F500 Problem Set 3 Oliver Linton University of Cambridge 1. Slowly varying expected returns. Suppose that rt+1 = xt + “t+1, where xt+1 = µ+ “xt + #t+1, !1 < " < 1 and "t, #s are mutually independent for all t, s and are individually iid with mean zero and variances $2" and $ 2 " . Calculate Ert, var(rt), and the autocorrelation function %(k) = cov(rt, rt!k) var(rt) . Is this consistent with the empirical evidence regarding the auto- correlation of return series? Also calculate the best linear predictor of rt+1 given rt. How does this change if cov("t, #t) = $"" 6= 0. 2. Blanchard and Watson (1982) model. Suppose that Bt+1 = ! 1+R # Bt + &t+1 with probability ' 0 with probability 1! ' where &t is iid with mean zero and variance one. What are the properties of the bubble process? What isEtBt+1?What is vartBt+1? What is the chance that the bubble lasts for more than 5 periods? Suppose that we observed prices that satisfy Pt = P " t +Bt P "t = P " t!1 + ut 1 where ut is normally distributed with mean zero and variance one. How would you test the null hypothesis of no bubble versus the alternative that there is a bubble in your sample {P1, . . . , PT}? What if instead Bt+1 = ! 1+R # Bt + &t+1 with probability ' &t+1 with probability 1! ' 3. Suppose that rt = %rt!1 + "t. We can write (1! %L)rt = "t Consider the process r2t . We have r2t = % 2r2t!1 + " 2 t + 2%rt!1"t so that (1! %2L)r2t = $2" + ut, where ut = "2t !E"2t +2%rt!1"t is a martingale di§erence sequence. It follows that r2t = $2" 1! %2 + 1X j=0 #jut, where # = %2. It follows that corr(r2t , r 2 t!j) = # j = %2j. It follows that for this process the autocorrelation of the squares dies out faster than the autocorrelation of the original series. This is not supported by the data. Suppose that returns follow a moving average process rt = "t + )"t!1, where "t # N(0,$2"). Recall that corr(rt, rt!j) = ! $ (1+$2) if j = 1 0 else. Suppose that returns are normally distributed, ie var("2t ) = 2$ 4 ". Then show that corr(r2t , r 2 t!j) = ( $2 (1+$2)2 if j = 1 0 else. 2 4. Consider the GARCH (1,1) model: rt = h 1/2 t &t ht = ! + +ht!1 + ,r2t!1 where ht is the conditional variance of time t returns. For nota- tional convenience assume that the asset’s expected return equals zero. a) Explain the restrictions on the parameters of the GARCH(1,1) model required to ensure that the long-run unconditional variance exists, b) Describe the unconditional variance in terms of these parameters, c) Discuss how the values of the parameters a§ect the persistence of the response of dynamic volatility to a return shock. 3
