Chapter 6 · Section 5 practice

ARCH, GARCH and Conditional Risk

0 of 6 answered

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

Question 1

For daily Dow Jones returns, which is more predictable from yesterday?

  • The sign (up or down)
  • The size (how big the move is)
  • Both equally
  • Neither — returns are i.i.d.

Question 2

Which pattern will acf(r) and acf(r**2) show at lags 1–5?

  • Both inside the ±0.052 band — returns are white noise, so are their squares
  • Both far outside the band — returns are strongly autocorrelated
  • \(r_t\) mostly inside the band; \(r_t^2\) far outside at every lag
  • \(r_t\) far outside; \(r_t^2\) inside

Question 3

Compared with ARCH(\(q\)), the \(\beta\) term makes the ACF of \(\varepsilon_t^2\)…

  • Cut off sharply after lag 1
  • Decay geometrically at rate \(\alpha+\beta\), never cutting off — GARCH(1,1) is ARCH(\(\infty\))
  • Zero at every lag
  • Constant at every lag

Question 4

Engle’s ARCH-LM test for order \(q\) regresses:

  • \(r_t\) on \(r_{t-1},\dots,r_{t-q}\) and reports the F-test
  • \(|r_t|\) on time and reports the slope
  • \(\hat\varepsilon_t^2\) on \(\hat\varepsilon_{t-1}^2,\dots,\hat\varepsilon_{t-q}^2\) and reports \(LM = nR^2 \sim \chi^2_q\)
  • \(\hat\varepsilon_t\) on \(\hat\varepsilon_t^2\) and reports \(R^2\)

Question 5

\(\hat\alpha = 0.1000\) and \(\hat\beta = 0.8800\) to four decimals. What is the most likely explanation?

  • The Dow really has \(\alpha=0.10\), \(\beta=0.88\) exactly
  • arch rounds to two decimals
  • These are the optimiser’s starting values — it never moved because the data were in decimals
  • Only 1 260 observations, so the estimate is coarse

Question 6

Over 2019–2020 the 1 % VaR should be breached about 5 times (473 × 0.01). Which is closer to 5?

  • Unconditional VaR — a fixed number is more robust
  • Both hit exactly 5
  • GARCH — it re-scales within days; unconditional VaR is breached far more, mostly in March 2020
  • Neither breaches at all