VARMA versus VAR for Macroeconomic Forecasting 1 VARMA versus VAR for Macroeconomic Forecasting George Athanasopoulos Department of Econometrics and Business Statistics Monash University Farshid Vahid School of Economics Australian National University
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VARMA versus VAR for Macroeconomic Forecasting 1
VARMA versus VAR forMacroeconomic Forecasting
George AthanasopoulosDepartment of Econometrics and Business Statistics
Monash University
Farshid VahidSchool of Economics
Australian National University
VARMA versus VAR for Macroeconomic Forecasting Introduction 2
Outline
1 Introduction
2 Canonical SCM
3 Forecast performance
4 Example
5 Simulation
6 Summary of findings and future research
VARMA versus VAR for Macroeconomic Forecasting Introduction 3
VAR models dominateWhy VARMA?
More parsimonious representationClosed with respect to linear transformations
Difficult to Identify“If univariate ARIMA modelling is difficult then VARMA modelling iseven more difficult - some might say impossible!” - Chatfield
Empirical Results:1. Average performance across many trivariate systems2. A four variable example3. Simulation: Why do VARMA models do better than VARs?
VARMA versus VAR for Macroeconomic Forecasting Canonical SCM 7
Empirical Results:1. Average performance across many trivariate systems2. A four variable example3. Simulation: Why do VARMA models do better than VARs?
VARMA versus VAR for Macroeconomic Forecasting Canonical SCM 8
Normalise diagonally (test for improper normalisations)
Reduce parameters of A to produce a Canonical SCM 1 0 0α21 1 0α31 α32 1
yt =
φ(1)11 φ
(1)12 φ
(1)13
φ(1)21 φ
(1)22 φ
(1)23
0 0 0
yt−1 + εt −
θ(1)11 θ
(1)12 0
0 0 00 0 0
εt−1
Empirical Results:1. Average performance across many trivariate systems2. A four variable example3. Simulation: Why do VARMA models do better than VARs
VARMA versus VAR for Macroeconomic Forecasting Canonical SCM 8
Normalise diagonally (test for improper normalisations)
Reduce parameters of A to produce a Canonical SCM
1 0 0α21 1 0α31 α32 1
yt =
φ(1)11 φ
(1)12 φ
(1)13
φ(1)21 φ
(1)22 φ
(1)23
0 0 0
yt−1 + εt −
θ(1)11 θ
(1)12 0
0 0 00 0 0
εt−1
Empirical Results:1. Average performance across many trivariate systems2. A four variable example3. Simulation: Why do VARMA models do better than VARs
VARMA versus VAR for Macroeconomic Forecasting Canonical SCM 8
Normalise diagonally (test for improper normalisations)
Reduce parameters of A to produce a Canonical SCM
1 0 0α21 1 0α31 α32 1
yt =
φ(1)11 φ
(1)12 φ
(1)13
φ(1)21 φ
(1)22 φ
(1)23
0 0 0
yt−1 + εt −
θ(1)11 θ
(1)12 0
0 0 00 0 0
εt−1
Empirical Results:1. Average performance across many trivariate systems2. A four variable example3. Simulation: Why do VARMA models do better than VARs
VARMA versus VAR for Macroeconomic Forecasting Canonical SCM 8
Normalise diagonally (test for improper normalisations)
Reduce parameters of A to produce a Canonical SCM 1 0 0α21 1 0α31 α32 1
yt =
φ(1)11 φ
(1)12 φ
(1)13
φ(1)21 φ
(1)22 φ
(1)23
0 0 0
yt−1 + εt −
θ(1)11 θ
(1)12 0
0 0 00 0 0
εt−1
Empirical Results:1. Average performance across many trivariate systems2. A four variable example3. Simulation: Why do VARMA models do better than VARs
VARMA versus VAR for Macroeconomic Forecasting Canonical SCM 8
Normalise diagonally (test for improper normalisations)
Reduce parameters of A to produce a Canonical SCM 1 0 0α21 1 0α31 α32 1
yt =
φ(1)11 φ
(1)12 φ
(1)13
φ(1)21 φ
(1)22 φ
(1)23
0 0 0
yt−1 + εt −
θ(1)11 θ
(1)12 0
0 0 00 0 0
εt−1
Empirical Results:1. Average performance across many trivariate systems2. A four variable example3. Simulation: Why do VARMA models do better than VARs
VARMA versus VAR for Macroeconomic Forecasting Forecast performance 9
Outline
1 Introduction
2 Canonical SCM
3 Forecast performance
4 Example
5 Simulation
6 Summary of findings and future research
VARMA versus VAR for Macroeconomic Forecasting Forecast performance 10
Forecasting: 40 monthly macroeconomic variables from 8 general
categories of economic activity, 1959:1-1998:12 (N=480)
50 × 3 variable systems
Test sample: N1 = 300
Estimated canonical SCM VARMAUnrestricted VAR(AIC) and VAR(BIC)Restricted VAR(AIC) and VAR(BIC)
Hold-out sample: N2 = 180
Produced N2 − h + 1 out-of-sample forecasts for eachh=1 to 15Forecast error measures: |MSFE | and tr(MSFE )Percentage Better: PBh
Relative Ratios:
RRdMSFE h = 150
∑50i=1
|MSFEVARi |
|MSFEVARMAi |
VARMA versus VAR for Macroeconomic Forecasting Forecast performance 10
Forecasting: 40 monthly macroeconomic variables from 8 general
categories of economic activity, 1959:1-1998:12 (N=480)
50 × 3 variable systems
Test sample: N1 = 300
Estimated canonical SCM VARMAUnrestricted VAR(AIC) and VAR(BIC)Restricted VAR(AIC) and VAR(BIC)
Hold-out sample: N2 = 180
Produced N2 − h + 1 out-of-sample forecasts for eachh=1 to 15Forecast error measures: |MSFE | and tr(MSFE )Percentage Better: PBh
Relative Ratios:
RRdMSFE h = 150
∑50i=1
|MSFEVARi |
|MSFEVARMAi |
VARMA versus VAR for Macroeconomic Forecasting Forecast performance 10
Forecasting: 40 monthly macroeconomic variables from 8 general
categories of economic activity, 1959:1-1998:12 (N=480)
50 × 3 variable systems
Test sample: N1 = 300
Estimated canonical SCM VARMAUnrestricted VAR(AIC) and VAR(BIC)Restricted VAR(AIC) and VAR(BIC)
Hold-out sample: N2 = 180
Produced N2 − h + 1 out-of-sample forecasts for eachh=1 to 15Forecast error measures: |MSFE | and tr(MSFE )Percentage Better: PBh
Relative Ratios:
RRdMSFE h = 150
∑50i=1
|MSFEVARi |
|MSFEVARMAi |
VARMA versus VAR for Macroeconomic Forecasting Forecast performance 10
Forecasting: 40 monthly macroeconomic variables from 8 general
categories of economic activity, 1959:1-1998:12 (N=480)
50 × 3 variable systems
Test sample: N1 = 300
Estimated canonical SCM VARMAUnrestricted VAR(AIC) and VAR(BIC)Restricted VAR(AIC) and VAR(BIC)
Hold-out sample: N2 = 180
Produced N2 − h + 1 out-of-sample forecasts for eachh=1 to 15Forecast error measures: |MSFE | and tr(MSFE )Percentage Better: PBh
Relative Ratios:
RRdMSFE h = 150
∑50i=1
|MSFEVARi |
|MSFEVARMAi |
VARMA versus VAR for Macroeconomic Forecasting Forecast performance 11
Relative Ratios
Forecast horizon (h)
%
2 4 6 8 10 12 14
1.05
1.10
1.15
1.20VAR(AIC) VAR(BIC)
PANEL A
RdMSFE for Unrestricted VAR
Forecast horizon (h)
%
2 4 6 8 10 12 14
1.05
1.10
1.15
1.20VAR(AIC) VAR(BIC)
PANEL B
RdMSFE for Restricted VAR
VARMA versus VAR for Macroeconomic Forecasting Forecast performance 12
Percentage Better: Unrestricted VAR
Forecast horizon (h)
%
2 4 6 8 10 12 14
20
30
40
50
60
70
80VARMA VAR(AIC)
PANEL A
PB counts for tr(MSFE) for VARMA versus Unrestricted VAR
Forecast horizon (h)
%
2 4 6 8 10 12 14
20
30
40
50
60
70
80
●
●
●
● ●●
●
● ●●
●● ● ● ●
●VARMA VAR(BIC)
VARMA versus VAR for Macroeconomic Forecasting Forecast performance 13
Percentage Better: Restricted VAR
Forecast horizon (h)
%
2 4 6 8 10 12 14
20
30
40
50
60
70
80VARMA VAR(AIC)
PANEL B
PB counts for tr(MSFE) for VARMA versus Restricted VAR
Forecast horizon (h)
%
2 4 6 8 10 12 14
20
30
40
50
60
70
80
●
●●
● ●
● ●
● ●●
●●
● ●●
●VARMA VAR(BIC)
VARMA versus VAR for Macroeconomic Forecasting Forecast performance 14
1 There are cases where VARMA significantly outperform VAR andvice versa
2 VARMA models significantly outperform VAR more than thereverse
3 As h increases the number significant differences decreases
4 Restrictions do not improve VAR performance when significantdifferences
VARMA versus VAR for Macroeconomic Forecasting Example 16
Outline
1 Introduction
2 Canonical SCM
3 Forecast performance
4 Example
5 Simulation
6 Summary of findings and future research
VARMA versus VAR for Macroeconomic Forecasting Example 17
Example:
Four variables (also six variables):
GDP growth rateinflation ratespread (10 yr gvt bill yield) − (3-month treasury bill rate)3-month treasury bill rate
- in line with term structure literature: Ang, Piazzesi, Wei (2006)- variations in New Keynesian DSGE - contributions in Taylor(1999)
Quarterly data
Message: We should start considering VARMA
VARMA versus VAR for Macroeconomic Forecasting Example 17
Example:
Four variables (also six variables):
GDP growth rateinflation ratespread (10 yr gvt bill yield) − (3-month treasury bill rate)3-month treasury bill rate
- in line with term structure literature: Ang, Piazzesi, Wei (2006)- variations in New Keynesian DSGE - contributions in Taylor(1999)
Quarterly data
Message: We should start considering VARMA
VARMA versus VAR for Macroeconomic Forecasting Example 17
Example:
Four variables (also six variables):
GDP growth rateinflation ratespread (10 yr gvt bill yield) − (3-month treasury bill rate)3-month treasury bill rate
- in line with term structure literature: Ang, Piazzesi, Wei (2006)- variations in New Keynesian DSGE - contributions in Taylor(1999)
Quarterly data
Message: We should start considering VARMA
VARMA versus VAR for Macroeconomic Forecasting Simulation 18
Outline
1 Introduction
2 Canonical SCM
3 Forecast performance
4 Example
5 Simulation
6 Summary of findings and future research
VARMA versus VAR for Macroeconomic Forecasting Simulation 19
Why do VARMA forecast better?
Estimated a VARMA(2,1): - SCM(2,0) - SCM(1,1)- SCM(1,0) - SCM(0,0)
1 0 0 00 1 0 00 0 1 0∗ ∗ ∗ 1
yt =
∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗0 0 0 0
yt−1 +
∗ ∗ ∗ ∗0 0 0 00 0 0 00 0 0 0
yt−2
−
0 0 0 0∗ ∗ ∗ 00 0 0 00 0 0 0
et−1 + et
Simulate from the benchmark estimated model assuming e ∼ N(0,Σ)
n = 164→ estimate VARMA(2,1), VAR(AIC), VAR(BIC)
nout = 42→ compute 1 to 12-step ahead forecasts
iterations = 100→ calculate |MSFE | for all models
Compare the percentage difference using |MSFEVARMA| as a base
Repeat by changing specific features and compare with the benchmark
VARMA versus VAR for Macroeconomic Forecasting Simulation 19
Why do VARMA forecast better?
Estimated a VARMA(2,1): - SCM(2,0) - SCM(1,1)- SCM(1,0) - SCM(0,0)
1 0 0 00 1 0 00 0 1 0∗ ∗ ∗ 1
yt =
∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗0 0 0 0
yt−1 +
∗ ∗ ∗ ∗0 0 0 00 0 0 00 0 0 0
yt−2
−
0 0 0 0∗ ∗ ∗ 00 0 0 00 0 0 0
et−1 + et
Simulate from the benchmark estimated model assuming e ∼ N(0,Σ)
n = 164→ estimate VARMA(2,1), VAR(AIC), VAR(BIC)
nout = 42→ compute 1 to 12-step ahead forecasts
iterations = 100→ calculate |MSFE | for all models
Compare the percentage difference using |MSFEVARMA| as a base
Repeat by changing specific features and compare with the benchmark
VARMA versus VAR for Macroeconomic Forecasting Simulation 20
Why do VARMA forecast better:%
2 4 6 8 10 12
010
2030
4050
60
● ●● ● ● ●
● ●● ● ● ●
DGP1: Estimated model VAR(AIC) VAR(BIC)
VARMA versus VAR for Macroeconomic Forecasting Simulation 21
Why do VARMA forecast better:%
2 4 6 8 10 12
010
2030
4050
60
● ●● ● ● ●
● ●● ● ● ●
DGP1: Estimated model
%2 4 6 8 10 12
010
2030
4050
60
●●
● ● ● ● ● ● ● ● ● ●
DGP2: No MA VAR(AIC) VAR(BIC)
VARMA versus VAR for Macroeconomic Forecasting Simulation 22
Why do VARMA forecast better:%
2 4 6 8 10 12
010
2030
4050
60
● ●● ● ● ●
● ●● ● ● ●
DGP1: Estimated model
%2 4 6 8 10 12
010
2030
4050
60
●
● ●●
● ● ●● ● ● ● ●
DGP3: 2 strong MAs VAR(AIC) VAR(BIC)
VARMA versus VAR for Macroeconomic Forecasting Simulation 23
Why do VARMA forecast better:%
2 4 6 8 10 12
010
2030
4050
60
● ●● ● ● ●
● ●● ● ● ●
DGP1: Estimated model
%2 4 6 8 10 12
010
2030
4050
60
●
● ●
●
● ● ●●
● ●●
●
DGP5: 3 strong MAs VAR(AIC) VAR(BIC)
VARMA versus VAR for Macroeconomic Forecasting Simulation 24
Why do VARMA forecast better:%
2 4 6 8 10 12
010
2030
4050
60
● ●● ● ● ●
● ●● ● ● ●
DGP1: Estimated model
%2 4 6 8 10 12
010
2030
4050
60
●
●● ● ● ● ● ● ● ● ● ●
DGP6: 3 weak MAs VAR(AIC) VAR(BIC)
VARMA versus VAR for Macroeconomic Forecasting Simulation 25
Why do VARMA forecast better:%
2 4 6 8 10 12
010
2030
4050
60
● ●● ● ● ●
● ●● ● ● ●
DGP1: Estimated model
%2 4 6 8 10 12
010
2030
4050
60
●●
●
●
●●
● ● ● ● ● ●
DGP7: Weak AR VAR(AIC) VAR(BIC)
VARMA versus VAR for Macroeconomic Forecasting Summary of findings and future research 26
Outline
1 Introduction
2 Canonical SCM
3 Forecast performance
4 Example
5 Simulation
6 Summary of findings and future research
VARMA versus VAR for Macroeconomic Forecasting Summary of findings and future research 27
Summary of findings:
1 We can obtain better forecasts for macroeconomic variables byconsidering VARMA models
2 With the methodological developments and the improvement incomputer power there is no compelling reason to restrict the classof models to VARs only
3 The existence of VMA components cannot be well-approximatedby finite order VARs
4 Are these favourable results specific the SCM methodology? No!Athanasopoulos, Poskitt and Vahid (2007) show that similarconclusions emerge when one uses the “Echelon” form approach
Future Research:
1 Developing a fully automated identification process
2 Developing an alternative estimation approach which avoids fittinga long VAR to estimate the lagged innovations
3 Move into the non-stationary world
VARMA versus VAR for Macroeconomic Forecasting Summary of findings and future research 27
Summary of findings:
1 We can obtain better forecasts for macroeconomic variables byconsidering VARMA models
2 With the methodological developments and the improvement incomputer power there is no compelling reason to restrict the classof models to VARs only
3 The existence of VMA components cannot be well-approximatedby finite order VARs
4 Are these favourable results specific the SCM methodology? No!Athanasopoulos, Poskitt and Vahid (2007) show that similarconclusions emerge when one uses the “Echelon” form approach
Future Research:
1 Developing a fully automated identification process
2 Developing an alternative estimation approach which avoids fittinga long VAR to estimate the lagged innovations
3 Move into the non-stationary world
VARMA versus VAR for Macroeconomic Forecasting Summary of findings and future research 27
Summary of findings:
1 We can obtain better forecasts for macroeconomic variables byconsidering VARMA models
2 With the methodological developments and the improvement incomputer power there is no compelling reason to restrict the classof models to VARs only
3 The existence of VMA components cannot be well-approximatedby finite order VARs
4 Are these favourable results specific the SCM methodology? No!Athanasopoulos, Poskitt and Vahid (2007) show that similarconclusions emerge when one uses the “Echelon” form approach
Future Research:
1 Developing a fully automated identification process
2 Developing an alternative estimation approach which avoids fittinga long VAR to estimate the lagged innovations
3 Move into the non-stationary world
VARMA versus VAR for Macroeconomic Forecasting Summary of findings and future research 27
Summary of findings:
1 We can obtain better forecasts for macroeconomic variables byconsidering VARMA models
2 With the methodological developments and the improvement incomputer power there is no compelling reason to restrict the classof models to VARs only
3 The existence of VMA components cannot be well-approximatedby finite order VARs
4 Are these favourable results specific the SCM methodology? No!Athanasopoulos, Poskitt and Vahid (2007) show that similarconclusions emerge when one uses the “Echelon” form approach
Future Research:
1 Developing a fully automated identification process
2 Developing an alternative estimation approach which avoids fittinga long VAR to estimate the lagged innovations
3 Move into the non-stationary world
VARMA versus VAR for Macroeconomic Forecasting Summary of findings and future research 27
Summary of findings:
1 We can obtain better forecasts for macroeconomic variables byconsidering VARMA models
2 With the methodological developments and the improvement incomputer power there is no compelling reason to restrict the classof models to VARs only
3 The existence of VMA components cannot be well-approximatedby finite order VARs
4 Are these favourable results specific the SCM methodology? No!Athanasopoulos, Poskitt and Vahid (2007) show that similarconclusions emerge when one uses the “Echelon” form approach
Future Research:
1 Developing a fully automated identification process
2 Developing an alternative estimation approach which avoids fittinga long VAR to estimate the lagged innovations
3 Move into the non-stationary world
VARMA versus VAR for Macroeconomic Forecasting Summary of findings and future research 28