as a ‘taste’ shock and is assumedto follow an AR(1) process

vmt = ρmvm

t−1 + εmt (10)

εmt ∼ N

(0, σ 2

m

)(11)

The exogenous taste shock vmt absorbs vari-

ations in imports that cannot be explained bychanges in relative prices, but ideally shouldonly explain a small portion of the dynamics ofimports.

2.4 The Domestic Budget Constraint andInternational Financial Flows

The representative household optimises theutility function, equation (1), subject to its flowbudget constraint

Bt+1 + B∗t+1 + Ct − ψ

2B2∗

t = Yt

+ (exp v

pxt − 1

)Xt

+ RtPt−1Pt

Bt + R∗t

St Pt−1St−1Pt

B∗t (12)

The variables on the left hand side are ex-penditure items and the terms on the right handside are income items. Bt(i) and B∗t (i) are do-mestic and foreign bonds, respectively, whereboth are expressed in real domestic terms. Theirrespective nominal returns are Rt and R∗t . St isthe nominal exchange rate defined such that anincrease in St implies a depreciation of the do-mestic currency. The term ψ

2 B2∗t is a cost paid

by domestic households when they are net bor-rowers in the aggregate (see Benigno 2001).This ensures that the net asset position of thedomestic economy is stationary and it implies



that, ceteris paribus, a highly indebted countywill have a higher equilibrium interest rate. Yt

on the right hand side is real GDP and the termexp v px

t Xt is export income adjusted for exoge-nous fluctuations in the price of exports (moreon this below).

Assuming a zero net supply of domesticbonds we can write the flow budget constraintas a difference equation describing the evolu-tion of the net foreign asset position

B∗t+1 = R∗

t

StPt−1

St−1Pt

B∗t − ψ

2B2∗

t

+ exp vpxt Xt − Cm

t(13)

where the change in the net foreign asset posi-tion is the difference between income receivedfor exports and expenditure on imports plusvaluation effects from inflation and changes inthe nominal exchange rate and the net debtorcost ψ

2 B2∗t . Households choose consumption

subject to the flow budget constraint given byequation (12). Optimally allocating consump-tion over time yields the standard consumptionEuler equation

UC(Ct ) = βEtRt

PtUC(Ct+1)

Pt+1(14)

where UC(Ct) is the marginal utility of con-sumption in period t. Households also choosebetween allocating their savings to bonds de-nominated in the domestic and foreign cur-rency. Equating the marginal expected returnon foreign and domestic bonds yields the un-covered interest rate parity (UIP) equilibriumcondition

Rt = (exp(vs

t

)Et

R∗t

ψB∗t

St

St+1(15)

where vst is a time varying ‘risk premium’ that

is assumed to follow the AR(1) process

vst = ρsv

st−1 + εs

t (16)

εst ∼ N

(0, σ 2

s

)(17)

The time varying and persistent risk premiumvs

t is necessary to account for the observed

deviations of the exchange rate from thatimplied by the UIP condition. There is no con-sensus in the literature on the causes of thedeviations and the interpretation of the risk pre-mium shock does not have to be literal.1

2.5 Firms

The domestic economy is populated by twotypes of firms: producers and importers. Do-mestic producers indexed by j use labour as thesole input to manufacture differentiated goodswith a linear technology

Yt (j ) = exp(at )Nt (j ) (18)

where at is a sector wide exogenous processthat augments labour productivity assumed tofollow

vat = ρav

at−1 + εa

t (19)

εat ∼ N

(0, σ 2

a

)(20)

In addition to the production sector, there isa sector that imports differentiated goods fromthe world and resells them domestically.

Firms have some market power over the priceof the goods that they are selling since con-sumers prefer a mixture of differentiated goodsrather than consuming just one variety. Unlikethe case when all goods are perfect substitutes,this means that consumers will not switch con-sumption away completely from a slightly moreexpensive good. In this monopolistically com-petitive environment firms charge a markupover marginal cost.

Quantities sold in a given period are demanddetermined in the sense that firms are assumedto set prices in domestic currency terms andthen supply the amount of goods that are de-manded by consumers at that price. Both im-porters and domestic producers set prices ac-cording to a discrete time version of the Calvo(1983) mechanism whereby a fraction θd offirms producing domestically and a fractionθm of importing firms do not change pricesin a given period. A fraction ω of both the do-mestic producers and importers that do changeprices, use a rule of thumb that links their price



to lagged inflation (in their own sector). This isa two-sector generalisation of Gali and Gertler(1999) that yields two Phillips curves of thefollowing form

πdt = μd

f πdt+1 + μd

bπdt−1 + λdmcd

t + επt (21)

and

πmt = μm

f πmt+1 + μm

b πmt−1 + λmmcm

t + επt (22)

where mcdt is the marginal cost of the domestic

producers and mcmt , defined as

mcmt = log

(StP

∗t

Pt

)(23)

is the real unit cost at the dock of importedgoods. The shock επ

t is a cost push shock com-mon to both sectors. The parameters in thePhillips curves are given by

μsf ≡ βθs

θs + ω (1 − θs (1 − β)),

μsb ≡ ω

θs + ω (1 − θs (1 − β))

λs ≡ (1 − ω) (1 − θs) (1 − βθs)

θs + ω (1 − θs (1 − β)), s ∈ {d,m}

and domestic CPI inflation is simply theweighted average of inflation in the two sec-tors

πt = (1 − α) πdt + απm

t (24)

2.6 Export Demand

As mentioned above, a large share of Australianexports are commodities that are traded in mar-kets where individual countries have little mar-ket power. The standard specification of exportdemand is amended to reflect the fact that Aus-tralian exports and export income depend onmore than just the relative cost of productionin Australia and the level of world output, aswould be the case in a standard open economymodel. Two shocks are added to the model.

The first shock, vxt captures variations in ex-

ports that are unrelated to the relative cost ofthe exported goods and the level of world out-put. Export volumes are then given by

Xt = (exp vx

t

) (P d

t

P ∗t

)δx

Y ∗t (25)

where Y ∗t is world output and vx

t is an exoge-nous shock that follows the AR(1) process

vxt = ρxv

xt−1 + εx

t (26)

εxt ∼ N

(0, σ 2

x

)(27)

We also want to allow for ‘windfall’ profitsdue to exogenous variations in the world marketprice of the commodities that Australia exports.We therefore add a shock to the export incomeequation, which in domestic real terms is givenby

Y xt = (

exp vpxt

)Xt (28)

The shock v pxt is thus a shock to real income

(expressed in real domestic currency terms) re-ceived for the goods that Australia exports. Itis assumed to follow the AR(1) process

vpxt = ρpxv

px

t−1 + εpxt (29)

εpxt ∼ N

(0, σ 2

px

)(30)

It is worth emphasising here the different im-plication of a shock to export demand, vx

t , as op-posed to a shock to export income, vpx

t : the for-mer leads to higher export incomes and higherlabour demand, while the latter improves thetrade balance without any direct effect on thedemand for labour by the exporting industry.

2.7 The World Economy

The log of world output, inflation and inter-est rates, denoted {y∗

t , π∗t , i∗t }, are assumed to

follow an unrestricted vector autoregression

⎡⎣ y∗

t

π∗t

i∗t

⎤⎦ = M

⎡⎣ y∗

t−1π∗

t−1i∗t−1

⎤⎦ + ε∗

t (31)



The rest of the world is assumed to be un-affected by the Australian economy and thecoefficients in M and the covariance matrix ofthe world shock vector ε∗

t can therefore be es-timated separately from the rest of the model.

2.8 Monetary Policy

A simple way to represent monetary policy thathas been found to empirically fit central bankbehaviour quite well is to let the short interestrate follow a variant of the Taylor rule, lettingthe interest rate be determined by a reactionfunction of lagged inflation, lagged output andthe lagged interest rate:

it = φyyt−1 + φππt−1 + εit (32)

where εit is a transitory deviation from the rule

with variance σ 2i . This completes the descrip-

tion of the structural model.2

3. Estimation Strategy

The parameters of the model are estimated us-ing Bayesian methods that combine prior in-formation and information that can be extractedfrom aggregate data series. An and Schorfheide(2007) provide an overview of the methodol-ogy. Conceptually, the estimation works in thefollowing way. Denote the vector of parame-ters to be estimated � ≡ {γ , η, ϕ . . . } andthe log of the prior probability of observinga given vector of parameters L(�). The func-tion L(�) summarises what is known about theparameters prior to estimation. The log likeli-hood of observing the data set Z for a givenparameter vector � is denoted L(Z|�). Theposterior estimate �̂ of the parameter vector isthen found by combining the prior informationwith the information in the estimation sample.In practise, this is done by numerically max-imising the sum of the two over �, so that�̂ = arg max(L(�) + L(Z|�))

The first step of the estimation process is tospecify the prior probability over the param-eters �. Prior information can take differentforms. For instance, for some parameters eco-nomic theory determines the sign. For other pa-rameters we may have independent survey data,

as is the case for the frequency of price changes,for example (see Bils and Klenow 2004;Alvarez et al. 2005). Priors can also be basedon similar studies where data for other coun-tries were used. The restrictions implied by thetheoretical model means that prior informationabout a particular parameter can also be usefulfor identifying other parameters more sharply.For instance, it is typically difficult to sepa-rately identify the degree of price stickiness θ

and the curvature of the disutility of supplyinglabour γ just by using information from aggre-gate time series. However, a combination of thetwo variables may have strong implications forthe likelihood function (that is, there may bea ‘ridge’ in the likelihood surface). Survey ev-idence suggests that the average frequency ofprice changes is somewhere between five and13 months. By choosing a prior probability forthe range of the stickiness parameter θ that re-flects this information, we may also identify γ

more sharply.Unfortunately, we do not have independent

information about all of the parameters of themodel. A cautious strategy when hard priorsare difficult to find is to use diffuse priors, thatis, to use prior distributions with wide disper-sions. If the data is informative, the dispersionof the posterior should be smaller than that ofthe prior. However, Fukac, Pagan and Pavlov(2006) point out that using informative priors,even with wide dispersions can affect the pos-teriors in non-obvious ways.

Arguably, hard prior information exists forthe discount factor β, the steady-state shareof imports/exports in GDP a and the aver-age duration of good prices θd and θm. Thefirst two can be deduced from the averagereal interest rate and the average share of im-ports and exports of GDP and are calibratedas {β, α} = {0.99, 0.18}. Calibration can beviewed as a very tight prior. The price stick-iness parameters θd and θm are assigned pri-ors that are centred around the mean durationfound in European data (see Alvarez et al.2005).

The prior distributions of the variances of theexogenous shocks are truncated uniform overthe interval [0,∞). It is common to use more re-strictive priors for the exogenous shocks, as for



example in Smets and Wouters (2003), Lubikand Schorfheide (forthcoming), Justiano andPreston (2005) and Kam, Lees and Liu (2006),but since most shocks are defined by the partic-ular model used, it is unclear what the sourceof the prior information would be.

The priors of the variances of the mea-surement error parameters are uniform distri-butions on the interval [ 0, σ 2

Zn) where σ 2Zn

is the variance of the corresponding time se-ries. Economic theory dictates the domains ofthe rest of the priors, but we have little in-formation about their modes and dispersions.These priors are therefore assigned wide dis-persions. Information about the prior distribu-tions for the individual parameters is given inTable 1.

Table 1 Prior and Posterior Distributions of Parameters

Prior distribution Posterior distribution

Parameter Type Mode Standard deviation Mode Standard deviation

Households and firmsγ normal 3 0.44 2.97 0.30η normal 2 0.66 1.48 0.13ϕ normal 2 0.44 1.35 0.31ω Beta 0.3 0.10 0.24 0.08δ normal 1 0.10 0.86 0.09δx normal 1 0.10 0.15 0.06θ beta 0.75 0.04 0.89 0.01θm beta 0.75 0.04 0.90 0.01ψ normal 0.01 0.02 0.07 0.02

Taylor ruleφ y normal 0.5 0.25 0.02 0.01φπ normal 1.5 0.29 0.41 0.04φ i beta 0.5 0.25 0.87 0.03

Exogenous persistenceρ a beta 0.5 0.28 0.71 0.05ρ s beta 0.5 0.28 0.81 0.08ρ px beta 0.5 0.28 0.81 0.05ρ x beta 0.5 0.28 0.90 0.07ρm beta 0.5 0.28 0.80 0.05

σ 2a uniform [0,∞) 9.13×10−5 2.30×10−5

σ 2s uniform [0,∞) 2.02×10−3 2.59×10−3

σ 2c uniform [0,∞) 1.79×10−5 5.81×10−6

σ 2π uniform [0,∞) 2.70×10−5 6.92×10−5

σ 2px uniform [0,∞) 6.82×10−5 3.39×10−5

σ 2x uniform [0,∞) 4.89×10−5 8.84×10−5

σ 2m uniform [0,∞) 2.12×10−5 5.51×10−6

σ 2i uniform [0,∞) 7.58×10−7 1.86×10−7

3.1 Mapping the Model into ObservableTime Series

The model of Section 2 is solved by first takinglinear approximations of the structural equa-tions around the steady state and then find-ing the rational expectations equilibrium lawof motion. The linearised equations are listedin the Appendix and the Soderlind (1999) algo-rithm was used to solve the model. The solutioncan be written in VAR(1) form

Xt = AXt−1 + Cεt (33)

where Xt is a vector containing the variables ofthe model and the coefficient matrices A andC are functions of the structural parameters �.



equation (33) is called the transition equation.The next step is to decide which (combinations)of the variables in Xt are observable. The map-ping from the transition equation to observabletime series are determined by the measurementequation

Zt = DXt + et (34)

The selector matrix D maps the theoreticalvariables in the state vector Xt into a vectorof observable variables Zt. The term et is avector of measurement errors. For theoreticalvariables that have clear counterparts in ob-servable time series, the measurement errorscapture noise in the data collecting process.The measurement errors may also capture dis-crepancies between the theoretical concepts ofthe model and observable time series. For in-stance, GDP, non-farm GDP and market sec-tor GDP all measure output, but none of thesemeasures corresponds exactly to the model’svariable yt. The measure of total GDP includesfarm output, which varies due to factors otherthan technology and labour inputs, most no-tably the weather. One may therefore want toexclude farm products. But in the model, moreabundant farm goods will lead to higher over-all consumption and lower marginal utility andperhaps also higher exports, so excluding italtogether is also not appropriate. Total GDPalso includes government expenditure whichis not determined by the utility maximisingagents of the model, but it will affect the aggre-gate demand for labour and therefore marketwages. The state space system, that is, the tran-sition equation (33) and the measurement equa-tion (34), is quite flexible and can incorporateall three measures of GDP, allowing the data todetermine how well each of them correspondto the model’s concept of output. This multi-ple indicator approach was proposed by Boivinand Giannoni (2005) who argue that not onlydoes this allow us to be agnostic about whichdata to use, but by using a larger informationset it may also improve estimation precision.

Some, but not all, of the observable time se-ries are assumed to contain measurement errorsand the magnitude of these are estimated to-gether with the rest of the parameters. Counting

both measurement errors and the exogenousshocks, the total number of shocks in the modelis more than is necessary to avoid stochasticsingularity. That is, the total number of shocksis larger than the total number of observablevariables in Zt. It is reasonable to ask whetheror not all of the shocks can be identified and theanswer is that it depends on the actual data gen-erating process. The measurement errors arewhite noise processes specific to the relevanttime series that are uncorrelated with other in-dicators as well as with their own leads andlags. To the extent that the cross-equation anddynamic implications that distinguish the struc-tural shocks from the measurement errors ofthe model are also present as observable cor-relations in the time series, it will be possibleto identify the structural shocks and the mea-surement errors separately. Incorrectly exclud-ing the possibility of measurement errors maybias the estimates of the parameters governingboth the persistence and variances of the struc-tural shocks. Also, by estimating the magnitudeof the measurement errors we can get an ideaof how well different data series match the cor-responding model concept.

3.2 Computing the Likelihood

The linearised model, equation (33), and themeasurement, equation (34), can be used tocompute the covariance matrix of the theoret-ical, one step ahead forecast errors implied bya given parameterisation of the model. That is,without looking at any data, we can computewhat the covariance of our errors would be ifthe model was the true data generating processand we used the model to forecast the observ-able variables. This measure, denoted �, is afunction of both the assumed functional formsand the parameters and is given by

� = DPD′ + Eete′t (35)

where P is the covariance matrix of the oneperiod ahead forecast errors of the state

P = A(P − PD′ (DPD′ + Eete

′t

)−1DPA′

)+ CEεtε

′tC

′ (36)



The covariance of the theoretical forecast er-rors � is used to evaluate the likelihood of ob-serving the time series in the sample, given aparticular parameterisation of the model. For-mally, the log likelihood of observing Z giventhe parameter vector � is

L (Z| �) = −.5T∑

t=0

[p ln(2π ) + ln |�|

+ u′t�

−1ut

](37)

where p×T are the dimensions of the observ-able time series Z and ut is a vector of theactual one step ahead forecast errors from pre-dicting the variables in the sample Z using themodel parameterised by �. The actual (sample)one step ahead forecast errors can be computedfrom the innovation representation

X̂t+1 = AX̂t + Kut (38)

ut = Zt − DX̂t (39)

where K is the Kalman gain

K = APD′ (DPD′ + Eete′t

)−1

The method is described in detail in Hansenand Sargent (2005).

To help understand the log likelihood func-tion intuitively, consider the case of only oneobservable variable so that both � and ut arescalars. The last term in the log likelihood func-tion, equation (37) can then be written as u2

t /�

so for a given squared error u2t the log likelihood

increases in the variance of the model’s fore-cast error variance. This term will thus make uschoose parameters in � that make the forecasterrors of the model large since a given erroris more likely to have come from a param-eterisation that predicts large forecast errors.The determinant term ln |�| (the determinantof a scalar is simply the scalar itself) countersthis effect; to maximise the complete likeli-hood function we need to find the parametervector � that yields the optimal trade-off be-tween choosing a model that can explain ouractual forecast errors ut while not making theimplied theoretical forecast errors too large.

Another way to understand the likelihoodfunction is to recognise that there are (roughlyspeaking) two sources contributing to the fore-cast errors ut, namely shocks and incorrect pa-rameters. The set of parameters � that max-imises the log likelihood function, equation(37), are those that reduce the forecast errorscaused by incorrect parameters as much as pos-sible by matching the theoretical forecast errorvariance � with the sample forecast error co-variance Eut u

′t , thereby attributing all remain-

ing forecast errors to shocks.

3.3 The Data

The data sample is from 1991:Q1 to 2006:Q2where the first eight observations are used asa convergence sample for the Kalman filter.13 time series were used as indicators for thetheoretical variables of the model, which ismore than that of most other studies estimat-ing structural small open economy models. Lu-bik and Schorfheide (2007) estimate a smallopen economy model on data for Canada, theUnited Kingdom, New Zealand and Australiausing terms of trade as the only observable vari-able relating to the open economy dimensionof the model. Similarly, in Justiano and Pre-ston (2005) the real exchange rate between theUnited States and Canada is the only data se-ries relating to the open economy dimension ofthe model. Neither of these studies use tradevolumes to estimate their models. This is alsotrue for Kam, Lees and Liu (2006), though thisstudy uses data on imported goods prices ratherthan only aggregate CPI inflation.

In this paper, data for the rest of theworld is based on trade weighted G7 out-put and inflation and an (unweighted) aver-age of US, Japanese and German/euro interestrates.3 Three domestic indicators that are as-sumed to correspond exactly to their respectivemodel concepts are the cash rate, the nominalexchange rate and trimmed mean quarterly CPIinflation. The rest of the domestic indicatorsare assumed to contain measurement errors.These are GDP, non-farm GDP, market sec-tor GDP, exports as share of GDP, the termsof trade (defined as the price of exports overthe price of imports) and labour productivity.



Table 2 Relative Magnitude of Measurement Errors

Data Model �ee/�zz

Interest rate it –Nominal exchange rate change �st –CPI trimmed mean inflation π t –Real GDP yt 0.03Real non-farm GDP yt 0.05Real market sector GDP yt 0.16Export share of GDP xt − yt 0.00Import share of GDP cm

t − yt 0.00Terms of trade vpx

t − mcmt 0.60

Labour productivity at 0.00

All real variables are linearly detrended and in-flation and interest rates were demeaned. Thecorrespondence between the data series and themodel concepts are described in Table 2.

4. Estimation Results

Table 1 reports the mode and standard devia-tion of the prior and posterior distributions ofthe structural parameters of the model. The pos-terior modes were found using Bill Goffe’s sim-ulated annealing algorithm. The posterior dis-tribution was generated by the Random-WalkMetropolis Hastings algorithm using 2 milliondraws, where the starting value for the param-eter vector is the mode of the posterior as es-timated by the simulated annealing algorithmand the first 100 000 draws are used as a burn-insample.

Ideally, the posterior distributions shouldhave a smaller variance than the prior distri-bution since this would indicate that the data isinformative about the parameters. For most ofthe parameters this is the case. Imports seem tobe more price elastic than exports, as evidencedby the significantly larger estimated value of δ

as compared to δx. The estimated frequency ofprice changes in the imported goods sector islower than that estimated for prices in the do-mestically produced goods sector.

The parameters in the Taylor rule suggestthat policy responses to inflation and outputare very gradual, with a high estimated valuefor the parameter on the lagged interest rate.The response of the short interest rate to outputdeviations is quite small, with the short interestrate appearing to respond mostly to inflation.

4.1 Model Fit

The in-sample fit of the model can be as-sessed by plotting the one period ahead fore-casts against the actual observed indicators (seeFigure 1).

The model provides a very good in-sampledescription of the dynamics of the cash rate,which is likely to be primarily because its per-sistence makes it easy to predict. The model isalso able to fit most of the other time series rea-sonably well, with the exception of the nominalexchange rate and the terms of trade.

The variances of the errors in the mea-surement, equation (34), are estimated jointlywith the structural parameters of the model.These variances capture series specific transi-tory shocks to the observable time series. Alow estimated measurement error variance in-dicates that the associated observable time se-ries matches the corresponding model conceptclosely. The ratios of the measurement errorsover the variance of the corresponding time se-ries are reported in Table 2.

The variance ratios for the various measuresof GDP are particularly interesting, since weused multiple indicators for this variable. Theestimated value of these ratios indicate that realGDP appears to conform slightly better to thedynamic and cross-equation implications of themodel than real non-farm GDP, but the differ-ence is small. The third indicator for output,domestic market sector GDP appears to pro-vide the poorest fit.

The terms of trade stands out as the timeseries that the model has the biggest problemfitting; more than half of the variance of theterms of trade is estimated to be due to mea-surement errors.

4.2 The Open Economy Dimension of theModel

Table 3 below reports the variance decompo-sition4 of the model evaluated at the estimatedposterior mode reported in Table 1. The firstrow contains the fraction of the variancesthat originate from outside Australia. Foreignshocks explain 27 per cent, 21 per cent and22 per cent respectively of the variance of



Figure 1 Actual Data and Model’s One Step Ahead Predictions

Table 3 Variance Decomposition

Output Inflation Exports �Exchange rate Interest rateShock/variable y π x �s i

Foreign 0.27 0.21 0.45 0.19 0.22(0.11–0.46) (0.10–0.37) (0.26–0.59) (0.10–0.22) (0.08–0.41)

Productivity 0.00 0.07 0.00 0.01 0.01(0–0.2) (0.02–0.16) (0–0.02) (0–0.01) (0–0.02)

UIP 0.06 0.04 0.06 0.56 0.05(0.01–0.18) (0.01–0.14) (0.02–0.23) (0.44–0.69) (0.01–0.12)

Demand 0.16 0 0.02 0.01 0.05(0.04–0.27) (0–0.01) (0.01–0.05) (0.01–0.01) (0.01–0.15)

Cost push 0.02 0.19 0 0.02 0.03(0–0.14) (0.06–0.34) (0–0.02) (0–0.15) (0–0.11)

Export demand 0.08 0.09 0.12 0.08 0.06(0.02–0.16) (0.03–0.17) (0.05–0.22) (0.03–0.13) (0.01–0.13)

Export price 0.23 0.34 0.27 0.11 0.25(0.09–0.48) (0.14–0.67) (0.11–0.51) (0.06–0.25) (0.05–0.60)

Import demand 0.03 0.03 0.06 0.02 0.02(0.01–0.05) (0.01–0.07) (0.03–0.09) (0.01–0.04) (0.01–0.06)

Taylor rule 0.15 0 0.02 0.01 0.31(0.03–0.29) (0–0.02) (0.01–0.06) (0.01–0.03) (0.11–0.56)

Note: Figures in brackets indicate 95% posterior probability intervals.



Figure 2 Impulse Responses to Monetary Policy Shock

domestic output, inflation and interest rates.This can be compared to what is suggestedby an unrestricted VAR(4) in world and do-mestic output, inflation and interest rates (withthe world variables assumed to be exogenousto the domestic variables). Such estimates sug-gest that foreign shocks are responsible for 49per cent, 32 per cent and 45 per cent of thedomestic variance of output, inflation and in-terest rates, respectively. The structural modelparameterised at the posterior mode thus at-tributes somewhat less of the variance of do-mestic variables to foreign shocks than the un-restricted VAR regressions; although for infla-tion, it is spanned by the 95 per cent probabilityinterval.

The fact that the model is close to match-ing the evidence of the influence of foreignshocks on the Australian economy from a lessrestricted model is reassuring, but at odds withsome previous studies. Justiano and Preston(2005), estimate a seemingly unrelated regres-sion model on Canadian and US data and find

that a sizeable fraction of domestic volatilitydoes indeed originate abroad. However, theirstructural model, which is similar to the onepresented here, attributes less than 1 per cent toforeign sources. They interpret this as a failureof their structural model to capture the openeconomy aspects of the data, in spite of itsability to replicate the cross correlations anddynamics of the Canadian variables.

Apart from the fact that the models are esti-mated using data for different countries, whatcan explain this difference in results? One rea-son may be that Justiano and Preston let theUnited States proxy for the world economywhile in this paper the rest of the world isrepresented by trade weighted data on a largerset of countries. Any shock that emanates fromoutside the United States, for instance from Eu-rope, will be attributed to the United Statesin their reduced form exercise, but it is notclear that a European shock will be appro-priately captured by the bilateral US–Canadadata.



Figure 3 Impulse Responses to Export Demand Shock

Another reason why the present model maybetter capture the impact of foreign shocks isthat it is estimated using data on trade vol-umes. Not using data on imports and exportsmakes it harder for any model to distinguishbetween domestic demand shocks and demandfor the domestically produced goods comingfrom abroad.

These results are not significantly affected bythe inclusion of measurement errors in someof the time series. The fraction of domesticvariance attributed to foreign shocks (evalu-ated at the mode) increases somewhat, but the95 per cent probability intervals hardly changeat all.5

4.3 The Impact of a Monetary Policy Shock

Figure 2 displays the impulse responses to aunit shock to the (annualised) cash rate for se-lected endogenous variables together with the95 per cent probability intervals.

Evaluated at the mode, an unanticipatedincrease in interest rates of one percentagepoint leads to a fall in output with the max-imum negative response of 2.5 percentagepoints occurring after six quarters. There aretwo factors contributing to the fall in out-put. First, the higher real interest rate leads toa fall in domestic consumption. Second, thehigher return on domestic bonds leads to ahigher demand for the domestic currency de-nominated assets, leading to a currency ap-preciation. Lower domestic consumption andless demand for labour both reduce the mar-ket real wage, causing a fall in inflation. Thisis reinforced by the appreciating exchange ratewhich makes imports cheaper and further de-creases inflation. (However, initially consumerprices of imported goods do not fall as muchas domestically produced goods which makesimported goods initially relatively more expen-sive.) The peak response of (annualised) infla-tion to the unit shock to the interest rate is a



Figure 4 Impulse Responses to Export Income Shock

fall of approximately 0.15 percentage pointstwo quarters after impact. The estimated max-imum response of inflation to a monetary pol-icy shock is faster than that which is found insome other studies, including those employingstructural VARs (see, for instance, Dungey andPagan 2000; Berkelmans 2005). Some of thisdifference may be explained by the relativelystringent restrictions imposed by the structuralmodel compared with an SVAR. Another fac-tor that could contribute to the relatively rapidresponse to a monetary policy shock in thepresent model may be that the sample used doesnot include the change to an inflation target-ing regime in the early 1990s. If the credibilityof the new monetary policy regime was estab-lished only gradually, then this could contributeto a relatively slow estimated responses of in-flation and output to an increase in the cashrate for studies that incorporate this transitoryperiod.

4.4 The Impact of Export Demand andIncome Shocks

The effects of an exogenous increase in thedemand for Australian exports are illustratedin Figure 3. A one percentage point increase inexport demand leads on impact to a 0.2 percent-age point increase in GDP (consistent with theshare of the export sector in GDP). It also leadsto an appreciation of the exchange rate andboosts imports. The appreciating exchange rateleads to a fall in inflation, though it is quanti-tatively small (less than 0.03 percentage pointsat the maximum impact). These effects can becontrasted with the estimated response to a pos-itive shock to the export price. Remember, themain difference between the export price anddemand shock is that a price shock does not putdirect pressure on the domestic labour market.Figure 4 shows that an income shock, like ademand shock, leads to an appreciation of the



Figure 5 Impulse Responses to Productivity Shock

exchange rate. The response of the endogenousvariables are very similar, with the exceptionof the volume of exports, which falls due to theappreciating exchange rate. Due to the low elas-ticity of export demand, the quantitative effectis small.

4.5 The Impact of a Productivity Shock

Figure 5 plots the impulse responses to a unitshock to Australian productivity. As expected,inflation falls and the nominal exchange rateappreciates. A less obvious effect is that theconsumption of imported goods falls in spite ofthe appreciating exchange rate. This is becausedomestic goods prices fall sufficiently so as tomake imports relatively more expensive. Theeffect of overall GDP growth is ambiguous.

5. Conclusion

This paper presents a small structural modelof the Australian economy estimated using

Bayesian techniques and based on a standardNew Keynesian small open economy specifi-cation similar to that used by numerous otherstudies. However, there are four aspects inwhich the estimation of the model deviatesfrom previous studies.

The first is that the export demand and ex-port income equations are amended with ex-ogenous shocks to control for the prominentrole played by commodities in the Australianexport sector. When the model is estimated, theexport demand shock appears to play a largerrole than the export income shock in explainingthe variance of domestic variables.

Second, a larger number of time series wereused to estimate the model. In particular, dataon import and export volumes were used inaddition to the standard aggregate variables toensure that the data spans the open economydimension of the model.

Third, flat prior distributions were used forthe variances of the structural shocks. This re-flects the fact that most of the structural shocks



are defined jointly by the model and the datawith little or no role for economic theory norindependent sources of information to help de-termine the magnitude of these shocks.

Fourth, the magnitude of measurement er-rors in some of the time series was estimatedtogether with the structural parameters of themodel. This acknowledges the fact that not onlyis error sometimes introduced through the datacollection process, but also that the model vari-ables do not always have clear cut counterpartsin observable time series.

The estimated model provides a good fitfor most of the observable variables and ap-pears to be able to capture the open econ-omy dimensions of the data reasonably well.The model produces estimates that are closeto those from studies using unrestricted VARson the importance of foreign shocks to the do-mestic variance of output, inflation and inter-est rates. Given the simplicity of the model,this result holds promise for the usefulness ofthese types of open economy models as ana-lytical tools. However, there are other dimen-sions in which the model performs less well.Particularly, movements in the terms of tradeare not well captured by the model and thereasons for this should be a subject of futureinvestigation.

First version received September 2007;final version accepted July 2008 (Eds).

Appendix 1: The Linearised Model

The consumption Euler Equation

ct = γ

γ − η − γ ηEtct+1 + −η (1 − γ )

γ − η − γ ηct−1

− 1γ−η−γ η

(it − Etπt+1) + εct (A1)

Import demand

cmt = ct − δτt + vm

t (A2)

Domestic consumption demand

cdt = ct + δτt (A3)

The relative price of imported goods for thedomestic consumer

τt = τt−1 + πmt − πt (A4)

Export demand

xt = −δxτ ∗t + Y ∗

t + vxt (A5)

The relative price of goods produced domes-tically sold to the world

τ ∗t = τ ∗

t−1 + πt − π∗t − �st (A6)

Domestic production (resource constraint)

yt = (1 − α) cdt + αxt (A7)

where total production is given by

yt = nt + at (A8)

where

at = ρaat−1 + εat (A9)

Inflation of domestically produced goods

πdt = μd

f πdt+1 + μd

bπdt−1 + λdmcd

t + επt

(A10)

Inflation of imported goods

πmt = μm

f πmt+1 + μm

b πmt−1 + λmmcm

t + επt

(A11)

CPI inflation

πt = (1 − α) πdt + απm

t (A12)

Uncovered interest rate parity condition

it − i∗t = �Etst+1 − ψb∗t + v∗

t (A13)

Flow budget constraint

b∗t+1 = b∗

t + xt − cmt + �st + v

pxt (A14)

Labour supply decision

wt − pt − γ (ct − ηct−1) = ϕnt (A15)



wt − pt = ϕ (yt − at ) + γ (ct − ηct−1) (A16)

Real domestic marginal cost (the realwage divided by marginal productivity oflabour)

mct = γ (ct − ηct−1) + ϕnt − at (A17)

or

mct = γ (ct − ηct−1) + ϕyt − (ϕ + 1) at

(A18)

Real marginal cost of imported goods

mcmt = st + p∗

t − pt (A19)

or

mcmt = �st + π∗

t − πt + mcmt−1 (A20)

The Taylor rule describing monetary policy

it = φyyt−1 + φππt−1 + φiit−1 + εit (A21)

Endnotes

1. See for instance Bacchetta and van Wincoop (2006) foran explanation based on information imperfections.

2. Readers who want a detailed derivation of openeconomy models are referred to Corsetti and Pesenti(2005).

3. Recently, China has emerged as one of Australia’s majortrading partners. However, there is no good quality quar-terly Chinese data on GDP and prices for the sample period.Some of the impact of the Chinese economy will insteadpresumably be absorbed by the exogenous demand shockfor Australian exports.

4. The variance decomposition is for the model variables,not the observable time series. For time series that areestimated to contain only a small measurement error com-ponent, the numbers in Table 3 are also a relatively accu-rate approximation to the variance decomposition of theobserved times series.

5. The model without measurement errors was esti-mated using real GDP as the only indicator for domes-tic output. More details of the model estimates withoutmeasurement errors are available from the author uponrequest.

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A Structural Model of Australia as a Small Open Economy

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