Prediction intervals for economic fixed-event forecasts Fabian Kr uger Karlsruhe Institute of TechnologyHendrik Plett

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Prediction intervals for economic ﬁxed-event forecasts∗

Fabian Kr¨uger

Karlsruhe Institute of Technology

Hendrik Plett

ETH Z¨urich

March 21, 2024

Abstract

The ﬁxed-event forecasting setup is common in economic policy. It involves a

sequence of forecasts of the same (‘ﬁxed’) predictand, so that the diﬃculty of the

forecasting problem decreases over time. Fixed-event point forecasts are typically

published without a quantitative measure of uncertainty. To construct such a measure,

we consider forecast postprocessing techniques tailored to the ﬁxed-event case. We

develop regression methods that impose constraints motivated by the problem at hand,

and use these methods to construct prediction intervals for gross domestic product

(GDP) growth in Germany and the US.

1 Introduction

Economic forecasts are often published in the ‘ﬁxed-event’ format. Consider Tagesschau

(2022), a German news website that maintains a list of recent GDP forecasts made by

various institutions. For example, in April 2022, the federal government predicted 2.2%

GDP growth for 2022, followed by 2.5% in 2023. In July 2022, the European Commission

predicted 1.4% growth for 2022, followed by 1.3% in 2023. These forecasts are called

‘ﬁxed-event’ because the quantity of interest – the GDP growth rate over the year 2022

or 2023 – remains ﬁxed, whereas the date at which the forecast is made moves forward

in time. This forecasting format is routinely used by institutions that make economic

forecasts, and by news media which comment on these forecasts.

Economic ﬁxed-event forecasts are typically published without a quantitative measure

of uncertainty. Such a measure would be particularly important in the current setup: By

construction, forecast uncertainty varies heavily depending on when the forecast is made

and which year it refers to. For example, in July 2022, the EU commission’s forecast for

∗We thank two anonymous reviewers, Andreas Eberl, Tilmann Gneiting, Malte Kn¨uppel as well as

participants of the MathSEE symposium (Karlsruhe, September 2023) and seminar participants at the

Universities of Amsterdam and Freiburg for helpful comments. We further thank Alexander Henzi for

sharing program code associated with Henzi (2023), and Eben Lazarus for sharing code related to Lazarus

et al. (2018). We acknowledge support by the state of Baden-W¨urttemberg through bwHPC.

arXiv:2210.13562v3 [econ.EM] 20 Mar 2024

2022 should be less uncertain than its forecast for 2023. But just how much less uncertain

depends on the time series properties of GDP, and is hard to grasp intuitively.

Motivated by this situation, the present paper constructs measures for the uncertainty

in ﬁxed-event point forecasts. Together with the point forecasts themselves, we can then

construct forecast distributions for GDP growth and other economic variables. The princi-

ple of forecast post-processing – using point forecasts and past forecast errors to construct

forecast distributions – is popular in meteorology (see e.g. Gneiting and Raftery 2005,

Gneiting et al. 2005, Rasp and Lerch 2018 and Vannitsem et al. 2021), economics (e.g.

Kn¨uppel 2014, Kr¨uger and Nolte 2016 and Clark et al. 2020) and other ﬁelds. State-of-the-

art point forecasts are often publicly available, so that using them as a basis for forecast

distributions is more practical than generating forecast distributions from scratch. Fur-

thermore, assessing forecast uncertainty based on past errors does not require knowledge

about how the point forecasts were generated. This is an important advantage in practice,

where the forecasting process may be judgmental, subject to institutional idiosyncracies,

or simply unknown to the public.

The vast majority of the postprocessing literature considers a ‘ﬁxed-horizon’ fore-

casting setup where the time between the forecast and the realization remains constant.

Examples of the ﬁxed-horizon case include daily forecasts of temperature 12 hours ahead,

or quarterly forecasts of the inﬂation rate between the current and next quarter. In eco-

nomics, Clements (2018) constructs a measure of ﬁxed-event forecast uncertainty that is

based on ﬁxed-horizon forecast errors, and thus requires that an appropriate database

of ﬁxed-horizon forecasts is available. This is the case in the US Survey of Professional

Forecasters analyzed by Clements, but not in other situations including the German GDP

example mentioned earlier. Existing approaches are thus not applicable to the ﬁxed-event

case. Instead, the latter requires diﬀerent tools which we develop in this paper.

The main idea behind our proposed approach is simple: We model quantiles of the

forecast error distribution as a function of the forecast horizon. The latter is deﬁned as

the time (measured in weeks) between the forecast and the end of the target year. For

example, forecasts made on July 1, 2022 for 2022 and 2023 correspond to horizons 26 and

78, respectively. To estimate regression models, we use a dataset of past forecast errors at

diﬀerent horizons. Pooling forecast errors across horizons allows us to estimate uncertainty

at any given horizon by considering uncertainty at neighboring horizons. This approach

is helpful in the ﬁxed-event case, where only a small number of past errors is typically

available for a given horizon. For example, the German data set we consider covers

forecast-observation pairs ranging from 1991 to 2022, and includes precise information

(daily time stamps) on the forecast horizon h. Speciﬁcally, the data contains 525 unique

values for h, ranging from h= 0 to h= 104. Note that hneed not be an integer; for

example, forecasts made on the seven days of the target year’s ﬁnal week correspond to

horizons h∈ {0,1/7,...,6/7}. Given that the data covers n= 1 307 observations in total,

the average number of forecast errors corresponding to each of the 525 diﬀerent horizons is

about 2.5. Using only forecast errors that correspond exactly to some horizon of interest

(h= 5 weeks, say) is hence not a promising strategy, and considering forecast errors from

neighboring horizons seems advisable. This aspect is not relevant when postprocessing

ﬁxed-horizon forecasts, which is typically based on a time series of past forecast errors for

the exact horizon of interest.

The statistical methods we consider incorporate constraints that are motivated by

the ﬁxed-event forecasting problem. In particular, it is plausible to assume that forecast

uncertainty increases monotonically across horizons, and levels oﬀ at some horizon. We

consider three main approaches that implement this idea: A Gaussian heteroscedastic

model, a decomposition approach that imposes a symmetry assumption, and a ﬂexible

approach that is (almost) nonparametric. The ﬁrst two approaches are parsimonious

in light of typically short samples of past forecast errors. Despite their simplicity, they

perform well in a cross-validated analysis of German and US data. In particular, the

prediction intervals attain coverage close to its nominal level, and they clearly outperform

benchmark predictions by a survey of professional economists for the US data.

To illustrate the quantitative implications of our results, consider a hypothetical fore-

caster who, as of mid-September, issues a point prediction of 2.1% GDP growth for the

current year, and 1.7% for the next year. Our results for the German data then suggest

that a plausible 80% prediction interval for the current year would be on the order of

2.1±0.35 = [1.75,2.45], compared to 1.7±2=[−.3,3.7] for the next year, i.e., the pre-

diction interval for the next year is more than ﬁve times as wide. We thus argue that

common statements along the lines of ‘we expect 2.1% GDP growth this year, and 1.7%

next year’ are not helpful: They implicitly put the current- and next-year forecasts on

an equal footing, and over-emphasize the next-year point forecast which is surrounded by

considerable uncertainty.

The rest of this paper is structured as follows. Section 2 introduces notation for the

ﬁxed-event setup. In order to illustrate the statistical implications of this setup, Sec-

tion 3 considers ﬁxed-event forecasting in an autoregressive time series model. Section

4 introduces the proposed regression methods for forecast postprocessing. Section 5 de-

scribes methodology for forecast evaluation and model selection. Section 6 studies the

performance of the proposed methods in a simulation experiment, and Section 7 consid-

ers empirical ﬁxed-event forecasts from Germany and the US. Section 8 concludes with a

discussion. The online supplement contains derivations and further analyses. Replication

materials are available at https://github.com/FK83/gdp_intervals.

2 Setup

We consider forecasting Yt, the real GDP growth rate in year t. The latter is deﬁned as

100 ×GDPt−GDPt−1

GDPt−1,where GDPtis the level of real GDP in year t, which in turn is taken

to be the average of the year’s four quarterly levels. For many countries, point forecasts

of Ytare available from various public and private forecasting institutions. We denote

the time between the origin date (on which the forecast is issued) and the end of year

tas the ‘forecast horizon’, denoted by the symbol hand measured in weeks. Let Xt,h

denote the point forecast of Ytat horizon h. For example, for year t= 2020, a forecast

issued on December 17, 2020 corresponds to horizon h= 2. We focus on forecast horizons

h∈[0,104], i.e., up to two years ahead, which covers most practical economic forecasts.

Even a forecast at horizon h= 0 need not be perfect in practice since a precise estimate

of the outcome may be available only after the end of year t. For simplicity, we treat each

year as having 52 weeks.

The forecast error of Xt,h is given by et,h =Yt−Xt,h. Let Ft,h denote a suitable

information set that is available hweeks before the end of year t. In order to obtain a

forecast distribution for Ytgiven Ft,h, or Yt|Ft,h in short, we model the distribution of

et,h|Ft,h. That is, we take the point forecast Xt,h as given, and focus on modeling the

error of this point forecast. Since

P(Yt≤y|Ft,h) = P(et,h ≤(y−Xt,h)|Ft,h),

and Xt,h is known given Ft,h, the distribution et,h|Ft,h can be used to construct the desired

distribution for Yt|Ft,h.

Our approach of modeling the error of a given point forecast, rather than constructing

a forecast distribution from scratch, is called ‘postprocessing’.1However, all postpro-

cessing studies we are aware of consider ﬁxed-horizon forecasting. Our ﬁxed-event setup

requires diﬀerent statistical tools in that only a small number of observations is typically

available for a given forecast horizon. We thus focus on modeling the properties of et,h

as a continuous function of h, thus interpolating across forecast horizons. By contrast,

postprocessing methods for ﬁxed-horizon forecasts typically treat each horizon separately,

using a time series (et,h)n

t=1 of past forecast errors at a single horizon h.

3 Illustrating ﬁxed-event forecasting via an autoregressive

time series model

In this section, we consider ﬁxed-event forecasting in an autoregressive Gaussian time

series model from the econometric literature. The stylized facts illustrated here will later

motivate our more general empirical methodology (described in Section 4).

3.1 Autoregressive model for weekly GDP growth

Our model is a modiﬁed version of the one in Patton and Timmermann (2011). It assumes

that ﬁxed-event forecasts are based upon noisy high-frequency observations, but make

correct use of these observations. That is, forecasts are equal to the true conditional mean

of the predictand, given their information base. While high-frequency data on GDP is

not literally available in practice (where GDP is measured at a quarterly frequency only),

there are various eﬀorts at measuring economic activity based on economic variables that

are available at monthly, weekly or daily frequency (Aruoba et al., 2009; Brave et al.,

1In meteorology, the term ‘ensemble postprocessing’ is more common since postprocessing is typically

applied to a collection (‘ensemble’) of point forecasts stemming from a numerical weather prediction model

(see Gneiting and Raftery, 2005). That said, the broader principle of modeling forecast errors readily

transfers to the case of a single point forecast that we consider here.

2019; Lewis et al., 2022; Eraslan and G¨otz, 2021). Furthermore, publication lags and

ex-post revisions in macroeconomic data (see e.g. Croushore and Stark, 2001) imply that

realizations become available with a delay. Taken together, the model’s assumption of

a noisy proxy observed at high frequency hence provides a plausible (albeit abstract)

representation of practical GDP forecasting.

We consider hypothetical weekly observations, whereas Patton and Timmermann use

hypothetical monthly observations. This increase in granularity is motivated by our em-

pirical data setup, where many forecasts are made within the month, so that a monthly

frequency may be too coarse. Section A.1 of the online supplement provides a detailed

comparison of our model to the one of Patton and Timmermann. We approximate Yt, the

GDP growth rate from year t−1 to year t, as the weighted sum of 103 weekly logarithmic

growth rates ranging from the beginning of year t−1 to the end of year t. As detailed in

the online supplement, this setup arises from the deﬁnitorial convention (’annual-average’)

that we use to compute Yt. We denote the weekly logarithmic growth rate by Y∗

w, with

the understanding that each year tcorresponds to a distinct set of 52 index values w(see

below for an example). Here and henceforth, we use the ‘star’ superscript notation for

weekly random variables. We further assume that Y∗

wfollows a ﬁrst-order autoregression,

so that

Y∗

w=ρ Y ∗

w−1+ε∗

w(1)

ε∗

iid

∼ N (0, σ2

ε) (2)

Yt≈

103

j=1

γjY∗

wlast(t)+1−j,(3)

where wlast(t) denotes the index of the last week of year t, and γj= 1 −|52−j|

52 . Note

that the coeﬃcients γjform a triangle when plotted against j. For example, suppose

that the sample starts in year t= 1 (containing weeks w= 1,2,...,52), so that week

w= 104 = wlast(2) is the last week of year 2. According to Equation (3), the approximate

annual growth rate Y2is a weighted sum of the weekly observations (Y∗

w)104

w=2. The greatest

weight of 1 is associated with Y∗

53, and the smallest weight of 1/52 is associated with Y∗

104

and Y∗

2. Furthermore, P103

j=1 γj= 52,in line with the fact that Y∗

wis measured at a weekly

frequency whereas Ytis measured annually. See Section A.2 of the online supplement for

a concise derivation of the approximation at (3), and Patton and Timmermann (2011,

Appendix B) for numerical evidence on the high precision of the approximation.

As noted earlier, we assume that forecasters observe a noisy version of Y∗

w. Specif-

ically, let ˜

Y∗

w=Y∗

w+η∗

w,where η∗

i.i.d.

∼ N (0, σ2

η) is an independent and identically dis-

tributed (IID) Gaussian measurement error. An h-week ahead mean forecast of Ytis then

based on the information set Ft,h generated by the sequence of noisy weekly observations

(˜

Y∗

w)wlast(t)−h

w=1 .Due to the presence of measurement error, the optimal forecast of Ytgiven

Ft,h has no simple closed-form expression. However, the optimal forecast can be computed

analytically by means of the Kalman ﬁlter. The latter uses the model’s linear state space

representation, which we describe in Section A.3 of the online supplement.

An important implication of the present model is that forecasts of Ytat horizon h= 0

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Predictionintervalsforeconomicfixed-eventforecasts∗FabianKr¨ugerKarlsruheInstituteofTechnologyHendrikPlettETHZ¨urichMarch21,2024AbstractThefixed-eventforecastingsetupiscommonineconomicpolicy.Itinvolvesasequenceofforecastsofthesame(‘fixed’)predictand,sothatthedifficultyoftheforecastingproblemdecrease...

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Prediction intervals for economic fixed-event forecasts Fabian Kr uger Karlsruhe Institute of TechnologyHendrik Plett

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