Question 1
a) Consider the linear regression model with a single regressor xt and an error term ut:
yt = βxt + ut
xt and ut are both mean zero, stationary processes. Data (yt, xt), t = 1, . . . T is observed. Assume T is large. It is known that xt and ut follow a periodic cycle. For 10 observations, from t = 1, E(ut) = 1 and E(xt) = −1, for the next 10 observations, from t = 11, E(ut) = −1 and E(xt) = 1 and so on. Apart from this cycle, xt and ut are independent.
i) [5 points] Will estimating equation (1) yield a consistent estimate of β? Explain your answer.
ii) [8 points] Imagine that you define a periodic dummy variable, zt, which takes a value of 1 for the first ten periods and -1 for the next ten periods, and so on. If you include zt as an additional regressor in equation (1), would your answer to the previous question change?
b) Consider the following ARMA(1,1) model:
yt = 0.8yt−1 + εt + 0.5εt−1. (2) where εt ∼ N (0, 1) and is serially independent..
i) [4 points] Explain why the model in equation (2) is stationary and invertible.
ii) [6 points] Derive the autocovariance function of the model in equation (2).
iii) [7 points] Derive an expression for the jth coefficient in the MA( ) version of the model in equation (2).
Question 2
Consider the following ARCH model:
rt = µ + φht + ηt;
ηt = htεt; εt ∼ iidN (0, 1)
2 = α0 + α1η2 + α2η2 + α3η2 + β1h2
where the coefficients are such that stationarity of ηt and non-negativity of h2 are satisfied. Furthermore, assume that the fourth moment E[η4] exists and does not depend
on time.
MSIN0106 1 TURN OVER
(a) [5 points] Compute the conditional mean and the conditional variance of rt.
(b) [5 points] Show that η2 follows an ARMA process.
(d) [5 points] Show that ηt features excess kurtosis, i.e.
E[η4]
κ = > 3.
E[η2]2
Which stylized fact in the data is consistent with this finding?
(e) [10 points] Set β1 = α2 = α3 = φ = 0.
i Derive an expression for the fourth moment E[η4] as a function of α1.
ii Show that α2 < 1/3 is a necessary condition for the existence of the fourth
moment.
(f) [10 points] Set β1 = α2 = α3 = φ = 0 and α2 < 1/3. i Derive the kurtosis of ηt as a function of α1.
ii Show that the kurtosis of ηt exceeds the kurtosis of εt by the quantity
6α2/{1 − 3α2}.
MSIN0106
Question 3
Define ft+τ|t as the forecast of the variable Yt+τ a the τ −step-ahead horizon, based on the information set Ωt. An economist wishes to produce optimal forecasts for a quadratic loss function and evaluate their optimality.
(a) [5 points] Suppose Yt = 0.5Yt−1 + 0.1Yt−2 + εt + 0.5εt−1 + 0.25εt−2. Derive the optimal one-step ahead forecast and corresponding forecast error for a quadratic loss.
(b) [5 points] Suppose Yt = 0.5Yt−1 + 0.1Yt−2 + εt + 0.5εt−1 + 0.25εt−2. Derive the optimal τ −step-ahead forecast and corresponding forecast error for a quadratic loss, for τ ≥ 2.
(c) Consider the regression
et+τ|t = α + ηt, (3)
where et+τ|t = Yt+τ ft+τ|t are the τ step-ahead forecast errors and ηt is an error term.
(i) [5 points] What is the implication of ft+τ|t being the best forecast (for a quadratic loss function) on the coefficient α in the regression (3)?
(ii) [5 points] How would you test the for this implication?
(iii) [5 points] Which estimator for the variance of αˆ should you use? Why?
(d) [5 points] Now consider the following *alternative* loss function: L(et+τ|t) = exp(et+τ|t) et+τ|t 1. Explain how you would modify equation (3) to test forecast optimality for this alternative loss function.
(e) [5 points] Consider the task of producing a time series of realized forecast errors et+τ|t to use in equation (3). Describe in detail how this be achieved in practice. Describe and discuss the advantages and disadvantages of a recursive versus a rolling estimation scheme for such exercise.
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