Nowcasting Euro Area GDP Growth Using QuantileRegression

Gian Luigi Mazzi and James Mitchell

September 2019

Nowcasting GDP

Offi cial quarterly GDP data are published with a delay

In order to form a view about the current state of the economy, awide range of more timely and higher frequency indicator variables areconsidered

This paper uses an application to explore the utility of quantileregression (QR) methods in producing (density) nowcasts of EuroArea GDP growth from these indicators

Our QR strategy accommodates the mixed-frequency and“ragged-edge”nature of the increasingly big datasets that typicallycharacterise recent nowcasting applications

Quantile Regression (QR) and Density Nowcasts

Because of their flexibility, in modelling the entire density, QR mayprove a useful, and a relatively simple, method of producing reliabledensity nowcasts

Traditionally when seeking to model, explain and indeed nowcast orforecast offi cial statistics using regression methods, focus tends to beon the conditional mean of the variable of interest

QR models the conditional τ-quantile of the dependent variable – forexample the first decile (τ = 0.1) or the ninth decile (τ = 0.9). QRdescribes the relationship at different points in the conditionaldistribution of the variable of interest

In an era of fan charts there is increased awareness of the importanceof modelling and understanding the risks associated with the centralestimate or nowcast/forecast. QR is therefore attractive in modellingthe full distribution

MIDAS QR

Our focus is nowcasting current quarter GDP growth usingwithin-quarter (specifically monthly) known information/data onindicators such as industrial production, retail sales and qualitativebusiness surveysWe follow Ghysels (2014) and consider a Mixed data sampling (orMIDAS) QR:

Qyt (τ|Ωt ) = β0(τ) + β1(τ)[B(L1/m ; θ)xmt

], τ ∈ (0, 1)

where t (t = 1, ...,T ) is the quarter, yt is quarterly GDP growth,

B(L1/m ; θ) = ∑Kk=1 b(k; θ)L

(k−1)/m , where K is the order of the lagpolynomial, Ls/mxmt = x

mt−s−m , x

mt denotes the (in principle, vector

of) indicators, where m = 1, 2, 3 denotes the month in the quarterThe B(L1/m ; θ) function provides a means of parsimoniously modellingthe monthly indicators

The estimated parameters may differ by quantile, τ

Unrestricted MIDAS QR

Following Foroni, Marcellino and Schumacher (2015), we useUnrestricted or UMIDAS:

Qyt (τ|Ωt ) = β0(τ) + β1(τ)x1t + β2(τ)x

2t + β3(τ)x

3t , τ ∈ (0, 1)

i.e. when nowcasting quarterly GDP using monthly indicators, theUMIDAS approach estimates the model at the quarterly frequencywith month 1 of the monthly indicator data forming one variable, x1t ,month 2 another, x2t , and month 3 the third, x

3t

We therefore have 3x as many parameters to estimate compared to amodel specified at the quarterly frequency, that of course would nottherefore exploit the within-quarter data as they accrue

The downside is parameter proliferation

Bayesian UMIDAS QR

We propose a Bayesian approach to estimate the UMIDAS QR

This offers a way to shrink the parameter space in the face ofparameter proliferation and has advantages when density forecastingas it accommodates parameter estimation uncertainties

Yu and Moyeed (2001) show that likelihood-based inference usingindependently distributed asymmetric Laplace densities (ALD) isdirectly related to the traditional quantile regression minimisationproblem

Bayesian UMIDAS QR: Details I

A random variable U follows ALD if its probability density is

fτ(u) = στ(1− τ) exp−σρτ(u)

where σ is the scale parameter and ρτ(u) is the traditional lossfunction minimised in estimation of QR

min(β0,β1:3)

T

∑t=1

ρτ(yt+h − β0 − β1x1t − β2x

2t − β3x

3t )

where

ρτ(u) = u(τ − I (u < 0))= u(τI (u > 0)− (1− τ)I (u < 0))

=|u|+ (2τ − 1)u

2

is the check loss function, and I (.) denotes the indicator function

Bayesian UMIDAS QR: Details II

When τ = 0.5, fτ(u) = (1/4) exp− |u| /2 which is the pdf of astandard symmetric Laplace distribution. For all other τ, fτ(u) isasymmetric

So minimisation is equivalent to maximising a likelihood functionunder the ALD with σ = 1:

L(y|β) = τT (1− τ)T exp

−

T

∑t=1

ρτ(yt − β0 − β1x1t − β2x

2t − β3x

3t )

Priors can be placed on the vector β and Bayesian estimation canproceed

Standard conjugate prior distributions are not available for the QRformulation. So analytical solutions are not available. However,Markov chain Monte Carlo (MCMC) methods can still be used toextract the posterior distributions of the unknown parameters. This,in fact, allows for the use of basically any prior distribution

Bayesian UMIDAS QR with Lasso I

To overcome the curse of dimensionality, a Lasso QR is used. LassoQR involves the following adapted minimisation:

min(β0,β1:3)

T

∑t=1

ρτ(yt − β0 − β1x1t − β2x

2t − β3x

3t ) + λ ‖β‖1

where λ is a nonnegative regularisation parameter. The new secondterm is a penalty. As λ increases, the Lasso continuously shrinks QRcoeffi cients towards zero

Lasso, while originally developed for least squares, is a natural methodto consider in QR too - as it performs both variable selection andregularisation. By forcing the absolute value of the coeffi cients βj tobe less than a fixed value it forces certain coeffi cients to zero

Thereby Lasso identifies a simpler model that does not include all thecoeffi cients.

Bayesian UMIDAS QR with Lasso II

The idea is similar to in ridge regression, where again the inclusion ofextra coeffi cients is penalised. But in ridge regression whilecoeffi cients are shrunk, they are not set to zero as in Lasso

Lasso QR imposes a Laplace prior on the βj

p(βj |σ,λ) =(

λ

2σ

)exp(−λ

∣∣∣βj ∣∣∣ /σ)

which can be rewritten into a mixture of the following hierarchicalpriors integrating out γj

βj |(σ,γj

)∼ N(0, σ2γj ) (1)

γj |σ(τ) ∼ exp(λ2/2) (2)

The Laplace distribution is sharply peaked at zero, compared to theGaussian density, explaining how Lasso sets some coeffi cients to zero

Bayesian UMIDAS QR with Lasso III

Following Li et al. (2010), gamma priors are placed on σ2 and(

λσ

)2,

leading to a Bayesian hierarchical model

Posterior computation then follows Li et al. (2010): sequentiallysample from the posteriors of each unknown parameter conditional onall other parameters using Gibbs

Density Nowcasts

Based on in-sample (t = 1, ...,T ) estimation, quantile nowcasts canbe computed given xT+1

QyT+1(τ|ΩjT+1)

r=βr0(τ)+β

r1(τ)x

1T+1+β

r2(τ)x

2T+1+β

r3(τ)x

3T+1

τ ∈ (0, 1), where βrk (k = 0, ..., 3) denotes the r -th draw from the

posterior parameter distribution, and ΩjT+1 denotes the j-th available

information set or conditioning information

As we will explain, the information set, j , increases as within-quarterindicator data accrue

Recall the quarter T + 1 values of the indicator variables arepublished ahead of the quarter T + 1 values for yt and can thereforebe exploited when nowcasting

The nowcasts, QyT+1(τ|ΩjT+1), can be evaluated when yT+1 is

subsequently published

Density Nowcasts: Construction

We collect together r = 1, ...,R draws from the quantile forecastQyT+1(τ|Ω

jT+1) across τ ∈ [0.05, 0.01, ..., 0.90, 0.95] and then

construct the full posterior density nowcast - using a Gaussian kernelto smooth

Quantiles in the extreme tails, less than 0.05 and greater than 0.95,are not modelled directly

This delivers the density nowcast:

pQR (yT+1 | ΩjiT+1)

From this density nowcast for GDP growth, the user is free to extractany feature of concern to them. This might be the conditional mean.But interest often focuses on tail events

Users of GDP growth forecasts are concerned about the probability of(a one-quarter) recession i.e. of negative GDP growth: the IMF nowpublish GDP at risk measures

Real-time Data

We consider quantitative (“hard”) and qualitative (“soft”) indicatorvariables, with the soft indicators typically published ahead of harddata

And we consider both EA (aggregate) and country-level(disaggregate) indicators

Data are real-time vintages

We focus on nowcasting the EA12 GDP growth aggregate, given dataavailability constraints

Real-time Data

Table: Quarterly and monthly data used

Real-time EA12 Country-level Seas. Adj. Source From

Quarterly GDP Yes Yes All 12 Yes Eurostat 1991q1

Monthly IndicIndustrial Prod Yes Yes All 12 Yes Eurostat 1980m1

ESI n/a Yes exc. GRE, IRE Yes EC 1985m1

Business Clim n/a No Germany Yes Ifo 1991m1

Int rate spread n/a Yes No No ECB 1991m1

Comparison against density combination

To evaluate the relative performance of QR we compare accuracyagainst the nowcast combination methodology of Mazzi et al. (2014).This approach has been previously applied successully to nowcast EAGDP growth

They construct density nowcasts for EA GDP growth by taking densitycombinations across a large number of competing ‘component’models using the same set of indicator variables discussed above

The information set as within-quarter data accrues I

For both QR and the density combination method, yt is related to Ωjt as

follows:

1 j=1. t-30: 30 days before the end of the quarter.

Ω1t =

(xmsoft ,t

2m=1 , xhard ,t−l

p1l=1 , yt−l

p2l=1

)N1 = no of elements of Ω1

t . p1 and p2 denote the number of lags ofthe quarterly variables xk ,t (k = hard) and yt ; and

xmsoft ,t

2m=1

meansthe m=1 (first) and m=2 (second) month’s soft data for quarter t areused

The information set as within-quarter data accrues II2 j=2. t-15: 15 days before the end of the quarter.

Ω2t =

(xmsoft ,t

2m=1 , x

1hard ,t , xhard ,t−l

p1l=1 , yt−l

p2l=1

)N2 = no of elements of Ω2

t . This means Ω2t now includes the first

month of within-quarter hard data, as well as Ω1t

3 j=3. t+0: 0 days after the end of the quarter.

Ω3t =

(xmsoft ,t

3m=1 , x

1hard ,t , xhard ,t−l

p1l=1 , yt−l

p2l=1

)N3 = no of elements of Ω3

t . Ω3t now includes the final month of

within-quarter soft data, as well as Ω2t

4 j=4. t+15: 15 days after the end of the quarter.

Ω4t =

(xmsoft ,t

3m=1 ,

xmhard ,t

2m=1 , xhard ,t−l

p1l=1 , yt−l

p2l=1

)N4 = no of elements of Ω4

t . Ω4t now includes the second month of

within-quarter hard data, as well as Ω3t

Results: out-of-sample

The evaluation period is fixed to match Mazzi et al. (2014):2003q2-2010q4 - to ensure replicability and comparisonThe nowcasts are evaluated by defining the ‘outturn’, yτ, as the first(Flash) GDP growth estimate from EurostatEvaluation using the logarithmic scores, log S , of the density nowcastsfrom QR and the combination approach, i.e. pQR (yT+1 | Ωj

T+1) and

plop(yT+1 | ΩjT+1), against the subsequent GDP growth outturn

yT+1The average log S over the evaluation period running from t = τ tot = τ are:

log SQRj =1

τ − τ ∑t=τ

t=τln pQR (yτ | Ωj

τ)

log S lopj =1

τ − τ ∑t=τ

t=τln plop(yτ | Ωj

τ)

for j = 1, ..., 4

N1 = 214; N2 = 293;N3 = 351;N4 = 430

Average log scores

Table: Average logarithmic score: 2003q2-2010q4

log SQRj log S lopj log SARjt − 30 : j = 1 −1.48 −0.85 −0.84t − 15 : j = 2 −1.44 −0.80 −0.87t + 0 : j = 3 −1.37 −0.79 −0.87t + 15 : j = 4 −1.41 −0.50 −0.84

Average log scores

2004 2005 2006 2007 2008 2009 2010 2011­2.5

­2.0

­1.5

­1.0

­0.5

0.0

0.5

1.0

j=1j=3Outturn (EA GDP Growth)

j=2j=4

GDP-at-Risk: Probability of Negative GDP Growth

2005 2010

0.5

1.0Prob RW_(­30_days) QR t­30 days

2005 2010

0.5

1.0 RW_(­15_days) QR t­15 days

2005 2010

0.5

1.0Prob

RW_(0_days) QR t+0 days

2005 2010

0.5

1.0 RW_(15_days) QR t+15 days

2005 2010

­2­101%

GDP_growth

2005 2010

­2­101

GDP_growth

PCA

These mixed results, and evidence that QR is overstating uncertainty,suggest the need to impose additional parameter or data parsimony inestimation

One option is to explore the use of different priors, that allow fordifferent types of dependencies between the indicators

Here we take up an alternative but pragmatic strategy, also designedto acknowledge that the Lasso prior may not be working well for oursample of indicators given that there is considerable dependencebetween them

From Nj>200 indicators recursively we select the first 30 principalcomponents (based on the size of the associated eigenvalue) and usethese in Bayesian QR

A large number of principal components is chosen deliberately, so theyexplain most of the variation in the underlying data

We also impose Gaussianity on QR

Average log scores with PCA

Table: Average logarithmic score from QR with PCA used for data shrinkage

log SQR ,PCAj log SQR ,PCA,Nj log SQRj log S lopj log SARjt − 30 : j = 1 -0.83 -0.61 −1.48 −0.85 −0.84t − 15 : j = 2 -0.85 -0.60 −1.44 −0.80 −0.87t + 0 : j = 3 -0.97 -0.80 −1.37 −0.79 −0.87t + 15 : j = 4 -0.90 -0.84 −1.41 −0.50 −0.84

Conclusion

We propose a UMIDAS strategy within a Bayesian QR toaccommodate a large mixed frequency dataset when nowcastingIn a real-time application to Euro Area GDP growth, using over 400mixed frequency indicators, we find that the QR approach does notproduce as accurate density nowcasts as the density combinationapproach unless indicators are orthogonalised and shrunk to a smallernumber prior to QRQR overstates the uncertainties associated with GDP growthHowever, when the nowcasts are formed t-30 or t-15 days before theend of the quarter, QR is better able to detect the ensuing recessionthan the density combination approachBut later in the quarter the density combination approach dominatesFuture work should consider the use of alternative priors, within theproposed Bayesian QR framework, to assess whether these helpimprove the performance of QR when producing density nowcasts

Lasso is in effect a variable selection method. Priors that allow for sets(or groups) of variables to be selected may help