Spline Garch as a Measure of
Unconditional Volatility and its
Global Macroeconomic Causes
Robert Engle and Jose Gonzalo Rangel
NYU and UCSD
HISTORY OF THE US EQUITY
MARKET VOLATILITY: S&P500
PLOT PRICES AND RETURNS
HOW MUCH DO RETURNS FLUCTUATE?
.1
.0
1600
-.1
800
-.2
400
-.3
200
100
50
65
70
75
80
SP500
85
90
95
SPRETURNS
00
.06
.04
.02
.00
-.02
400
-.04
-.06
200
100
50
1965
1970
SP500
1975
1980
SPRETURNS
1985
.08
.04
.00
-.04
1600
-.08
800
400
200
1988
1990
1992
SP500
1994
1996
1998
SPRETURNS
2000
.08
.04
.00
1600
-.04
1400
-.08
1200
1000
800
1998
1999
2000
SP500
2001
2002
SPRETURNS
2003
MEAN REVERSION QUOTES

“Volatility is Mean Reverting”
– no controversy

“The long run level of volatility is constant”
– very controversial

“Volatility is systematically higher now than it
has been in years”
– Very controversial. Cannot be answered by
simple GARCH
DEFINITIONS
 rt
is a mean zero random variable
measuring the return on a financial
asset
 CONDITIONAL VARIANCE
ht  Et 1  rt
2

 UNCONDITIONAL VARIANCE
2
2
t
t
  E r

GARCH(1,1)
ht     r   ht 1
2
t 1
 The
unconditional variance is then
2
2
2
E
r

E
h









 t


  t 

 
1     
2
GARCH(1,1)
h     r   ht 1
2
t
t
t 1
 If omega is slowly varying, then
E  rt
2
  E h   
t
t
2
t
   t  j    
2
t

 t      
j
j 0
This is a complicated expression to interpret
2
t 1
SPLINE GARCH
 Instead,
use a multiplicative form
rt   t gt  t , where  t |  t 1
N (0,1)
 rt21 
gt  (1     )   
   gt 1
  t 1 
 Tau
is a function of time and exogenous
variables
UNCONDITIONAL
VOLATILTIY
 Taking
unconditional expectations
2

E  rt     t E ( gt )   t
 Thus
we can interpret tau as the
unconditional variance.
SPLINE
 ASSUME
UNCONDITIONAL
VARIANCE IS AN EXPONENTIAL
QUADRATIC SPLINE OF TIME
k


2
 t  c exp  w0t   wi  (t  ti 1 )    zt 
i 1


THIS IS EASY TO COMPUTE
 For
K knots equally spaced, construct
new regressors
log 
2
t

0
K
 1t  2t   k  max  t  tk ,0  
2
k 1
2
ESTIMATION
 FOR A GIVEN
K, USE GAUSSIAN MLE
2

1
rt 
L     log  t gt  

2 t 1 
 t gt 
T
 CHOOSE
K TO MINIMIZE BIC FOR K
LESS THAN OR EQUAL TO 15
EXAMPLES FOR US SP500
 DAILY DATA FROM
1963 THROUGH
2004
 ESTIMATE WITH 1 TO 15 KNOTS
 OPTIMAL NUMBER IS 7
RESULTS
LogL: SPGARCH
Method: Maximum Likelihood (Marquardt)
Date: 08/04/04 Time: 16:32
Sample: 1 12455
Included observations: 12455
Evaluation order: By observation
Convergence achieved after 19 iterations
Coefficient
Std. Errorz-Statistic Prob.
C(4)
-0.000319 7.52E-05 -4.246643 0.0000
W(1)
-1.89E-08 2.59E-08 -0.729423 0.4657
W(2)
2.71E-07 2.88E-08 9.428562 0.0000
W(3)
-4.35E-07 3.87E-08 -11.24718 0.0000
W(4)
3.28E-07 5.42E-08 6.058221 0.0000
W(5)
-3.98E-07 5.40E-08 -7.377487 0.0000
W(6)
6.00E-07 5.85E-08 10.26339 0.0000
W(7)
-8.04E-07 9.93E-08 -8.092208 0.0000
C(5)
1.137277 0.043563 26.10666 0.0000
C(1)
0.089487 0.002418 37.00816 0.0000
C(2)
0.881005 0.004612 191.0245 0.0000
Log likelihood
-15733.51 Akaike info criterion
Avg. log likelihood -1.263228 Schwarz criterion
Number of Coefs. 11
Hannan-Quinn criter.
2.528223
2.534785
2.530420
1.2
1.0
0.8
0.6
0.4
0.2
0.0
60
65
70
75
UVOL
80
85
90
CVOL
95
00
Italy, 1
.8
.7
.6
.5
.4
.3
.2
.1
.0
60
65
70
75
80
CVOL
85
90
UVOL
95
00
India, 5
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
60
65
70
75
80
CVOL
85
90
UVOL
95
00
Japan, 4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
60
65
70
75
80
CVOL
85
90
UVOL
95
00
Brazil, 6
3.0
2.5
2.0
1.5
1.0
0.5
0.0
60
65
70
75
80
CVOL
85
90
UVOL
95
00
South Africa, 3
.9
.8
.7
.6
.5
.4
.3
.2
.1
.0
60
65
70
75
80
CVOL
85
90
UVOL
95
00
Poland, 1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
60
65
70
75
80
CVOL
85
90
UVOL
95
00
PATTERNS OF
VOLATILITY
 ASSET CLASSES
–
–
–
–
–
–
EQUITIES
EQUITY INDICES
CURRENCIES
FUTURES
INTEREST RATES
BONDS
VOLATILITY BY ASSET CLASS
70.00%
IBM
General Electric
Citigroup
McDonalds
Wal Mart Stores
60.00%
S&P500
80.00%
50.00%
Penn Virginia Corp
Norfolk Southern Corp
Airgas Inc
G T S Duratek Inc
Metrologic Instruments Inc
40.00%
30.00%
3 month
5 year
20 year
20.00%
10.00%
0.00%
Volatility
$/AUS
$/CAN
$/YEN
$/L
Annualized Historical Volatilities November 2004; CBOE
500
Series: VOLS
Sample 1 2000
Observations 1653
400
300
200
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
33.07719
28.22500
284.2990
1.060000
21.24304
3.222794
26.42289
Jarque-Bera
Probability
40648.46
0.000000
100
0
0
40
80
120
160
200
240
280
PATTERNS OF EQUITY VOLATILITY

COUNTRIES
–
–
–
–
–
–

DEVELOPED MARKETS
EUROPE
TRANSITION ECONOMIES
LATIN AMERICA
ASIA
EMERGING MARKETS
Calculate Median Annualized Unconditional
Volatility 1997-2003 using daily data
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Median Annual Unconditional Volatility
UK
CHILE
NEWZEALAND
AUSTRALIA
LITHUANIA
CANADA
AUSTRIA
PORTUGAL
BELGIUM
ITALY
DENMARK
SWISS
IRELAND
COL
NORWAY
SOUTHAFRICA
ISRAEL
USSP
NETHERLANDS
FRANCE
MALAYSIA
CZECHREP
SPAIN
SWEDEN
GERMANY
GREECE
INDIA
MEXICO
INDONESIA
PHILIPPINES
SLOVAKREP
HUNGARY
CROATIA
JAPAN
TAIWAN
SINGAPORE
POLAND
VENEZUELA
THAILAND
FINLAND
ECUADOR
RUSSIA
KOREA
ARG
BRAZ
HONGKONG
TURKEY
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Median Annual Unconditional Emerging
Market Volatility
UK
CHILE
NEWZEALAND
AUSTRALIA
LITHUANIA
CANADA
AUSTRIA
PORTUGAL
BELGIUM
ITALY
DENMARK
SWISS
IRELAND
COL
NORWAY
SOUTHAFRICA
ISRAEL
USSP
NETHERLANDS
FRANCE
MALAYSIA
CZECHREP
SPAIN
SWEDEN
GERMANY
GREECE
INDIA
MEXICO
INDONESIA
PHILIPPINES
SLOVAKREP
HUNGARY
CROATIA
JAPAN
TAIWAN
SINGAPORE
POLAND
VENEZUELA
THAILAND
FINLAND
ECUADOR
RUSSIA
KOREA
ARG
BRAZ
HONGKONG
TURKEY
MACRO VOLATILITY
 Macro
volatility variables measure the
size of the surprises in macroeconomic
aggregates over the year.
 If y is the variable (cpi, gdp,…), then:
 log  yt   c  ut , ut   ut 1  et
 y2,t
1

4
t 1

j t  2
ej
0.06
0.05
0.04
0.03
0.02
0.01
0
Median Annual Volatility of GDP
UK
CHILE
NEWZEALAND
AUSTRALIA
LITHUANIA
CANADA
AUSTRIA
PORTUGAL
BELGIUM
ITALY
DENMARK
SWISS
IRELAND
COL
NORWAY
SOUTHAFRICA
ISRAEL
USSP
NETHERLANDS
FRANCE
MALAYSIA
CZECHREP
SPAIN
SWEDEN
GERMANY
GREECE
INDIA
MEXICO
INDONESIA
PHILIPPINES
SLOVAKREP
HUNGARY
CROATIA
JAPAN
TAIWAN
SINGAPORE
POLAND
VENEZUELA
THAILAND
FINLAND
ECUADOR
RUSSIA
KOREA
ARG
BRAZ
HONGKONG
TURKEY
0.25
0.2
0.15
0.1
0.05
0
Median Annual Volatility of CPI
UK
CHILE
NEWZEALAND
AUSTRALIA
LITHUANIA
CANADA
AUSTRIA
PORTUGAL
BELGIUM
ITALY
DENMARK
SWISS
IRELAND
COL
NORWAY
SOUTHAFRICA
ISRAEL
USSP
NETHERLANDS
FRANCE
MALAYSIA
CZECHREP
SPAIN
SWEDEN
GERMANY
GREECE
INDIA
MEXICO
INDONESIA
PHILIPPINES
SLOVAKREP
HUNGARY
CROATIA
JAPAN
TAIWAN
SINGAPORE
POLAND
VENEZUELA
THAILAND
FINLAND
ECUADOR
RUSSIA
KOREA
ARG
BRAZ
HONGKONG
TURKEY
0.25
MACRO VOL
0.2
0.15
GDP VOL
CPI VOL
0.1
0.05
0
0
0.2
0.4
VOLATILITY
0.6
EXPLANATORY VARIABLES
Table (2)
Explanatory Variables
Name
Description
emerging Indicator of Market Development (1=Emerging, 0=Developed)
Transition Indicator of Transition Economies (Central European and Baltic Countries)
log(mc)
log Market Capitalization ($US)
log(gdp_dll) Log Nominal GDP in Current $US
nlc
Number of Listed Companies in the Exchange
grgdp
GDP Growth Rate
gcpi
Inflation Growth Rate
vol_irate Volatility of Short Term Interest Rate*
vol_forex Volatility of Exchange Rates*
vol_grgdp Volatility of GDP*
vol_gcpi Volatility of Inflation*
*Volatilities are obtained from the residuals of AR(1) models
ESTIMATION
Volatility is regressed against explanatory
variables with observations for countries and
years.
 Within a country residuals are auto-correlated
due to spline smoothing. Hence use SUR.
 Volatility responds to global news so there is
a time dummy for each year.
 Unbalanced panel

ONE VARIABLE
REGRESSIONS
Table (5)
Individual SUR Regressions
emerging
Transition
log(mc)
log(gdp_dll)
log(mc/gdp_dll)
nlc
grgdp
gcpi
vol_irate
vol_forex
vol_grgdp
vol_gcpi
Coefficient
0.0853
-0.0146
-0.0092
-0.0034
-0.0274
0.0000
-0.7150
0.5631
0.0085
0.5644
1.0974
0.9115
Std. Error
0.0187
0.0184
0.0032
0.0052
0.0050
0.0000
0.1350
0.0446
0.0006
0.0434
0.1097
0.0895
t-Statistic
4.5588
-0.7927
-2.8495
-0.6626
-5.5075
-2.4753
-5.2965
12.6113
14.1663
13.0083
10.0080
10.1836
Prob.
0.0000
0.4282
0.0045
0.5078
0.0000
0.0136
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
Det residual
covariance
2.74E-38
5.52E-38
1.37E-37
9.68E-37
1.65E-36
4.76E-37
1.46E-37
8.13E-38
4.80E-38
7.24E-38
4.04E-38
1.03E-37
MULTIPLE REGRESSIONS
Table (6)
emerging
transition
log(mc)
log(gdpus)
nlc
grgdp
gcpi
vol_irate
vol_gforex
vol_grgdp
vol_gcpi
Estimation Results: SUR Models for Lon
M1
M2
M3
M4
0.0307
0.0312
0.0351
0.0350
(0.0147) **
(0.0146) **
(0.0138) **
(0.0136)
-0.0187
-0.0187
-0.0195
-0.0184
(0.0184)
(0.0184)
(0.0181)
(0.0178)
-0.0036
-0.0037
(0.0062)
(0.0062)
0.0198
0.0201
0.0167
0.0170
(0.0077) **
(0.0076) **
(0.0051) **
(0.0051)
-1.81E-05
-1.82E-05
-1.75E-05
-1.78E-05
(0.000006) ** (0.000006) ** (0.000005) ** (0.000005)
-0.1779
-0.1625
-0.1444
(0.1999)
(0.1954)
(0.1839)
0.3992
0.3693
0.3470
0.3523
(0.1975) **
(0.1821) **
(0.1725) **
(0.1643)
0.0022
0.0022
0.0025
0.0025
(0.0008) **
(0.0008) **
(0.0008) **
(0.0008)
-0.0332
(0.0882)
0.9003
0.9054
0.9120
0.9119
(0.1543) **
(0.1536) **
(0.1492) **
(0.1457)
1.0485
1.0260
0.9406
1.0306
(0.3512) **
(0.3460) **
(0.3321) **
(0.3279)
Time Effects
0.2
0.15
0.1
0.05
0
1990 1994 1998 2002
ANNUAL REALIZED
VOLATILITY
CONCLUSIONS AND
IMPLICATIONS
Unconditional volatility changes in systematic
ways.
 Macro volatility is an important determinant of
financial volatility
 Potential justification for inflation targeting
monetary policy as well as stabilization.
 Big swings in global financial volatility are
associated with US volatility.
