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Forecasting GDP growth
Torsten Lisson (contact: [email protected])Emanuel Gasteiger (contact: [email protected])
Note: This talk was given at the class „Economic Forecasting“ of Prof. Robert. M. Kunst at the University of Vienna, Austria on January the 9th 2007.
Exhibit 2
Introduction
Model-free forecast
Model-based univariate forecast
Model-based multivariate forecast
Discussion of results
Exhibit 3
Basic idea to this study steams from the Keynesian economy
How close are our forecasts of model-free and model-based procedures to forecasts ofleading research institutions?
!
Y = C +G + I + (EX " IM)
How do forecasts develope if we use the components of the GDP according to a Keynesianeconomy?
Exhibit 4
We use public data of the Austrian GDP and its components onquarterly basis
Data source: Statistical Office of the European Communities (EUROSTAT)http://ec.europa.eu/eurostat
Data set: Country: Austria Period: 1988q01 to 2006q03 (75 observations) Variables:
GDP (Yt) Household and non-profit sector consumption (Ct) Government expenditures (Gt) Gross investment (It) Exports of goods and services (EXt) Imports of goods and services (IMt)
Unit: mio. EURO fixed prices (base year is 1995)
Exhibit 5
GDP over time shows clear cyclical patterns and trending
Cyclical patterns:
q1 to q2 ⇑
q2 to q3 ⇑
q3 to q4 ⇑
q4 to q1 ⇓
Trends: More ressources Technological progress (Inflation in nominal
GDP)
Exhibit 6
The time series of GDP growth rate appears to be stable butcyclical patterns remain
!
ˆ y t =(Yt "Yt"1
)
Yt"1
The growth rate:
Exhibit 7
Introduction
Model-free forecast
Model-based univariate forecast
Model-based multivariate forecast
Discussion of results
Exhibit 8
We choose the Holt-Winters Seasonal Smoothing method
Why Holt-Winters method?
Which parameter shall one choose?
Which form shall one choose?
Exhibit 9
Recall the Holt-Winters method
Multiplicative version: Additive version:
!
Lt="(X
t# S
t#s) + (1#")(Lt#1 + T
t#1)
!
Tt
= "(Lt# L
t#1) + (1#")Tt#1
!
St= "
Xt
Lt
+ (1# ")St#s
!
X
^
N (h) = LN
+ TNh + S
N +h"s
!
Lt="
Xt
St#s
+ (1#")(Lt#1 + T
t#1)
!
Tt
= "(Lt# L
t#1) + (1#")Tt#1
!
St= "
Xt
Lt
+ (1# ")St#s
!
X
^
N (h) = (LN
+ TNh)S
N +h"s
Note: Stata derives the starting value from the mean of the first half of the samples’ observations by default
Exhibit 10
We evaluate the procedures by the predicted mean squarederror (PRMSE)
!
ˆ Y t
!
Yt
88q1 05q3!
PRMSE =1
n( ˆ Y
i
i= 72
n
" #Yi)
2
06q3
used observations forprediction
Evaluation method:
benchmarkobservations
forecast
predictions
Idea: we want a good forecast, not the best model fit
Note: For all further analysis we use the sub-sample of 71 observations
Exhibit 11
Evaluation results favour the additive (0.3; 0.3; 0.3) method
0,012500,011380,01214
Multiplicative Additive(0,3; 0,1; 0,1)(0,3; 0,3; 0,3)(0,?; 0,?; 0,?)
0,266340,012300,01312
Course
Evaluation results:
StataLiterature maximum
We choose (0,3; 0,3; 0,3)
Exhibit 12
Best multiplicative Holt-Winters prediction appears to beexplosive
Exhibit 13
Best additive Holt-Winters prediction appears to be morerealistic
Exhibit 14
Introduction
Model-free forecast
Model-based univariate forecast
Model-based multivariate forecast
Discussion of results
Exhibit 15
We perform a Dickey-Fuller test:
H0: is I(1) vs. H1: is stationary
The time series is not first order integrated
1%-level: -3,548
5%-level: -2,912
10%-level: -2,591
> -11,811
Critical values: Dickey-Fuller Value:
is stationary
!
ˆ y t
!
ˆ y t
!
ˆ y t
!
ˆ y t
Exhibit 16
We use 4 methods to find the right lag-order
Correlation functions
Information criteria
Residual diagnostics
Hypotheses testing
Exhibit 17
The correlation functions do not converge to zero and point inthe direction of seasonal adjustment
Exhibit 18
We can not be sure about a seasonal pattern from thecorrelation functions of ∆4
We focus on ARMA models
!
ˆ y t
Exhibit 19
Akaike and Schwarz Information Criterion both recommend aARMA (3,3) model
Akaike criterion
0 1 2 3
0
AR(p)
MA(q)
-225,2912 -230,4085 -246,6321 -385,382
1 -260,3076 -258,3298 -276,264 -414,176
2 -275,2224 -256,7264 -310,5073 -419,3524
3 -281,6136 -279,678 -336,7877 -435,191
4 -281,1353 -283,1412 -272,3503 -434,2746
0 1 2 3
0
AR(p)
MA(q)
-220,7943 -223,663 -237,6381 -374,1395
1 -253,5621 -249,3358 -265,0216 -400,6856
2 -266,2284 -245,484 -297,0163 -403,6129
3 -270,3711 -266,187 -323,2968 -417,203
4 -267,6443 -269,9202 -254,3624 -414,0382
Schwarz criterion
Exhibit 20
We analyse the residuals by the Portmanteau Test
We perform the test for all of the 20 models
We plot the 4 models where the Q-Statistic is lower than the critical value of the chi-squareddistribution
We choose our favourite model by eyeball analysis
Exhibit 21
Residuals of ARMA (2,4)
Exhibit 22
Residuals of ARMA (3,2)
Exhibit 23
Residuals of ARMA (3,4)
Exhibit 24
Residuals of ARMA (3,3)
Exhibit 25
The general to specific approach of hypothesis testingrecommends a ARMA (3,3) model
!
ˆ y t ="*
1ˆ y t#1
+"*
2ˆ y t#2
+"*
3ˆ y t#3
+ $t +%*
1$t#1+%
*
2$t#2+%
*
3$t#3+%
4$t#4
!
ˆ y t ="*
1ˆ y t#1
+"*
2ˆ y t#2
+"*
3ˆ y t#3
+ $t +%*
1$t#1+%
*
2$t#2+%
*
3$t#3
Exhibit 26
The ARMA (3,3) comes close to the data
Exhibit 27
Calculate the growth rate
We receive similar models for the sum of the components of theGDP
!
Yt= C
t+G
t+ I
t+ (EX
t" IM
t)Sum up the components
+
Receive similar models!
ˆ y t =(Yt "Yt"1
)
Yt"1
Iterate procedures
+Model-free / model-based univariate approach
Exhibit 28
Yt and the sum of its components is almost the same
Exhibit 29
Introduction
Model-free forecast
Model-based univariate forecast
Model-based multivariate forecast
Discussion of results
Exhibit 30
Multivariate analysis is based on the components of the GDP
!
ln(Ct)
ln(Gt)
ln(It)
ln(EXt)
ln(IMt)
"
#
$ $ $ $ $ $
%
&
' ' ' ' ' '
= A1
ln(Ct(1)
ln(Gt(1)
ln(It(1)
ln(EXt(1)
ln(IMt(1)
"
#
$ $ $ $ $ $
%
&
' ' ' ' ' '
+ A2
ln(Ct(2)
ln(Gt(2)
ln(It(2)
ln(EXt(2)
ln(IMt(2)
"
#
$ $ $ $ $ $
%
&
' ' ' ' ' '
+ A3
ln(Ct(3)
ln(Gt(3)
ln(It(3)
ln(EXt(3)
ln(IMt(3)
"
#
$ $ $ $ $ $
%
&
' ' ' ' ' '
+ A4
ln(Ct(4 )
ln(Gt(4 )
ln(It(4 )
ln(EXt(4 )
ln(IMt(4 )
"
#
$ $ $ $ $ $
%
&
' ' ' ' ' '
+ )t
We estimate a multivariate VAR model of the components of Yt:
Exhibit 31
The information criterias suggest a VAR(4) model
Akaike SchwarzVAR (0)
VAR (2)VAR (1)
-13,5647-22,0562-22,7482-22,9124-24,0597*-23,6622
HQ
VAR (3)VAR (4)
We compare VAR(.) models:
-13,6285-22,4389-23,4499-23,9331-25,3994*-25,3208
-13,4679-21,4753-21,6832-21,3634-22,0266*-21,1451VAR (5)
We check whether VAR(4) is stable by the stability condition below:
!
det(IK" #z) $ 0
det(IK" #1z " #2z
2" #3z
3" #4z
4) $ 0, for |z|≤1
Exhibit 32
Before calculating the growth rate one has to remove thelogarithm and sum up the predicted values
, where B:= (EX-IM)
Finally we calculate the growth rate:
!
ˆ y t +h =( ˆ C t +h + ˆ G t +h + ˆ I t +h + ˆ B t +h " ( ˆ C t +h"1
+ ˆ G t +h"1+ ˆ I t +h"1
+ ˆ B t +h"1))
( ˆ C t +h"1+ ˆ G t +h"1
+ ˆ I t +h"1+ ˆ B t +h"1
)
!
ˆ y t +h =( ˆ Y t +h "
ˆ Y t +h"1)
ˆ Y t +h"1
Exhibit 33
The VAR(4) seems to be close to the data
Exhibit 34
Introduction
Model-free forecast
Model-based univariate forecast
Model-based multivariate forecast
Discussion of results
Exhibit 35
The differences in the forecast results are quiet high
2,1%- 7,3% 5,1% 3,0% 2,1% 2,7%
1,1%-8,8% 7,5% 1,9% 0,9% 1,1% 2,7% 2,6%
H.-W. M. Univariate Multivar. Wifo (21.12.06) IHS (21.12.06)
2006q42007q12007q22007q32007q4
2007
4,4%- 11,3% 6,1% 4,5% 4,9% 5,7%
3,1%- 9,3% 6,2% 3,0% 3,1% 2,3%
H.-W. M. Univariate
2,5%-7,4%4,1%2,9%2,5%2,2%
Yt Yt = Ct+Gt+It+EXt-IMt
The univariate ARMA (3,3) model is in line with the forecasts of the researchinstitutions
Exhibit 36
Thank you!
Exhibit 37
Back up
Exhibit 38
Nominal versus real GDP
Why to correct for inflation?
Uncover the true trendof GDP
Stabilize variance, i.e.eliminate price-shocks
Exhibit 39
Complete output for ACF and PACF of Yt
Exhibit 40
Complete output for ACF and PACF of ∆4Yt
