8-11 Another Way of writing arch models For illustration, consider an ARCH(I). Instead of the above, we can write y,=Bi +B2x2t t.+Bkkt +ut, u, v,ot =Va+ax4-1,v~N(0,1) The two are different ways of expressing exactly the same model. The first form is easier to understand while the second form is required for simulating from an arCH model, for example 非负性约束:条件方差必须严格是正的,从而通常要求条件方 差关系式中的系数是非负的
8-11 Another Way of Writing ARCH Models • For illustration, consider an ARCH(1). Instead of the above, we can write yt = 1 + 2x2t + ... + kxkt + ut , ut = vtt , vt N(0,1) • The two are different ways of expressing exactly the same model. The first form is easier to understand while the second form is required for simulating from an ARCH model, for example. • 非负性约束：条件方差必须严格是正的，从而通常要求条件方 差关系式中的系数是非负的。 t = 0 +1 ut −1 2
Testing for " ARCH Effects run any postulated linear regression of the form given in the equation above, e.g. D,=Bi+ Bx2t+.+Bkxkt+ut saving the residuals, ii Then square the residuals, and regress them on g own lags to test for ARCH of order i.e. run the regression where v, is iid Obtain r2 from this regression 3. The test statistic is defined as TRZ from the last regression and is distributed as a x(@
8-12 Testing for “ARCH Effects” 1. run any postulated linear regression of the form given in the equation above, e.g. yt = 1 + 2x2t + ... + kxkt + ut saving the residuals, . 2. Then square the residuals, and regress them on q own lags to test forARCH of order q, i.e. run the regression where vt is iid. Obtain R2 from this regression 3. The test statistic is defined as TR2 from the last regression, and is distributed as a 2 (q). t u ˆ t t t q t q t u = + u + u + + u + v − − − 2 2 2 2 2 0 1 1 2 ˆ ˆ ˆ ... ˆ
8-13 Testing for“ ARCH Effects,(cont’d) 4. The null and alternative hypotheses are M1=0 and r2=0 and 13=0 and . and ra=0 H1:y≠00ry2≠0or≠0or…0ry≠0 If the value of the test statistic is greater than the critical value from the x distribution, then reject the null hypothesis Note that the arch test is also sometimes applied directly to returns instead of the residuals from Stage I above. Testing for ARCH effects in S&p500 index using Eviews p449
8-13 Testing for “ARCH Effects” (cont’d) 4. The null and alternative hypotheses are H0 : 1 = 0 and 2 = 0 and 3 = 0 and ... and q = 0 H1 : 1 0 or 2 0 or 3 0 or ... or q 0. If the value of the test statistic is greater than the critical value from the 2 distribution, then reject the null hypothesis. • Note that the ARCH test is also sometimes applied directly to returns instead of the residuals from Stage 1 above. • Testing for ARCH effectsin S&P500 index using Eviews p449
8-14 Problems with arCH(g Models How do we decide on g? One approach: Likelihood ratio test The required value of g might be very large to capture all of the dependence in the conditional variance, so model would be not parsimonious. Engle 1982, p452 Non-negativity constraints might be violated. When we estimate an ArCH model, we require a >0V i=1, 2,. g(since variance cannot be negative) A natural extension of an ArCH(q model which gets around some of these problems is a garCH model. GarCh models are widely employed in practice
8-14 Problems with ARCH(q) Models • How do we decide on q? – One approach :Likelihood ratio test • The required value of q might be very large to capture all of the dependence in the conditional variance, so model would be not parsimonious. – Engle 1982, p452 • Non-negativity constraints might be violated. – When we estimate an ARCH model, we require i >0 i=1,2,...,q (since variance cannot be negative) • A natural extension of an ARCH(q) model which gets around some of these problems is a GARCH model. GARCH models are widely employed in practice
8-15 4 Generalised ARCH GARCH Models Due to Bollerslev(1986). Allow the conditional variance to be dependent upon previous own lags The variance equation is now Bo This is a garCH(l, 1)model, which is like an ARMA(l, 1) model for the variance equation.(p453) We could also write t-1 2 +Bot O-2 Bo1-3 Substituting into(D)for o: G=0+a111+B(+an12+B02) 00+a1l1-1+f+a1B12+Bo12
8-15 4 Generalised ARCH (GARCH) Models • Due to Bollerslev (1986). Allow the conditional variance to be dependent upon previous own lags • The variance equation is now (1) • This is a GARCH(1,1) model, which is like an ARMA(1,1) model for the variance equation. (p453) • We could also write • Substituting into (1) for t-1 2 : t 2 = 0 + 1 2 ut−1 +t-1 2 t-1 2 = 0 + 1 2 ut−2 +t-2 2 t-2 2 = 0 + 1 2 ut−3 +t-3 2 t 2 = 0 + 1 2 ut −1 +(0 + 1 2 ut−2 +t-2 2 ) = 0 + 1 2 ut−1 +0 + 1 2 ut−2 +t-2 2 (2)