spline garch as a measure of unconditional volatility and its global macroeconomic causes robert...
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Spline Garch as a Measure of Unconditional Volatility and its Global Macroeconomic Causes
Robert Engle and Jose Gonzalo RangelNYU and UCSD
GOALS
ESTIMATE THE DETERMINANTS OF GLOBAL EQUITY VOLATILITY– How are long run volatility forecasts
affected by macroeconomic conditions?– What volatility can be expected for a newly
opened financial market?MEASURE AND MODEL CHANGING
UNCONDITIONAL VOLATILITY
WHAT MOVES ASSET PRICES AND VOLATILITY?
NEWS vs OTHER THINGSRESEARCH STRATEGIES
– VOLATILITY MODELSe.g.Officer(1973), Schwert(1989)
– ANNOUNCEMENT + NEWS MODELSe.g.Roll(1988), Cutler Poterba and
Summers(1990)
In all cases, macro effects appear small
A MODEL
CAMPBELL(1991), CAMPBELL& SHILLER(1988) LOG LINEARIZATION
Decompose into Innovations to the present discounted value of future dividends or expected returns
1d r
t t t t tr E r
MULTIPLICATIVE EFFECTS
The impact of a news event may depend upon the macro economy.
Eg. News about a firm will have a bigger effect in a recession or close to bankruptcy
NEWS EVENTS
Return is a function of news times its impact
– e = observable news– z = macro or deterministic events
if news is not observable, then there is just an innovation, u
1 ,1
K
t t t i i i ti
r E r z e
1 1t t t t tr E r z u
NEWS VARIANCE
The variance of the news also depends upon macro and other deterministic elements both through the intensity and the magnitude of the news.
2t tV u z
1 1 2
1
,
1
t t t t t t t
t t t
r E r z z g
E g V
REALIZED VARIANCE
Realized Variance is the unconditional variance plus an error. Assuming mean zero returns:
2 2
1 1
1
T T
t t j t j t j t jj j
T
t j tj
RV r z g
z U
HISTORY OF THE US EQUITY MARKET VOLATILITY: S&P500
PLOT PRICES AND RETURNS
HOW MUCH DO RETURNS FLUCTUATE?
50
100
200
400
800
1600
-.3
-.2
-.1
.0
.1
65 70 75 80 85 90 95 00
SP500 SPRETURNS
50
100
200
400
-.06
-.04
-.02
.00
.02
.04
.06
1965 1970 1975 1980 1985
SP500 SPRETURNS
200
400
800
1600-.08
-.04
.00
.04
.08
1988 1990 1992 1994 1996 1998 2000
SP500 SPRETURNS
800
1000
1200
1400
1600
-.08
-.04
.00
.04
.08
1998 1999 2000 2001 2002 2003
SP500 SPRETURNS
MEAN REVERSION QUOTES
“Volatility is Mean Reverting”– no controversy
“The long run level of volatility is constant”– very controversial
“Volatility is systematically lower 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
UNCONDITIONAL VARIANCE
21t t th E r
2 2t tE r
GARCH(1,1)
The unconditional variance is then
21 1t t th r h
2 2 2
2
1
t tE r E h
GARCH(1,1)
If omega is slowly varying, then
This is a complicated expression to interpret
21 1t t tth r h
2 2 21
2
0
t t t t t
tj
t t jj
E r E h
SPLINE GARCH
Instead, use a multiplicative form
Tau is a function of time and exogenous variables
1, where | (0,1)t t t t t tr g N
21
11
(1 ) tt t
t
rg g
UNCONDITIONAL VOLATILTIY
Taking unconditional expectations
Thus we can interpret tau as the unconditional variance.
2 ( )t t t tE r E g
SPLINE
ASSUME UNCONDITIONAL VARIANCE IS AN EXPONENTIAL QUADRATIC SPLINE OF TIME
For K knots equally spaced
22 20 1 2
1
log max ,0K
t k kk
t t t t
ESTIMATION
FOR A GIVEN K, USE GAUSSIAN MLE
CHOOSE K TO MINIMIZE BIC FOR K LESS THAN OR EQUAL TO 15
2
1
1log
2
Tt
t tt t t
rL g
g
EXAMPLES FOR US SP500
DAILY DATA FROM 1963 THROUGH 2004
ESTIMATE WITH 1 TO 15 KNOTSOPTIMAL NUMBER IS 7
RESULTSLogL: SPGARCHMethod: Maximum Likelihood (Marquardt)
Date: 08/04/04 Time: 16:32Sample: 1 12455Included observations: 12455Evaluation order: By observationConvergence achieved after 19 iterations
Coefficient Std. Errorz-Statistic Prob. C(4) -0.000319 7.52E-05 -4.246643 0.0000W(1) -1.89E-08 2.59E-08 -0.729423 0.4657W(2) 2.71E-07 2.88E-08 9.428562 0.0000W(3) -4.35E-07 3.87E-08 -11.24718 0.0000W(4) 3.28E-07 5.42E-08 6.058221 0.0000W(5) -3.98E-07 5.40E-08 -7.377487 0.0000W(6) 6.00E-07 5.85E-08 10.26339 0.0000W(7) -8.04E-07 9.93E-08 -8.092208 0.0000C(5) 1.137277 0.043563 26.10666 0.0000C(1) 0.089487 0.002418 37.00816 0.0000C(2) 0.881005 0.004612 191.0245 0.0000Log likelihood -15733.51 Akaike info criterion 2.528223Avg. log likelihood -1.263228 Schwarz criterion 2.534785Number of Coefs. 11 Hannan-Quinn criter. 2.530420
0.0
0.2
0.4
0.6
0.8
1.0
1.2
60 65 70 75 80 85 90 95 00
CVOL UVOL
S&P500
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
90 92 94 96 98 00 02
CVOL UVOL ANNUAL RV
India,5
0
1
2
3
4
90 92 94 96 98 00 02
CVOL UVOL ANNUAL RV
Argentina, 3
0.0
0.2
0.4
0.6
0.8
1.0
90 92 94 96 98 00 02
CVOL UVOL ANNUAL RV
Japan,4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
90 92 94 96 98 00 02
CVOL UVOL ANNUAL RV
Brazil,6
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
90 92 94 96 98 00 02
CVOL UVOL ANNUAL RV
South Africa,3
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
90 92 94 96 98 00 02
CVOL UVOL ANNUAL RV
Poland,1
PATTERNS OF VOLATILITY
ASSET CLASSES– EQUITIES– EQUITY INDICES– CURRENCIES– FUTURES– INTEREST RATES– BONDS
VOLATILITY BY ASSET CLASS
0.00%
10.00%
20.00%
30.00%
40.00%
50.00%
60.00%
70.00%
80.00%
Volatility
IBMGeneral ElectricCitigroupMcDonaldsWal Mart Stores
S&P500
Penn Virginia CorpNorfolk Southern CorpAirgas IncG T S Duratek IncMetrologic Instruments Inc
3 month5 year20 year
$/AUS$/CAN$/YEN$/L
0
100
200
300
400
500
0 40 80 120 160 200 240 280
Series: VOLSSample 1 2000Observations 1653
Mean 33.07719Median 28.22500Maximum 284.2990Minimum 1.060000Std. Dev. 21.24304Skewness 3.222794Kurtosis 26.42289
Jarque-Bera 40648.46Probability 0.000000
Annualized Historical Volatilities November 2004; CBOE
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
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Median Annual Unconditional Volatility
UKCHILEAUSTRALIANEWZEALANDLITHUANIACANADAAUSTRIAPORTUGALBELGIUMITALYDENMARKSWISSIRELANDPERUCOLNORWAYSOUTHAFRICAISRAELUSSPNETHERLANDSFRANCEMALAYSIASWEDENCZECHREPSPAINCHINAGERMANYGREECEINDIAMEXICOINDONESIAPHILIPPINESHONGKONGSLOVAKREPRUSSIAHUNGARYCROATIAJAPANTAIWANSINGAPOREPOLANDVENEZUELATHAILANDFINLANDECUADORKOREAARGBRAZTURKEY
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Median Annual Unconditional EmergingMarket Volatility
UKCHILEAUSTRALIANEWZEALANDLITHUANIACANADAAUSTRIAPORTUGALBELGIUMITALYDENMARKSWISSIRELANDPERUCOLNORWAYSOUTHAFRICAISRAELUSSPNETHERLANDSFRANCEMALAYSIASWEDENCZECHREPSPAINCHINAGERMANYGREECEINDIAMEXICOINDONESIAPHILIPPINESHONGKONGSLOVAKREPRUSSIAHUNGARYCROATIAJAPANTAIWANSINGAPOREPOLANDVENEZUELATHAILANDFINLANDECUADORKOREAARGBRAZTURKEY
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:
1
12
,2
log ,
1
4
t t t t t
t
y t jj t
y c u u u e
e
0
0.01
0.02
0.03
0.04
0.05
0.06
Median Annual Volatility of GDP
UKCHILEAUSTRALIANEWZEALANDLITHUANIACANADAAUSTRIAPORTUGALBELGIUMITALYDENMARKSWISSIRELANDPERUCOLNORWAYSOUTHAFRICAISRAELUSSPNETHERLANDSFRANCEMALAYSIASWEDENCZECHREPSPAINCHINAGERMANYGREECEINDIAMEXICOINDONESIAPHILIPPINESHONGKONGSLOVAKREPRUSSIAHUNGARYCROATIAJAPANTAIWANSINGAPOREPOLANDVENEZUELATHAILANDFINLANDECUADORKOREAARGBRAZTURKEY
0
0.05
0.1
0.15
0.2
0.25
Median Annual Volatility of CPI
UKCHILEAUSTRALIANEWZEALANDLITHUANIACANADAAUSTRIAPORTUGALBELGIUMITALYDENMARKSWISSIRELANDPERUCOLNORWAYSOUTHAFRICAISRAELUSSPNETHERLANDSFRANCEMALAYSIASWEDENCZECHREPSPAINCHINAGERMANYGREECEINDIAMEXICOINDONESIAPHILIPPINESHONGKONGSLOVAKREPRUSSIAHUNGARYCROATIAJAPANTAIWANSINGAPOREPOLANDVENEZUELATHAILANDFINLANDECUADORKOREAARGBRAZTURKEY
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6
VOLATILITY
MA
CR
O V
OL
GDP VOL
CPI VOL
EXPLANATORY VARIABLES
Name Descriptionemerging 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 $USnlc Number of Listed Companies in the Exchange
grgdp GDP Growth Rategcpi 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
Explanatory Variables
Table (2)
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
Coefficient Std. Error t-Statistic Prob. Det Residual Covariance
emerging 0.0957 0.0176 5.4528 0.0000 6.45E-39transition -0.0077 0.0180 -0.4284 0.6685 1.53E-38log(mc) -0.0093 0.0032 -2.9345 0.0035 3.76E-38
log(gdp_dll) 0.0015 0.0055 0.2740 0.7842 2.18E-37nlc -1.29E-05 0.0000 -2.3706 0.0181 1.23E-37
grgdp -0.6645 0.1255 -5.2945 0.0000 3.89E-38gcpi 0.6022 0.0418 14.4181 0.0000 1.64E-38
vol_irate 0.0089 0.0006 14.4896 0.0000 8.59E-39vol_forex 0.5963 0.0399 14.9468 0.0000 2.47E-38vol_grgdp 1.1192 0.1008 11.1056 0.0000 8.71E-39vol_gcpi 0.9364 0.0848 11.0375 0.0000 2.84E-38
Individual SUR Regressions
Table (5)
MULTIPLE REGRESSIONS
0
0.05
0.1
0.15
0.2
0.25
1990 1994 1998 2002
Time EffectsAll Countries
emerging 0.0376( 0.0131 )**
transition -0.0178( 0.0171 )
log(mc) -0.0092( 0.0055 )*
log(gdpus) 0.0273( 0.0068 )**
nlc -1.8E-05( 5.4E-06 )**
grgdp -0.1603( 0.1930 )
gcpi 0.3976( 0.1865 )**
vol_irate 0.0020( 0.0008 )**
vol_gforex 0.0222( 0.0844 )
vol_grgdp 0.8635( 0.1399 )**
vol_gcpi 0.9981( 0.3356 )**
-4
-2
0
2
4
6
8
5 10 15 20 25 30 35 40 45
Emerging
-4
-2
0
2
4
6
8
5 10 15 20 25 30 35 40 45
Transition
-4
-2
0
2
4
6
8
5 10 15 20 25 30 35 40 45
LOG(MC)
-4
-2
0
2
4
6
8
5 10 15 20 25 30 35 40 45
LOG(GDP_US)
-4
-2
0
2
4
6
8
5 10 15 20 25 30 35 40 45
NLC
-4
-2
0
2
4
6
8
5 10 15 20 25 30 35 40 45
GRGDP
-4
-2
0
2
4
6
8
5 10 15 20 25 30 35 40 45
GCPI
-4
-2
0
2
4
6
8
5 10 15 20 25 30 35 40 45
VOL_IRATE
-4
-2
0
2
4
6
8
5 10 15 20 25 30 35 40 45
VOL_FX
-4
-2
0
2
4
6
8
5 10 15 20 25 30 35 40 45
VOL_GRGDP
-4
-2
0
2
4
6
8
5 10 15 20 25 30 35 40 45
VOL_GCPI
Figure 4T-Statistics for Unconditional Volatility: Droping One Country at a Time
CPI VOLATILITY T-STAT
-1
0
1
2
3
4
5
5 10 15 20 25 30 35 40 45
VOL_CPI
DROP ARGENTINA?
OUTLIER?
HIGHLY INFORMATIVE?
ESTIMATE BOTH WAYS.
PANEL ESTIMATE
RANDOM COUNTRY EFFECTS
AR(1) DYNAMIC COUNTRY EFFECTS
TIME FIXED EFFECTS
Panel SpecificationAll Countries Opt. Reduction Logs Without Arg Random Country Effects
emerging 0.0376 0.0387 0.2079 0.0322 0.0478( 0.0131 )** ( 0.0128 )** ( 0.0592 )** ( 0.0128 )** ( 0.0212 )**
transition -0.0178 -0.0164 -0.0332 -0.0147 -0.0258( 0.0171 ) ( 0.0167 ) ( 0.0741 ) ( 0.0163 ) ( 0.0304 )
log(mc) -0.0092 -0.0085 -0.0345 -0.0083 -0.0046( 0.0055 )* ( 0.0053 ) ( 0.0235 ) ( 0.0054 ) ( 0.0067 )
log(gdpus) 0.0273 0.0271 0.1156 0.0245 0.0175( 0.0068 )** ( 0.0066 )** ( 0.0302 )** ( 0.0067 )** ( 0.0099 )*
nlc -1.8E-05 -1.8E-05 -8.1E-05 -1.4E-05 -1.7E-05( 5.4E-06 )** ( 5.3E-06 )** ( 2.3E-05 )** ( 5.2E-06 )** ( 8.6E-06 )**
grgdp -0.1603 0.0962 -0.4046 -0.2094( 0.1930 ) ( 0.7474 ) ( 0.1984 )** ( 0.2258 )
gcpi 0.3976 0.3915 1.1459 0.5985 0.6114( 0.1865 )** ( 0.1641 )** ( 0.7755 ) ( 0.1939 )** ( 0.2229 )**
vol_irate 0.0020 0.0022 0.0061 0.0032 0.0034( 0.0008 )** ( 0.0008 )** ( 0.0031 )* ( 0.0008 )** ( 0.0009 )**
vol_gforex 0.0222 0.0185 0.0068 -0.0221( 0.0844 ) ( 0.3383 ) ( 0.0878 ) ( 0.0959 )
vol_grgdp 0.8635 0.8373 2.5808 0.9392 0.9019( 0.1399 )** ( 0.1352 )** ( 0.6138 )** ( 0.1371 )** ( 0.1862 )**
vol_gcpi 0.9981 1.0983 3.1467 -0.2243 -0.0849( 0.3356 )** ( 0.3208 )** ( 1.3431 )** ( 0.3627 ) ( 0.3917 )
SUR Models
ANNUAL REALIZED VOLATILITY
Panel SpecificationAll Countries Opt. Reduction Logs Without Arg Random Country Effects
emerging 0.0434 0.0408 0.0964 0.0413 0.0373( 0.0134 )** ( 0.0124 )** ( 0.0317 )** ( 0.0136 )** ( 0.0199 )*
transition -0.0013 -0.0084 -0.0007 0.0018( 0.0182 ) ( 0.0417 ) ( 0.0183 ) ( 0.0282 )
log(mc) -0.0116 -0.0112 -0.0256 -0.0107 -0.0042( 0.0055 )** ( 0.0052 )** ( 0.0130 )** ( 0.0056 )* ( 0.0074 )
log(gdpus) 0.0314 0.0309 0.0730 0.0292 0.0245( 0.0068 )** ( 0.0066 )** ( 0.0162 )** ( 0.0069 )** ( 0.0101 )**
nlc -1.5E-05 -1.4E-05 -3.8E-05 -1.3E-05 -1.3E-05( 6.4E-06 )** ( 6.2E-06 )** ( 1.5E-05 )** ( 6.2E-06 )** ( 8.8E-06 )
grgdp -0.6222 -0.6568 -0.9639 -0.5400 -1.0773( 0.2442 )** ( 0.2322 )** ( 0.5277 )* ( 0.2517 )** ( 0.2939 )**
gcpi 0.1598 0.2366 0.2286 0.4299( 0.2159 ) ( 0.4840 ) ( 0.2312 ) ( 0.2630 )
vol_irate 0.0040 0.0043 0.0059 0.0048 0.0056( 0.0010 )** ( 0.0008 )** ( 0.0021 )** ( 0.0010 )** ( 0.0011 )**
vol_gforex 0.1329 0.1649 0.2807 0.1120 0.1040( 0.1057 ) ( 0.0894 )* ( 0.2247 ) ( 0.1105 ) ( 0.1203 )
vol_grgdp 0.6500 0.7002 1.3278 0.6414 0.6728( 0.1437 )** ( 0.1277 )** ( 0.3378 )** ( 0.1463 )** ( 0.1989 )**
vol_gcpi -0.0432 -0.1124 -0.4683 -0.5073( 0.3978 ) ( 0.9042 ) ( 0.4700 ) ( 0.4799 )
SUR Models
Unconditional Vol Realized Vol1990 0.5816 0.40191991 0.6435 0.57861992 0.7293 0.36401993 0.6463 0.51021994 0.5798 0.55771995 0.6689 0.49821996 0.7040 0.72181997 0.5700 0.41721998 0.5608 0.48351999 0.4481 0.38782000 0.3908 0.24422001 0.3477 0.25562002 0.3636 0.09852003 0.3968 0.2026
Average 0.5451 0.4087
R-Squared Statistics for Each Equation in the SUR
Table 8
System
CONCLUSIONS AND IMPLICATIONS
Unconditional volatility changes in systematic ways.
Macro volatility and growth are important determinants of financial volatility.
Unconditional volatility and realized volatility give similar results but the former fits better.
Big swings in financial volatility are common across the globe.