Two hundred observations of a time series, X1, ..., X200, gave the following sample statistics: sample mean=3.82, sample variance=1.15. The sample ACF: ??1 = 0.427, ??2 = 0.475, ??3 = 0.169. (a) Based on these sample statistics, is it reasonable to suppose that {Xt ? } is white noise? (b) Assuming that {Xt ? } can be modeled as the AR(2) process Xt ? ? ?1(Xt?1 ? ) ? ?2(Xt?2 ? ) = Zt , {Zt} ? IID(0, ?2 ) Find estimates of , ?1, ?2, and ? 2 . (c) Would you conclude that = 0? (d) Construct 95% confidence intervals for ?1 and ?2. (e) Assuming that the data were generated from an AR(2) model, derive estimates of the PACF for all lags h ? 1.

Two hundred observations of a time series, X1, ..., X200, gave the following sample statistics: sample mean=3.82, sample variance=1.15. The sample ACF: P1 = 0.427, P2 = 0.475, p3 = 0.169. (a) Based on these sample statistics, is it reasonable to suppose that {Xt -} is white noise? (b) Assuming that {Xt - u} can be modeled as the AR(2) process Xt - M - $1(Xt-1 - M) - $2(Xt-2 - 1) = Zt, {Z+} ~ IID(0, 02) Find estimates of M, $1, $2, and 62. (c) Would you conclude that u = 0? (d) Construct 95% confidence intervals for $1 and d2. (e) Assuming that the data were generated from an AR(2) model, derive estimates of the PACF for all lags h 2 1