# Time Series Models Cointegration Granger Causality and ARCH Interbank Rates

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```					Stochastic Volatility Models

Course: Applied Econometrics
Lecturer: Zhigang Li
Why Study Stochastic Volatility?
 Heteroskedasticity can affect the standard error
estimates of model parameters
 We can address this using robustness estimators.
 Risk is a key factor in many decision process (e.g.
investment). Risk changes over time and this often
reflect on instable variance in the disturbance of
models with price measures as dependent variables.
 Capital Asset Pricing Model (CAPM) suggests a
natural relationship between expected returns and
volatility of returns.
 Option pricing depends critically on the risks of
assets.
 News in the financial market may reflect on changes
in the volatility of price measures.
Autoregressive Conditional
Heteroskedasticity (pp. 416 or 438)
 First-order ARCH (or ARCH(1))model
ut2=α0+α1ut-12+vt
 U is the error term of a typical regression
model and v is the “error term of the error
term u”.
 The model implies that the variance of the
error term u is correlated over time, so
called “volatility clustering”.
 ARCH is just one particular form of
heteroskedasticity.
Generalized ARCH Model
 GARCH(p,q)
 Yt=μ+ut
 ut=ht1/2vt
 v is i.i.d. with standard normal
distribution
 ht=α0+Σ1qαiu2t-i+Σ1pβjht-j
 GARCH(0,q) model is an ARCH (q)
model.
Why ARCH-Type Models?
 ARCH-Type models are easy to
estimate and interpret. (There are
other volatility models, but the ARCH-
Type is generally much easier to
implement)
 Knowledge of future risk is useful for
optimal decision at the current stage.
Currency Board Reforms and Interbank
Market of Hong Kong (Tse and Yip, 2003)
 A CBS scheme is introduced in Hong Kong
on 17 October 1983, fixing the exchange
rates between HK\$ and US\$. Seven major
reforms have been made to the CBS
system since them.
 What are the effects of the reforms on the
quality and stability of the financial sector
of Hong Kong?
 The impact of the mean and variance of the
interest rate differential between Hong Kong and
the US.
Shocks to CBS
 Oct 1983: Account Aggrangements
introduced to limit the ability of HSBC to
create money.
 Jul 1992: Liquidity Adjustment Facility
introduced to cap the variation of interest
rates.
 Mar 1994: Change target from interbank
liquidity to interbank interest rate.
 Oct 1997: Financial Crisis
 Sept 1998: Reform packages
Empirical Strategy
 Data
 Daily observations of the HK and US
interbank interest rates (from
Datastream).
 Model
yt=δ1D1t+…+δ7D7t+φ1yt-1+…+φ1yt-p+εt
ζt2=γ1D1t+…+γ7D7t+αε2t-1+βζ2t-1
Meteor showers or Heat Waves?
(Engle et al., 1990)
 Heat waves: Domestic news affects
only local financial markets.
 Meteor showers: Domestic news can
affect foreign financial markets.
 Market failure
 Need time to absorb news into trading
prices
Test Framework
 N nonoverlapping markets within a day
with market 1 open first.
 Let εi,t be the intra-day exchange rate
change divided by the square root of
business hours in market i on date t.
 A modified GARCH model
 Hi,t=ωαi+βjihi,t-1+Σ1i-1αijε2j,t+Σinαijε2j,t-1
 If αij=0 for i different from j, then the heat
waves model is supported. Otherwise, the
meteor shower model is supported.
Empirical Strategy
 Intra-day yen/dollar exchange rate from
Oct 3, 1985 to Sep 26, 1986.
 Four markets
   Tokyo
   Pacific
   New York
   Europe
 Three tests
 Heat waves
 Meteor showers with foreign news
 Meteor showers with country-specific news
Computer Exercise
 Example 12.9 (Using NYSE weekly
stock returns data)

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