[2019] STAT 153 Introduction to Time Series - Midterm Exam - Q4 Estimate Trend Function
4. A scientist considers the model Xt = mt + st + Wt for some time series data, where mt = a t + b is a linear trend function with parameters a, b and st is a seasonal component with period 2, that is, st = st+2 for all t. Wt is some zero mean stationary process.
(a) First, the scientist wants to estimate the trend function mt using a filter of the form 1 + αB + βB2 + γB3, where B denotes the backshift operator and α, β, γ are parameters. How should she chose the parameters α,β,γ such that the filtered time series is an unbiased estimator of the trend mt, that is, E((1 + αB + βB2 + γB3)Xt) = mt?
Hint: First, argue that without loss of generality you can assume that s1 + s2 = 0. (5 Points)
(b) Is Xt a stationary process? Explain. s (1 Points)
(c) Propose a transformation using differencing to make the process stationary. Explain. (3 Points)
(d) For the stationary process Wt the scientist considers two different models:
• an MA(1) model,
• an AR(1) model.
For both of these choices identify the transformed data from (4c) as some ARMA model.
Hint: It is enough to state the orders of the respective ARMA models with explanation. (6 Points)
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