Modelling & Controlling Monetary and Economic Identities ...people.ok.ubc.ca/gpandher/ISR07.pdfof the series (NFA and NDA) into unobserved finite-variation components for level, trend

International Statistical Review (2007), 75, 2, 150–169 doi:10.1111/j.1751-5823.2007.00012.x

Modelling & Controlling Monetary andEconomic Identities with ConstrainedState Space Models

Gurupdesh S. Pandher

Department of Finance, DePaul University, Chicago, IL 60604, USAE-mail: [email protected]

Summary

The paper presents a method for modelling and controlling time series with identity structures. Theapproach is presented in the context of monetary targeting where the monetary identity (e.g. reservemoney equals net foreign assets plus domestic credit) is modelled using a constrained state spacemodel and next-period changes in domestic credit (policy variable) are estimated to reach the targetlevel of reserve money. The constrained modelling ensures that aggregation and identity relationsamong items are dynamically satisfied during estimation, leading to more accurate forecasting andtargeting.

Applications to Germany, UK and USA show that the constrained state space model providessignificant improvements in targeting and forecasting performance over the AR(1) benchmark andthe unconstrained model. Reduction in the mean square error of targeting over AR(1) is in the range of76–95% for the three countries while the gain in targeting efficiency over unconstrained modelling isbetween 21% and 55%. Beyond monetary targeting, the method has wide application to the dynamicmodelling and control of economic and financial time series with identity and aggregation constraints(e.g. balance of payment, national income, purchasing power parity, company balance sheet).

Key words: Structural state space models; constrained Kalman filter; economic identities.

1 Introduction

An identity structure is frequently present in economic and financial data where certain itemsmust, in the aggregate, equal other quantities. Examples of this include national balance ofpayments accounts, monetary identities, national income accounts, and the balance sheet of acompany. In the monetary identity of a central bank, for instance, reserve money in the monetarysystem equals net foreign assets and net domestic assets (domestic credit). Furthermore, certaincomponents of economic identities are more amenable to influence by policy planners than others.For example, monetary authorities may modify domestic credit to control the level of reservemoney consistent with policy objectives such as controlling inflation or promoting economicrecovery.

The paper presents a method for modelling and controlling multivariate time series withidentity and aggregative structures in the constrained state space framework. The methodology ispresented in the context of monetary targeting where reserve money is controlled by determining

C© 2007 The Author. Journal compilation C© 2007 International Statistical Institute. Published by Blackwell Publishing Ltd, 9600 Garsington Road,

Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.

Modelling and Controlling Monetary and Economic Identities 151

interventions to domestic credit. Items of the identity are first decomposed into dynamicsignals for level, trend, and seasonal effects (Harrison & Stevens, 1976; Harvey, 1984) andthen dynamically constrained to satisfy the aggregation and identity structure. This leads tomore accurate forecasting and targeting. While we present the methodology in the context ofmonetary targeting, the method has wide application to the time series modelling and control ofother economic and financial identities.

For the purpose of this paper, we view a country’s monetary identity series as having thestructure

RMt ≡ NFAt + NDAt , t = 1, . . . , T (1)

where RM is reserve money, NFA is net foreign assets and NDA is net domestic assets. TheIMF’s financial programming model is also frequently described in terms of the above monetaryrepresentation (Lorie, 1989; Tarp, 1993; Taylor, 1994; Fischer, 1995; Polak, 1998; Agenor &Montiel, 1999; Blejer et al., 2001; Easterly, 2004 and others). The identity (1) could apply to thecentral bank, in which case RM is high-powered money, or it could apply to the entire banking orfinancial system, in which case RM is broad money. The above monetary identity can be furtherdecomposed into finer components. For example, NDA includes credit to commercial bankswhile RM includes currency outstanding and bank deposits. To keep the exposition simple, wemodel the monetary account at a higher level of aggregation given by (1) knowing that, withoutloss of generality, the model easily extends to more detailed representations of the monetaryaccount.

The proposed method is evaluated using quarterly monetary data for Germany, UK and USAfrom the International Financial Statistics (IFS) database maintained by the IMF. A counterfactualforecasting and targeting approach is undertaken where we estimate the next period of changes innet domestic assets required to be on the actual observed reserve money trajectory. Results fromour empirical study show a significant improvement in targeting and forecasting performancefor the constrained state space model over the AR(1) benchmark, as well as, the unconstrainedmodel. For example, the mean square error (MSE) from targeting with the constrained state spacemodel is lower than AR(1) by 95%, 76% and 88% for Germany, UK and USA, respectively.Furthermore, the gain in targeting efficiency of the constrained model over the unconstrainedmodel for the three countries is 55%, 21% and 34%, respectively.

The related literature on constrained estimation is briefly reviewed below. Some of the earlierwork considers estimating time-varying regression parameters with dynamic constraints usingthe Kalman filter. Doran (1992) estimates a Cobb–Douglas production function where theconstant return-to-scales return is dynamically imposed and Doran & Rambaldi (1997) considerthe estimation of constrained demand equations. Leybourne (1993) considers the estimation ofrestricted random coefficients in a linear regression and reformulates the problems in terms of anunconstrained model. The work of Durbin & Quenneville (1997) on benchmarking addresses theproblem of adjusting a less reliable monthly series to be consistent with more reliable values of anannual series. Their estimation approach combines the two series and fits the state space model tothe single series. Pandher (2002) presents a state space framework for modelling multivariate timeseries with multiple linear constraints and shows that the constrained model is identifiable andefficient. This paper follows this approach to modelling the monetary identity and extends it tocontrolling and targeting the aggregate level of reserve money (the endogenous or target variable)with changes to net domestic assets (the policy variable). The remaining items in the monetaryidentity, such as net foreign assets, are viewed as exogenous variables that may be separatelymodelled but constrained to satisfy the aggregation and identity restrictions. Beyond monetarytargeting, the combined targeting-forecasting framework of the paper provides rich applicationsto the dynamic control of multivariate economic and financial time series with identity and

International Statistical Review (2007), 75, 2, 150–169C© 2007 The Author. Journal compilation C© 2007 International Statistical Institute

152 G.S. PANDHER

aggregation constraints (e.g. national income identity, balance of payment, purchasing powerparity and company balance sheet).

The remainder of the paper is organized as follows. Section 2 describes the constrained statespace model for the monetary identity. The targeting procedure is described in section 3. Section4 describes the data used in the targeting applications and section 5 reports the results for thethree countries. Conclusions follow in section 6.

2 Modelling the Monetary Identity

This section considers the modelling of the monetary identity structure using constrained statespace models. The targeting methodology is discussed in section 3.

2.1 The Constrained Structural State Space Model

The identity structure RMt = NFAt + NDAt is modelled by first decomposing the free itemsof the series (NFA and NDA) into unobserved finite-variation components for level, trend andseasonal effects using the Basic Structural Model (Harrison & Stevens, 1976; Harvey, 1989).These components are then dynamically constrained to satisfy the monetary identity throughan additional equation in the measurement system. This ensures that the estimated level andseasonal signals for NFA and NDA sum to equal the constraining item RMt . For estimation, theconstrained monetary model is cast into state space form and estimated with the Kalman filter(unconstrained estimation entails modelling the independent components separately without theidentity constraint).

For the monetary identity (1), the free items and constraining items are given by Xt =(NFAt , NDAt )

/ and RMt , respectively. Let xt ε {NFAt , NDAt} represent a univariate elementof the free items; then its representation under the basic structural model (BSM) is given by thefollowing set of equations:

xt = Lt + St + εt , εt ∼ N(0, σ 2

ε

),

Lt = Lt−1 + Dt−1 + ηLt , ηL

t ∼ N(0, σ 2

L

),

Dt = Dt−1 + ηDt , ηD

t ∼ N(0, σ 2

D

),

where Lt , Dt and St are the level, trend and seasonal signals, respectively. In addition, the randomseasonal effects satisfy a stochastic additivity constraint given by

St + St−1 + . . . + St−(q−1) = ηSt , ηS

t ∼ N(0, σ 2

S

),

where q is the order of seasonality equal to the number of seasonal periods. For instance, aquarterly time series with quarterly seasonals will have q = 4. The BSM has a state spacerepresentation and may be estimated iteratively with the Kalman filter (Harvey, 1989).

For the monetary account, the state vector is defined by αt = (αNFAt , αNDA

t )/ where αNFAt =

(LNFAt , DNFA

t , SNFAt , SNFA

t−1 , SNFAt−2 ); likewise for αNFA

t . The free items of the monetary identity maythen be cast in state space form as

Xt = Z Xt αt + εX

t , εXt ∼ N (0, V X ), (2)

αt = Ttαt−1 + ηt , ηt ∼ N (0, Q), (3)



where ZXt is the measurement matrix for the BSM with level, trend and quarterly sea-

sonal effects; Tt is the transition matrix for the state variables and εXt = (εNFA

t , εNDAt )

and ηt = (ηL ,NFAt , η

R,NFAt , η

S,NFAt , 0, 0, η

L ,NDAt , η

R,NDAt , η

S,NDAt , 0, 0) are the correspond-

ing zero mean error vectors with covariance matrices V X = diag(σ 2εN F A , σ

2εN D A ) and Q =

diag(σ 2ηL,N F A , σ

2ηR,N F A , σ

2ηS,N F A , 0, 0, σ 2

ηL,N D A , σ2ηR,N D A , σ

2ηS,N D A , 0, 0), respectively. All errors are as-

sumed to be i.i.d., leading to diagonal covariance matrices.The structure of the monetary identity RMt = NFAt + NDAt , t = 1, . . . , T, is modelled by

dynamically restricting the level and seasonal signals of NFA and NDA to stochastically add upto reserve money,

RMt = L N F At + SN F A

t + L N D At + St

N D A + εRt , (4)

where εRt is a zero mean error and σ 2

εR is the corresponding variance. In addition, the identityrestriction (4) may be expressed in terms of the state vector αt = (αNFA

t , αNDAt )/ as

RMt = [ 1 0 1 0 0 1 0 1 0 0 ]αt + εRt , εR

t ∼ N(0, σ 2

εR

). (5)

Note that the stochastic constraint (5) does not impose the identity restriction on the observedmonetary items Xt = (NFAt , NDAt )

/ but on unobserved signals (level and seasonal terms). Thisrepresentation parallels the measurement error regression model where the dependent variableis related to a “true” unobserved variable that is observed with the measurement error. Forrestrictions to hold exactly, one may set σ 2

εRM = 0; however, allowing the variance to be stochasticleads to a stochastic constraint (e.g. as per se stochastic seasonals of the BSM). It was foundthat this “looser” formulation of the identity constraint improves the empirical performance ofmonetary forecasting and targeting (Section 4).

The constrained state space model (2)–(5) for the monetary identity can be combined into asingle system for estimation. Define Yt = (Xt , RMt )

/, Zt = [ZXt , ZR

t ]/ and εt = (ZXt , ZR

t )/ to be theaugmented observation vector, measurement matrix and measurement error vector, respectively,with ZR

t = [1 0 1 0 0 1 0 1 0 0]. Then, the constrained monetary model may be represented as

Yt ≡[

Xt

RMt

]= Ztαt +

(εX

t

εRt

),

(εX

t

εRt

)∼ N (0, V ), (6)

αt = Ttαt−1 + ηt , ηt ∼ N (0, Q), (7)

where

Zt =

⎡⎢⎣

1 0 1 0 0 0 0 0 0 0

0 0 0 0 0 1 0 1 0 0

1 0 1 0 0 1 0 1 0 0

⎤⎥⎦ , Tt =

[T 05,5

05,5 T

],

T =

⎡⎢⎢⎢⎢⎢⎢⎣

1 1 0 0 0

0 1 0 0 0

0 0 −1 −1 −1

0 0 1 0 0

0 0 0 1 0

⎤⎥⎥⎥⎥⎥⎥⎦

and V =

⎡⎢⎣

σ 2εN F A 0 σ 2

εN F A

0 σ 2εN D A σ 2

εN D A

σ 2εN F A σ 2

εN D A σ 2εN F A + σ 2

εN D A

⎤⎥⎦ .


154 G.S. PANDHER

2.2 Estimation

The first step in the implementation of the monetary state space model (6)–(7) is the estimationof the hyperparameters V and Q. The state vector and its covariance matrix are recursively updatedusing the Kalman filtering equations.

Let αt = E(αt | Y t ) be the best linear unbiased predictor of the state vector αt given theinformation set Y t = {Y1, . . . , Yt}. Similarly, let αt/t−1 = E(αt | Y t−1) be the efficient forecastof αt based on the information available at time t − 1. Define Pt = E[(αt − αt )(αt − αt )

/]as the covariance matrix of αt and let Pt/t−1 = E[(αt/t−1 − αt )(αt/t−1 − αt )

/] = Tt Pt−1T /t + Q

represent the covariance matrix of the conditional one-step-ahead state vector estimator αt/t−1.Finally, the one-step-ahead forecast error (innovation) is It = Yt − Zt αt/t−1 with covariance

matrix Ft = Zt Pt/t−1 Z/t + Q.

The hyperparameters in V and Q—stacked into the vector θ—are estimated by numericallymaximizing the log-likelihood below (the estimation is implemented in GAUSS). Due to theMarkovian structure of the state space model, it is convenient to work with the conditional formof the likelihood function

L(YT , . . . , Y1; θ ) = L(YT /YT −1)L(YT −1/YT −2) . . . L(Y1),

where, with normal errors, Yt/Yt−1 ∼ N (Zt αt/t−1, Ft ). It follows that the log-likelihood may beexpressed as

L(YT , . . . , Y1; θ ) = −T

2ln(2π ) − 1

2

T∑t=1

{ln |Ft | + I /

t F−1t It

}.

Given initial starting values for the state vector α0, its covariance matrix P0 and the hyperparam-eters (V and Q), the Kalman filtering and updating equations generate subsequent estimates forαt , Pt , α t/t−1 , and Pt/t−1, t = 1, . . . , T, through the following iterations (Brockwell & Davis,

1991):

αt/t−1 = Tt αt−1, (8)

Pt/t−1 = Tt Pt−1T /t + Q, (9)

αt = αt/t−1 + Pt/t−1 Z /t (Zt Pt/t−1 Z /

t + V )−1(Yt − Zt αt/t−1), (10)

Pt = Pt/t−1 − Pt/t−1 Z /t F−1

t Zt Pt/t−1. (11)

To start the Kalman filter, a common approach (De Jong, 1991) is to estimate the starting valuesfor the state vector α0 from the “front end” of the data series and set the covariance matrix P0 toa large matrix (diffuse prior). We follow the same procedure and estimate the initial level states(L0) as the average of the first year quarterly values for NDA and NFA. Similarly, the initial trendstates (D0) are estimated as the average difference in quarterly values and the starting seasonalstates (S0, S−1 and S−2) are all initialized to zero.

3 Targeting the Monetary Identity

We now consider targeting the desired level of reserve money RM in the monetary identitywith changes to NDA. The first step of the targeting methodology is to forecast the monetaryidentity (RMt = NFAt + NDAt ) into the next period (e.g. quarter) and then to determine the



required monetary intervention to NDA needed to achieve the target level of reserve money inthe following period.

The simplest possible application of targeting economic identities recognizes three types ofvariables in the identity relation. First, there is a target or endogenous variable(s) that is to bechanged by the desired amount based on policy considerations. In our context, the target variablewill be reserve money RM . Second, there is a policy variable(s) such as net domestic credit orloan disbursements (components of NDA) through which the monetary authority will attemptto control the target variable. Third, there are other elements in the identity that are projectedexogenously or with econometric equations and may be related to other variables in the economy.In a monetary system with capital mobility, net foreign assets are not directly under the controlof the monetary authority and are viewed largely as the exogenous variable of the identity. Theexogoneity here refers to the assumption that the policy variable NDA affects the endogenousvariable RM , but not the exogenous variable NFA (Easterly, 2004).

The assumption that NFA is perfectly exogenous in the monetary identity (while NDAis the policy variable) is made for statistical convenience. In reality, the central bank mayinfluence NFA through domestic interest rates as this affects net capital flows. Such com-plexities in the monetary identity can be incorporated by separately modelling the relationbetween interest rates and net capital inflows and adjusting NFA forecasts during the targetingstep.

3.1 Monetary Forecasting—Constrained and Unconstrained

The forecasting of the monetary identity under the constrained and unconstrained state spacemonetary models is discussed below. To distinguish between the unconstrained model (2)–(3)and constrained model (6)–(7), we will use β t to represent the state vector of the unconstrainedmodel. Upon estimation of the state vector, the next period forecasts of the monetary accountare given by Yt+1/t = E(Yt+1 | Y t ).

(i) Unconstrained (UC) ForecastingIn unconstrained forecasting, the free items Xt = (NFAt , NDAt )

/ are first modelled usingthe state space model

Xt = Z Xt αt + εX

t , εXt ∼ N (0, V X ), (12)

αt = Ttαt−1 + ηt , ηt ∼ N (0, Q). (13)

Reserve money RMt is then forecasted ex post under the identity restriction (1). This leadsto the forecast

Y UCt+1/t =

[XUC

t+1/t

RMUCt+1/t

]=

[Z X

t+1βt+1/t

Z Rt+1βt+1/t

], (14)

with ZRt+1 = [1 0 1 0 0 1 0 1 0 0]. This yields RMt+1/t = L N F A

t+1/t + SN F At+1/t + L N D A

t+1/t + SN D At+1/t .

(ii) Constrained (C) ForecastingUnder the constrained monetary model (6)–(7), the state vector αt is estimated underthe monetary identity constraint (5) using the free items Xt = (NFAt , NDAt )

/ and theconstraining item RMt . The constrained state space model differs from its unconstrainedcounterpart in that the identity restriction is included in the measurement matrix Zt and


156 G.S. PANDHER

the estimated state vectors αt satisfy the aggregation and identity structure. Constrainedmonetary forecasts are represented by

Y Ct+1/t =

[XC

t+1/t

RMCt+1/t

]= Zt+1αt+1/t =

[Z X

t+1αt+1/t

Z Rt+1αt+1/t

]. (15)

Given that ZRt+1 = [1 0 1 0 0 1 0 1 0 0] defines the last row of the augmented matrix Zt+1

in (6), the next period forecast for reserve money is given by

RMt+1/t = Z Rt+1αt+1/t . (16)

It is easy to show by using (9) that the variance of the next-period predictor of the monetaryaccount under the state space representation (6)–(7) is

Var(Yt+1/t ) = Zt+1(Tt+1 Pt/t T/

t+1 + Q)Z /

t+1. (17)

Pandher (2002) shows that the forecasts from the constrained state space model areefficient in the mean square error sense (V (αt+1/t ) < V (βt+1/t )).

3.2 Targeting

Let RMt+1/t = RMt+1/t − RMt be the next period forecast of the change in reserve moneywith respect to its current estimate under the monetary state space model. Then, the forecastedchange in the monetary identity over the next period with respect to its current estimated state is

RMt+1/t = (NFAt+1/t − NFAt ) + (NDAt+1/t − NDAt )

= NFAt+1/t + NDAt+1/t . (18)

Further, let {RM∗t }, t = 1, . . . , T, be the trajectory of planned reserve money targets that the

monetary authorities plan to pursue. At the start of each period, the targeting exercise requiressetting the desired change in reserve money equal to the forecasted change to determine thechange in net domestic assets (NDA∗) required to reach the planned trajectory:

RM∗t+1 = RMt+1/t

RM∗t+1 − RMt = NFAt+1/t + NDA∗

t+1/t . (19)

Note that the change in reserve money above is with respect to the current modelled stateof reserve money RMt as opposed to its actual level RMt . This is done because forecasts of allmonetary variables are made from the current estimated monetary model. Therefore, estimatesof the monetary intervention to NDA are only consistent with the desired level of reserve moneyrelative to its current estimated value.

From (19), the monetary intervention to NDA required to track the monetary target is givenby

NDA∗t+1/t = RM∗

t+1 − RMt − NFAt+1/t

= RM∗t+1 − RMt − (NFAt+1/t − NFAt )

= RM∗t+1 − NFAt+1/t − NDAt . (20)



It can be seen from (20) that the required monetary intervention may be stated in terms of thelevels of planned next-period RM , the forecasted NFA and the current estimate of NDA. In termsof the monetary state space model (6)–(7) of section 2, the change in net domestic assets (20)required to track the monetary target is given by

NDA∗t+1/t = RM∗

t+1 − NFAt+1/t − NDAt

= RM∗t+1 − Z (1)

t+1αt+1/t − Z (2)t αt , (21)

where the superscript in Zt+1 refers to its row (see (6)–(7)), and αt+1/t and αt are the predictedand filtered state vectors, respectively. It is easy to show using (9) that the variance of the nextperiod monetary intervention NDA∗

t+1/t is given by

Var(NDA∗

t+1/t

) = (Z (1)

t+1Tt+1 + Z (2)t

)Pt

(Z (1)

t+1Tt+1 + Z (2)t

)/. (22)

3.3 Performance and Benchmarks

The targeting performance of the models is measured by the proximity of the predictedchanges in NDA (denoted NDA∗

t+1/t ) to actual observed changes (NDAt+1). Similarly, the

distance between the monetary forecasts Yt+1/t = E(Yt+1 | Y t ) and the actual observed valuesYt+1 indicates the model’s accuracy in forecasting the monetary identity.

The forecasting and targeting performance of the various models is measured by the followingforecasting mean square errors (FMSE):

FMSE(Yt+1/t ) =

T −1∑t=T −n

(Yt+1 − Yt+1/t )2

n, (23)

FMSE(NDA∗

t+1/t

) =

T −1∑t=T −n

(NDAt+1 − NDA∗

t+1/t

)2

n. (24)

The quarterly out-of-sample period for the above FMSEs is [T − n + 1, T] and NDAt+1 =NDAt+1 − NDAt is the actual change in net domestic assets estimated by the monetaryintervention NDA∗

t+1/t while counter-factually targeting reserve money. Targeting is performedover the latter end of the time series to allow the state space model estimates to approach a steadystate. FMSEs over three-year and two-year out-of-sample periods are reported, leading to 12 and8 out-of-sample observations, respectively.

To further evaluate the constrained and unconstrained state space models, two benchmarkmodels are estimated and their forecasting and targeting performance is also reported. Theseinclude an auto-regressive model (AR (1)) and a “simple” model. The AR(1) model separatelyfits the NFA and NDA time series and forecasts RM as the sum of their individual forecasts.An ARMA(1,1) model was also considered as a potential benchmark; however, its performancerelative to the AR(1) model was very similar for Germany and the USA while its forecastingand targeting FMSEs for the UK deteriorated dramatically. The “simple” benchmark model usesthe last period value of the monetary account to forecast the current period value. The rationale


158 G.S. PANDHER

behind this benchmark model is that a more complicated estimation model should at the veryleast outperform the simple random walk model.

To aid in the relative comparison of models, a relative efficiency gain ratio (RG) is computed.It is defined as

RGM =(

1 − FMSEM

FMSEAR

)100. (22)

RG summarizes the relative gain in forecasting the efficiency achieved by model M over theAR(1) benchmark model.

4 Data and Estimation

Our monetary targeting applications for Germany, UK and USA use quarterly monetary datafrom the IMF’s International Financial Statistics (IFS) database. The data used to construct theseries for reserve money (RM), net foreign assets (NFA) and net domestic assets (NDA) is fromthe “Monetary Authorities” grouping of the IFS. For Germany, the pre-euro period over 1987–1997 is used to exclude the impact of policies related to monetary integration on the monetaryaccount. For UK and USA, the monetary data covers the 1996–2006 period. Summary statisticsand correlations for monetary aggregates (and their quarterly changes) are reported in Table 1.The reported figures are in billions of Deutsche mark (DM), pounds sterling (GBP) and the USdollar (USD) and reveal some interesting dynamics among the monetary items.

For Germany, the average value of RM over the 41 quarters is 265.4 billion DM and thesame for NFA and NDA is 97.1 and 168.3, respectively. There is high positive correlationbetween changes in RM and NDA (0.67), which is indicative of the use of net domestic assetsto modify reserve money. Also, changes in NDA are in the opposite direction as quarterlychanges in NFA (correlation of −0.78). This again suggests attempts to manage the level ofRM .

The average values of RM , NFA and NDA for UK are 15.1, 16.7 and 31.7 billion GBP,respectively. Again, the strong negative correlation between changes in NDA and NFA (−0.98)and positive correlation between changes in RM and NDA (0.41) are suggestive of the use of netdomestic assets to manage reserve money.

For the USA, NDA is a much bigger component of RM in the monetary account of the USAThis leads to a strong positive correlation between changes in RM and NDA (0.99). Further,the correlation between changes in NDA and NFA for the USA is virtually non-existent (−0.07)while the same is strongly negative for both Germany and the UK

In all the three countries, the skewness of NFA and NDA is in the opposite direction whilethe skewness in RM is more moderate. This is again suggestive of an attempt to manage/hedgechanges in NFA through NDA, over which monetary authorities have greater influence in an openexchange rate regime.

Lastly, estimated hyperparameters for the monetary state space model (6)–(7) are reportedin Table 2 along with the AR(1) parameters for the differenced series for NFA and NDA. Thehyperparameters are estimated from the monetary time series excluding the out-of-sample period(e.g. the last 12 quarters). The estimation is performed in GAUSS. The AR(1) model is estimated(in SAS) using a moving estimation window that advances the endpoint of the time series by eachquarter for the next quarter forecast. This leads to 12 different estimates for the AR parameters



Table 1Descriptive StatisticsSummary statistics and correlations for quarterly monetary aggregates for Germany, UK and USA are reported. The “MonetaryAuthorities” grouping of the International Financial Statistics (IFS) database maintained by the IMF is used to constructobservations for reserve money (RM), net foreign assets (NFA) and net domestic assets (NDA). The pre-euro period from1987-I to 1997-I is used in the targeting application for Germany while the 1997-I to 2006-I period is used for UK and USAQuarterly changes in monetary items are prefixed by “”. Monetary items are expressed in billions of Deutsche mark (DM),pounds sterling (GBP) and US dollars (USD).

Germany Obs Mean Median Std Kurt Skew

NFA 41 97.1 105.3 3.9 −0.78 0.00NDA 41 168.3 179.3 6.5 −0.38 −0.84

RM 41 265.4 287.5 6.9 −1.36 −0.48NFA 40 0.4 −0.1 2.7 15.10 2.85NDA 40 2.7 2.3 3.7 6.87 −1.44RM 40 3.0 2.6 2.3 −0.42 −0.24

Correlations NFA NDA RM NFA NDA RM

NFA 1NDA −0.21 1

RM 0.37 0.83 1NFA 0.35 −0.13 0.09 1NDA −0.34 0.22 0.01 −0.78 1RM −0.13 0.20 0.11 −0.06 0.67 1

UK Obs Mean Median Std Kurt Skew

NFA 37 15.1 9.2 2.4 10.33 2.80NDA 37 16.7 23.4 2.8 8.12 −2.52

RM 37 31.7 31.7 0.8 −0.10 0.36NFA 36 −0.5 0.3 2.2 5.46 −0.26NDA 36 0.8 0.0 2.4 4.68 0.17RM 36 0.3 0.4 0.5 0.31 0.04


NFA 1NDA −0.96 1

RM −0.38 0.61 1NFA 0.46 −0.42 −0.08 1NDA −0.45 0.43 0.15 −0.98 1RM −0.10 0.20 0.38 −0.22 0.41 1

USA Obs Mean Median Std Kurt Skew

NFA 37 57.1 57.0 0.9 −0.80 −0.01NDA 37 557.9 569.9 16.9 −1.18 −0.02

RM 37 615.0 625.4 17.0 −1.27 −0.06NFA 37 −0.3 −0.2 0.5 0.20 −0.30NDA 37 8.9 6.7 3.5 7.62 0.42RM 37 8.6 8.0 3.6 6.44 0.29


NFA 1NDA 0.15 1

RM 0.20 1.00 1NFA 0.37 −0.20 −0.18 1NDA 0.06 0.15 0.15 0.07 1RM 0.11 0.11 0.12 0.21 0.99 1

over the out-of-sample period and averages of these parameter values and their standard errorsare reported.

In the targeting application of the next section (5), non-seasonal variants of (6)–(7) are alsoused. These models have a smaller state vector without seasonal terms and the transition equation


160 G.S. PANDHER

Table 2Parameter EstimatesEstimated hyperparameters of monetary state space model (6)–(7) are reported along with the AR(1) parameters for thedifferenced NFA and NDA series. The last 12 quarters of the out-of-sample period are excluded in the estimation of thehyperparameters. For the AR(1) model, averaged parameter values and their standard errors are reported over the out-of-sample period.

Germany USA UK

Model Parameter Estimate SE Estimate SE Estimate SE

Monetary σεN F A 10.6490 3.7360 1.4970 0.237 7.0519 3.0472State Space σεN D A 12.4940 3.0070 7.1050 5.611 7.9685 2.8294

σηL,N F A 10.5890 3.9620 0.0000 1.119 2.1064 0.6392σηR,N F A 0.0001 0.4810 0.8680 0.212 0.0003 0.2726σηS,N F A 11.9160 3.2220 10.4580 4.039 2.3608 1.2093σηL,N D A 0.0000 1.2720 1.5650 0.787 0.1002 0.3490σηR,N D A 0.8860 1.4380 0.0000 0.206 0.0001 0.1832σηS,N D A 1.6270 1.0310 5.5720 2.092 0.5265 0.2751

AR(1): NFA Mean 0.4446 3.0070 0.3008 0.5317 −0.5387 3.3480AR1 −0.1267 0.1794 0.0231 0.1980 0.1317 0.1924

AR(1): NDA Mean 2.9200 2.7250 9.3930 2.7470 0.7462 3.4750AR1 −0.4534 0.1627 −0.4600 0.1730 0.1067 0.1930

does not include σηS,N F A and σηS,N D A . These hyperparameters are not reported in the interest ofbrevity.

5 Targeting Applications

The monetary forecasting and targeting methodology of section 3 is now illustrated forGermany, UK and USA and its performance is evaluated relative to the two benchmark models.

5.1 Germany

As discussed earlier, the first task of monetary programming methodology is the issue offorecasting the monetary identity (RM = NFA + NDA) into the next planning period. The secondstep involves targeting the desired trajectory for reserve money and estimating the monetaryintervention to domestic assets required to be on this path. To evaluate the targeting performanceof the models, we employ a counterfactual targeting approach that forecasts changes in netdomestic assets (the policy intervention) needed to track the observed level of reserve money(the “target”).

Although Germany is known to have engaged in monetary targeting over the sample period,an advantage of the counterfactual approach is that it does not require the assumption that actualchanges in reserve money are a direct consequence of monetary targeting by the Bundesbank.The method simply investigates how well the various models are able to predict changes in netdomestic assets corresponding to actual changes in reserve money. Comparison of forecastedcounterfactual changes in NDA to observed changes then provides a basis for evaluating thevarious models.

Table 3 reports the quarterly forecasting and targeting performance of the various statespace and benchmark models for Germany. These include the constrained and unconstrainedmonetary state space model (with seasonal and non-seasonal variants) and the benchmarkAR(1) and “simple” models. In the unconstrained model, forecasts of NFA and NDA undermodel (2)–(3) are aggregated to obtain the forecast for RM using (14). Reserve money does notappear in the measurement equation of the unconstrained model and, hence, the estimated state



Table 3Forecasting and Targeting Performance—GermanyThe quarterly monetary forecasting and targeting performance of the various state space and benchmark models is reportedfor Germany in the pre-euro period from 1987 to 1997. These include the constrained and unconstrained monetary state spacemodel (with seasonal and non-seasonal variants) and the benchmark AR(1) and “simple” models. FMSE is the next-periodforecasting mean square error for the monetary item (NFA, NDA, RM). The targeting FMSE is the forecast mean squareerror for the policy intervention NDA required to track changes in RM. RG is the relative efficiency ratio of the model andmeasures the reduction in FMSE relative to the AR(1) benchmark model. The FMSEs are based on quarterly forecasts overthe last three years of the monetary time series (1994-I to 1997-I). Monetary items are expressed in billions of Deutsche mark.

Monetary Forecasting TargetingFMSE FMSE

Model All NFA NDA RM NDA

AR(1) 114.6 21.3 194.0 128.6 236.1Simple (S) 164.6 20.3 266.5 207.1 333.5

FMSE Unconstrained (U) 111.1 19.6 182.5 131.2 48.2Constrained (C) 128.5 22.8 198.0 164.6 25.6Seasonal Constrained (SU) 71.1 12.1 107.4 93.4 25.0Seasonal Unconstrained (SC) 52.2 16.1 77.0 63.7 11.9

Simple (S) −43.6 4.9 −37.4 −61.0 −41.3Unconstrained (U) 3.1 7.8 5.9 −2.0 79.6

RG(%) Constrained (C) −12.1 −6.9 −2.1 −28.0 89.1Seasonal Constrained (SU) 38.0 43.4 44.4 27.4 89.4Seasonal Unconstrained (SC) 54.4 24.5 60.3 50.5 94.9

vector is not restricted to satisfy the aggregation and identity constraint. Constrained monetaryforecasts (15) are generated under model (7)–(8) where the monetary identity restriction appearsin the measurement equation. AR(1) forecasts are based on the aggregation of the NFA andNDA forecasts, and the “simple” model uses the previous lagged value to predict currentvalues.

The predictive performance of the various models is reported by the FMSE for NFA, NDAand RM (see (23)). The “All” column is the overall pooled FMSE for the whole monetaryaccount. The targeting performance is reported in the last column by the FMSE of the policyintervention NDA (24) required to track RM . RG is the relative efficiency gain of the model(25) with respect to the AR(1) benchmark model and measures the percentage reduction inFMSE.

The results in Table 3 provide a number of interesting insights. First, the seasonal monetarystate space models—both constrained (SC) and unconstrained (SU)—dominate the AR(1) andsimple (S) models in forecasting and targeting efficiency. The forecasting efficiency of the SCand SU models is 54.4% and 38.0% higher than the AR(1) benchmark, respectively. Similarly, thetargeting efficiency of SC and SU is 94.9% and 89.4% higher. Second, the seasonal constrainedmodel (SC) provides superior forecasting and targeting relative to its unconstrained counterpart(SU). The overall FMSE of SC and SU is 52.2 and 71.1 (in billions of Deutche marks),respectively, while the targeting FMSE of SC and SU is 11.9 and 25.0, respectively. Third,we find that the AR(1) model outperforms the simple model in both forecasting and targetingefficiency (RG of −43.6% and −41.3%, respectively).

Quarterly predictions of changes in NDA needed to target RM for the best monetary statespace model (SC) and the AR(1) and simple models are reported in Table 4 and plotted inFigure 1. Forecasts of reserve money (RM) under these models are graphed in Figure 2. Thetable and graphs show that the forecasts from the SC model are closest to actual values for RMand NDA.


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Table 4Monetary Targeting with NDA—GermanyPredictions of quarterly changes in NDA needed to target actual RM are reported for the seasonal constrained state spacemodel (SC) and the benchmark AR(1) and simple models. The figures are in billions of DM .

Forecasted NDA

Forecast Actual ActualPeriod NDA SC AR(1) Simple NDA

94-II 8.7 14.8 −7.7 −11.5 183.394-III −13.9 −16.1 −22.0 −5.1 169.494-IV 29.3 22.1 2.3 3.7 198.795-I −20.2 −19.0 −0.3 11.7 178.595-II −4.0 −0.8 −11.3 −20.1 174.595-III −0.6 −0.3 −16.0 −2.1 173.995-IV 23.6 26.4 15.4 21.2 197.596-I −14.9 −14.9 5.4 10.0 182.696-II 1.9 1.2 −6.5 −12.8 184.596-III 6.9 4.9 −2.1 9.2 191.496-IV 21.0 23.7 22.8 26.1 212.497-I −12.7 −15.3 5.7 7.0 199.7

Figure 1. Targeting performance—GermanyPredicted changes in NDA required to target RM are plotted along with actual changes in NDA. The series is in billionsof DM.

5.2 United Kingdom

The targeting methodology of section 3 is now applied to the UK using monetary data over1997–2006. The quarterly forecasting and targeting performance of the seasonal state spacemodels (SC and SU) and the benchmark models is reported in Table 5 (the non-seasonal statespace models C and U are not reported because of the much superior performance of the seasonalconstrained and unconstrained model as for Germany). Again, both the two monetary state spacemodels dominate the AR(1) and simple (S) benchmarks and the constrained model (SC) offersthe best performance. The monetary targeting FMSEs of SC, SU, S and AR(1) are 13.2, 16.8,45.0 and 25.2 billion GBP, respectively, while the overall forecasting FMSEs are 7.3, 10.8, 16.0and 18.1, respectively.



Figure 2. Forecasting reserve money—GermanyRM and forecasts for RM from the SC, AR(1) and simple models are plotted.

Table 5Forecasting and Targeting Performance—UKThe quarterly forecasting and targeting performance of the state space and benchmark models is reported for UK over 1997–2006. These include the constrained and unconstrained seasonal state space models (SC and SU) and the benchmark AR(1)and “simple” models. The forecasting mean square errors (FMSEs) are based on quarterly forecasts over the last three yearsof the monetary time series (2003-I to 2006-I). Monetary items are expressed in billions of pounds (GBP).

Monetary Forecasting TargetingFMSE FMSE


AR(1) 18.1 16.0 28.1 10.2 25.2FMSE Simple (S) 16.0 13.7 25.2 9.2 45.0

Seasonal Constrained (SU) 10.8 15.5 14.1 2.7 16.8Seasonal Unconstrained (SC) 7.3 10.0 10.4 1.5 13.2

Simple (S) 11.4 13.9 10.4 9.8 17.0RG (%) Seasonal Constrained (SU) 40.4 2.6 50.0 73.3 69.0

Seasonal Unconstrained (SC) 59.6 37.0 63.0 85.4 75.7

Targeting predictions for changes in NDA under the SC, AR(1) and simple models areplotted in Figure 3 and reported in Table 6. Figure 4 plots the forecasts for reserve money(RM).

5.3 United States

The last application considers targeting the monetary identity of the USA over 1997–2006.From Table 7, the monetary targeting FMSEs of SC, SU, S and AR(1) are 10.7, 16.2, 120.9and 89.6 billion USD while their overall forecasting FMSEs are 40.4, 41.2, 88.2 and 49.3,respectively. Again, both the SC and SU models dominate the benchmark AR(1) model intargeting and forecasting. Furthermore, the constrained SC model outperforms its unconstrainedcounterpart (SU) in targeting; however, their performance in forecasting the monetary identityis similar.


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Figure 3. Targeting performance—UKPredicted changes in NDA required to target RM are plotted along with actual changes in NDA. The series are in billionsof GBP.

Table 6Monetary Targeting with NDA—UKPredictions of quarterly changes in NDA needed to target actual RM are reported for the seasonal constrained state spacemodel (SC) and the benchmark AR(1) and simple models. The figures are in billions of GBP.

Forecasted NDA


2003-II 0.6 0.1 −3.5 −2.7 23.02003-III 0.9 1.4 1.0 0.8 23.92003-IV 3.6 3.1 4.6 4.7 27.52004-I −2.5 −2.2 0.7 1.0 25.02004-II 0.4 0.3 −2.5 −1.8 25.52004-III 0.5 2.2 1.5 1.4 26.02004-IV 6.0 6.1 6.4 6.7 32.02005-I −6.7 −2.7 0.9 1.1 25.22005-II −1.5 −1.1 −8.8 −7.5 23.82005-III −2.8 4.0 2.5 2.1 20.92005-IV 13.0 4.1 −1.1 0.0 34.02006-I −1.7 −0.9 13.2 11.7 32.3

Targeting forecast from the SC, AR(1) and simple models are plotted in Figure 5 and reportedin Table 8. Figure 6 graphs the forecasts for reserve money (RM).

5.4 Two-Year Out-of-Sample Performance

As a robustness check, the forecasting period is shortened from three years to two years. Thisleads to eight out-of-sample forecast quarters, as opposed to the earlier twelve. The forecastingand targeting performance for all three countries is reported in Table 9. The results identifiedearlier continue to hold as the seasonal constrained state space model (SU) provides the lowestFMSEs for targeting and forecasting relative to the AR(1) benchmark and the unconstrained



Figure 4. Forecasting reserve money—UKRM and forecasts for RM from the SC, AR(1) and simple models are plotted.

Table 7Forecasting and Targeting Performance—USAThe quarterly forecasting and targeting performance of the state space and benchmark models is reported for the US over1997–2006. These include the constrained and unconstrained seasonal state space models (SC and SU) and the benchmarkAR(1) and “simple” models. The forecasting mean square errors (FMSE) are based on quarterly forecasts over the last threeyears of the monetary time series (2003-I to 2006-I). Monetary items are expressed in billions of dollars.

Monetary Forecasting Targeting


AR(1) 49.3 9.6 75.0 63.4 89.6FMSE Simple (S) 88.2 9.7 132.2 122.7 120.9

Seasonal Constrained (SU) 41.2 7.6 68.3 47.8 16.2Seasonal Unconstrained (SC) 40.5 7.6 67.2 46.7 10.7

Simple (S) −78.7 −0.7 −76.3 −93.5 −35.0RG (%) Seasonal Constrained (SU) 16.5 21.1 9.0 24.6 81.9

Seasonal Unconstrained (SC) 18.0 21.6 10.4 26.5 88.1

model (SU). For Germany, the targeting FMSEs for SC, SU and AR(1) are 5.0, 25.0 and 180.0billion DM, respectively, and the same for the UK is 20.4, 25.6 and 81.0 billion GBP, respectively.The targeting FMSEs for the USA are 16.1, 24.2 and 165.5 billion USD, respectivly.

6 Conclusion

This paper presents a method for modelling and controlling multivariate time series withidentity structures using the constrained state space framework. The approach is presented in thecontext of monetary targeting where the monetary identity (e.g. reserve money equals net foreignassets plus domestic credit) is modelled using constrained state space models and next-periodchanges in domestic credit (policy variable) are estimated to reach the target level of reservemoney. Constrained modelling ensures that the aggregation and identity relations among itemsare dynamically satisfied during estimation, leading to more accurate forecasting and targeting.While we present the methodology in the context of monetary targeting, the method has wide


166 G.S. PANDHER

Figure 5. Targeting performance—USAPredicted changes in NDA required to target RM are plotted along with actual changes in NDA. The series are in billions ofUSD.

Table 8Monetary Targeting with NDA— USAPredictions of quarterly changes in NDA needed to target actual RM are reported for the seasonal constrained state spacemodel (SC) and the benchmark AR(1) and simple models. The figures are in billions of USD.

Forecasted NDA


2003-II 6.9 6.5 5.8 9.0 623.72003-III 3.9 4.3 −1.2 12.8 627.62003-IV 19.9 18.7 12.9 24.0 647.52004-I −6.0 −8.6 0.9 13.1 641.52004-II 23.3 20.2 10.1 15.3 664.82004-III 0.3 1.0 7.0 23.4 665.12004-IV 14.0 17.7 13.8 17.0 679.22005-I 6.7 −0.7 −0.4 13.6 685.92005-II 9.3 9.3 7.0 14.7 695.22005-III 6.1 2.7 −0.4 10.3 701.32005-IV 32.0 29.5 23.3 32.4 733.32006-I −6.3 −1.5 15.9 25.6 727.1

application to the dynamic modelling and control of other economics and financial time serieswith identity and aggregation constraints (e.g. national income identity, balance of payment,purchasing power parity, company balance sheet).

The proposed method is evaluated using the quarterly monetary series for Germany, UK andUSA. Two core components of monetary targeting are: (i) forecasting the monetary account intothe next planning period and (ii) determining the required monetary intervention (e.g., change indomestic credit) needed to achieve the target level of reserve money. We apply a counterfactualtargeting approach where the next quarter changes in net domestic assets are estimated, whichwould move reserve money along the actual observed trajectory.



Figure 6. Forecasting reserve money—USARM and forecasts for RM from the SC, AR(1) and simple models are plotted.

Table 9Forecasting and Targeting over Two YearsThe quarterly monetary forecasting and targeting performance of the state space and benchmark models is reported over thelast two years of the monetary series for Germany, UK and USA

Monetary Forecasting Targeting

All NFA NDA RM NDA

Panel A: Germany

AR(1) 119.9 5.2 198.0 156.4 180.0Simple (S) 125.4 4.5 199.0 172.8 212.2Seasonal Constrained (SU) 78.8 4.5 116.1 115.8 25.0Seasonal Unconstrained (SC) 58.7 7.1 86.6 82.5 5.0

Panel B: UK


Panel C: USA


Empirical results show that the constrained state space model provides significant gains intargeting and forecasting performance over the AR(1) benchmark and unconstrained modelling.For example, the constrained model yields gains in targeting efficiency of 95%, 76% and 88%for Germany, UK and USA, respectively, over AR(1) modelling. Furthermore, the targetingefficiency gain of the constrained model over its unconstrained counterpart is 55%, 21% and34%, respectively, for the three countries. Lastly, reductions in FMSEs for the monetary identityunder the constrained model over AR(1) are 54%, 60% and 18% for Germany, UK and USA,respectively (the same for the unconstrained model are 38%, 40% and 17%, respectively).


168 G.S. PANDHER

Acknowledgement

The author would like to acknowledge IMF support for this research and thank Paul Cotterell,Yongmiao Hong, David Marston, Gennady Samorodnitsky and Peter Wickham for helpfuldiscussions and comments on this paper. The paper has also benefited greatly from the veryconstructive suggestions and comments provided by the Editor and two referees.

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Resume

L’article presente une methode de modelisation et de controle des series temporelles avec des structures d’identite.L’approche est presentee dans le contexte de ciblage monetaire ou l’identite monetaire (c. a d. monnaie de reserve egaleavoirs etrangers plus credit interieur) est modelisee en utilisant un modele spatial sous contrainte et ou les variationsdu credit interieur a la periode suivante (variable de politique) sont estimes pour atteindre le niveau vise de monnaie dereserve. La modelisation sous contrainte assure que les relations d’agregation et d’identite entre items sont satisfaitesen dynamique dans l’estimation, ce qui conduit a des previsions et ciblages plus precis. L’application a l’Allemagne, leRoyaume-Uni et les USA montrent que le modele contraint apporte des ameliorations importantes dans la performancede ciblage et de prevision par rapport a l’etalonnage auto-regressif (1) et au modele sans contrainte. La reductiond’erreur du moindre carre par rapport a l’AR est comprise entre 76 et 95% pour les trois pays tandis que le gain en



efficacite de ciblage sur le modele sans contrainte se situe entre 21 et 55%. Par dela le ciblage monetaire, la methodea une large application a la modelisation dynamique et au controle des series temporelles economiques et financieresavec des contraintes d’identite et d’agregation (par ex. la balance des paiements, le revenu national, la parite de pouvoird’achat, le bilan d’une compagnie).

[Received July 2005, accepted February 2007]


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Modelling & Controlling Monetary and Economic Identities ...people.ok.ubc.ca/gpandher/ISR07.pdfof the series (NFA and NDA) into unobserved finite-variation components for level, trend