Home
/
Today's Focus
Time Series for Quants: ARIMA and GARCH
Medium
·
22 min read
Statistical / ML for Quants
Time Series
ARIMA
GARCH
Volatility Forecasting
1
Article
2
Quiz
Quick Quiz
1.
For a stationary AR(1)
r
t
=
c
+
ϕ
r
t
−
1
+
ε
t
r_t=c+\phi r_{t-1}+\varepsilon_t
r
t
=
c
+
ϕ
r
t
−
1
+
ε
t
with
∣
ϕ
∣
<
1
|\phi|<1
∣
ϕ
∣
<
1
, what are the unconditional variance and the lag-
k
k
k
autocorrelation as
k
→
∞
k\to\infty
k
→
∞
?
Variance
=
σ
ε
2
/
(
1
−
ϕ
2
)
=\sigma_\varepsilon^2/(1-\phi^2)
=
σ
ε
2
/
(
1
−
ϕ
2
)
; autocorrelation
=
ϕ
k
→
0
=\phi^k\to0
=
ϕ
k
→
0
Variance
=
σ
ε
2
/
(
1
−
ϕ
2
)
=\sigma_\varepsilon^2/(1-\phi^2)
=
σ
ε
2
/
(
1
−
ϕ
2
)
; autocorrelation
=
k
ϕ
k
→
0
=k\phi^k\to0
=
k
ϕ
k
→
0
Variance
=
σ
ε
2
/
(
1
−
ϕ
)
=\sigma_\varepsilon^2/(1-\phi)
=
σ
ε
2
/
(
1
−
ϕ
)
; autocorrelation
=
ϕ
k
→
0
=\phi^k\to0
=
ϕ
k
→
0
Variance
=
σ
ε
2
=\sigma_\varepsilon^2
=
σ
ε
2
; autocorrelation
=
ϕ
k
→
0
=\phi^k\to0
=
ϕ
k
→
0
2.
ADF on log-prices gives statistic
−
1.8
-1.8
−
1.8
(p = 0.38); on log-returns it gives
−
28.4
-28.4
−
28.4
(p < 0.001). What do you conclude for model selection?
Both series are stationary — fit ARIMA(0,1,0) on log-prices
Log-prices are stationary; log-returns are not — fit ARIMA(p,1,q) on log-prices
Prices are I(1), returns I(0) — fit ARMA (ARIMA(p,0,q)) on returns
Both series are non-stationary — fit GARCH on log-prices
3.
GARCH(1,1) on daily S&P returns:
ω
^
=
2
×
10
−
6
\hat\omega=2\times10^{-6}
ω
^
=
2
×
1
0
−
6
,
α
^
=
0.08
\hat\alpha=0.08
α
^
=
0.08
,
β
^
=
0.90
\hat\beta=0.90
β
^
=
0.90
. Find (a) unconditional daily variance, (b) shock half-life, (c) the 10-day-ahead conditional variance forecast if
σ
T
2
=
4
×
10
−
4
\sigma_T^2=4\times10^{-4}
σ
T
2
=
4
×
1
0
−
4
.
(a)
2
×
10
−
5
2\times10^{-5}
2
×
1
0
−
5
, (b) 17 days, (c)
1.7
×
10
−
4
1.7\times10^{-4}
1.7
×
1
0
−
4
(a)
1
×
10
−
4
1\times10^{-4}
1
×
1
0
−
4
, (b) 69 days, (c)
4
×
10
−
4
4\times10^{-4}
4
×
1
0
−
4
(a)
1
×
10
−
4
1\times10^{-4}
1
×
1
0
−
4
, (b) 34 days, (c)
2.45
×
10
−
4
2.45\times10^{-4}
2.45
×
1
0
−
4
(a)
1
×
10
−
4
1\times10^{-4}
1
×
1
0
−
4
, (b) 34 days, (c)
≈
3.45
×
10
−
4
\approx3.45\times10^{-4}
≈
3.45
×
1
0
−
4
4.
In GARCH(1,1) with
α
^
+
β
^
=
1
\hat\alpha+\hat\beta=1
α
^
+
β
^
=
1
(IGARCH), the unconditional variance is infinite (non-stationary in variance), yet conditional variance forecasts remain finite and usable for short-horizon forecasting.
True
False
5.
GJR-GARCH adds
γ
1
ε
t
−
1
<
0
ε
t
−
1
2
\gamma\mathbf 1_{\varepsilon_{t-1}<0}\varepsilon_{t-1}^2
γ
1
ε
t
−
1
<
0
ε
t
−
1
2
; equities show
γ
^
>
0
\hat\gamma>0
γ
^
>
0
. What mechanism explains this, and which spec would you use for VaR on a long equity book?
It is a statistical artefact; use Gaussian GARCH for simplicity and regulatory acceptability
Leverage effect: down moves raise vol; use GJR-GARCH with Student-t innovations
Volatility mean-reverts faster after negative shocks; use EGARCH because it guarantees positivity
Positive news raises vol more than negative; use standard GARCH since tail risk is symmetric
6.
BIC selects ARIMA(0,0,0) (white noise) for daily returns; a colleague prefers ARIMA(3,0,2) on lower AIC. Who is right, and what does this reveal?
You are right: AIC is always wrong for financial time series because returns are non-normal
The colleague is right: lower AIC always indicates a better model
You: BIC's
k
ln
T
k\ln T
k
ln
T
penalty exceeds AIC's
2
k
2\,k
2
k
, favouring parsimony
The colleague is right: more parameters always improve out-of-sample forecasting
Submit
←
Article