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# 时间序列分析代写 Time Series Analysis代写 Exam代写

157

## Final Examination

Duration: 3hours

Examination aids allowed: Non-programmable scientific calculator, open book/notes

### Instructions:

Answer all questions directly on the examination paper; use the last pages if you need more space, and provide clear pointers to your work.

Show your intermediate work, and write clearly and legibly.

### 1. 时间序列分析代写

Consider the time series Xt = 5 + Wt + .5Wt−1 − .25Wt−2, where Wt ∼ WN(0, 1).

(a) (4 points) The series follows an ARMA(p, q) model. Find the order of the model (i.e. p, q) and determine whether it is stationary and/or invertible.

(b) (4 points) Find the ACF of the series.

### 3. 时间序列分析代写

The time series {Xt} follows a zero-mean SARIMA(0, 1, 0) × (1, 0, 0)[3] model with a single parameter Φ and an i.i.d. Normal(0, σ2 ) white noise sequence {Wt}.

(a) (4 points) Write down the linear equation describing the evolution of Xt based on its past (Xt1, Xt2, . . .) and the white noise (Wt , Wt1, . . .).

(b) (4 points) Can you find a causal representation for Xt ? (If yes, provide the representation; if no, explain why not.)

(c) (4 points) Find the ACF of Xt = Xt Xt1 .

(d) (8 points) Write the conditional likelihood, given X0 = 0, of the first 4 observations of the series (x1, . . . , x4), expressed as a function of the parameters Φ, σ2 and the values x1, . . . , x4.

### 4. 时间序列分析代写

Consider two jointly stationary time series {Xt , Yt}, with individual auto-covariance functions γX(h),γY (h), ∀h ≥ 0 and cross-covariance function γX,Y (h), ∀h ∈ Z.

(a) (7 points) Find the Best Linear Predictor (BLP) of Yt given Xt , and its Mean Square Prediction Error (MSPE), expressed in terms of γX(h),γY (h), γX,Y (h).

(b) (13 points) Find the BLP of Yt given Xt , Xt1, Yt1, and its MSPE, expressed in terms of γX(h),γY (h), γX,Y (h).

(Note: you don’t need to solve the system of equations defining the BLP coefficients.)

### 5. 时间序列分析代写

Consider a zero-mean (i.e. no drift) random walk Xt = Xt−1 + Wt , t ≥ 1, where X0 = 0 and Wt ∼WN(0,σ2 ).

The prev:

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