1. MODELING VOLATILITY BY ARCH- GARCH MODELS 1

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. 2

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 3

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 4

5. VARIANCE Stylized Facts of asset returns i.Thick tails: they tend to be leptokurtic ii.Volatility clustering: Mandelbrot, “large changes tend to be followed by large changes of either sign” iii.Leverage Effects: the tendency for changes in stock prices to be negatively correlated with changes in volatility. iv.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. v.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. vi.Volatility and serial correlation: There is a suggestion of an inverse relationship between the two. vii.Co-movements in volatility: There is considerable evidence that volatility is positively correlated across assets in a market and even across markets 5

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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. 8

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. 9

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

11. ARCH MODEL Figure 2: ACF of the squared returnsFigure 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 11

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. 12

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

14. ARCH(q) MODEL 14

15. ARCH(q) MODEL The equation of h t 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. 15

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. 16

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. 17

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. 18

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, 19 where I  and I  matrices are information matrices.

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

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

22. 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 . 22 STEPS IN ESTIMATION

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. 23

24. 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 nR 2, where R 2 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. 24 TESTING FOR ARCH DISTURBANCES

25. 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. 25

26. 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. GARCH (Bollerslev,1986) In empirical work with ARCH models high q is often required, a more parsimonious representation is the Generalized ARCH model 26

27. 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 27

28. 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. 28

29. INTEGRATED GARCH When the process has a unit root.  Use Integrated GARCH or IGARCH process 29

30. 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. 30

31. 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. 31

32. 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. 32 TARCH(1,1)

33. 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. 33

34. Non-linear ARCH model NARCH This then makes the variance depend on both the size and the sign of the variance which helps to capture leverage type effects. 34

35. 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. 35

36. Non normality assumptions 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. 36

37. 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 37

38. 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). 38

39. 39 S&P COMPOSITE STOCK MARKET RETURNS

40. proc autoreg data=returns maxit=50; model ret = ret_1 inf_1 dt_bill_1/ archtest; Ordinary Least Squares Estimates SSE 6023.21652 DFE 567 MSE 10.62296 Root MSE 3.25929 SBC 2991.08622 AIC 2973.69666 MAE 2.44852035 AICC 2973.76733 MAPE 258.49597 HQC 2980.48101 Durbin-Watson 1.9457 Regress R-Square 0.1011 Total R-Square 0.1011 Parameter Estimates Standard Approx Variable Variable DF Estimate Error t Value Pr > |t| Label Intercept 1 1.1771 0.2100 5.60 <.0001 ret_1 1 0.2080 0.0410 5.07 <.0001 Inf_1 1 -1.1795 0.4480 -2.63 0.0087 Inf_1 DT_bill_1 1 -1.2501 0.2914 -4.29 <.0001 DT-bill_1 40 S&P COMPOSITE STOCK MARKET RETURNS

41. Tests for ARCH Disturbances Based on OLS Residuals Order Q Pr > Q LM Pr > LM 1 6.3810 0.0115 6.9874 0.0082 2 6.5739 0.0374 7.0149 0.0300 3 8.5749 0.0355 9.0576 0.0285 4 8.7780 0.0669 9.0917 0.0588 5 10.8979 0.0534 11.2837 0.0460 6 17.7423 0.0069 16.7401 0.0103 7 18.4102 0.0103 16.9397 0.0178 8 18.5711 0.0173 16.9607 0.0305 9 18.7501 0.0274 17.0048 0.0486 10 18.7956 0.0429 17.0058 0.0742 11 19.2071 0.0575 17.1694 0.1030 12 19.8498 0.0700 17.5120 0.1313 41 S&P COMPOSITE STOCK MARKET RETURNS

42. proc autoreg data=returns maxit=50; model ret = ret_1 inf_1 dt_bill_1/ garch=(q=9);run; GARCH Estimates SSE 6048.16958 Observations 571 MSE 10.59224 Uncond Var 12.094028 Log Likelihood -1466.4833 Total R-Square 0.0973 SBC 3021.82996 AIC 2960.96652 MAE 2.4465657 AICC 2961.72191 MAPE 258.891059 HQC 2984.71174 Normality Test 33.6859 Pr > ChiSq <.0001 42 ARCH(9)

43. Parameter Estimates Standard Approx Variable Variable DF Estimate Error t Value Pr > |t| Label Intercept 1 1.1960 0.1897 6.30 <.0001 ret_1 1 0.2146 0.0443 4.84 <.0001 Inf_1 1 -0.8773 0.3807 -2.30 0.0212 Inf_1 DT_bill_1 1 -0.9640 0.2491 -3.87 0.0001 DT-bill_1 ARCH0 1 4.7466 0.9261 5.13 <.0001 ARCH1 1 0.1207 0.0477 2.53 0.0114 ARCH2 1 0.005981 0.0356 0.17 0.8667 ARCH3 1 0.1341 0.0548 2.45 0.0144 ARCH4 1 1.021E-19 0 Infty <.0001 ARCH5 1 0.1266 0.0553 2.29 0.0220 ARCH6 1 0.1499 0.0510 2.94 0.0033 ARCH7 1 0.0179 0.0402 0.45 0.6563 ARCH8 1 -5.49E-21 0 -Infty <.0001 ARCH9 1 0.0524 0.0486 1.08 0.2812 43

44. ARCH(9) 44

45. ARCH(8) GARCH Estimates SSE 6053.67303 Observations 571 MSE 10.60188 Uncond Var 11.1188142 Log Likelihood -1133.4449 Total R-Square 0.0965 SBC 2355.75315 AIC 2294.8897 MAE 2.44715477 AICC 2295.6451 MAPE 259.470044 HQC 2318.63492 Normality Test 50.4365 Pr > ChiSq <.0001 45

46. ARCH(8) Parameter Estimates Standard Approx Variable DF Estimate Error t Value Pr > |t| Variable Label Intercept 1 1.2017 0.1926 6.24 <.0001 ret_1 1 0.1964 0.0443 4.43 <.0001 Inf_1 1 -0.7688 0.3936 -1.95 0.0508 Inf_1 DT_bill_1 1 -1.0448 0.2853 -3.66 0.0003 DT-bill_1 ARCH0 1 6.1876 1.3140 4.71 <.0001 ARCH1 1 0.0793 0.0533 1.49 0.1372 ARCH2 1 0.001457 0.0401 0.04 0.9710 ARCH3 1 0.0588 0.0542 1.08 0.2787 ARCH4 1 0.0145 0.0580 0.25 0.8030 ARCH5 1 0.1349 0.0715 1.89 0.0591 ARCH6 1 0.1230 0.0614 2.01 0.0449 ARCH7 1 0.0316 0.0521 0.61 0.5441 ARCH8 1 2.88E-20 0 Infty <.0001 TDFI 1 0.1365 0.0502 2.72 0.0065 Inverse of t DF 46

47. ARCH(8) 47

48. 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 6038.45467 Observations 571 MSE 10.57523 Uncond Var 10.9965281 Log Likelihood -1136.1108 Total R-Square 0.0988 SBC 2323.00074 AIC 2288.22162 MAE 2.44725923 AICC 2288.47785 MAPE 260.336638 HQC 2301.79032 Normality Test 54.3846 Pr > ChiSq <.0001 48

49. GARCH(1,1) Parameter Estimates Standard Approx Variable DF Estimate Error t Value Pr > |t| Variable Label Intercept 1 1.2560 0.1917 6.55 <.0001 ret_1 1 0.1858 0.0446 4.16 <.0001 Inf_1 1 -1.0253 0.3834 -2.67 0.0075 Inf_1 DT_bill_1 1 -1.1092 0.2768 -4.01 <.0001 DT-bill_1 ARCH0 1 1.2423 0.8819 1.41 0.1589 ARCH1 1 0.0841 0.0416 2.02 0.0432 GARCH1 1 0.8029 0.1085 7.40 <.0001 TDFI 1 0.1508 0.0499 3.02 0.0025 Inverse of t DF 49

50. GARCH(1,1) 50

51. EGARCH(1,1) Exponential GARCH Estimates SSE 6033.42807 Observations 571 MSE 10.56642 Uncond Var. Log Likelihood -1461.7473 Total R-Square 0.0995 SBC 2974.27381 AIC 2939.49469 MAE 2.44961323 AICC 2939.75092 MAPE 246.96188 HQC 2953.06339 Normality Test 21.0253 Pr > ChiSq <.0001 51

52. Parameter Estimates Standard Approx Variable Variable DF Estimate Error t Value Pr > |t| Label Intercept 1 1.1517 0.1978 5.82 <.0001 ret_1 1 0.1903 0.0469 4.06 <.0001 Inf_1 1 -1.0770 0.3677 -2.93 0.0034 Inf_1 DT_bill_1 1 -1.0312 0.2466 -4.18 <.0001 DT-bill_1 EARCH0 1 0.3394 0.1176 2.88 0.0039 EARCH1 1 0.2360 0.0598 3.94 <.0001 EGARCH1 1 0.8553 0.0504 16.95 <.0001 THETA 1 -0.6143 0.2099 -2.93 0.0034 52 EGARCH(1,1)

53. 53 EGARCH(1,1)

54. IGARCH(1,1) Integrated GARCH Estimates SSE 6034.97631 Observations 571 MSE 10.56914 Uncond Var Log Likelihood -1140.055 Total R-Square 0.0993 SBC 2324.54181 AIC 2294.11008 MAE 2.44746877 AICC 2294.30902 MAPE 259.845571 HQC 2305.98269 Normality Test 49.0458 Pr > ChiSq <.0001 54

55. Parameter Estimates Standard Approx Variable DF Estimate Error t Value Pr > |t| Variable Label Intercept 1 1.2475 0.1879 6.64 <.0001 ret_1 1 0.1851 0.0465 3.98 <.0001 Inf_1 1 -1.0481 0.3879 -2.70 0.0069 Inf_1 DT_bill_1 1 -1.1441 0.2988 -3.83 0.0001 DT-bill_1 ARCH0 1 0.3691 0.2422 1.52 0.1275 ARCH1 1 0.1357 0.0450 3.01 0.0026 GARCH1 1 0.8643 0.0450 19.20 <.0001 TDFI 1 0.1865 0.0559 3.34 0.0008 Inverse of t DF 55 IGARCH(1,1)

56. 56 IGARCH(1,1)

57. S&P COMPOSITE RETURNS VS FITTED DATA 57

58. ESTIMATED CONDITIONAL VARIANCE OF S&P COMPOSITE RETURNS FROM IGARCH(1,1) MODEL 58

59. REFERENCES Bollerslev, Tim. 1986. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics. April, 31:3, pp. 307–27. Engle, Robert F. 1982. “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. 1993. “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. 1991. “Conditional Heteroscedasticity in Asset Returns: A New Approach.” Econometrica. 59:2, pp. 347–70. 59