Estimation technique

Few techniques are applied in the literature when estimating the effects of determi-

nants on fiscal policy variables: (i) VAR technique and a number of its modifications,

however, VAR estimates are subject to various critiques (seeAfonso,2005) because

of their reliance on causality or the effect of variables’ ordering. (ii) Panel techniques

can help address some further empirical challenges (short fiscal series, quarterly in particular), even though the implicit assumption of one regime across the panel

may be too strict. (iii) Markov-switching models allow for several states of the

dependent variable, however, they are subject to issues both with the choice of the number of states (usually two or three) and their rather data-demanding (sensitive)

features. (iv) A rolling regression and/or the Kalman filter is employed to estimate

similar models since they allow for the time-variation of parameters. However, both the former and the latter method works with ‘fiscal flexibility’ in a particular man-

ner, in comparison to more flexible TVP models.49 (v) Quasi-Bayesian methods in

the form of a ‘classical’ maximum likelihood estimation (MLE), combined with the

Kalman filter represent a more recent approach.50

In this chapter, I utilize a state-space representation of two models of fiscal rules (FR I and FR II), and their parameters are estimated using TVP model estima- tion combined with Bayesian methods. Bayesian methods consider all parameters to be random variables, and all to-be-estimated parameters are treated as those stemming from jointly distributed random variables. Their estimation takes into

account the uncertainty associated with all the others (seeKim and Nelson, 1999)

and as a result, it enables the computation of the so-called credible intervals using

the posterior distribution (for details see for exampleKoop et al.,2007). Although

it is not the only possibility that is available in this case, it ensures that the estima- tion proceeds in a way that helps to eliminate issues associated with applications of

MLE in combination with the Kalman filter (see footnote50).

Since the choice (specification) of priors, that is the formulation of beliefs

on prior distributions, is of key importance, I follow recommendations inPrimiceri

(2005) orBlake and Mumtaz(2012) and use the beginning of my sample to gener-

49_{Some authors raise the issue of rolling window regression estimates’ sensitivity to outliers, in}

small samples in particular, seeZivot and Wang(2006) or the specification of a window utilized.

50_{There are two potential problem of this combination (for details seeKim and Nelson,1999): 1)}

there is a risk of accumulation of estimation errors since there can be numerous likelihood functions, especially in large models where one state variable estimation is carried out with conditioning upon the MLE estimates of the remaining parameters in the system, and 2) the initialization of the Kalman filter requires specifying correct (objective) priors. While the latter can be solved easily (for example using a training sample), (the former) obtaining efficient estimates of parameters is non-trivial.

CHAPTER 3. INSTITUTIONAL CHANGES AND FISCAL POLICY BEHAVIOUR

ate the necessary information for country-specific models (parameters and variance), each of the same length for all countries. This choice means that country-specific characteristics are taken into account at the start of the estimation procedure and thus, they are aimed at alleviating problems with a fixed choice of some arbitrary values. This approach is also assumed to lead to more accurate time-varying esti- mates (reduced variance) because of the natural shrinkage contained in the likelihood (Byrne et al.,2016).

My novel quarterly dataset allows to carry out simultaneously country-specific estimations and capture time variation, owing to the use of Bayesian methods for time-varying parameter estimation in the state-space model framework. Follow- ing Blake and Mumtaz (2012), for the simulation of draws from the parameters’ posterior distribution, the Gibbs sampler is employed (belonging to Markov Chain

Monte Carlo (MCMC) methods), complemented with theCarter and Kohn (1994)

algorithm. In order to do that, one has to specify (see Byrne et al.,2016): (a) the

unknown parameters of the model to be estimated and (b) their posterior condi-

tional distributions, subsequently allowing (c) the algorithm to draw samples from

them.

The TVP regression model allowing for specified time-variation of coefficients

with the measurement (also called observation) equation being the (3.8a), and the

transition equation being the (3.8b) below) takes the following form:

yt=ℵb0+βtXt+t,

(3.8a)

βt =ω+Λβt−1+εt,

(3.8b)

where the error terms take the form

t∼N[0,Ξ] (iid),

(3.8c)

εt∼N[0,Θ] (iid).

(3.8d)

yt is univariate (T ×1), Xt consists of p > 2 fiscal rule determinants (including

the first lag of dependent variable), βt is a p×1 matrix of coefficients, Λ is the

p×pmatrix (∼Ikidentity matrix), andE[t, εt] = 0 (independence). When setting

Ξ = Ik and parameter ω = 0), the regression coefficients evolve according to a

random walk with innovations εt. ℵb0 represents any time invariant variables –

none in my case – and the coefficient (matrix (k×k) of their coefficients) included

in the model specification. In this case it is set to zero.

CHAPTER 3. INSTITUTIONAL CHANGES AND FISCAL POLICY BEHAVIOUR

values for time-varying parametersβt (observable state variables),β(0|0), and their

variances (V(0|0)) need to be specified before the Kalman filter can be initialized.

This is done alongside selecting initial values for the variance of the measurement

(observation) equation – Ξ – and the variance-covariance matrix of the transition

equation – Θ. A full derivation of the Kalman filter can be found for example in

Fr¨uhwirth-Schnatter (2006, Ch. 13) or inBlake and Mumtaz (2012, Ch. 3).

FollowingPrimiceri(2005) to find the initial values (β₍₀|0),V(0|0)) for the TVP

model, a simple time-invariant OLS regression is run over a training period (T P) of

five years at the beginning of the sample; because of one-quarter lag the actual period

is: TT P = 1980q2–1984q4. Even though the length of this training sample period is

rather short, it should provide some information for the estimation. The potentially

limited information content is reflected in the value of the scaling parameter ν

(see below), which accords withByrne et al. (2016) suggestion for country-specific

estimations. As a result, the starting values for the Kalman filter are (the initial

state and the initial variance): β₍₀|0) ≡ βOLS and V(0|0) ≡ VOLS. This, however,

reduces the sample period for estimation of fiscal responses to 1985q1–2015q4 (i.e. 124 quarters).

βOLS = (B0T P, t· BT P, t)−1(B0T P, t·yT P, t),

(3.9a)

VOLS = ΩT P ⊗(BT P, t0 · BT P, t)−1,

(3.9b)

whereβOLS is the vector of OLS coefficients andVOLS is the OLS covariance matrix

with

ΩT P = (yT P, t− BT P, t·βOLS)0(yT P, t− BT P, t·βOLS)·(TT P −r)−1,

(3.9c)

wherer is the number of parameters to be estimated and β_(.) andV_(.) are priors to

be calculated.

The priors for the measurement (observation) and transition equation are respectively represented by the inverse Gamma distribution and inverse Wishart

distribution, for a country-specific estimation. Firstly, inverse Gamma (IGa−1) for

the measurement equation,P(Ξ)∼IGa−1[ΞT P,(TT P −r)] with the degree of free-

dom from the training sample (TT P −r) and the scale parameter ΞT P = ΩT P.

The prior for the transition (updating) equation is the inverse Wishart distribu-

tion: P(Θ) ∼IW[ΘT P, TT P] with the training sample variance VT P = ΘT P (with

a scaling factor, see below) and degrees of freedom of the training sample TT P.

CHAPTER 3. INSTITUTIONAL CHANGES AND FISCAL POLICY BEHAVIOUR

tents of the matrix ΘT P since it affects the variation of coefficients in my model

(larger values lead to large dispersion), and the starting period provides limited information on individual variables. Therefore, the calculated variance is re-scaled

via: ΘT P =VOLS×TT P ×ν, where ν is the factor of proportionality. I set it to a

relatively small numberν = 3.510·10−5 for the base model – FR I – followingBlake

and Mumtaz (2012); a similar treatment of variance for a single equation model is

justified inByrne et al.(2016). That choice also affects the speed of adjustment for

parameters of my model.51 Parameters (β’s) are drawn from multivariate normal

distribution, in accordance with the recommendation in the Bayesian literature (see

Kim and Nelson,1999). Since my both models include the lagged dependent vari- able, I restrict the generated set of random draws of coefficients for that variable to

those that lie only between zero and one in absolute value, that is, for Ψ ={%_bt,%b

0 t}

in models (3.4) or (3.5) above so that Ψ∈(−1; +1) holds.52

Also for the initialization of the Gibbs sampler some initial values are required

– Ξ and Θ. These are set similarly to the previous case: Ξ0 = VT P and Θ0 =

VT P×T0×ν. In this model I do not consider the possibility of allowing for changing

volatility over time (stochastic variability, see for exampleBlake and Mumtaz,2012)

that can be added, when estimating policy rules. This decision is primarily driven by the length of available time series, which limits the amount of information that can be obtained for the identification of individual parameters.

Two statistics were calculated to verify that the algorithm meets the neces- sary conditions for convergence. Firstly, a statistic that contains information about the necessary number of draws to achieve a given level of numerical accuracy for

a particular simulation (for details see Geweke,1992). It is the relative numerical

efficiency (RNE) of the Gibbs sampler. The other (ibid.), is a statistic (convergence

diagnostic, CD) that aims to capture the behaviour of the generated sample by com- paring its two sub-samples, usually at 10% and 50% of the retained Gibbs draws. I

reduced the lower threshold to 20%, which is similar to its application inByrne et

al.(2016); for details seeBlake and Mumtaz(2012). Using the calculation suggested

by Raftery and Lewis (1992), I arrive at the minimum number of draws (∼ 4000) for both models, but I utilize 30000 draws and 10000 draws are stored and used for

further inference. Blake and Mumtaz(2012) recommend analysing autocorrelation

51_{On the one hand, one could argue that fiscal measures and changes in fiscal policy are not}

very frequently adopted because of institutional dynamics. On the other hand, there have been periods of time, when fiscal policy have responded rather quickly. Therefore, alternative values of the scaling factorν were tried, see the robustness section below.

52

This step eliminates non-stationary processes. For some countries this condition is met without difficulties, while for others a larger number of draws is needed to generate the stored sample (for example, the Netherlands or Denmark).

CHAPTER 3. INSTITUTIONAL CHANGES AND FISCAL POLICY BEHAVIOUR

functions and recursive means of the retained draws from the Gibbs sampler, to

detect any irregularities in the simulation exercise.53 Therefore, I conduct a recom-

mended visual inspection of the simulated series and for all models autocorrelation

and values of the CD statistic are calculated.54

In document Essays in debt sustainability, effects of institutional changes on fiscal policy in the euro area and consumption responses to a shock in public salaries (Page 94-98)