FA_Unit 4 (Time Series Analysis E-views).pdf

Background to Time Series
Analysis
MBAE 0049, Financial Analytics (Instructor - Dr. Ankit Saxena)
GLA University, Mathura
2

Volatility vs.
Risk
• Volatility refers to the upward and downward trend
swings in market indices / interest rates, on which an
investor will have little or no control at all.
• Whereas Risk, on the other hand, is a personal matter
like possibility or chance of an injury, loss or hazard, or
how much financial uncertainty an investor can
tolerate.

• Volatility
• Risk
Volatility vs.
Risk
MBAE 0049, Financial Analytics (Instructor - Dr. Ankit Saxena) 4

Volatility vs.
Risk
• A volatility in stock market can be measured in
multiple ways. Standard deviation indicates how
much a stock market index varies from its
average, both on the upside and the downside.
• But in case of risk there is no such measure to
calculate.
• In financial terminology risk refers to the
potential permanent loss of money. Technically
risk tolerance means different things to
different people.

Is volatility good? 6
• Volatility finds its extensive application in the domain of equity investing.
• The risk-averse usually opts for a diversified equity portfolio as against the risk-
seeker who prefers a portfolio inclined towards small caps.
• In order to maximize returns, volatility can be extremely helpful. But how one takes
it makes all the difference.
• Closely following market valuations and instances of high volatility may help you to
make wise decisions as regards diversification, asset allocation, and rebalancing.
• Volatility does not imply risk of loss. Volatility simply refers to the price fluctuation.

Types of volatility

1. Historic
Volatility
Volatility in its most basic form represents daily
changes in stock prices.
We call this historical volatility (or historic
volatility) and it is the starting point for
understanding volatility in the greater sense.
Historic volatility is the standard deviation of
the change in price of a stock or other financial
instrument relative to its historic price over a
period of time.

2. Implied Volatility
The options market is a bid and offer system in which buyers and sellers come together in an
auction environment to actuate price discovery and execute trades. These prices are quoted in
dollars and cents.
From these prices, knowing all of the other Black-Scholes variables and using the Black-Scholes
formula, we can calculate the volatility, which is implicit from a traded price or the bid and
offer.
This is referred to as the option's implied volatility.
Whereas historic volatility is static for a fixed given period of time, please note that implied
volatility will vary for a stock based on different options strike prices. This is referred to as the
volatility skew.
9

3. Volatility Indices 10
• This concept is taken one step further. For many indices, a volatility index has been
created and is commonly quoted in the financial media. The three most common
ones:
• S&P 500 Volatility Index (VIX)
• S&P 100 Volatility Index (VXO)
• Nasdaq 100 Volatility Index (VXN)
• These volatility indices are a weighted average of the implied volatilities for several
series of options (puts and calls). Many market participants and observers will use
these indices as a gauge of market sentiment.

4. Intraday Volatility 11
• Finally, we have intraday volatility.
• This represents the market swings during the course of a trading day and
is the most noticeable and readily available definition of volatility.
• A common mistake is equating intraday volatility with the implied
volatility index. Both of these forms of volatility are not interchangeable,
but do carry their own importance in ascertaining investor sentiment and
expectations.

Summary
• the movement of an asset or asset class
relative to itself;
Historical
Volatility
• volatility that is embedded in an option
price;
Implied
Volatility
• a weighted average of implied volatilities
for options on a particular index;
Volatility Index
• the price movements in a stock or index
on or during a given trading day.
Intraday
Volatility
12

Statistical Stationarity
A stationary time series is one
whose statistical properties such
as mean, variance,
autocorrelation, etc. are all
constant over time.
Most statistical forecasting
methods are based on the
assumption that the time series
can be rendered approximately
stationary (i.e., "stationarized")
through the use of mathematical
transformations.
A stationarized series is relatively
easy to predict: you simply
predict that its statistical
properties will be the same in the
future as they have been in the
past!
The predictions for the
stationarized series can then be
"untransformed," by reversing
whatever mathematical
transformations were previously
used, to obtain predictions for
the original series.
Thus, finding the sequence of
transformations needed to
stationarize a time series often
provides important clues in the
search for an appropriate
forecasting model.

Unit root
(Dickey-Fuller)
and Stationarity
tests on time
series
• A time series Y_t (t=1,2...) is said to
be stationary (in the weak sense) if
its statistical properties do not vary
with time (expectation, variance,
autocorrelation).
15

Stationarity Tests
Stationarity tests allow verifying whether a series is stationary or
not.
There are two different
approaches:
Stationarity Tests such as the KPSS test that consider as
•null hypothesis H0 that the series is stationary, and
Unit Root Tests, such as the Dickey-Fuller test and its augmented
version, the augmented Dickey-Fuller test (ADF), or the Phillips-Perron
test (PP),
•for which the null hypothesis is on the contrary that the series
possesses a unit root and hence is not stationary.
16

Interpreting the results of an ADF test, a
PP test and a KPSS test (example on
stationary series)
MBAE
0049,
Financial
Analytics
(Instructor
-
Dr.
Ankit
Saxena)
17

• We can see that the three
tests agree for these series. For
the first series, both the ADF
and the PP rejects the null
hypothesis that the series is
autocorrelated with (r=1) and
retains the alternative
hypothesis that it is stationary,
and the KPSS test keeps the null
hypothesis that the series is
stationary.

• For the second series, the p-
values are not as low (ADF test)
or high (KPSS test) as it was
with the first sample but the
same conclusions are retained.

Interpreting the results of an ADF
test, a PP test and a KPSS test
(example on non-stationary series)

Performing Unit
Root Tests In
EViews
• To begin,
• double click on the series
name to open the series
window,
• and
• choose View/Unit Root
Test…

• For the time series in
columns F (see results on sheet
Dickey-Fuller|Phillips-Perron3),
we change the alternative
option to explosive for the ADF
test and to trend for the KPSS
test. The KPSS tests lead to the
conclusion that the series is not
stationary while both unit root
tests reject the hypothesis of a
unit root in the data generation
mechanism.

Correlogram & Q-
Statistics

Correlogram
• This view displays the
autocorrelation and partial
autocorrelation functions up to the
specified order of lags.
• These functions characterize the
pattern of temporal dependence in
the series and typically make sense
only for time series data.
• When you
select View/Correlogram… the Cor
relogram Specification dialog box
appears.

Johansen Co Integration
Test

Johansen Co-integration Test
Johansen’s test is a way to determine if three or more time series are
cointegrated. More specifically, it assesses the validity of a cointegrating
relationship, using a maximum likelihood estimates (MLE) approach. It is also
used to find the number of relationships and as a tool to estimating those
relationships (Wee & Tan, 1997).
There are two types of Johansen’s test: one uses trace (from linear algebra), the
other a maximum eigenvalue approach (an eigenvalue is a special scalar; When
you multiply a matrix by a vector and get the same vector as an answer, along
with a new scalar, the scalar is called an eigenvalue).
26

MBAE 0049, Financial Analytics (Instructor - Dr.
Ankit Saxena)
27
Causality granger test
- A test to check
interdependency

Granger Causality Test
The Granger causality test is a statistical hypothesis test for determining whether
one time series is useful in forecasting another
Ordinarily, regressions reflect "mere" correlations, but Clive Granger argued that
causality in economics could be reflected by measuring the ability of predicting the
future values of a time series using past values of another time series.
Since the question of "true causality" is deeply philosophical, econometricians
assert that the Granger test finds only "predictive causality"

Explanation
• Granger defined the causality relationship
based on two principles
• The cause happens prior to its effect.
• The cause has unique information
about the future values of its effect.
MBAE
0049,
Financial
Analytics
(Instructor
-
Dr.
Ankit
Saxena)
29

Granger Causality Test: Y = f(X)
Model Res.DF Diff. DF F p-value
Complete model 2
Reduced model 3 -1 0.318 0.629
Granger Causality Test: X = f(Y)
Model Res.DF Diff. DF F p-value
Complete model 2
Reduced model 3 -1 1.280 0.375
30

Autoregressive-
Integrated-
Moving
Average
31

ARIMA
• The term Arima and Box-Jenkin are use
interchangeably
• The purpose of ARIMA modeling is to establish a
relationship between the present value of a time
series and its past values so that forecasts can be
made on the basis of the past values alone.
• ARIMA stands for Autoregressive- Integrated-
Moving Average. The letter "I” (Integrated) indicates
that the modeling time series has been transformed
into a stationary time series.
• ARIMA represents three different types of models: It
can be an AR (autoregressive) model, or a MA
(moving average) model, or an ARMA which includes
both AR and MA terms.
32

Modeling Process
MODEL
IDENTIFICATION
MODEL
ESTIMATION
DIAGNOSTIC
CHECKING
FORECASTING

Modeling in
Time series
forecasting
34

Economic Forecasting
Based on past and current information, the
objective of forecasting is to provide
quantitative estimate(s) of the likelihood of
the future course of the object of interest
(e.g. personal consumption expenditure).
We develop econometric models and use
one or more methods of forecasting its
future course.
35

Economic Forecasts
Economics [GDP, Unemployment, Consumption, Investment, Interest Rates]
Financial Asset Management [Asset Returns, Exchange Rates and Commodity Prices]
Financial Risk Management [Asset Return Volatility]
Marketing [Response of Sales to Different marketing Schemes]
Business and Government [Revenue Forecasts]
Crisis Management [Probabilities of default, currency devaluations]
36

Point &
Interval
Forecasts:
In point forecasts we provide a
single value for each forecast
period.
In interval forecasts we obtain a
range, or an interval, that will
include the realized value with
some probability.
The interval forecast provides a
margin of uncertainty about the
point forecast.

Volatility
Clustering
• Financial time series, such as stock prices,
interest rates, foreign exchange rates
exhibit volatility clustering.
• Period of Turbulence – Price show
wide swings
• Period of Tranquility – There is a
relative calm
38

Volatility
Clustering
• Various sources of news and other
economic events may have an impact on
the time series pattern of asset prices
• News can lead to various
interpretations,
• economic events may occur
• Normally, large positive and large
negative observations in financial time
series appears in clusters
39

Real and
Financial
Effect
40
Such swings in prices have serious
effects
Investors are concerned about the
• Rate of return on their investment
• Risk of investment and variability / volatility of
risk
It is important to measure asset returns
volatility

Measuring
Volatility
41
A simple measure of asset return volatility
is its variance over time
Variance by itself does not capture
volatility clustering
• Subtract the mean value from individual value,
square the difference and divide it by number of
observations
• A measure of unconditional variance
• A single number of a given sample
• Does not take into account the past history (time-
series volatility)

The ARCH Model
A measure that takes into
account the past history
(conditional / time varying
volatility)
In Time Series, involving
asset returns, such as
returns on stocks or
foreign exchange, we
observe auto-correlated
heteroscedasticity

Heteroscedasticity
The basic version of the least
squares model assumes that the
expected value of all error terms,
when squared, is the same at any
given point. This assumption is
called homoskedasticity.
Data in which the variances of the
error terms are not equal, in
which the error terms may
reasonably be expected to be
larger for some points or ranges of
the data than for others, are said
to suffer from heteroskedasticity.

Auto-correlated
Heteroscedasticity
• Heteroscedasticity, or unequal variance, in
cross section (multiple company, industry,
economy) data because of heterogeneity
among individual cross-section units.
• In time series data, we usually observe auto-
correlation
• In financial data, we observe Auto-correlated
Heteroscedasticity (Heteroscedasticity
observed over different periods is auto-
correlated)
• In Literature, this phenomenon is called ARCH
Effect
45

ARCH IN
EViews
• To estimate an ARCH or
GARCH model,
• Open the equation
specification dialog by
selecting Quick/Estimate
Equation…, by
selecting Object/New
Object.../Equation….
• Select ARCH from the
method dropdown menu at
the bottom of the dialog.

Interpretation
ARCH Test
• Null Hypothesis: There is no
Arch Effect
• Alternate Hypothesis: There
is Arch Effect.
• In above illustration, Null
Hypothesis is rejected.

FA_Unit 4 (Time Series Analysis E-views).pdf

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