Lecture 16 – Formulating and Estimating Time Series Models of Conditional Heteroskedasticity References: Hamilton, Chapter 21 Enders, Chapter 3Estimation and inference in regressions typically rely on the assumption that the error term is conditionally homoskedastic: E(εt2│εt-1, εt-2,…) = σ2 [Refer, e.g., to Econ 672:II; also, e.g., estimation and inference in TAR models.] Note that if E(εt2│εt-1, εt-2,…) = σ2, then E[E(εt2│εt-1, εt-2,…)] = E(σ2) = σ2. But Var(εt) = E(εt2) = E[E(εt2│εt-1, εt-2,…)]. Therefore, conditional homoskedasticity implies unconditional homoskedasticity. Does conditional heteroskedasticity imply unconditional heteroskedasticity? No. It is possible that, e.g., E(εt2) = σ2 but E(εt2│εt-1, εt-2,…) is a function of time. (Just like it is possible that a time series yt has an unconditional mean that constant but its conditional mean is time varying. Can you think of such a series?)An i.i.d. sequence is conditionally homoskedastic (provided the unconditional variance is finite). [“identically distributed” means the unconditional variance is constant; “independent” means that the conditional and uncondional variances are the same.] Stationary sequences, including w.n. sequences and AR sequences, are unconditionally homoskedastic but can be conditionally homoskedastic or conditionally heteroskedastic. [ARCH processes, e.g., are stationary, conditionally heteroskedastic processes.]It is often argued that many economic time series, particularly financial time series, are conditionally heteroskedastic: these series seem to display persistent periods of unusually high (or low) volatility (even though the level of series might be serially uncorrelated). • That, the squared deviation of y from its mean is serially correlated (even though the deviation from the mean itself might be serially uncorrelated).If conditional heteroskedasticity is a relatively common feature of economic time series, it would be useful to explicitly and parametrically model the condtional heteroskedasticity, for the same reasons it is useful to parametrically model the form of serial correlation – • Efficient estimation of a regression model’s parameters • Forecasting volatility • More complete characterization of the dynamic structure of a time series • The conditional variance of xt may be an explanatory variable in a regression model of yt. The most common approach to modeling conditional heteroskedasticity – ARCH models.The AR model is a simple way to capture persistence in the conditional mean of a time series. The ARCH model is a natural extension of the AR model designed to capture persistence in the conditional variance of a time series. The original development of the ARCH model: – Engle, Econometrica, 1982 Assume that yt is a stationary process with an AR(p) representation: yt = β0 + β1yt-1+…+ βpyt-p + εt where εt is a w.n. process with variance σ2.Assume, too, that: ∑++=−mtititw122εδηε where the δi’s satisfy the stationarity condition and the wt’s are white noise (and uncorrelated with past ε2’s). That is, assume that εt is an ARCH(m) process. Then Var(εt │εt-1,…) = E(εt2 │εt-1,…) = ∑−+miti12εδηand Var(εt) = η/(1-δ1-…-δm) (How do we know 1-δ1-…-δm > 0?) Problem – How do we assure that εt2 > 0?An alternative formulation (that turns out to be more useful for econometric purposes): Assume that εt = ht1/2vt where vt ~ i.i.d. (0,1) vt is independent of εt-1,… and ht = ∑−+miti12εδη Then, εt2 = htvt2 (vs. εt2 = ht + wt) Note that E(εt) = E(ht1/2vt) = E(ht1/2)E(vt) = 0. So:Var(εt │εt-1,…) = E(εt2 │εt-1,…) = E(htvt2 │εt-1,…) = E(ht│εt-1,…) E(vt2 │εt-1,…) = (∑)E(v−+miti12εδηt2) = ∑ −+miti12εδηNotes: 1. εt2 is stationary provided the δ’s satisfy the stationarity condition. In this case, Var(εt) = η/(1-δ1-…-δn) (Why?) and Var(εt+s│εt,…) converges to Var(εt) as t goes to infinity 2. Since εt2 cannot be negative for any t, conditions must be added to the model to guarantee that {εt2} is a nonnegative sequence for all realizations of {vt}. Sufficient conditions: η > 0 and δ’s >