Variables? George Athanasopoulos Monash University Clayton, - PDF document

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

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

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

• 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 y t   y 1 t ,....., y kt   y t   1 y t  1  ....   p y t  p   t   1  t  1  ....   q  t  q where  t  N  0,    and   is positive definite • Identification problem – Echelon form • Lütkepohl and Poskitt (1996) – Scalar Components • Simplifying Underlying Structures

Identification Problem • Consider a k = 2 y t ~ VARMA(1,1) y t   y t  1   t   t  1  11   12   11   12  0 • y 1, t   1, t y 2, t   21 y 1, t  1   22 y 2, t  1   21  1, t  1   22  2, t  1   2, t •  21 and θ 21 are not separately identified • “Rule of Elimination” Scalar Component Methodology (Tiao and Tsay, JRSS, 1989) • Definition: y t  VARMA  p , q  z t    y t  SCM  p 1 , q 1     p 1  0 T where 0  p 1  p , if  satisfies    l  0 T for l  p 1  1,..., p ,    q 1  0 T where 0  q 1  q ,    q 1  0 T for l  q 1  1,..., q .

• Find k-linearly independent vectors A    1 ,  ,  k   which transform y t   1 y t  1   p y t  p   t   1  t  1     q  t  q into  z t  1   p  u t  1     q  z t  p  u t   1  u t  q z t   1   A  i A  1 , u t  A  t and  i   A  i A  1 where z t  Ay t ,  i • A series of C/C tests:  1     2    k Let be the squared C/C   and Y h , t  j  1    ,  y t  m    between Y m , t   y t ,  y t  j  1  h y t  j  1 then the LR test statistic:   i C  s     n  h  j   i  1 a s ln 1  ~  s   h  m  k  s  2 d i tests for s SCM s of order (m,j) versus the alternative of less than s SCM s of this order. d i is a correction factor accounting for the cases that the canonical covariates can be MA(j)

• EXAMPLE: y t = (y 1,t , y 2,t ,…, y k,t ) T • sequence of C/C tests • Underlying WN process z i,t ~ SCM(0,0) – Is there a linear combination of y t which has zero correlation with y t-1 ,…? • MA(1) process z i,t ~ SCM(0,1) – Is there a linear combination of y t which has zero correlation with y t-2 ,… given that it has one period serial correlation? • AR(1) process z i,t ~ SCM(1,0) – Is there a linear combination of y t and y t-1 which has zero correlation with y t-1 ,…? • ARMA(1,1) process z i,t ~ SCM(1,1) – Is there a linear combination of y t and y t-1 which has zero correlation with y t-2 ,…? – and so on … • Up to i = 1 ,…, k such combinations • “Criterion” and “Root” tables

Identified Model (for k = 3)  1    1    1    1    1    1    11  12  13  11  12  13  1    1    1   z t  z t  1  u t   21  22  23 u t  1 0 0 0 0 0 0 0 0 0 z 3, t  u 3, t  z 3, t  1  u 3, t  1 2  1 j 2  1 j  1    1    1    1   u 3, t  1   j  1 z j , t  1  u 1, t   j  1 z 1, t   13 z 3, t  1   13 u j , t  1  1    12  1    13  1    12  1   0  1    11  11  1    22  1    23  1   z t  z t  1  u t   21 u t  1 0 0 0 0 0 0 0 0 0 or Ay t   1 y t  1   t   1  t  1 Concerns: (Hannan, Reinsel, Chatfield, Tunnicliffe-Wilson, Ord etc.)  Transformed series z t  # of parameters in A  Real reduction in parameters is less than 10  C/C estimates not most efficient  No standard errors in A

Extension to Tiao and Tsay • Keep the model in terms of y t  1   12  1   13  1   12  1  0  1   11  11 a 11 a 12 a 13  1   22  1   23 y t   1  y t  1   t   t  1  21 a 21 a 22 a 23 0 0 0 a 31 a 32 a 33 0 0 0 0 0 0 • Normalise  1   12  1   13  1   12  1  0  1   11  11 1 a 12 a 13  1   22  1   23 y t   1  y t  1   t   t  1  21 1 a 21 a 23 0 0 0 1 a 31 a 32 0 0 0 0 0 0 • Set of rules for determining the free parameters in A – 3 rd equation uniquely identified – SCM(0,0) is nested in SCM(1,0) and SCM(1,1)  1   12  1   13  1   12  1  0  1   11  11 1 a 12 0  1   22  1   23 y t   1  y t  1   t   t  1 1 0  21 a 21 0 0 0 a 31 a 32 1 0 0 0 0 0 0 – SCM(1,0) is nested in SCM(1,1)  1   12  1   13  1   12  1  0  1   11  11 1 0 0  1   22  1   23 y t   1  y t  1   t   t  1  21 1 0 a 21 0 0 0 a 31 a 32 1 0 0 0 0 0 0 • Number of parameters reduced by: 10-3 = 7

Checking for Correct Normalisations • Safeguard against normalising a zero parameter to 1. • Apply the C/C test to subsets of variables Example (continued)  1   12  1   13  1   12  1  0  1   11  11 1 0 0  1   22  1   23 y t   1  y t  1   t   t  1 1 0  21 a 21 0 0 0 a 31 a 32 1 0 0 0 0 0 0 Check for individual y 1,t , y 2,t or y 3,t ~ SCM(0,0)  If yes set that coefficient to one  If not check for combinations of two and so on… • The number of parameters to be estimated can potentially be further reduced Summary of Extension • Keep model in terms of y t • 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

• Example: US flour price data Tiao and Tsay (1989), Grubb (1992), Lutkepohl and Poskitt (1996) 5.5 5.3 5.1 4.9 4.7 4.5 4.3 y1t Buffalo y2t Minneapolis y3t Kansas • Stage I : Overall Tentative Order Criterion Table j 0 1 2 3 4 m 0 34.17 5.8 3.0 2.11 1.68 1 2.38 0.44 0.49 0.22 0.34 2 0.58 0.60 0.49 0.46 0.25 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 VAR(2) or VARMA(1,1)

• Stage II : Individual SCMs Root Table j 0 1 2 3 4 m 0 0 0 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 2 ~ SCM(1,0) and 1 ~ SCM(1,1)  1   12  1   13  1   12  1   13  1   1   11  11 1 a 12 a 13  1   22  1   23  1  y t  y t  1   t   t  1  21 1 a 21 a 23 0 0 0  1   32  1   33  1  1  31 a 31 a 32 0 0 0  1   12  1   13  1   12  1   13  1   1   11  11 1  1   22  1   23  1  y t  y t  1   t   t  1  21 1 0 0 0  1   32  1   33  1  1  31 0 0 0 • Normalisations check finds y 3,t ~ SCM(1,0)  1   12  1   13  1   12  1   13  1   1   11  11 1 0 0  1   22  1   23  1  y t  y t  1   t   t  1  21 a 21 1 0 0 0 0  1   32  1   33  1  0 0 1  31 0 0 0

• 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: N 1 = 300 – Hold-out: N 2 = 180 – h = 1 to 15 step ahead forecasts – PB of |FMSE| and tr(FMSE) | FMSE  VAR  i | M  i  1 – Ratio h  M 1 | FMSE  VARMA  i |