where vi is just the difference between the random variable a and its expectation Since Ei is unobservable, we would use the OlS residual, for which x1(-B)=E1+u1 But in large sample, as B-6, terms in ui will become negligible, so that at least approximatel The procedure suggested is to treat the variance function as a regression and use the squares of the OLS residual as the dependent variable. For example, if 02=ao+d a1, then a consistent estimator of a will be the OLS in the model z a1+U In this model. uf is both heteroscedastic and autocorrelated. so a is consistent but inefficient. But, consistency is all that is required for asymptotically efficient estimation of B using (a) The two-step estimator may be iterated by recomputing the residuals after computing the FGLS estimate and then reentering the computation (OLS, B- e→a→B→e→ Exercise Reproduce the results at Table 11.2 on p 231 4.2 Maximum Likelihood estimation 5 Autoregressive Conditional Heteroscedastic- ity(ARCHLet ε 2 i = σ 2 i + vi , where vi is just the difference between the random variable ε 2 i and its expectation. Since εi is unobservable, we would use the OLS residual, for which ei = εi − x 0 i (βˆ − β) = εi + ui . But in large sample, as βˆ p −→ β, terms in ui will become negligible, so that at least approximately, ei = σ 2 i + v ∗ i . The procedure suggested is to treat the variance function as a regression and use the squares of the OLS residual as the dependent variable. For example, if σ 2 i = α0 + z 0 iα1, then a consistent estimator of α will be the OLS in the model e 2 i = α0 + z 0 iα1 + v ∗ i . In this model, v ∗ i is both heteroscedastic and autocorrelated, so αˆ is consistent but inefficient. But, consistency is all that is required for asymptotically efficient estimation of β using Ω(αˆ). The two-step estimator may be iterated by recomputing the residuals after computing the FGLS estimate and then reentering the computation (OLS, βˆ → e → αˆ → βˇ → eˇ →....). Exercise: Reproduce the results at Table 11.2 on p.231. 4.2 Maximum Likelihood Estimation 5 Autoregressive Conditional Heteroscedastic￾ity (ARCH) 7