SAAPM

Session 7
26th May, 2025

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ARDL model for mixed Variables

https://www.rdocumentation.org/packages/ARDL/versions/0.
2.4

Natsiopoulos, Kleanthis, & Tzeremes, Nickolaos G. (2022). ARDL

bounds test for cointegration: Replicating the Pesaran et al. (2001)
results for the UK earnings equation using R. Journal of Applied
Econometrics, 37(5), 1079-1090. https://doi.org/10.1002/jae.2919
Pesaran, M. H., Shin, Y., & Smith, R. J. (2001). Bounds testing
approaches to the analysis of level relationships. Journal of Applied
Econometrics, 16(3), 289-326

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 Volatility Within the Context of Our Forecasting Problem

The Forecasting Problem

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 Time-Varying Dispersion: Empirical Evidence

U.S. Real GDP with Volatility Bands

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Conditional Heteroscedasticity:ARCH-GARCH
 Volatility
Although volatility is not directly observable, it
has some characteristics that are commonly seen in
financial variables, for example.
First, there exist volatility clusters (i.e.,
volatility is high for certain time periods and low
for other periods).
Second, volatility evolves over time in a continuous
manner–that is, volatility jumps are rare.
Third, volatility does not diverge to infinity–that
is, volatility varies within some fixed range.
Statistically speaking, this means that volatility
is often stationary.

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 Is There Time Dependence in Volatility?

Figure Time Series of SP500 Index, Yen/Dollar Exchange Rate, and

10-year Treasure Note Yield

Table Unit Root Testing: Value of the Dickey-Fuller Test

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 Figure Transformations of Weekly Returns to the SP500 Index
and Their Autocorrelations

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 Figure Transformations of Daily Returns to Yen/Dollar Exchange Rate
and Their Autocorrelations

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 Figure Transformations of Daily Returns to the 10-Year Treasury Notes
and Their Autocorrelations

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Naked Eye Observations
• Plots of all indices show that volatility
clustering.
• Large (Small) shocks followed by large (Small)
shocks.
• Lots of large observations implying lots of
observations are on the tail of the corresponding
distribution. So Distributions are of THICK TAILS.
• High kurtosis coefficients---thick tails.
• suggest Non-Normality of these distributions.
• Squared variables are auto-correlated---non-linear
dependence.

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Implications
• As non-linear dependence is there, there is
opportunity to improve the underlying model.

• Motivation: the linear structural (and time series) models cannot

explain a number of important features common to much of the real
world data, particularly true for financial data
- leptokurtosis
- volatility clustering or volatility pooling

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Time dependence of volatility
we need to determine what models we could propose to
capture time dependence of variances.
use a model which does not assume that the variance is constant:
Recall the definition of the variance of ut:

We usually assume that E(u ) = 0

t
t =Var(ut  ut-1, ut-2,...) = E[ut  ut-1, ut-2,...].
2 2

What could the current value of the variance of the errors plausibly depend
upon?
Previous squared error terms.
This leads to the autoregressive conditionally heteroscedastic model for the
variance of the errors:

t2 = 0 + 1 ut21
This is known as an ARCH(1) model.
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 Autoregressive Conditionally Heteroscedastic
(ARCH) Models (cont’d)

• The full model would be

yt = 1 + 2x2t + ... + kxkt + ut , ut  N(0, t )
2

where t = 0 + 1 ut 1
2 2

• We can easily extend this to the general case where the error variance
depends on q lags of squared errors:
t2 =  +  ut21+ ut  2+...+ ut2 q
2
0 1 2 q
• This is an ARCH(q) model.

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Testing for “ARCH Effects”

1. First, run any postulated linear regression of the form given in the equation
above, e.g. yt = 1 + 2x2t + ... + kxkt + ut
saving the residuals, û.t

2. Then square the residuals, and regress them on q own lags to test for ARCH
of order q, i.e. run the regression
uˆt2   0   1uˆt21   2uˆt2 2  ...   quˆt2 q  vt
where vt is iid.
Obtain R2 from this regression

3. The test statistic is defined as TR2 (the number of observations multiplied by the
coefficient of multiple correlation) from the last regression, and is distributed as
a 2(q).

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Testing for “ARCH Effects” (cont’d)

4. The null and alternative hypotheses are

H0 : 1 = 0 and 2 = 0 and 3 = 0 and ... and q = 0
H1 : 1  0 or 2  0 or 3  0 or ... or q  0.

If the value of the test statistic is greater than the critical value from the 2 distribution, then reject
the null hypothesis.

• Note that the ARCH test is also sometimes applied directly to the variables instead of the residuals
from Stage 1 above.

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MLE for ARCH parameters along
with mean equation
OLS is possible, though MLE is preferred because of joint estimation.

𝑚𝑎𝑥𝜃 log f(𝑟𝑇 , 𝑟𝑇−1 , … … . , 𝑟2 , 𝑟1, 𝜃) 𝑤ℎ𝑒𝑟𝑒 𝜃 = {𝜇, 𝛼0 , 𝛼1 ) 𝑓𝑜𝑟 𝐴𝑅𝐶𝐻(1)

𝐴𝑠𝑠𝑢𝑚𝑖𝑛𝑔 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑡𝑦 𝑜𝑓 𝜀𝑡 ~𝑁(𝜇𝑡 |𝑡 − 1, 𝜎𝑡2 |𝑡 − 1

𝑚𝑎𝑥𝜃 log 𝑓( f(𝑟𝑇 , 𝑟𝑇−1 , … … . , 𝑟2 |𝑟1; 𝜃) =𝑚𝑎𝑥𝜃 σ𝑇𝑡=2 log 𝑓( 𝑟𝑡 |𝐼𝑡−1 )

{𝑟𝑡 −𝜇𝑡 |𝑡−1 }2

= 𝑚𝑎𝑥𝜃 . [-(T-1)/2 log 2𝜋-1Τ2 log 𝜎𝑡2 |𝑡−1 − 1Τ2 σ𝑇𝑡=2( )
𝜎𝑡2 |𝑡−1

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Forecasting with ARCH
with an information set up to time t, the 1-step-ahead
variance forecast is

For any h > 1, using backward substitution we find that the

h-step-ahead forecast of the conditional variance is

Unconditional

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Problems with ARCH(q) Models

• How do we decide on q?
• The required value of q might be very large
• Non-negativity constraints might be violated.
• When we estimate an ARCH model, we require  i /𝛼𝑖 >0 (whatever notation
you use)  i=1,2,...,q (since variance cannot be negative)

• A natural extension of an ARCH(q) model which gets around some of these problems is a GARCH
model.

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Generalised ARCH (GARCH) Models

• Due to Bollerslev (1986). Allow the conditional variance to be dependent upon

previous own lags
• The variance equation is now
(1)
t = 0 + 1 ut 1 +t-1
2 2 2

• This is a GARCH(1,1) model, which is like an ARMA(1,1) model for the

variance equation.
• We could also write t-12 = 0 + 1ut22 +t-22
t-22 = 0 + 1 ut23 +t-32

• Substituting into (1) for t-12 :

 t2 = 0 + 1 ut21 +(0 + 1ut22 +t-22)

= 0 + 1 ut21 +0 + 1ut22 +t-22

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 Generalised ARCH (GARCH) Models (cont’d)

• Now substituting into (2) for t-22

t2 = 0 + 1 ut21 +0 + 1ut22 +2(0 + 1ut23 +t-32)

t2 = 0 + 1 ut21 +0 + 1ut22 +02 + 12ut23 +3t-32
t2 = 0 (1++2) + 1 ut21 (1+L+2L2 ) + 3t-32
• An infinite number of successive substitutions would yield

t2 = 0 (1++2+...) + 1 ut21 (1+L+2L2+...) + 02

• So the GARCH(1,1) model can be written as an infinite order ARCH model.

• We can again extend the GARCH(1,1) model to a GARCH(p,q):

q p
t2 =  0   i u    j t  j
2 2
t i
i 1 j 1

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 Generalised ARCH (GARCH) Models (cont’d)

• In general a GARCH(1,1) model will be sufficient to capture the volatility clustering in the data.

• Why is GARCH Better than ARCH?

- more parsimonious - avoids over-fitting
- less likely to breech non-negativity constraints

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 The Unconditional Variance under the GARCH
Specification 2
t = 0 + 1 ut 1 +t-1
2 2

• The unconditional variance of ut is given by

0
Var(ut) =
1  (1   )

when 1   < 1

1    1
• is termed “non-stationarity” in variance

1   = 1
is termed intergrated GARCH

• For non-stationarity in variance, the conditional variance forecasts will

not converge on their unconditional value as the horizon increases.

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 • If the AR polynomial of the GARCH
representation has a unit root,
I-GARCH then we have an IGARCH model.
• Thus, IGARCH models are unit-root
GARCH models. Similar to ARIMA
models, a key feature of IGARCH
models is that the impact of past
squared shocks is persistent.
• The I-GARCH model is therefore
written as:

The unconditional variance of at, is not defined under the above

IGARCH(1,1) model.. From a theoretical point of view, the IGARCH
phenomenon might be caused by occasional level shifts in volatility.

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 Forecasting Variances using GARCH Models

If the forecast horizon is very large, and for α + β < 1, the h-step-ahead
forecast becomes

UNCONDITIONAL
VARIANCE
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GARCH-in Mean

• We expect a risk to be compensated by a higher return. So why not let the return of a security be
partly determined by its risk?

• Engle, Lilien and Robins (1987) suggested the ARCH-M specification. A GARCH-M model would
be
yt = m + t-1+ at , at  N(0,t )
2

t2 = 0 + 1 at21 +t-12

•  can be interpreted as a sort of risk premium.

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