Download Approximating Fixed-Horizon Forecasts Using Fixed-Event

Approximating Fixed-Horizon Forecasts Using
Fixed-Event Forecasts
Malte Knüppel and Andreea L. Vladu
Deutsche Bundesbank
9th ECB Workshop on Forecasting Techniques
04 June 2016
This work represents the authors’ personal opinions and does not necessarily reflect the views of the Bundesbank
1 / 27
What we do
I
Construct approximations for fixed-horizon forecasts from
fixed-event forecasts;
I
Approximation uses optimal weighting of fixed-event forecasts
derived by minimizing the mean-squared approximation error;
I
Explore gains in approximating one-year-ahead mean inflation
and GDP growth for the 13 countries from the Consensus
Economics (CE) survey...
I
and the corresponding cross-sectional forecast disagreement.
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Example: Fixed-event forecasts from Consensus Economics
3 / 27
Inflation forecasts from Consensus Economics (CE)
Question: What is the forecast of 11-month-ahead year-on-year
inflation rate (fixed-horizon), standing at beginning of month t?
pt+11 − pt−1
≈ gt,t−1 + gt+1,t + · · · gt+11,t+10 .
gt+11,t−1 =
pt−1
Information at time t: Two CE fixed-event forecasts:
I Annual growth rate of price index for current calendar year:
(12)
(12)
g1,0 =
p1
(12)
(12)
I
− 1,
p0
(12)
1
1
where p1 = 12
(p1 + · · · + p12 ), p0 = 12
(p−11 + · · · + p0 ).
Annual growth rate of price index for next calendar year:
(12)
(12)
g2,1 =
(12)
where p2
=
1
12 (p13
p2
(12)
− 1,
p1
+ · · · + p24 ).
Drawbacks of fixed-event forecasts
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Annual growth rates and monthly growth rates
CE fixed-event forecasts are well approximated by linear function of
monthly growth rates (Patton and Timmermann, 2011):
(12)
g1,0 ≈ ω1 g12,11 + ω2 g11,10 + · · · + ω24 g−11,−12
(12)
g2,1 ≈ ω1 g24,23 + ω2 g23,22 + · · · + ω24 g1,0
where wk = 1 −
|k−12|
12 .
PT−weights current year
PT−weights next year
1
0.8
0.6
0.4
0.2
0
Jan
Apr
Jul
Oct
previous year
Jan
Apr
Jul
current year
Oct
Jan
Apr
Jul
next year
Oct
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Ad-hoc approximation of fixed-horizon forecasts at time t
Approximate gt+11,t−1 by weighting the two CE fixed-horizon
forecasts according to:
(12)
(12)
ĝt+11,t−1 ≈ wtadhoc g1,0 + 1 − wtadhoc g2,1 ,
with
13 − t
.
12
Example: In beginning of January (t = 1) one wants to forecast
inflation Dec(current year)-to-Dec(previous year):
wtadhoc =
(12)
ĝ12,0 ≈ g1,0
since
wtadhoc = 1.
Note: Information contained in the second fixed-event forecast
(12)
g2,1 is ignored in this case.
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Optimal approximation of fixed-horizon forecasts at time t
We propose to determine the weight wt in:
(12)
(12)
g̃t+11,t−1 = wt g1,0 + (1 − wt ) g2,1 ,
by minimizing the expected squared approximation error
h
i
min E (gt+11,t−1 − g̃t+11,t−1 )2 .
w
Novel approximation accounts correctly for monthly growth rates
(12)
(12)
embedded in gt+11,t−1 and g̃t+11,t−1 = wt g1,0 + (1 − wt ) g2,1 .
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Derive optimal approximation of fixed-horizon forecasts
By introducing the vectors



G=

g24,23
g23,22
..
.
g−11,−12





013−t






1
,
A
=
,
B
=

 t,12
12
1


011+t

012
ω1
ω2
..
.
ω24


ω1
ω2
..
.







,
B
=
 2 


 ω24

012






we write the approximation error as linear expression of weight wt :
gt+11,t−1 − g̃t+11,t−1 = A0t,12 G − wt B01 G + (1 − wt ) B02 G




= A0t,12 − B02 + wt B02 − B01  G.
| {z }
| {z }
:=Mt
:=N
h
i
⇒ E (gt+11,t−1 − g̃t+11,t−1 )2 is a quadratic expression in wt .
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Optimal weights formula
The optimal weight on the current year fixed-event forecast is:
wt∗ = −Mt ΩN0 / NΩN0 .
Ω is the covariance matrix of vector G of monthly growth rates
Ω = E (G − E [G]) G0 − E G0
.
Ω needs to account for known and forecasted monthly growth
rates in vector G.
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Characteristics of the optimal weighting approach
Optimal weighting approach can account for:
I
type of growth rates embedded in the fixed-event forecasts
(annual growth rates, quarterly growth rates, ...);
I
(assumptions about) the data generating process;
I
the (assumed) covariance matrix of the data generating
process (realized and forecast);
I
discretionary forecast horizons (3-month, 6-month, 24-month);
I
information from additional fixed-event forecasts,
but does not consider information contained in previous or later
forecasts or from other variables.
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Properties of the optimal weighting approach
Example:
I
Assume data-generating process for the monthly growth rate
of the variable:
with εt+1
gt+1,t − µ = ρ (gt,t−1 − µ) + εt+1 ,
iid N 0, σε2 .
I
Assume that prior’s month growth rate is observed, but not
the one for the current month.
I
The forecaster makes optimal forecasts.
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Weight on current-year forecast from two methods
1.2
w∗, ρ = 0
w∗, ρ = 0.5
w∗, ρ = 0.8
1
∗
w , ρ = 0.99
adhoc
w
0.8
0.6
0.4
0.2
0
−0.2
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Optimal weights w ∗ depending on ρ and ad-hoc weights w adhoc for current-year
(12)
forecast g1,0 .
12 / 27
Relative approximation errors
1
0.9
ρ=0
ρ = 0.5
ρ = 0.8
ρ = 0.99
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Ratio of expected squared approximation error with optimal weights to expected
squared approximation error with ad-hoc weights for different values of ρ.
13 / 27
Empirical application based on Consensus Economics data
I
Approximate each forecaster’s unobservable forecast for CPI
and GDP four-quarters-ahead year-on-year quarterly growth
rate:
(3)
pt+4
(3)
gt+4,t = (3) − 1,
pt
(3)
where pt = 13 (p3(t−1)+1 + p3(t−1)+2 + p3(t−1)+3 ),
t = 1, 2, 3, 4.
I
using their individual forecasts for annual growth rates of CPI
(12)
(12)
and GDP for current and next year, i.e. g1,0 and g2,1 ;
I
in each March, June, September and December.
14 / 27
Assumptions of the optimal approximation method
I
inflation data is known up to the previous month;
I
GDP growth data is available up to previous quarter;
I
both inflation and GDP growth are assumed to be monthly
processes;
I
growth rates of both variables are assumed to be iid, so that
0
0
Ω=
0 σε2 · I
where I is the identity matrix.
15 / 27
Comparison of optimal and ad-hoc method weights
March
June
September
December
inflation
0.04
−0.05
−0.07
0.08
w∗
GDP growth
0.00
−0.03
−0.08
−0.03
w adhoc
0.75
0.50
0.25
0.00
Weights for current-year forecasts
I
In the first three quarters, the ad-hoc approach places (much)
larger weight on the current-year forecasts
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Optimal and ad-hoc approximations for Eurozone (EZ)
Euro Zone - Inflation Mean
Euro Zone - Inflation Disagreement
3
0.5
2.5
0.4
2
0.3
1.5
0.2
1
0.1
0.5
0
2002
2008
2014
0
2007
Euro Zone - GDP Mean
3
1
2
0.8
1
0.6
0
0.4
-1
-2
2002
2009
2010
2012
2013
2015
Euro Zone - GDP Disagreement
True forecast
Optimal appr.
Adhoc appr.
0.2
2008
2014
0
2007
2009
2010
2012
2013
2015
Actual four-quarter-ahead Consensus mean forecasts and the approximations based on optimal and ad-hoc weights
for inflation and GDP growth (left panels) and time series of disagreement (measured by the standard deviation)
among the actual individual four-quarter-ahead Consensus forecasts and the approximations based on optimal and
ad-hoc weights for inflation and GDP growth (right panels).
17 / 27
Relative approximation errors - Mean forecast inflation
Inflation Mean, Forecast horizon = 4q
2
1.8
1.6
1.4
1.2
USA
Japan
Germany
France
UK
Italy
Canada
Netherlands
Norway
Spain
Sweden
Switzerland
Euro Zone
Average
1
0.8
0.6
0.4
0.2
0
Q1
Q2
Q3
Q4
Ratios of the average squared approximation errors using the optimal weights to the average squared approximation
errors using the ad-hoc weights for inflation mean forecasts four quarters ahead.
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Relative approximation errors - Mean forecast GDP growth
GDP Mean, Forecast horizon = 4q
2
1.8
1.6
1.4
1.2
USA
Japan
Germany
France
UK
Italy
Canada
Netherlands
Norway
Spain
Sweden
Switzerland
Euro Zone
Average
1
0.8
0.6
0.4
0.2
0
Q1
Q2
Q3
Q4
Ratios of the average squared approximation errors using the optimal weights to the average squared approximation
errors using the ad-hoc weights for GDP growth mean forecasts four quarters ahead.
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Relative approximation errors - mean forecast
Avg
US
JP
GE
FR
UK
MSE ratio
0.42
0.3
0.7
0.4
0.3
0.4
MSE ratio
0.38
0.4
0.3
0.3
0.3
0.4
IT
CA
NL
NO
ES
SE
CH
EZ
Inflation
0.8 0.3
0.3
0.4
0.3
0.5
0.4
0.2
GDP growth
0.3 0.4 0.4
0.5
0.9
0.2
0.5
0.2
Ratio of the mean-squared approximation error obtained with optimal weights to the
mean-squared approximation error obtained with ad-hoc weights in all quarters.
20 / 27
Relative approx. errors - disagreement inflation forecasts
Inflation Disagreement, Forecast horizon = 4q
2
1.8
1.6
1.4
1.2
USA
Japan
Germany
France
UK
Italy
Canada
Netherlands
Norway
Spain
Sweden
Switzerland
Euro Zone
Average
1
0.8
0.6
0.4
0.2
0
Q1
Q2
Q3
Q4
Ratios of the average squared approximation errors using optimal weights to the average squared approximation
errors using ad-hoc weights for the standard deviations among individual inflation forecasts four quarters ahead.
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Relative approx. errors - disagreement future GDP growth
GDP Disagreement, Forecast horizon = 4q
2
1.8
1.6
1.4
1.2
USA
Japan
Germany
France
UK
Italy
Canada
Netherlands
Norway
Spain
Sweden
Switzerland
Euro Zone
Average
1
0.8
0.6
0.4
0.2
0
Q1
Q2
Q3
Q4
Ratios of the average squared approximation errors using optimal weights to the average squared approximation
errors using ad-hoc weights for the standard deviations among individual GDP growth forecasts four quarters
ahead. Nr. Forecasters
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Relative approximation errors - forecast disagreement
MSE ratio
Bias
optim.
ad-hoc
Corr. true
with optim
with ad-hoc
MSE ratio
Bias
optim.
ad-hoc
Corr. true
with optim
with ad-hoc
Avg
US
JP
GE
FR
UK
IT
CA
NL
NO
ES
SE
CH
EZ
0.52
0.6
0.5
0.5
0.3
0.3
Inflation
0.5
0.6
0.5
0.7
0.6
0.6
0.6
0.3
-0.07
-0.13
-0.2
-0.3
-0.0
-0.1
-0.1
-0.1
-0.0
-0.1
-0.1
-0.2
-0.0
-0.1
-0.0
-0.1
-0.1
-0.2
-0.1
-0.1
-0.0
-0.1
-0.0
-0.1
-0.1
-0.1
-0.0
-0.1
0.73
0.63
0.7
0.6
0.7
0.8
0.7
0.6
0.9
0.8
0.7
0.5
0.8
0.7
0.8
0.7
0.6
0.4
0.6
0.5
0.7
0.6
0.6
0.6
0.7
0.6
0.9
0.7
0.62
0.4
0.4
0.5
0.4
0.5
GDP growth
0.5
0.8
0.9
1.1
0.3
1.2
0.8
0.4
-0.08
-0.15
-0.1
-0.2
-0.1
-0.2
-0.1
-0.1
-0.1
-0.1
-0.2
-0.3
-0.1
-0.1
0.0
-0.1
-0.1
-0.2
-0.1
-0.1
-0.1
-0.2
0.0
-0.1
-0.1
-0.2
-0.1
-0.1
0.73
0.74
0.8
0.8
0.9
0.7
0.8
0.8
0.9
0.9
0.8
0.9
0.8
0.8
0.8
0.8
0.4
0.5
0.4
0.6
0.9
0.8
0.4
0.6
0.8
0.8
0.9
0.9
Ratio of the mean-squared approximation error obtained with optimal weights to the
mean-squared approximation error obtained with ad-hoc weights in all quarters,
together with bias and correlation between true disagreement and disagreement based
on the two approximations.
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Conclusions
Optimal approximation...
I
gives easily-computable weights for constructing fixed-horizon
forecasts from fixed-event forecasts;
I
is flexible with respect to fixed-horizon forecasts of interest
and availability of fixed-event forecasts;
I
significantly decreases the mean-squared approximation error
for mean forecasts of inflation and GDP growth empirically;
I
is a first step towards a better measurement of disagreement
among forecasters.
24 / 27
Background slides
25 / 27
Annual growth rates for EZ HICP - forecasts and
realizations
Euro Zone Consumer Prices Nowcast
4
3.5
3
2.5
2
1.5
1
0.5
0
−0.5
Realised CPI (at year end)
Mean forecast for the year−end event
−1
Jan2008
Jan2009
Jan2010
Jan2011
Jan2012
Jan2013
Black line represents realized annual growth rate of Eurozone’s HICP known (values known just at year end). Blue
dots represent monthly individual forecasts of the current year’s annual growth rate. Green line represents the mean
of these fixed-event forecast. Main
26 / 27
Average number of respondents for fixed-event and
fixed-horizon forecasts
US
JP
GE
FR
UK
IT
fixed horizon
fixed event
26
28
16
21
17
29
14
21
12
25
6
15
fixed horizon
fixed event
27
28
17
21
19
29
15
21
12
25
CA
NL
NO
ES
SE
CH
EZ
inflation
10
7
16 10
5
9
10
16
6
14
10
15
17
27
GDP growth
7 10
8
15 16 10
5
9
11
16
6
14
10
15
18
27
Average number of forecasters for Consensus Economics fixed-horizon
(four-quarter-ahead) and fixed-event forecasts. The number displayed for the
fixed-event forecasts is the average of the numbers for current- and next-year forecasts
which tend to be virtually identical.
main
27 / 27