BMAN71122: Time Series Econometrics Review Questions: III
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BMAN71122: Time Series Econometrics
Review Questions: III
1 Estimation and Testing
1. Prove that the conditional log-likelihood of an ARMA(p,q) model can be expressed as:
where 1 (ξ; x) = - n log 2π - - log 2 , (1)
Interpret this result. 2 = tn21 \2 . (2)
2. Consider a stationarry AR(p) process:
Xt = φ 1 Xt − 1 + . . . + φp Xt −p + Zt , (3)
where {Zt } ~ WN (0, σ2 ).
● Let γ(h) = t [Xt Xt −h ], for h ≥ 0. Derive the Yule-Walker equations:
γ(1) = φ1 γ(0) + φ2 γ(1) + . . . + φp γ(p - 1)
γ(2) = φ1 γ(1) + φ2 γ(0) + . . . + φp γ(p - 2)
(4)
. . . = . . .
γ(p) = φ1 γ(p - 1) + φ2 γ(p - 2) + . . . + φp γ(0)
● Explain, how the Yule-Walker equations can be used for estimating the unknown pa- rameters.
3. Suppose you were trying to differentiate between an AR(1) and an MA(1) but could not estimate any regressions. What would you do?
4. Outline the steps needed to perform a Diebold-Mariano test that two models for the condi- tional mean are equivalent (in the MSE sense).
5. Outline the steps needed to perform a unit root test on as time-series of FX rates. Be sure to detail the any important considerations that may affect the test.
6. Let {Xt } be a trend-stationary time series defined by
where {Yt } is a stationary series. Defined two new time-series {Zt } and {Wt } by:
Zt = Xt - a` - a1 t, and Wt = (1 - B)Xt , (6)
where B denotes the back-shift operator.
● Are {Zt } and {Wt } stationary?
● If yes, which time series ({Zt } or {Wt }) has larger variance? Interpret this result.
2023-05-18