Arch Model.ppt

Time Series Econometrics:

ARCH/GARCH Models Measuring volatility: Conditional heteroscedastic Models K.R. Shanmugam, MSE

Features of Financial Data (e.g Stock returns) 1. Volatility Clustering: Some periods are highly volatile while others are less. Big shocks (residuals) tend to follow big shocks in either direction and small shocks follow small. This implies strong AC. In addition, constant variance assumption is inappropriate (If unconditional (or long-run) variance is constant but there are periods in which the variance is relatively high. Such series are called conditionally heteroscedastic).

Features…. 2. Leverage Effect: Volatility is higher in a falling market than it is in a rising market (there is a tendency for volatility to rise more following a large price fall than following a price rise of the same magnitude).

Features 3. Leptokurtosis: They have distributions which exhibit flatter tails and excess peakedness (due to a large number of excessive values). (a normal distribution is skewed (3rd moment) and has a coefficient of kurtosis of 3 (it is symmetric and said to be meso-kurtic) • Leptokurtic is one which has flatter tails and is more peaked at mean than normal with same mean and variances.

NSE (2/1/1996 to 20/12/2002) 1800

.12

1600

.08

1400

.04

1200

.00

1000

-.04

800

-.08

600 1996 1997 1998 1999 2000 2001 2002 CLOSE

-.12 1996 1997 1998 1999 2000 2001 2002 RETURNS

Impact of special characteristics • They lead to violations of homoscedasticity as well as autocorrelation assumptions of OLS • (consequences of heteroscedasticity: estimates of parameters are unbiased, but SEs are large, CIs are very narrow, and precision is affected) • Linear Models are unable to explain these special features

Tasks of the Asset Holder • He may be interested in forecasting the rate of return and the variance of the stock asset over the holding period. • ARMA models are useful to forecast mean returns. But it ignores the risk factor (variance or SD is a measure of risk or volatility) • Some series are subject to fat tails, volatility clustering and leptokurtic, the task is to specify and forecast both mean and variance of the series conditional on past information.

Historical Volatility • Calculating the variance of the series in the usual way over some historical period (e.g. In option pricing model, historical average variance is used as volatility measure) • Rolling standard deviation (RSD): Studies such as Tauchen and Pitts (1987) used this. They calculate standard deviation using fixed number of observations. That is, first calculate it using most recent (say) 22 days data. Then drop first day and add 23rd day data. This is also called Officier’s approach.

Problems with RSD • It uses equal weight for all cases (more recent observations should be more relevant and be given higher weight) • Use of overlapping observations lead to correlation issues • Zero weight for other observations are unattractive

Summary of Problems • unconditional forecast has a greater variance than the conditional forecast, plus overlapping problem and usage of equal weighting. • So conditional forecast (since they take into account the known current and past realization of series) are preferable.

Engle (1982) ARCH Model • It says that the variance of the error term at time t depends on the squared error terms from previous periods: Rt =mt + et and et = N(0, t2 ) ……………….(1) (or et = vt t ; vt ~ N(0, 1)) where 2t =0 + 1 et-12 + 2 et-22+……+ p et-p2…..(2) (here the AC in volatility is modeled) • this is an ARCH (p) model • ARCH(1) Model: 2t =0 + 1 et-12

ARCH TEST ARCH (joint) TEST:

To choose number of lagged terms. If we start with one lag, this test will tell us whether we need to add any additional lag term. Steps: (i) Run mean regression Rt =mt + et and save residuals et (ii) Squared the residuals and regress it on its own lagged term to test for ARCH (e2t =0 + 1 et-12 + 2 et-22+……+ p et-p2 (iii) Define TR2 ~ 2 (p), where p is number of lagged term on the right has side of second equation (iv) Null Hypothesis: H0: 1=0; 2=0…. p =0 Alternate Hypothesis: H1 : 1≠0; 2 ≠ 0…. p ≠ 0 (v) If test value is greater than critical value, reject the null

Properties of ARCH (1) process et = vt t and 2t =0 + 1 et-12 t = (0 + 1 et-12)1/2 Since E vt =0; E et = 0 Since E vt vt-i = 0, E et et-i =0 Unconditional variance: E et2 = E [vt2 (0 + 1 et-12)] Since E vt2 =1, E et2 = 0 + 1Eet-12 2 = 0 + 1 2 = 0 /(1- 1) • • • • •

Conditional Mean and Variance • • • •

E [et | et-1, et-2……..] =0 E [et2 | et-1, et-2……..] = E [vt 2 (0 + 1et-12)] 2t = 1. (0 + 1et-12) Thus, the conditional variance depends on realized value of et-12. • If et-12is larger, the conditional variance in t will be larger as well.

ARCH (1) Model… • Unconditional variance: • E et2 = 0 /(1- 1) ; we restrict that 0>0 and |1|<1 to make it positive and finite. • Conditional variance • E [et2 | et-1….] = 0 + 1 et-12 • That is, et is conditionally heteroscedastic

Conditional Forecast Vs Unconditional Forecast • Engle (1982): conditional is better than unconditional • Example: Consider an AR(1) Model: yt = a0 + a1 yt-1 + et • Conditional Forecast of yt+1: E (yt+1|t) = a0 + a1 yt • Forecast Error Variance is: E [(yt+1- a0 - a1 yt)2] = E et+12 =2

Unconditional Forecast of y yt= a0 + a1 yt-1 + e1

yt (1-a1L)= a0 + et yt = a0/( 1-a1L) + et/(1-a1L) yt+1 = a0 /(1-a1L) + et+1 / (1-a1L)

E yt+1 = a0/ (1-a1L) = a0 / (1-a1): Long run mean

Unconditional Forecast of Error Variance

E [yt+1- Eyt+1]]2 = E (et+1/(1-a1)) = 2/ (1-a1)

Estimation • ML Estimation Method • Log-Likelihood function using a normality assumption for the disturbance is: ln L = -T/2 ln (2)-(1/2) ln t2 – (1/2)[(yt – a0-a1 yt-1)2 (1/t2)] • It is an updating formula: Observed variance of the residuals is taken for 1st observation; then it calculates variance for the second and so on for any given set of parameter values; thus the entire time series of variance forecast is constructed; then the likelihood function provides a systematic way to adjust the parameter to give the best fit.

Generalized ARCH • Developed independently by Bollerslev (1986) and Taylor (1986) • Bollerslave’s GARCH Specification: Rt =mt + et, where et ~ N(0, t2) • 2t =0 + 1 et-12 + 2 et-22+…+ 1 t-12+… • It allows conditional variance to be dependent on previous own lags • It is a weighted average of long term average (variance), information about volatility during previous years and fitted variance in previous years. • Such an updating formula is a simple description of adaptive or learning behaviour. • Estimation is same as ARCH • However, we use parsimonious model; GARCH (1,1) is sufficient

GARCH (1,1) Vs. ARCH • • • •

Proof: Let 2t =0 + 1 et-12 +  t-12......(1) 2t-1 =0 + 1 et-22 +  t-22….................(2) 2t-2 =0 + 1 et-32 +  t-32………………(3) Substitute (2) in (1) and then (3) in…

• • • • • •

2t =0 (1+ + 2) + 1 et-12 (1+  L+ 2L2) + 3 t-32 An infinite number of successive substitutions leads to 2t =0 (1+ + 2+….) + 1 et-12 (1+  L+ 2L2+….) +  02 The last term is almost zero The rest is similar to ARCH infinite specification Although GARCH (1,1) has 3 parameters in conditional variance equation (parsimonious), it allows an infinite number of past squared errors

(1) GARCH (1,1) rt    t  t2

   0      2 2 t

2 1 t 1

2 t 1

where t-1 and t-1 are ARCH and GARCH terms respectively.

The (G)ARCH-M Model

rt       t 2 t

   0  1 2 t

2 t 1



2 t

  2

2 t 1

In this application, the dependent variable in the mean equation is determined by its conditional variance. For instance, the expected return on an asset is related to expected asset risk.

The Asymmetric ARCH Models • Often, we notice that the downward movement in the market are highly volatile than upward movement of the same magnitude. In such case, symmetric ARCH model undermine the true variance process. Engle and Ng (1993) provide a news impact curve with asymmetric response to good and bad news. Volatility

Bad News

Good news

two types of asymmetric ARCH models: TARCH and EGARCH The TARCH or Threshold ARCH due to Zakoian (1990) can be defined as: rt    t  t2

 t2   0  1 t21   2 t21   t21dt 1 where d t  1if ε t < 0 and 0 otherwise ε t 0  good news, ε t 0  bad news They have differential effects on conditional var. Good (bad) news has an effect of 1 (1   ) If  0, there is leverage effect (bad news increases volatility). if   0, the news impact is asymmetric.

The EGARCH Model (by Nelson 1991) log    0  1 log  2 t

2 t 1

 t 1  t 1  3   t 1  t 1

If  0, there is leverage effect. if   0, the impact is asymmetric Log of conditional variance equation. This means that leverage effect is exponential than quadratic. This guarantees that forecasts of conditional variances are positive;

Power GARCH (PARCH) • Taylor (1986) and Schwert (1989) developed S.D GARCH model. The conditional variance is not a linear function in lagged squared residuals 2  t



( )   0    i (  t i   t i )    j ( ) 2 t 1



• where >0. • For symmetric model =0; if ≠0, asymmetric effects are present • If =2 and =1 for all i, then the model is standard GARCH