Question 1. Consider a general time series model of the form

Y_t = m_t + s_t + e_t, (1)

where {m_t} is the trend component of the form mt = θ₀+θ₁ (t-12) + θ₂ (t-12)² with θ_i ≠ 0 (i = 0, 1, 2) being unknown parameters, {s_t} is the seasonal component satisfying s_t+12 = s_t and ∑_j=1¹² s_j = 0, and {e_t} is a sequence of stationary residuals with E[e₁] = 0 and E[e₂] = 1.

(a) Is {Y_t} stationary? Give your reasoning.

(b) Is the first-order differenced version of Y_t, Z_t = ∇₁₂Y_t = Y_t - Y_t-12, stationary ? Give your detailed reasoning.

(c) Is the second-order differenced version of Y_t, W_t = ∇²₁₂ Y_t = (1 - B¹²)²Y_t, stationary ? Give your detailed reasoning.

Question 2. The stationary process {Z_t} is said to be white noise with mean 0 and variance σ², written

Z_t ~ WN (0, σ²),

if and only if {Z_t} has zero mean and covariance function given by

            σ²   if h = 0

Υ(h) =

            0    Otherwise.

(a) Consider an ARMA(1,1) model of the form

X_t = φX_t-1 + Z_t + θZ_t-1,

where |φ| < 1 and |θ| < 1.

Find the auto-correlation coefficient function (ACF) of {X_t}, ρ(k), for

k = 1, 2, .....

(b) Consider an MA(1) model with drift of the form X_t = µ + Z_t + θZ_t-1.

Find the ACF. Does it depend on µ?

(c) Consider a time series model of the form: (1 - B)(1 - 0.2B)X_t = (1 - 0.5B)Z_t.

Classify the model as an ARIMA(p,d,q) model (i.e., give your reasoning for the specification of (p, d, q)).

Question 3. Consider an auto-regressive model of order one (AR(1)) of the form

X_t = φX_t-1 + Z_t, (2)

where {Z_t} is a sequence of white noises with E[Z_t] = 0 and 0 < σ² = E[Z_t²], and |φ| < 1 is an unknown parameter.

(a) Derive the autocorrelation function ρ(k) for all k ≥ 1.

(b) State the necessary and sufficient condition such that {X_t} is stationary.

(c) Give some detailed description for each of the possible estimation methods you have learned.

(d) Write down the corresponding code functions from R for the possible estimation methods to be implemented in R.

(e) Using at least one of the estimation methods, write down some detailed formulae for the estimators of the unknown parameters φ and σ².

Question 4. (a) Let {Z_t} be a sequence of random errors satisfying

E[Z_t|F_t-1] = 0

In addition, we allow for a heteroscedastic structure of the form

Z_t² = α₀ + α₁Z²_t-1 + u_t,

where {u_t} is a sequence of white noises, and α₀ > 0 and α₁ ≥ 0.

The process {Z_t} satisfying (3)-(4) is called an auto-regressive conditional heteroscedastic model of order one, simply, ARCH(1).
- Rewrite model (4) as an auto-regressive model of order one (AR(1)).

- Give some detailed description for each of the possible estimation methods you have learned.

- Write down the corresponding code functions from R for the possible estimation methods to be implemented in R.

- Under the conditions: 0 < α₁ < 1 and 3α₁² < 1, find the second and fourth moments:

E[Z_t²] and E[Z_t⁴].

(b) Let {Z_t} be a sequence of random errors satisfying

E[Z_t|F_t-1] = 0.

In addition, we allow for a heteroscedastic structure of the form

E [Z_t²|F_t-1] = h_t = α₀ + Σ_i=1^r1α_iZ²_t-i + Σ_j=1^r2β_jh_t-j.

The process {Z_t} satisfying (5)-(6) is called a generalized auto-regressive conditional heteroscedastic model of order (r₁, r₂), simply, GARCH(r₁, r₂). Consider a GARCH(1,1) model of the form

Z_t² = α₀ + (α₁ + β₁)Z²_t-1 + u_t - β¹u_t-1

where ut ~ WN (0, σ²).

- Find the conditions such that Z_t² is stationary and 0 < E[Z_t²] < ∞.

Question 5. (a) Consider a seasonal ARIMA (SARIMA) model of the form

φ₂(B)Φ₃(B¹²)Y_t = θ₁(B)Θ₂(B¹²)Z_t, (8)

where B denotes the backward shift operator, φ₂, Φ₃, θ₁ and Θ₂ are polynomials of order 2, 3, 1 and 2, respectively, {Z_t} ∼ WN (0, σ²), and Y_t = ∇²∇₁₂² X_t = (I - B)²(I - B¹²)²X_t. This model is called a SARIMA model of order (2, 2, 1) × (3, 2, 2)₁₂ for {X_t}.

- Does {Y_t} follow an ARIMA model of ARIMA(b₁, b₂, b₃) ? If so, can you specify the values of bi for i = 1, 2, 3 ?

- Does Wt = ∇₁₂² X_t follow an ARIMA model of ARIMA(c₁, c₂, c₃) ? If so, can you specify the values of ci for i = 1, 2, 3 ?

- Based on your own understanding and experience, write down the main steps for you to identify and then estimate a seasonal ARIMA model of the form X_t ∼ SARIMA(2, 2, 1) × (3, 2, 2)₁₂.

(b) The real data set USAccDeaths was fitted by a seasonal ARIMA model with the following summary:

> USAccDeaths
> usa.arima1<-arima(USAccDeaths, order=c(0,1,1), seasonal = list(order=c(0,1,1), period =12))
> usa.arima1

Call:
arima(x = USAccDeaths, order = c(0, 1, 1),

seasonal = list(order = c(0, 1, 1), period = 12))

Coefficients:

ma1          sma1

-0.4303     -0.5528

s.e.        0.1228      0.1784

sigma^2 estimated as 99347:

log  likelihood = -425.44,      aic = 856.88

> usa.fore<-predict(arima(USAccDeaths, order =

c(0,1,1),   seasonal   =   list(order=c(0,1,1),

period =12)),       n.ahead = 12)

$pred

Jan               Feb               Mar                  Apr               May              Jun

8336.061   7531.829        8314.644      8616.868      9488.912      9859.757

Jul                  Aug                Sep               Oct               Nov              Dec

10907.470   10086.508      9164.958     9384.259      8884.973      9376.573

$se

Jan               Feb               Mar             Apr               May              Jun

315.4481   363.0054     405.0164     443.0618    478.0891     510.7197

Jul                Aug               Sep              Oct               Nov              Dec

541.3871     570.4081    598.0224     624.4167    649.7397     674.1121

> ts.plot(window(USAccDeaths,1973-1978), usa.fore$pred, usa.fore$pred + 2*usa.fore$se, usa.fore$pred - 2*usa.fore$se)

Using the summarized information given above, answer the following questions:

- Which seasonal ARIMA model was used ? Give your identification of (p, d, q) × (P, D, Q)s.

- Write down an explicit expression for the fitted model.