[PDF] - Variables? George Athanasopoulos Monash University Clayton, PDF Document

SLIDE 1

Are VAR Models Good Enough for Forecasting Macroeconomic Variables?

George Athanasopoulos Monash University Clayton, Victoria 3800 Australia and Farshid Vahid Australian National University Canberra, ACT 2601 Australia Corresponding author e-mail: George.Athanasopoulos@BusEco.monash.edu.au

SLIDE 2

In this paper:

Present and complete the “Scalar

Component Methodology” of Tiao and Tsay (1989) for “developing” VARMA models

Study the properties of this

methodology through simulations

Extensive application to Macro-

Economic Data

Overall we find evidence of

superiority of VARMA models for forecasting

SLIDE 3

VAR v VARMA

VAR(p)

– Dominate the field of macro- econometric modelling – Good forecasting record

MOTIVATION for VARMA(p,q)

– More general – Parsimonious representations

any invertible VARMA can be

represented by an infinite order VAR

effects on forecasting

– Aggregation:

induces MA dynamics

– See Lütkepohl (1987) where a linearly transformed:

VARMA process has a finite

VARMA(p,q) representation

However not necessarily the case with

VAR

SLIDE 4

VARMA difficulties

– Chatfield

“If univariate ARIMA modelling is difficult then VARMA modelling is even more difficult - some might say impossible”

General VARMA(p,q) model
Identification problem

– Echelon form

Lütkepohl and Poskitt (1996)

– Scalar Components

Simplifying Underlying Structures

yt  1yt1 ....pytp  t  1t1 ....qtq yt  y1t,.....,ykt where t  N0, and  is positive definite

SLIDE 5

Identification Problem

Consider a k = 2 yt ~ VARMA(1,1)
21 and θ21 are not separately identified
“Rule of Elimination”

Scalar Component Methodology

(Tiao and Tsay, JRSS, 1989)

Definition:

yt  yt1  t  t1 11  12  11  12  0 y1,t  1,t y2,t  21y1,t1  22y2,t1  211,t1  222,t1  2,t

zt  yt  SCMp1,q1

if  satisfies

yt  VARMAp,q

p 1  0T where 0  p1  p, l  0T for l  p1  1,...,p, q 1  0T where 0  q1  q, q 1  0T for l  q1  1,...,q.

SLIDE 6

Find k-linearly independent vectors

which transform into

A series of C/C tests:

Let be the squared C/C then the LR test statistic: tests for s SCMs of order (m,j) versus the alternative of less than s SCMs of this order. di is a correction factor accounting for the cases that the canonical covariates can be MA(j)

A  1,,k

yt  1yt1 pytp  t  1t1 qtq zt  1

zt1 p ztp  ut  1 ut1 q utq

where zt  Ayt, i

  AiA1,ut  At and i   AiA1

 1   2   k

between Ym,t  yt

,ytm 

 and Yh,tj1  ytj1



,ytj1h

 

Cs  n  h  ji1

s

ln 1 

 i di a

~ shm ks

2

SLIDE 7

EXAMPLE: yt = (y1,t, y2,t ,…, yk,t)T
sequence of C/C tests
Underlying WN process zi,t ~ SCM(0,0)

– Is there a linear combination of yt which has zero correlation with yt-1,…?

MA(1) process zi,t ~ SCM(0,1)

– Is there a linear combination of yt which has zero correlation with yt-2,… given that it has

ne period serial correlation?
AR(1) process zi,t ~ SCM(1,0)

– Is there a linear combination of yt and yt-1 which has zero correlation with yt-1,…?

ARMA(1,1) process zi,t ~ SCM(1,1)

– Is there a linear combination of yt and yt-1 which has zero correlation with yt-2,…? – and so on …

Up to i = 1 ,…, k such combinations
“Criterion” and “Root” tables

SLIDE 8

Identified Model (for k = 3)

r

Concerns:

(Hannan, Reinsel, Chatfield, Tunnicliffe-Wilson, Ord etc.)

Transformed series zt
# of parameters in A
Real reduction in parameters is less than 10
C/C estimates not most efficient
No standard errors in A

Ayt  1yt1  t  1t1

zt  11

1

12

1

13

1

21

1

22

1

23

1

zt1  ut  11

1

12

1

13

1

ut1

z1,t  13

1

z3,t1  13

1

u3,t1  j1

2 1j 1

zj,t1  u1,t  j1

2 1j 1

uj,t1 zt  11

1 12 1 13 1

21

1 22 1 23 1

zt1  ut  11

1 12 1 0

ut1

z3,t  u3,t  z3,t1  u3,t1

SLIDE 9

Extension to Tiao and Tsay

Keep the model in terms of yt
Normalise
Set of rules for determining the free parameters

in A

– 3rd equation uniquely identified – SCM(0,0) is nested in SCM(1,0) and

SCM(1,1)

– SCM(1,0) is nested in SCM(1,1)

Number of parameters reduced by: 10-3 = 7

a11 a12 a13 a21 a22 a23 a31 a32 a33 yt  11

1 12 1 13 1

21

1 22 1 23 1

yt1  t  11

1 12 1 0

t1 1 a12 a13 a21 1 a23 a31 a32 1 yt  11

1 12 1 13 1

21

1 22 1 23 1

yt1  t  11

1 12 1 0

t1 1 a12 0 a21 1 a31 a32 1 yt  11

1 12 1 13 1

21

1 22 1 23 1

yt1  t  11

1 12 1 0

t1 1 a21 1 a31 a32 1 yt  11

1 12 1 13 1

21

1 22 1 23 1

yt1  t  11

1 12 1 0

t1

SLIDE 10

Checking for Correct Normalisations

Safeguard against normalising a zero

parameter to 1.

Apply the C/C test to subsets of variables

Example (continued) Check for individual y1,t , y2,t or y3,t ~ SCM(0,0)

If yes set that coefficient to one
If not check for combinations of two and so
n…
The number of parameters to be estimated can

potentially be further reduced

Summary of Extension

Keep model in terms of yt
Provide set of rules for determining the free

parameters parameters in A

Safeguard against normalising a zero

parameter to 1

Estimate the model by FIML

1 a21 1 a31 a32 1 yt  11

1 12 1 13 1

21

1 22 1 23 1

yt1  t  11

1 12 1 0

t1

SLIDE 11

Example: US flour price data

Tiao and Tsay (1989), Grubb (1992), Lutkepohl and Poskitt (1996)

Stage I: Overall Tentative Order

VAR(2) or VARMA(1,1)

4.3 4.5 4.7 4.9 5.1 5.3 5.5 y1t Buffalo y2t Minneapolis y3t Kansas

Criterion Table j m 1 2 3 4 34.17 5.8 3.0 2.11 1.68 1 2.38 0.44 0.49 0.22 0.34 2 0.25 0.58 0.60 0.49 0.46 3 0.37 0.46 0.67 0.53 0.58 4 0.73 0.62 0.57 0.70 0.77 The statistics are normalised by the corresponding 5% 2 critical values

SLIDE 12

Stage II: Individual SCMs

2 ~ SCM(1,0) and 1 ~ SCM(1,1)

Normalisations check finds y3,t ~ SCM(1,0)

Root Table j m 1 2 3 4 1 1 1 1 2 3 3 3 3 2 3 5 6 6 6 3 3 6 8 9 9 4 3 6 9 11 12

1 a12 a13 a21 1 a23 a31 a32 1 yt  11

1 12 1 13 1

21

1 22 1 23 1

31

1 32 1 33 1

yt1  t  11

1 12 1 13 1

t1 1 1 1 yt  11

1 12 1 13 1

21

1 22 1 23 1

31

1 32 1 33 1

yt1  t  11

1 12 1 13 1

t1

1 0 0 a21 1 0 0 1 yt  11

1 12 1 13 1

21

1 22 1 23 1

31

1 32 1 33 1

yt1  t  11

1 12 1 13 1

t1

SLIDE 13

Application to Macro Data
Data

– Stock and Watson (1999) & (2001) – 40 monthly series 1959:1 – 1998:12 (N=480) – 8 major categories:

Output and Real Income
Employment and Unemployment
Consumption, Retail Sales and Housing
Real Inventories and Sales
Prices and Wages
Money and Credit
Interest Rates
Exchange Rates

– 70 3 variable systems

VARMA
VAR selected by AIC and SC
Restricted and Unrestricted
Forecasting and Forecast Evaluation

– Test sample: N1 = 300 – Hold-out: N2 = 180 – h = 1 to 15 step ahead forecasts – PB of |FMSE| and tr(FMSE) – Ratioh 

1 M i1 M |FMSEVARi| |FMSEVARMAi|

SLIDE 14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 (%) Forecast horizon (h) Var(AIC) Var(BIC) Varma

PB counts for |FMSE| for the Unrestricted Models

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0

(%) Forecast horizon (h) Var(BIC) Var(AIC) Varma

PB counts for |FMSE| for the Restricted Models

SLIDE 15

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 (%) Forecast horizon (h) Var(AIC) Var(BIC) Varma

PB counts for trFMSE for the Unrestricted Models

1 3 5 7 9 11 13 15 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0

(%) forecast horizon (h) Var(BIC) Var(AIC) Varma

PB counts for trFMSE for the Restricted Models

SLIDE 16

Summary
Completed the methodology
Addressed issues raised in the

discussion of Tiao and Tsay (1989) and

Simulations
Small number of repetitions
Good insight
Empirical Application
VARMA in many cases outperform

their VAR counterparts

Future direction
Consider larger dimensions
Look at Echelon form
Study the implications for