Modelling conditional means and variances:

differences between macro and financial time

series

Computational and Financial Econometrics (CFE’11)

London 16th December 2011London 16 December 2011

Esther Ruiz

Dpt. Statistics

Universidad Carlos III

• Basic concepts in time series– Stationary stochastic processes

– Marginal and conditional distributions: Gaussian processes

– Uncorrelated processes: white noise, strict white noise, martingale difference

– Linear models

• Models for macroeconomic time series– Linear dependence in conditional means– Linear dependence in conditional means

– ARMA-GARCH models

• Models for financial time series– Models for conditional variances

– Getting conditional distributions: VaR

1. Basic concepts in time series

– Stationary stochastic processes

– Marginal and conditional distributions: Gaussian

processes

– Uncorrelated processes: white noise, strict white– Uncorrelated processes: white noise, strict white

noise, martingale difference

– Linear models

1.1 Stationary stochastic processes

Time series are characterized by:

i) Dependent observations

ii) The context in which each observation is obtainedii) The context in which each observation is obtained

changes over time

Some empirical examples

Macroeconomic variables Financial variables

600

800

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Daily S&P500 from 4/1/1993 to 20/9/2011

10,000

11,000

12,000

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17,000

Euro area unemployment May 1992-October 2011

400500 1000 1500 2000 2500 3000 3500 4000 4500

0.8

0.9

1.0

1.1

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500 1000 1500 2000 2500 3000

Daily Dollar/Euro from 4/1/1999 to 10/11/2011

10,0001992 1994 1996 1998 2000 2002 2004 2006 2008 2010

-1

0

1

2

3

4

5

97 98 99 00 01 02 03 04 05 06 07 08 09 10 11

Euro area inflation January 1997-October 2011

Consequently:

They are represented by stochastic process: in

each moment of time, we have a random

variable. The random variables in the

stochastic process can be dependent.

When dealing with economic or financial data,

one single realization of the stochastic process

is available.

Consequently, we need to assume stationarity in orderto carry out inference.

There are two main concepts of stationarity:

· Weak stationarity: The process is weaklystationary if the second order moments are constantover time

{ })(tY

ttYEi y ∀= ,))(() µ

· Strict stationarity: The process is weaklystationary if any joint distribution is constant undertime translations.

thhtYtYCoviii

ttYVarii y

∀=−

∀∞<=

),())(),(()

,))(() 2

γσ

{ })(tY

hkhtYhtYhtYftYtYtYf kiiikiii ,)),(),...,(),(())(),...,(),(( 11 ∀+++= ++++

It is important to note that weak stationarityrefers to the marginal means and variancesbeing constant with the latter being alsofinite.

It is possible to find examples of processeswhich are strictly stationary without beingwhich are strictly stationary without beingweakly stationary because the marginalvariance is not finite. For example, considerthe IGARCH model which is strictly stationarybut it is not weakly stationary; see Nelson(1990).

1.2 Marginal and conditional distributions

• The full description of a stochastic process isgiven by the joint distribution. However, inpractice, it is not possible to deal with a jointdistribution with hundreds or even thousands ofvariables.

• Consequently, we usually summarize thestochastic process using two different scalar

• Consequently, we usually summarize thestochastic process using two different scalardistributions:– Marginal distributions: The distribution of each

random variable integrating over all possible values ofthe other variables in the process.

– Conditional distributions: The distribution of interestis

11,...,| yyy tt −

• When a process is Gaussian, its joint distribution is

multivariate Normal. In this case, all marginal and

conditional distributions are Normal.

• It is possible to have conditional distributions being

Normal without the marginal (and consequently, the

joint) being Normal. Consider, for example, an

ARCH(1) process given byARCH(1) process given by

In this case, the conditional distribution is Normal but

the marginal distribution has excess kurtosis and,

consequently, is not Normal.

)1,0(

21

2

NID

y

y

t

tt

ttt

→+=

=

−

εαωσ

σε

It is also important to distinguish between

marginal and conditional moments. It is

common to find processes with constant

marginal moments (the marginal mean) but

with conditional moments that change over

time (the conditional mean). Consider, for

example, the following AR(1) process:example, the following AR(1) process:

where at is a martingale difference. Then, the

marginal mean is

while the conditional mean is

ttt ayy += −1φ

0)( =tyE

111 ),...,|( −− = ttt yyyyE φ

Another important issue involves the relationshipbetween marginal and conditional moments.Consider, for example, that the conditional meanis zero. Then, using the law of iteratedexpectations, we can show that the marginalmean is also zero:

[ ] 0)0()()( === EyEEyE

However, the implication in the other direction isnot true. By assuming that the marginal mean iszero, we cannot ensure that the conditional meanis also zero.

[ ] 0)0()()(1

===−

EyEEyE tt

t

1.3 Uncorrelated processes: white noise, strict white

noise, martingale difference

The final objective when analysing a time series

is to separate its “predictable” component

from the unexpected component.

ayygy += ),...,(

Therefore, should not be related with the

past. There are alternative ways of defining

the lack of relationship with the past.

ttt ayygy += − ),...,( 11

ta

• White noise: A process at is a white process if it isstationary with E(at)=0 and cov(at,at-h)=0.

(There are not linear dependencies in the conditionalmean)

• Strict white noise: A process at is a strict white processif it is an independent sequence of variables withE(a )=0 and finite constant variances.if it is an independent sequence of variables withE(at)=0 and finite constant variances.

• Martingale difference: A process at is a martingaledifference if E(at|at-1,…,a1)=0.

(There are not nonlinear dependencies in the conditionalmean)

Example of strict white noise: Gaussian white noise

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SWN

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-3 -2 -1 0 1 2 3

Series: SWNSample 1 3000Observations 3000

Mean 0.015796Median -0.006261Maximum 3.403402Minimum -3.214966Std. Dev. 0.998508Skewness 0.069527Kurtosis 2.984615

Jarque-Bera 2.446572Probability 0.294262

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Example of white noise: Stationary GARCH(1,1)

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500 1000 1500 2000 2500 3000

Y

0

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-6 -4 -2 0 2 4 6 8

Series: YSample 1 3000Observations 3000

Mean 0.020783Median 0.003036Maximum 9.408676Minimum -5.632596Std. Dev. 1.012755Skewness 0.174772Kurtosis 7.066620

Jarque-Bera 2082.447Probability 0.000000

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Example of martingale difference: IGARCH(1,1)

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24

Y

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0

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-15 -10 -5 0 5 10 15 20

Series: YSample 1 3000Observations 3000

Mean 0.001829Median 0.026865Maximum 20.66286Minimum -15.41837Std. Dev. 2.375031Skewness 0.132546Kurtosis 9.830231

Jarque-Bera 5840.290Probability 0.000000

White noise

Strict white noiseStrict white noise

Martingale difference

+ Stationarity

“One clear conclusion that can be reached is

that one can never be sure that a white noise

is not forecastable, either from some

nonlinear or time-varying model or from the

use of a wider information set, so one should

never stop trying to find superior models”never stop trying to find superior models”

Granger (1981)

1.4 Linear models

• A linear model is defined as yt being obtainedas a linear combination of a strict white noise.

• The Wald theorem states that any stationaryprocess can be obtained as a linearcombination of a white noise: the door toprocess can be obtained as a linearcombination of a white noise: the door tononlinear dependencies is left open.

• If the process is linear and Gaussian, thenMaravall (1983) shows that

[ ]222 ),(),( htthtt yyCorryyCorr −− =

2. Models for macroeconomic time series

- Linear dependence in conditional means

- Univariate ARMA-GARCH models

2.1 Linear dependence in conditional means

When working with macroeconomic time series,the main objective usually is to represent thedependence in conditional means.

To simplify the presentation, consider the followingAR(1) process

ttt aycy ++= −1φ

If at is assumed to be martingale difference, then

Consequently, if we assume stationarity, themarginal mean is constant and the conditionalmean evolves over time.

ttt aycy ++= −1φ

11 ),...,|( −− += tttt ycyyyE φ

-2

0

2

4

6

-6

-4

-2

100 200 300 400 500 600 700 800 900 1000

YConditional meanMarginal mean

If there is a stochastic trend

The conditional mean is given by

tttt

ttt

ayycy

aycy

+−++=+∆+=∆

−−

−

21

1

)1( φφφ

2111 )1(),...,|( −−− −++= tttt yycyyyE φφ

Because the process is non-stationary, both the

marginal and the conditional means change

over time.

• In any case, if the noise is a martingale

difference, the conditional variance is always

given by

If further, we assume that at is a strict white

noise, then

),...,|(),...,|( 111 yyaVaryyyVar ttttt −− =

noise, then

2211

2 )(),...,|( attt aEyyaE σ==−

-2

0

2

4

6

8

-6

-4

100 200 300 400 500 600 700 800 900 1000

95% Conditional PI95% Conditional PI95% Marginal PI95% Marginal PIY

2.2 Univariate ARMA-GARCH models

• The main interest when incorporating GARCH

disturbances to ARIMA models is to allow the

uncertainty to evolve over time in such a way

that:

– The prediction intervals are more adequate toThe prediction intervals are more adequate to

represent the uncertainty in each moment of time

– Measures of the relationship between the level

and uncertainty are obtained; see Broto and Ruiz

(2006)

Empirical example

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.00

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14,000

16,000-.02

.00

1992 1994 1996 1998 2000 2002 2004 2006 2008 2010

Euro-area Unemployment May 1992-October 2011Unemployment rate

• The second order significative autocorrelation ofsquared residuals can be due to the presence ofoutliers. Carnero et al. (2006) show that theautocorrelations of squares can be biased by largeoutliers.

• Maravall (1983) shows that this pattern can also beattributed to a bilinear process. He also points out theattributed to a bilinear process. He also points out therelationship between the presence of outliers andbilinear processes.

• In any case, the GARCH model is fitted to represent thepresence of conditional heterocedasticity; see Ling(2007) for the asymptotic distribution of the QMLestimator of ARMA-GARCH models.

• Neither the constant nor the ARCH parameter

is not significantly different from zero; see

Nelson (1990) for implications on the constant

beign zero.

• To analyse the effects of outliers on the• To analyse the effects of outliers on the

estimated model, a dummy variable for

January 2001 is introduced and the

parameters are estimated by maximizing the

Student likelihood (robustifying the estimator;

see Carnero et al. (2012)).

3. Models for financial time series

– Models for conditional variances

– Getting the conditional distribution: VaR

3.1 Models for conditional variances

When dealing with time series of financial returns:

· The mean is zero.

· The covariances are zero.

· The squares are positively correlated.

· The marginal distribution is leptokurtic· The marginal distribution is leptokurtic

· The cross-correlations between returns and

future squared returns are negative.

Therefore, returns are white noise (as far as they are

assumed to have constant and finite marginal

variance).

-.08

-.04

.00

.04

.08

.12

SP500 daily returns from 4/1/1993 to 20/9/2011

-.12500 1000 1500 2000 2500 3000 3500 4000 4500

0

400

800

1,200

1,600

2,000

-0.10 -0.05 0.00 0.05 0.10

Series: RETURN_DAILYSample 1 4716Observations 4715

Mean 0.000215Median 0.000594Maximum 0.109572Minimum -0.094695Std. Dev. 0.012214Skewness -0.244155Kurtosis 11.57832

Jarque-Bera 14503.73Probability 0.000000

Volatility clustering

Leverage effect

The interest is not only to obtain more accurate

prediction intervals for future returns but also to

estimate the volatility itself as this is a fundamental

element of many asset valuation models, portfolio

management strategies, risk measures etc.

Consider first the basic GARCH(1,1) process given by

=y σε

If εt is a (Gaussian) strict white noise with variance 1,

then

21

21

2−− ++=

=

ttt

ttt

y

y

βσαωσσε

),0(,...,| 211 ttt Nyyy σ→−

Some important issues:

· The parameters have to be restricted to guaranteethe positivity. These restrictions can be violated inthe presence of outliers; see Carnero et al.(2006).

· The constant has to be strictly positive for theprocess not to degenerate; see Nelson (1990).process not to degenerate; see Nelson (1990).

· The ARCH parameter has to be strictly positive forthe GARCH parameter to be identified. If α iszero, the process is homoscedastic.

· The unit root, α+β=1, can be interpreted ashomoscedasticity when α=0.

12

1222

12

12222 )()1()1( −−−− −+++=−+++=−+= ttttttttttt yyy βννβαωεσβσαωεσσ

Obviously, the GARCH model is not able to represent

the leverage effect which is still present in the

residuals

There have been several proposals to

incorporate the leverage effect. Consider, for

example, the EGARCH model which is given by

[ ] 22 )log(|)(|||)log( −−−− ++−+=

= ttt

E

y

γεσβεεαωσσε

[ ] 1111 )log(|)(|||)log( −−−− ++−+= ttttt E γεσβεεαωσ

.0012

.0016

.0020

.0024

.0028

.0000

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.0008

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GARCH EGARCH

-.10

-.05

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1000 2000 3000 4000

RETURN_DAIF ± 2 S.E.

Forecast: RETURN_DAIFActual: RETURN_DAILYForecast sample: 1 4716Adjusted sample: 2 4716Included observations: 4715Root Mean Squared Error 0.012215Mean Absolute Error 0.008168Mean Abs. Percent Error 116.2537Theil Inequality Coefficient 0.966663 Bias Proportion 0.000284 Variance Proportion NA Covariance Proportion NA

.0000

.0005

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.0020

.0025

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Forecast of Variance

3.2 Getting the conditional distribution: VaR

One interesting application of modelling the

conditional variance is to measure the risk

which is given by the VaR which is given by

When assuming a particular conditional

distribution for returns, the VaR can be

computed as

tt qm σ01.0+

Conditional mean

Quantil of the conditional

distribution

Conditional standard deviation

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-.05

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SP500 Daily returns VaR 0.01

References

• Broto, C. and E. Ruiz (2006), Unobserved component models withasymmetric conditional variances, Computational Statistics and DataAnalysis, 50(9), 2146-2166.

• Carnero, M.A., D. Peña and E. Ruiz (2006), Effects of outliers on theidentification and estimation of GARCH models, Journal of Time SeriesAnalysis, 28(4), 471-497.

• Carnero, M.A., D. Peña and E. Ruiz (2012), Estimating and forecastingGARCH volatility in the presence of outliers, Economics Letters, 114, 86-90.

• Granger, C.W.J. (1981), Forecasting white noise, in Zellner, A. (ed.),• Granger, C.W.J. (1981), Forecasting white noise, in Zellner, A. (ed.),Proceedings of the Conference on Applied Time Series Analysis ofEconomic Data, U.S. Department of Commerce, Bureau of the Census.Reproduced by Ghysels, E., R. Swanson and M.W. Watson (eds.), 2001,Essays in Econometrics, Cambridge University Press.

• Ling, S. (2007), Self-weighted and local quasi-maximum likelihoodestimators for ARMA-GARCH/IGARCH models, Journal of Econometrics,140(2), 849-873.

• Nelson, D.B. (1990), Stationarity and persistence in the GARCH(1,1) model,Econometric Theory, 6, 318-334.

• Maravall, A. (1983), An application of nonlinear time series forecasting,Journal of Business & Economic Statistics, 1(1), 66-74