1. MODELING VOLATILITY BY ARCH-GARCH MODELS

2. VARIANCE
A time series is said to be heteroscedastic, if its variance changes over time, otherwise it is called homoscedastic.
When the variance is not constant ( it will follow mixture normal distribution), we can expect more outliers than expected from normal distribution. i.e. when a process is heteroscedastic, it will follow heavy-tailed or outlier-prone probability distributions.

3. VARIANCE
Until the early 80s econometrics had focused almost solely on modeling the means of series, i.e. their actual values. Recently however researchers have focused increasingly on the importance of volatility, its determinates and its effects on mean values.
A key distinction is between the conditional and unconditional variance.
The unconditional variance is just the standard measure of the variance
Var(X) =E(XE(X))2

4. VARIANCE
The conditional variance is the measure of our uncertainty about a variable given a model and an information set. Cond Var(X) =E(X-E(X| ))2
 This is the true measure of uncertainty
mean
variance
Conditional variance

5. VARIANCE Stylized Facts of asset returns
Thick tails: they tend to be leptokurtic
Volatility clustering: Mandelbrot, “large changes tend to be followed by large changes of either sign”
Leverage Effects: the tendency for changes in stock prices to be negatively correlated with changes in volatility.
Non-trading period effects: when a market is closed, information seems to accumulate at a different rate to when it is open. e.g. stock price volatility on Monday is not three times the volatility on Tuesday.
Forecastable events: volatility is high at regular times such as news announcements or other expected events, or even at certain times of day, e.g. less volatile in the early afternoon.
Volatility and serial correlation: There is a suggestion of an inverse relationship between the two.
Co-movements in volatility: There is considerable evidence that volatility is positively correlated across assets in a market and even across markets

8. ARCH MODEL Stock market’s volatility is rarely constant over time.
In finance, portfolios of financial assets are held as functions of the expected mean and variance of the rate of return. Since any shift in asset demand must be associated with changes in expected mean and variance of rate of return, ARCH models are the best suitable models.
In regression, ARCH models can be used to approximate the complex models.

9. ARCH MODEL
Even though the errors may be serially uncorrelated they are not independent, there will be volatility clustering and fat tails.
If there is no serial correlation of the series but there is of the squared series, then we will say there is weak dependence. This will lead us to examine the volatility of the series, since that is demonstrated by the squared terms.

10. ARCH MODEL
Figure 1: Autocorrelation for the log returns for the Intel series

11. ARCH MODEL
Figure 2: ACF of the squared returns
Figure 3 PACF for squared returns
Combining these three plots, it appears that this series is serially uncorrelated but dependent. Volatility models attempt to capture such dependence in the return series

12. ARCH MODEL
Engle(1982) introduced a model in which the variance at time t is modeled as a linear combination of past squared residuals and called it an ARCH (autoregressive conditionally heteroscedastic) process.
Bolerslev (1986) introduced a more general structure in which the variance model looks more like an ARMA than an AR and called this a GARCH (generalized ARCH) process.

13. Engle(1982) ARCH Model Auto-Regressive Conditional Heteroscedasticity
an AR(q) model for squared innovations.

14. ARCH(q) MODEL

15. ARCH(q) MODEL
The equation of ht shows that if t-1 is large, then the conditional variance of t is also large, and therefore t tends to be large. This behavior will spread through out the process and unusual variability tends to persist, but not always.
The conditional variance will revert to unconditional variance provided
so that the process will be stationary with finite variance.

16. ARCH(q) MODEL
If the ARCH processes have a non-zero mean which can be expressed as a linear combination of exogenous and lagged dependent variables, then a linear regression frame work is appropriate and the model can be written as,
This model is called a “ Linear ARCH(q) Regression “ model.

17. ARCH(q)
If the regressors include no lagged dependent variables and can be treated as fixed constants then the ordinary least square (OLS) estimator is the best linear unbiased estimator for the model. However, maximum-likelihood estimator is nonlinear and is more efficient than OLS estimator.

18. ESTIMATION OF THE LINEAR ARCH( q ) REGRESSION MODEL
Let the log likelihood function for the model is
The likelihood function can be maximized with respect to the unknown parameters  and ’s.

19. ESTIMATION OF THE LINEAR ARCH( q )
To estimate the parameters, usually we use Scoring Algorithm.
Each iteration for parameters  and  produces the estimates based on the previous iteration according to,
where I and I matrices are information matrices.

20. STEPS IN ESTIMATION
STEP 1: Estimate  by OLS and obtain residuals. STEP 2: Compute where
residuals
Conditional variance

21. STEPS IN ESTIMATION
STEP 3: Using (i+1), compute

22. STEPS IN ESTIMATION
STEP 4: Obtain residuals by using (i+1). Go to Step 2. This iterative procedure will be continued until the convergence of the estimation of .

23. TESTING FOR ARCH DISTURBANCES
Method 1. The autocorrelation structure of residuals and the squared residuals can be examined. An indication of ARCH is that the residuals will be uncorrelated but the squared residuals will show autocorrelation.

24. TESTING FOR ARCH DISTURBANCES
Method 2. A test based on Lagrange Multiplier ( LM ) principle can be applied. Consider the null hypothesis of no ARCH errors versus the alternative hypothesis that the conditional error variance is given by an ARCH(q) process. The test approach proposed by Engle is to regress the squared residuals on a constant and q lagged residuals. From the residuals of this auxiliary regression, a test statistic is calculated as nR2, where R2 is coming from the auxiliary regression. The null hypothesis will be rejected if the test statistic exceeds the critical value from a chi-square distribution with q degree of freedom.

25. TESTING FOR ARCH DISTURBANCES
White’s test
Breush-Pagan test
The Goldfeld-Quandt test
Likelihood ratio test
LM tests: the Glejser test; the Harvey-Godfrey test, and the Park test

26. TESTING FOR HETEROSCEDASTICITY
Popular heteroscedasticity LM tests: - Breusch and Pagan (1979)’s LM test (BP). - White (1980)’s general test.
Both tests are based on OLS residuals. That is, calculated under H0: No heteroscedasticity.
The BP test is an LM test, based on the score of the log likelihood function, calculated under normality. It is a general tests designed to detect any linear forms of heteroskedasticity.
The White test is an asymptotic Wald-type test, normality is not needed. It allows for nonlinearities by using squares and crossproducts of all the x’s in the auxiliary regression.

27. TESTING FOR HETEROSCEDASTICITY
Drawbacks of the Breusch-Pagan test: - It has been shown to be sensitive to violations of the normality assumption. - Three other popular LM tests: the Glejser test; the Harvey-Godfrey test, and the Park test, are also sensitive to such violations.
Drawbacks of the White test - If a model has several regressors, the test can consume a lot of df’s. - In cases where the White test statistic is statistically significant, heteroscedasticity may not necessarily be the cause, but model specification errors. - It is general, but does not give us a clue about how to model heteroscedasticity to do FGLS. The BP test points us in a direction.

28. PROBLEMS IN ARCH MODELING
In most of the applications of the ARCH model a relatively long lag in the conditional variance is often called for, and this leads to the problem of negative variance and non-stationarity. To avoid this problem, generally a fixed lag structure is typically imposed. So it is necessary to extent the ARCH models to a new class of models allowing for a both long memory and much more flexible lag structure.
Bollerslev introduced a Generalized ARCH (GARCH) models which allows long memory and flexible lag structure.

29. GARCH (Bollerslev,1986)
In empirical work with ARCH models high q is often required, a more parsimonious representation is the Generalized ARCH model
which is an ARMA(max(p,q),p) model for the squared innovations.
GARCH (p, q) process allows lagged conditional variances to enter as well.

30. GARCH MODEL The GARCH (p, q) process is stationary iff
The simplest but often very useful GARCH process is the GARCH (1,1) process given by

31. TESTING FOR GARCH DISTURBANCES
METHOD 1: Use the previous LM test for ARCH. If the null hypothesis is rejected for long disturbances, GARCH model is appropriate.
METHOD 2: A test based on Lagrange Multiplier (LM) principle can be applied. Consider the null hypothesis of ARCH (q) for errors versus the alternative hypothesis that the errors are given by a GARCH (p, q) process.

32. INTEGRATED GARCH When the process has a unit root.
Use Integrated GARCH or IGARCH process

33. SYMMETRYCITY OF GARCH MODELS
In ARCH, GARCH, IGARCH processes, the effect of errors on the conditional variance is symmetric, i.e., positive error has the same effect as a negative error.
However, in finance, good and bad news have different effects on the volatility.
Positive shock has a smaller effect than the negative shock of the same magnitude.

34. EGARCH
For the asymmetric relation between many financial variables and their volatility changes and to relax the restriction on the coefficients in the model, Nelson (1991) proposed EGARCH process.

35. THRESHOLD GARCH (TGARCH) OR GJR-GARCH
Glosten, Jaganathan and Runkle (1994) proposed TGARCH process for asymmetric volatility structure.
Large events have an effect but small events not.
TARCH(1,1)

36. TGARCH
When 1>0, the negative shock will have larger effect on the volatility.
THE LEVERAGE EFFECT: The tendency for volatility to decline when returns rise and to rise when returns falls.
TEST FOR LEVERAGE EFFECT: Estimate TGARCH or EGARCH and test whether 1=0 or =0.

37. NONLINEAR ARCH (NARCH) MODEL
This then makes the variance depend on both the size and the sign of the variance which helps to capture leverage type effects.

38. ARCH in MEAN (G)ARCH-M
 Many classic areas of finance suggest that the mean of a relationship will be affected by the volatility or uncertainty of a series. Engle Lilien and Robins(1987) allow for this explicitly using an ARCH framework.
typically either the variance or the standard deviation are included in the mean relationship.

39. NORMALITY ASSUMPTION
While the basic GARCH model allows a certain amount of leptokurtic behaviour, this is often insufficient to explain real world data. Some authors therefore assume a range of distributions other than normality which help to allow for the fat tails in the distribution.
t Distribution
The t distribution has a degrees of freedom parameter which allows greater kurtosis.

40. THE GARCH ZOO
• QARCH = quadratic ARCH • TARCH = threshold ARCH • STARCH = structural ARCH • SWARCH = switching ARCH • QTARCH = quantitative threshold ARCH • vector ARCH • diagonal ARCH • factor ARCH

41. S&P COMPOSITE STOCK MARKET RETURNS
Monthly data on the S&P Composite index returns over the period 1954:1–2001:9. Lags of the inflation rate and the change in the three-month Treasury bill (T-bill) rate are used as regressors, in addition to lags of the returns. We begin by modeling the returns series as a function of a constant, one lag of returns (Ret_l), one lag of the inflation rate (Inf_l) and one lag of the first-difference of the three-month T-bill rate (DT-bill_l).

42. S&P COMPOSITE STOCK MARKET RETURNS

43. S&P COMPOSITE STOCK MARKET RETURNS
proc autoreg data=returns maxit=50; model ret = ret_1 inf_1 dt_bill_1/ archtest; Ordinary Least Squares Estimates SSE DFE 567 MSE Root MSE SBC AIC MAE AICC MAPE HQC Durbin-Watson Regress R-Square Total R-Square Parameter Estimates Standard Approx Variable Variable DF Estimate Error t Value Pr > |t| Label Intercept <.0001 ret_ <.0001 Inf_ Inf_1 DT_bill_ <.0001 DT-bill_1

44. S&P COMPOSITE STOCK MARKET RETURNS
Tests for ARCH Disturbances Based on OLS Residuals Order Q Pr > Q LM Pr > LM

45. ARCH(9)
proc autoreg data=returns maxit=50; model ret = ret_1 inf_1 dt_bill_1/ garch=(q=9);run; GARCH Estimates SSE Observations 571 MSE Uncond Var Log Likelihood Total R-Square SBC AIC MAE AICC MAPE HQC Normality Test Pr > ChiSq <.0001

46. ARCH(9)
Parameter Estimates Standard Approx Variable Variable DF Estimate Error t Value Pr > |t| Label Intercept <.0001 ret_ <.0001 Inf_ Inf_1 DT_bill_ DT-bill_1 ARCH <.0001 ARCH ARCH ARCH ARCH E-19 0 Infty <.0001 ARCH ARCH ARCH ARCH E Infty <.0001 ARCH

47. ARCH(9)

48. ARCH(8)
GARCH Estimates SSE Observations 571 MSE Uncond Var Log Likelihood Total R-Square SBC AIC MAE AICC MAPE HQC Normality Test Pr > ChiSq <.0001

49. ARCH(8)
Parameter Estimates Standard Approx Variable DF Estimate Error t Value Pr > |t| Variable Label Intercept <.0001 ret_ <.0001 Inf_ Inf_1 DT_bill_ DT-bill_1 ARCH <.0001 ARCH ARCH ARCH ARCH ARCH ARCH ARCH ARCH E-20 0 Infty <.0001 TDFI Inverse of t DF

50. ARCH(8)

51. GARCH(1,1)
proc autoreg data=returns maxit=100; model ret = ret_1 inf_1 dt_bill_1/ garch=( q=1, p=1 ) dist = t ;run; GARCH Estimates SSE Observations 571 MSE Uncond Var Log Likelihood Total R-Square SBC AIC MAE AICC MAPE HQC Normality Test Pr > ChiSq <.0001

52. GARCH(1,1)
Parameter Estimates Standard Approx Variable DF Estimate Error t Value Pr > |t| Variable Label Intercept <.0001 ret_ <.0001 Inf_ Inf_1 DT_bill_ <.0001 DT-bill_1 ARCH ARCH GARCH <.0001 TDFI Inverse of t DF

53. GARCH(1,1)

54. EGARCH(1,1)
Exponential GARCH Estimates SSE Observations 571 MSE Uncond Var . Log Likelihood Total R-Square SBC AIC MAE AICC MAPE HQC Normality Test Pr > ChiSq <.0001

55. EGARCH(1,1)
Parameter Estimates Standard Approx Variable Variable DF Estimate Error t Value Pr > |t| Label Intercept <.0001 ret_ <.0001 Inf_ Inf_1 DT_bill_ <.0001 DT-bill_1 EARCH EARCH <.0001 EGARCH <.0001 THETA

56. EGARCH(1,1)

57. IGARCH(1,1)
Integrated GARCH Estimates SSE Observations 571 MSE Uncond Var Log Likelihood Total R-Square SBC AIC MAE AICC MAPE HQC Normality Test Pr > ChiSq <.0001

58. IGARCH(1,1)
Parameter Estimates Standard Approx Variable DF Estimate Error t Value Pr > |t| Variable Label Intercept <.0001 ret_ <.0001 Inf_ Inf_1 DT_bill_ DT-bill_1 ARCH ARCH GARCH <.0001 TDFI Inverse of t DF

59. IGARCH(1,1)

60. S&P COMPOSITE RETURNS VS FITTED DATA

61. ESTIMATED CONDITIONAL VARIANCE OF S&P COMPOSITE RETURNS FROM IGARCH(1,1) MODEL

62. REFERENCES
Bollerslev, Tim “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics. April, 31:3, pp. 307–27.
Engle, Robert F “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation.” Econometrica. 50:4, pp. 987–1007.
Glosten, Lawrence R., Ravi Jagannathan and David E. Runkle “On the Relation Between the Expected Value and the Volatility of the Nominal Excess Returns on Stocks.” Journal of Finance. 48:5, pp. 1779–801.
Nelson, Daniel B “Conditional Heteroscedasticity in Asset Returns: A New Approach.” Econometrica. 59:2, pp. 347–70.