Question

Consider the model Yt = ΦYt−3 + et − θet−1, where et has variance σ 2 .

(a) Identify Yt as a certain SARIMA(p, d, q) × (P, D, Q)s model. That is, specify each of p, d, q, P, D, Q, and s. You may assume that Φ < 1.

(b) Find the variance of Yt .

(c) What are the forecasts for Yt+1 and Yt+4?

(d) What are the error variances for your forecasts above?

(e) If σ 2 = 1, Φ = .7, and θ = −.5, find 95% limits for your forecasts above. You may assume that et are normally distributed. Also, the four most recent values are yt−3 = 0.13, yt−2 = −0.50, yt−1 = 0.38, and yt = 1.53. Similarly, the four most recent et values are 0.08, −0.60, 0.75, and 0.95.

Answer 1

Answer a

SARIMA(p, d, q) × (P, D, Q)S with

1. p = non-seasonal AR order,
2. d = non-seasonal differencing,
3. q = non-seasonal MA order,
4. P = seasonal AR order,
5. D = seasonal differencing,
6. Q = seasonal MA order, and
7. S = time span of repeating seasonal pattern.

Without differencing operations, the model could be written more formally as

Φ(B^S)φ(B)(x_t - μ) = Θ(B^S)θ(B)w_t

Now here we are having

=>

Hence here we are having SARIMA((1,3), 0, 1) × (0, 0, 0)0 that means it is an AMRA model with 1st and 3rd oreder of AR and MA(1)

Answer b

remaining cross product terms will be zero because of definition of Yt and et that et are independently distributed and observations and error terms are independent of each other.

=> by using variance of et

=>

Answer c

When t=t+1 then the equation would be

=>

When t=t+4 then the equation would be

=>

Answer d

When t=t+1,

were mean of et is zero

Now Variance of forecast error would be

Similarly hen t=t+4, and Variance of forecast error would be

Answer d

we are having σ 2 = 1, Φ = .7, and θ = −.5

a 95% prediction interval for the h-step forecast is

where σ^2_h is an estimate of the standard deviation of the h-step forecast distribution.

Yt =0.7 Yt-3 + et+0.5 et-1

				Confedence interval
time	Yt	et	Var(yt)	Lower bound	Upper bound
t-3	0.13	0.08	0.84	-1.5143	1.774295
t-2	-0.50	−0.60	0.84	-2.1443	1.144295
t-1	0.38	0.75	0.84	-1.2643	2.024295
t	1.53	0.95	0.84	-0.1143	3.174295