Chapter 5 - Seasonality

Question 1

Write the stationary AR(1) model with |φ1| < 1 and dummies.

Answer 1

y_t − μ_t = φ1(y_(t−1) − μ_t ) + εt

with μ_t = μ1D_(1,t) + μ2D_(2,t) + ··· + μSD_(S,t)
where D_(s,t) = 1 if time t corresponds with season s, and 0 otherwise.

Question 2

What is the unconditional mean E[y_t] = ?

Answer 2

μ_s, when t corresponds with season s.

Question 3

write the AR(1) model with L

Answer 3

(1 − φ1L)(yt − μ1D_(1,t) − μ2D_(2,t) − ··· − μSD_(S,t)) = εt

Question 4

When |φ1| < 1, what can we say about the unconditional mean, the unconditional variance and the autocovariances of y_t over time?

Answer 4

E[y_t] = μ_s,
constant, and constant
=> time Series y_t has deterministic seasonality (in mean), stable seasonal pattern is stable over time.

Question 5

what happens when φ1 = 1in
(1 − φ1L)(yt − μ1D_(1,t) − μ2D_(2,t) − ··· − μSD_(S,t)) = εt
?
How do we call it?

Answer 5

we obtain:
y_t =y_(t−1) + SUM{ μs(D_(s,t) − D_(s,t−1) ; 1:S)+δ+εt

= y_(t−1) + μs − μ_(s-1) + δ + εt,

Random walk with seasonal drift