VARMA versus VAR for Macroeconomic Forecastingusers.monash.edu.au/~gathana/slides/isf07.pdf ·...

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

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

Identification Problem[y1,t

[φ11 φ12

φ21 φ22

] [y1,t−1

y2,t−1

[ε1,t

]−[θ11 θ12

θ21 θ22

] [ε1,t−1

ε2,t−1

]y2,t = ε2,t ⇒ y2,t−1 = ε2,t−1 ⇒ (φ12, θ12)

- SCM framework: Tiao & Tsay (1989) completed by Athanasopoulos &Vahid (2006)- Echelon form: Hannan & Kavalieris (1984); Poskitt (1992); Lutkepohl& Poskitt (1996), Athanasopoulos, Poskitt & Vahid (2007)

[φ11 φ12

φ21 φ22

] [y1,t−1

y2,t−1

[ε1,t

]−[θ11 θ12

θ21 θ22

] [ε1,t−1

ε2,t−1

]y2,t = ε2,t ⇒ y2,t−1 = ε2,t−1 ⇒ (φ12, θ12)

[φ11 φ12

φ21 φ22

] [y1,t−1

y2,t−1

[ε1,t

]−[θ11 θ12

θ21 θ22

] [ε1,t−1

ε2,t−1

]y2,t = ε2,t ⇒ y2,t−1 = ε2,t−1 ⇒ (φ12, θ12)

[φ11 φ12

] [y1,t−1

y2,t−1

[ε1,t

]−[θ11 θ12

] [ε1,t−1

ε2,t−1

y2,t = ε2,t ⇒ y2,t−1 = ε2,t−1 ⇒ (φ12, θ12)

[φ11 φ12

] [y1,t−1

y2,t−1

[ε1,t

]−[θ11 θ12

] [ε1,t−1

ε2,t−1

]y2,t = ε2,t ⇒ y2,t−1 = ε2,t−1

⇒ (φ12, θ12)

[φ11 φ12

] [y1,t−1

y2,t−1

[ε1,t

]−[θ11 θ12

] [ε1,t−1

ε2,t−1

]y2,t = ε2,t ⇒ y2,t−1 = ε2,t−1 ⇒ (φ12, θ12)

[φ11 φ12

] [y1,t−1

y2,t−1

[ε1,t

]−[θ11 θ12

] [ε1,t−1

ε2,t−1

]y2,t = ε2,t ⇒ y2,t−1 = ε2,t−1 ⇒ (φ12, θ12)

VARMA versus VAR for Macroeconomic Forecasting Canonical SCM 5

Outline

1 Introduction

2 Canonical SCM

4 Example

5 Simulation

Definition of a SCM: For a given K -dimensional VARMA(p, q)

yt = Φ1yt−1 + . . .+ Φpyt−p + ηt −Θ1ηt−1 − . . .−Θqηt−q (1)

zr ,t = αr′yt ∼ SCM(pr , qr )

if αr satisfies αr′Φpr 6= 0T where 0 ≤ pr ≤ p

αr′Φl = 0T for l = pr + 1, ..., p

αr′Θqr 6= 0T where 0 ≤ qr ≤ q

αr′Θl = 0T for l = qr + 1, ..., q

SCM Methodology: Find K-linearly independent vectors

A = (α1, . . . , αK )′ which transform (1) into

Ayt = Φ∗1yt−1 + . . .+ Φ∗pyt−p + εt −Θ∗1εt−1 − . . .−Θ∗qεt−q (2)

where Φ∗i = AΦi , εt = Aηt and Θ∗i = AΘiA−1

Series of C/C tests: E

[y1,t−1

y2,t−1

] [y1,t y2,t

][α1] = 0

α′1yt ∼ SCM(0, 0)

αr′Φl = 0T for l = pr + 1, ..., p

αr′Θl = 0T for l = qr + 1, ..., q

[y1,t−1

y2,t−1

] [y1,t y2,t

][α1] = 0

α′1yt ∼ SCM(0, 0)

αr′Φl = 0T for l = pr + 1, ..., p

αr′Θl = 0T for l = qr + 1, ..., q

[y1,t−1

y2,t−1

] [y1,t y2,t

][α1] = 0

α′1yt ∼ SCM(0, 0)

αr′Φl = 0T for l = pr + 1, ..., p

αr′Θl = 0T for l = qr + 1, ..., q

[y1,t−1

y2,t−1

] [y1,t y2,t

][α1] = 0

α′1yt ∼ SCM(0, 0)

Example: K = 3 α′1yt ∼ SCM(1, 1)α′2yt ∼ SCM(1, 0)α′3yt ∼ SCM(0, 0)

α11 α12 α13

α21 α22 α23

α31 α32 α33

φ(1)11 φ

(1)12 φ

φ(1)21 φ

(1)22 φ

yt−1+εt−

θ(1)11 θ

(1)12 0

0 0 00 0 0

εt−1

Reduce parameters of A to produce a Canonical SCM

1 0 0α21 1 0α31 α32 1

φ(1)11 φ

(1)12 φ

φ(1)21 φ

(1)22 φ

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?

α11 α12 α13

α21 α22 α23

α31 α32 α33

φ(1)11 φ

(1)12 φ

φ(1)21 φ

(1)22 φ

yt−1+εt−

θ(1)11 θ

(1)12 0

0 0 00 0 0

εt−1

1 0 0α21 1 0α31 α32 1

φ(1)11 φ

(1)12 φ

φ(1)21 φ

(1)22 φ

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?

1 α12 α13

α21 1 α23

α31 α32 1

φ(1)11 φ

(1)12 φ

φ(1)21 φ

(1)22 φ

yt−1+εt−

θ(1)11 θ

(1)12 0

0 0 00 0 0

εt−1

Normalise diagonally (test for improper normalisations)

Reduce parameters of A to produce a Canonical SCM 1 0 0α21 1 0α31 α32 1

φ(1)11 φ

(1)12 φ

φ(1)21 φ

(1)22 φ

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

1 α12 α13

α21 1 α23

α31 α32 1

φ(1)11 φ

(1)12 φ

φ(1)21 φ

(1)22 φ

yt−1+εt−

θ(1)11 θ

(1)12 0

0 0 00 0 0

εt−1

1 0 0α21 1 0α31 α32 1

φ(1)11 φ

(1)12 φ

φ(1)21 φ

(1)22 φ

yt−1 + εt −

θ(1)11 θ

(1)12 0

0 0 00 0 0

εt−1

1 α12 α13

α21 1 α23

α31 α32 1

φ(1)11 φ

(1)12 φ

φ(1)21 φ

(1)22 φ

yt−1+εt−

θ(1)11 θ

(1)12 0

0 0 00 0 0

εt−1

1 0 0α21 1 0α31 α32 1

φ(1)11 φ

(1)12 φ

φ(1)21 φ

(1)22 φ

yt−1 + εt −

θ(1)11 θ

(1)12 0

0 0 00 0 0

εt−1

1 α12 α13

α21 1 α23

α31 α32 1

φ(1)11 φ

(1)12 φ

φ(1)21 φ

(1)22 φ

yt−1+εt−

θ(1)11 θ

(1)12 0

0 0 00 0 0

εt−1

φ(1)11 φ

(1)12 φ

φ(1)21 φ

(1)22 φ

yt−1 + εt −

θ(1)11 θ

(1)12 0

0 0 00 0 0

εt−1

1 α12 α13

α21 1 α23

α31 α32 1

φ(1)11 φ

(1)12 φ

φ(1)21 φ

(1)22 φ

yt−1+εt−

θ(1)11 θ

(1)12 0

0 0 00 0 0

εt−1

φ(1)11 φ

(1)12 φ

φ(1)21 φ

(1)22 φ

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

1 α12 α13

α21 1 α23

α31 α32 1

φ(1)11 φ

(1)12 φ

φ(1)21 φ

(1)22 φ

yt−1+εt−

θ(1)11 θ

(1)12 0

0 0 00 0 0

εt−1

φ(1)11 φ

(1)12 φ

φ(1)21 φ

(1)22 φ

yt−1 + εt −

θ(1)11 θ

(1)12 0

0 0 00 0 0

εt−1

VARMA versus VAR for Macroeconomic Forecasting Forecast performance 9

Outline

1 Introduction

2 Canonical SCM

4 Example

5 Simulation

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 |

Relative Ratios:

RRdMSFE h = 150

∑50i=1

|MSFEVARi |

|MSFEVARMAi |

Relative Ratios:

RRdMSFE h = 150

∑50i=1

|MSFEVARi |

|MSFEVARMAi |

Relative Ratios:

RRdMSFE h = 150

∑50i=1

|MSFEVARi |

|MSFEVARMAi |

Relative Ratios

Forecast horizon (h)

2 4 6 8 10 12 14

1.20VAR(AIC) VAR(BIC)

PANEL A

RdMSFE for Unrestricted VAR

2 4 6 8 10 12 14

1.20VAR(AIC) VAR(BIC)

PANEL B

RdMSFE for Restricted VAR

Percentage Better: Unrestricted VAR

2 4 6 8 10 12 14

80VARMA VAR(AIC)

PANEL A

PB counts for tr(MSFE) for VARMA versus Unrestricted VAR

2 4 6 8 10 12 14

● ●●

●● ● ● ●

●VARMA VAR(BIC)

Percentage Better: Restricted VAR

2 4 6 8 10 12 14

80VARMA VAR(AIC)

PANEL B

PB counts for tr(MSFE) for VARMA versus Restricted VAR

2 4 6 8 10 12 14

●●

● ●

● ●●

●●

● ●●

●VARMA VAR(BIC)

Diebold-Mariano test for tr(MSFE):

VARMA sign better (5%, 25%, 25%,5%) VARMA sign worst

Forecast horizon (h)1 4 8 12 15

VAR(AIC) - Unrest 24,46,20,14 16,38,10,0 14,32,6,4 10,22,10,0 10,22,8,0VAR(BIC) - Unrest 24,50,22,12 10,38,18,10 12,26,22,8 12,18,16,4 12,18,16,2

VAR(AIC) - Rest 34,54,26,12 14,36,16,0 14,26,8,2 12,22,10,0 10,24,10,0VAR(BIC) - Rest 32,54,22,14 10,38,18,8 12,26,22,6 10,20,20,6 10,16,14,2

Messages:

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

dummy space

Messages:

dummy space

Messages:

dummy space

Messages:

dummy space

Messages:

dummy space

Messages:

4 Restrictions do not improve VAR performance when significantdifferences

VARMA versus VAR for Macroeconomic Forecasting Example 16

Outline

1 Introduction

2 Canonical SCM

4 Example

5 Simulation

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

Example:

Quarterly data

Example:

Quarterly data

VARMA versus VAR for Macroeconomic Forecasting Simulation 18

Outline

1 Introduction

2 Canonical SCM

4 Example

5 Simulation

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

∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗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

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

∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗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

Why do VARMA forecast better:%

2 4 6 8 10 12

● ●● ● ● ●

DGP1: Estimated model VAR(AIC) VAR(BIC)

2 4 6 8 10 12

● ●● ● ● ●

DGP1: Estimated model

%2 4 6 8 10 12

●●

● ● ● ● ● ● ● ● ● ●

DGP2: No MA VAR(AIC) VAR(BIC)

2 4 6 8 10 12

● ●● ● ● ●

%2 4 6 8 10 12

● ●●

● ● ●● ● ● ● ●

DGP3: 2 strong MAs VAR(AIC) VAR(BIC)

2 4 6 8 10 12

● ●● ● ● ●

%2 4 6 8 10 12

● ●

● ● ●●

● ●●

DGP5: 3 strong MAs VAR(AIC) VAR(BIC)

2 4 6 8 10 12

● ●● ● ● ●

%2 4 6 8 10 12

●● ● ● ● ● ● ● ● ● ●

DGP6: 3 weak MAs VAR(AIC) VAR(BIC)

2 4 6 8 10 12

● ●● ● ● ●

%2 4 6 8 10 12

●●

● ● ● ● ● ●

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

4 Example

5 Simulation

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

Future Research:

Thank you!!!!!

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