An Unobserved Components Model

The European System of National and Regional Accounts (ESA 2010) contains quarterly data for both GDP and the government deficit for at least from 2000 onwards for most coun-tries. This allows straightforward estimation and interpretation of quarterly, cyclically-adjusted deficit series. However, prior to the year 2000 there is a shortage in the availability of quarterly government spending data for a variety of countries. Using a model first proposed by Camba-Mendez and Lamo (2004)⁶, the signal-extraction problem is augmented with an update step from annual to quarterly frequency. I will show below that quarterly states can be extracted even when new information is only available every fourth quarter, i.e. at annual frequency. This allows a longer-term evaluation of the fiscal policy stance in the European Union.

Real GDP is modeled as the sum of a stochastic trendµ_t,q^y , a cyclical component ψt,q and a measurement errorζ_1t,q. The budget balance ratio follows a stochastic trendµ^d_t,q, a cyclical componentΦd(L)ψt,qproportional to the business cycle, and a presumably negative level effect of the European Debt crisis between 2008Q4 and 2012Q4η^d_t,q. In addition, the budget balance ratio is affected by gross debt over nominal GDP plus a componentζ2t,qaccounting for measure-ment errors. The model assumes that trend processes are independent, as economic theory tells us that potential or long-run output is usually determined by supply-side factors that concern the structure of the economy and are not altered by government spending or monetary policy.

On the contrary, the structural component of the budget balance is at the discretion of policy makers and depends mainly on legislative and other political factors. The nature of the propor-tionality between the business cycle and the cyclical budget balance lies in automatic stabilizers.

Common examples are tax revenues and unemployment benefits which fluctuate proportional to the business cycle. During a recession, tax revenues decrease which supports disposable in-come and dampens the pass through to aggregate demand. In times of boom, higher revenues are collected which diminishes the effect on aggregate demand. The more progressive the tax code, the stronger the stabilizing nature of taxation. In addition, higher unemployment bene-fits during recessions absorb shocks to aggregate demand, whereas the converse holds during booms. Stated otherwise, these variables work as automatic stabilizers on the business cycle. A very recent contribution to this otherwise extensively researched field has been made by McKay and Reis (2016).

In order to account for measurement errors,ζ_{t i,q}is aniid, normally-distributed process with standard deviationσ_ζi,∀i∈{1, 2}. A measurement error of zero implied that the interpolated series is exactly the sum of its unobserved components.

The observation equation of the state-space system can be described as a two-dimensional vector-autoregression where:

y_t,q = µ_t^y_,q+αψt,q+ζ_1t,q

d_t,q = µ^d_t,q+(α₁+α₂L)ψ_t,q+λfη_t,q^f +λdη^d_t,q+ζ_2t,q, (3.1)

6Whereas the original paper documented quarterly, structural deficit series of Germany and Italy from 1970 to 2000, the present chapter supplies the decomposition of the government budget balance for a set of 30 countries for the years 1995 to 2015 or even earlier.

3.4 An Unobserved Components Model | 83

wheret,qindicate the specific year and quarter. Outputy_t,qis described by the log of quarterly real GDP, whereasd_t,qis nominal government net lending over nominal GDP. The joint, cyclical component is modeled as a second-order autoregressive process:

ψ_t,q = ρ₁ψ_t,q–1+ρ₂ψ_t,q–2+ζ_3t,q, ζ_3t,q ∼ N(0, 1). (3.2)

The polynomial lag operatorΦd(L)=(α₁+α₂L)onψ_t,qaccounts for the proportionality with the business cycle. The stochastic trend components evolve according to the following two equa-tions, wherei= {y,d}:

µⁱ_t,q = β_tⁱ_,q+µⁱ_t,q–1+"ⁱ_t,q, "ⁱ_t,q ^iid∼ N(0,σ²_"i)

β_t,qⁱ = β_tⁱ_,q–1+νⁱ_t,q. ν_t,q^{y iid}∼ N(0,σ²_νy) (3.3) The trend component of both GDP and budget balance ratio are assumed to follow independent random walks. In addition, Camba-Mendez and Lamo (2004) model the slope of this trend,β_t,q as a second random walk. If however, the variance affecting this slope-random walk is near zero, the entire trend component converges against a random walk with drift. Moreover, I include exogenous variables η_t,q^f and η^d_t,q to model changes in the level of the budget balance ratio after 2008Q4 until the end of 2012 (indicator variable) and the reaction of the budget balance to the debt-to-GDP balance.

3.4.2 State-Space System

In what follows I describe the state-space system for the fully observable model. In an additional step I explain how it is possible to interpolate the missing 3 quarters in case annual budget balance data has been used to extend the sample to a longer time horizon. To keep things comparable, the notation still closely follows Camba-Mendez and Lamo (2004). A more detailed and comprehensive overview of all the steps and equations can be found in Appendix B.2.

Fully-observable Model

The two-dimensional vector-autoregressive process in equation (3.1) is summarized by the fol-lowing equation, representing the measurement equation of the state-space system:

x_t,q = As_t,q+Bz_t,q+"_t,q, (3.4)

whereAis a selection matrix on the state-vectors_t,q,Bis a vector capturing the additional level effect on the budget balance ratio between 2008Q4-2012Q4 and the effect of the gross debt over GDP ratio and"t,qallows to control for measurement errors. The vectorx_t,qstacks the two observables,(y_t,q,d_t,q)⁰. The state-vectors_t,qcollects the unobservable components discussed above. The underlying transition equation is given by:

s_t,q = Cs_t,q–1+e_t,q. (3.5)

If quarterly data is available for both real GDP and the budget balance ratio at all times no partial-updating steps are needed. The state vector is augmented such that the new state vector contains both the unknown statess_t,q and the known observables, x_t,q represented inq_t,q= (x_t,q,s_t,q)⁰. This results in the following state-space system at quarterly frequency:

x_t,q = Z⁰q_t,q (3.6)

q_t,q = Nq_t,q–1+Mz_t,q+Rv_t,q. (3.7)

In eight out of 31 cases all data is fully available at quarterly frequency and the structural com-ponents of the model can be recovered directly without any further interpolation steps. In case annual budget balance data is used to help obtaining the desired sample length, an interpolation based on the full sample requires the aggregation to an annual frequency. A subsequent partial updating step then recovers the structural components also at quarterly frequency. Exploiting the recursive structure of equation 3.7, the annualized representation at quarterly frequency looks as follows:

x_t,4 = Z⁰q_t,4 (3.8)

q_t,4 = N_t⁴q_t–1,4 +M⁴_t(z_t1. . .z_t4)+ξ⁴_t,4 (3.9)

L⁴_t = Var(ξ⁴_t,4). (3.10)

The annualized representation is used to estimate the unknown parameters of the model con-ditional on the full sample length. The model’s equations are now time-dependend due to the construction of an auxiliary budget balance-ratio series for the missing quarters. Whenever quar-terly data is available, the later will be used during the updating procedure. The state-space system allows to construct the joint log-likelihood function of the model via the prediction error decomposition. Prediction errors and the prediction error variance stem from a Kalman Filter-ing algorithm robust to outliers which evaluates the log-likelihood function at every point in time, t. Maximum Likelihood estimates are obtained through numerical minimization of the negative of the joint log-likelihood function. In addition, the standard errors of the estimated parameters are obtained from the inverse Hessian matrix. Finally, performing Ljung-Box Tests on the correlation of standardized residuals ensures that the model is well-specified.

Interpolation Step

In case quarterly data on government deficits are only partially available, annual deficit series are used to interpolate the unobservable states to a quarterly frequency. Camba-Mendez and Lamo (2004) overcome this problem by introducing auxiliary variables represented in a vector x_t^a_,qwhich is fully observable even at quarterly frequency.

x^a_t,j =

j

X

i=1

W_t,ix_t,i,