3 Autoregressive Conditionally 8-9 Heteroscedastic (ARCH) Models a model which does not assume that the variance is constant the definition of the conditional variance of u. ariu, u,i u t-1 )=El(u E(uD)2 u,p, ur2 We usually assume that e(up=0 SO var(u2|u1,2…)=E t1,“t2 What could the current value of the variance of the errors plausibly depend upon? Previous squared error terms. This leads to the autoregressive conditionally heteroscedastic model for the variance of the errors. This is known as an arCH(1) model.8-9 3 Autoregressive Conditionally Heteroscedastic (ARCH) Models • a model which does not assume that the variance is constant. • the definition of the conditional variance of ut : = Var(ut ut-1 , ut-2 ,...) = E[(ut -E(ut ))2 ut-1 , ut-2 ,...] We usually assume that E(ut ) = 0 so = Var(ut  ut-1 , ut-2 ,...) = E[ut 2 ut-1 , ut-2 ,...]. • What could the current value of the variance of the errors plausibly depend upon? Previous squared error terms. • This leads to the autoregressive conditionally heteroscedastic model for the variance of the errors: = 0 + 1 • This is known as an ARCH(1) model. t 2 t 2 t 2 ut−1 2