Chapter 6 · Section 1 practice

Stationarity, Dependence and ARIMA

These questions review the examples in the companion slides. Choose an answer, then check the explanation.

Question 1

Which series looks stationary — constant mean and variance over the day?

The close price — it stays between 108 and 111 all day
The minute return — it oscillates around zero with a stable spread
Both — the day is too short for either to drift
Neither — intraday data is never stationary

Question 2

\(y_t = 0.5\,t + 3\varepsilon_t\). You difference it instead of detrending. What do you get?

A random walk — differencing creates a unit root
A stationary series, but with a larger variance than the detrended one (over-differenced)
Exactly the same residual as detrending
A series with a deterministic trend of slope 1

Question 3

For the close price: KPSS(c) p = 0.01, ADF(c) p = 0.035. What is the honest verdict?

Stationary — ADF rejects at 5 %
Unit root — KPSS rejects
Conflict: near-unit-root; do not model the level, model the returns
The tests are broken on intraday data

Question 4

For monthly airline passengers (144 months, 1949–1960), which is larger: the ACF at lag 6 or at lag 12?

Lag 6 — closer in time is always more correlated
Lag 12 — same month last year beats opposite season
They are equal — the series is a random walk
Both are ≈ 0 — passengers are unpredictable

Question 5

\(p, q \in \{0,1,2\}\), \(d=0\), chosen by AIC on the Dow returns above. Which order wins?

(0, 0, 0) — a constant; the market has no linear memory worth two parameters
(1, 0, 0) — yesterday’s return matters
(2, 0, 2) — the richest model always fits best
(0, 0, 1) — bid–ask bounce shows up as an MA(1)

Question 6

Best by AIC: (2, 1, 1), AIC −228.9. RMSE over 1960: ARIMA(2,1,1) versus the seasonal-naive forecast (1959’s value for the same month). Who wins?

ARIMA, easily — it has a likelihood
Roughly a tie
Seasonal-naive, by a wide margin — the ARIMA has no lag-12 term
Neither can forecast 12 months ahead