Applied Econometrics

Applied Econometrics
Seminar: ARIMA and GARCH models
Jaromír Baxa
Jaromir.baxa@centrum.cz

Outline
● ARMA and ARIMA models
● Box-Jenkins method and fitting ARIMA models
● Diagnostics of ARIMA models – ACF, PACF,
formal tests and information criteria
● GARCH models and modelling volatility:
intuition
● ARCH-LM test, fitting GARCH models

ARMA and ARIMA Models
● Just the essence from the lecture:
● AR(p)...y t = μ + Φ 1 y (t-1) +...+ Φ p y (t-p) + ε t ;
● MA(q)...y t = μ + ε t – θ 1 ε t-1 – … – θ q ε t-q
● ARMA (p,q):
y t = μ + Φ 1 y (t-1) +...+ Φ p y (t-p) + ε t - θ 1 ε t-1 -…- θ q ε t-q
● ARIMA (p,d,q):
Δ d y t = μ + Φ 1 Δ d y (t-1) +...+ Φ p Δ d y (t-p) + ε t - θ 1 ε t-1 -…- θ q ε t-q

Box-Jenkins methodology
● Recall: purpose of ARIMA models – univariate
decomposition of time series, goal: extract all
statistically significant components
● Box-Jenkins methodology:
● Identification: stationarity and order of
integration, identify p and q.
● Estimation: use software package and do not
care about the details (usually non-linear least
squares or maximum likelihood method).
● Model Validation: ACF and PACF insignificant,
minimized information criteria.

ARIMA models in JMulti
● Data: PX/PX-50 index and CEZ stock 2001-06
● In a lecture: does the PX50 time series follows
the random walk? The result is that it is not.
● Stationarity? Order of integration to achieve
stationarity? Use the ADF test.
● ACF and PACF of transformed series. What
number of p and q?
● Estimation and analysis of residuals.
● Note: automatic model selection with the
Hannan-Rissanen procedure: h...ĺogT

Diagnostics of ARIMA models
● Resulting residuals should be white noise
● White noise:
=> zero mean and no autocorrelations, thus
Ljung-Box statistics or Portmanteau statistics
insignificant and ACF and PACF bars within
„bands“ - 95% confidence intervals, that the
true value equals zero, usually approximated as
+/- 2/√T.
● Information criteria: to find the best in the tradeoff
between parsimony and R-sq.

Comments to the results
● Still some siginificant autocorrelations
● Reason: heteroscedasticity, non-normality and
other assumptions violated.

ARCH and GARCH models
● Motivation: homoscedasticity is to restrictive
assumption.
● See the plot of squared residuals of CEZ stock
from ARIMA (1,1,1):

GARCH and ARCH - Summary
● ARCH – Autoregressive conditional
heteroscedasticity: variance depends on past
variance
● GARCH – generalized autoreg. cond. het.: variance
is a weighted average of long-run variance (ω), the
last period forecasted var. for this period (σ) and
variance implied by most recent square residual (ε).
● Formally:
● ARCH(q): σ 2 = E(u t ) 2 = ω + α 1 (ε t - 1 ) 2 + …+ α q (ε t - q ) 2
● GARCH(p,q):
σ 2 = ω+α 1 (ε t - 1 ) 2 +…+α q (ε t - q ) 2 + β 1 σ 2
t – 1 + … + β p σ2
t - p.

GARCH - Motivation again...
● ACFand PACF of log returns (logs of first
differences) – almost none significant (very
common for financial time series) => ARIMA
useless
● Conclusion: price development not predictable.

● But: Squares of log returns – variance (under
the assumption of zero mean, which usually
holds) – has significant autocorrelations:
● Clustered volatility, which is persistent and thus
predictable

GARCH in JMulti
● Window ARCH Analysis.
● But first, save the residuals from the ARIMA
model: Model Checking => Plot/Add Residuals
=>Add to dataset.
● Switch to ARCH panel
● Select GARCH or ARCH or TARCH models,
parameters and execute the test.
● Results should be white noise, have no
autocorrelations... (see from diagnostics)
● Note: this two equation method less efficient

● Plots for residuals of CEZ ARIMA (1,1,1)

Some closing considerations
● TGARCH – see the lecture for the formulas –
whether bad piece of news influences volatility.
Here insignificant.
● Several autocorrelations still significant. Why?
1) Always some difficulties with real data. 2)
Perhaps some seasonalities – day of the week
effect. However for this dataset not applicable
as days in the week with no trade were (by me
some months ago – unfortunatelly) deleted.