Transcript 2. Vector Autoregressino (VAR)

Ch8 Time Series Modeling
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1. Autoregressive Integrated Moving Average
Model (ARIMA Model), popularly known as the
Box-Jenkins Methodology
2. Vector Autoregression Model (VAR Model)
3. Autoregressive Conditional Heteroscedasticity
(ARCH Model) or Generalized Autoregressive
Conditional Heterscedasticity (GARCH Model)
1. ARIMA Model
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Time series models “explain” the movement of a
time series by relating it to its past values and to a
weighted sum of current and lagged random
disturbances.
For stationary process:
1. Moving average models
2. Autoregressive models
3. Autoregressive-moving average models
For non-stationary process:
4.Autoregressive integrated moving average models
Moving Average Model: A Special Case of ARIMA
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Moving average process of order q: MA(q)
yt     t  1 t 1   2 t 2     q t q
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Sample autocorrelation function can be used to
specify the order of moving average process.
Autoregressive Model: A Special Case of ARIMA
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Autoregressive models: AR(p)
yt  1 yt 1  2 yt 2     p yt  p     t
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Sample partial-autocorrelation function can be used
to specify the order of autoregressive process.
Mixed Autoregressive-Moving Average Models
(ARMA): A Special Case of ARIMA
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Mixed autoregressive-moving average process of
order (p,q): ARMA(p,q)
yt  1 yt 1     p yt  p     t  1 t 1     q t q
Non-stationary processes
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For non-stationary series, we can adopt
ARIMA(p,d,q) models .
Firstly, we can use ADF test or directly observe
its autocorrelogram to determine the degree of
homogeneity d.
After d is determined, we can work with the
stationary series wt  d yt , and examine both its
autocorrelation function and its partial
autocorrelation function to determine possible
values for p and q.
The Box-Jenkins (BJ) Methodology
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Step 1. Identification
Chosing p,d,q with the help of correlogram and
ADF test
Step 2. Estimation
Parameter estimation of the chosen model
Step 3. Diagnostic checking
Are the estimated residuals white noise?
Step 4. Forecasting.
2. Vector Autoregression Model (VAR):
The Sims Methodology
Yt  1  11Yt 1  ...  1 pYt  p  11 X t 1  ...  1 p X t  p   11Z t 1  ...  1 p Z t  p  e1t
X t  1  21Yt 1  ...  2 pYt  p   21 X t 1  ...   2 p X t  p   21Z t 1  ...   2 p Z t  p  e2t
Z t  1  31Yt 1  ...  3 pYt  p   31 X t 1  ...   3 p X t  p   31Z t 1  ...   3 p Z t  p  e3t
where the u’s are the stochastic error terms, called
impulses or innovations or shocks in the language of VAR.
2. VAR vs. Structural Model
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A structural model means that the specific
relationships between variables are based ( either
formally or informally) on economic and finance
theories
A VAR makes minimal theoretical demands on
the structure of a model. With a VAR, one needs
to specify only two things:
(1) the variables (endogenous and exogenous) and
(2) the largest number of lags.
2. Technical Problems of VAR
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A purist may argue all variables are assumed to be
endogenous.
Both endogenous and exogenous variables assume have
the same lag length.
One way of deciding the lag length is to use a criterion
like the Akaike or Schwaz and choose that model gives
the lowest values of these criteria. There is no question
that “trial and error” is inevitable.
  eˆi2  2k
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AIC  log
 N  N
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  eˆi2  k log N
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SC  log
 N 
N
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2. Advantages of VAR
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VAR is introduced as an alternative approach to multiequation modeling through the work of C. A. Sims (1980)
VAR provides a framework for testing Granger causality
between each set of variables
VAR is useful for understanding the relationships between
several series.In a VAR model, the explanatory variables
might influence the dependent variable, but there is no
possibility that the dependent variable influences the
explanatory variable.
VARs usually have better forecasting ability than
sophisticated economic/financial models.
2. Drawbacks of VAR
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VAR models are atheoretical; that is, do not draw heavily
on economic and finance theories
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VAR has a lag length selection problem.One must trade
off having a sufficient number of lags and having a
sufficient number of free parameters. In practice, one
often finds it necessary to constrain the number of lags to
be less than what is ideal given the nature of the dynamics.
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Because of its emphasis on forecasting, VAR models are
less suited for policy analysis.
2. VAR and VECM
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If all the variables in the VAR are stationary, OLS
can be used to estimate each equation and
standard statistical methods can be employed.
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If the variables under study have unit roots and
are cointegrated, a variant on the VAR called the
vector error correction model, or VECM, should
be used.
3. ARCH and GARCH Models
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ARCH: autoregressive conditional
heterscedasticity
ARCH models were introduced by Engle (1982)
and generalized as GARCH (Generalized ARCH)
by Bollerslev (1986).
ARCH and GARCH are specially designed to
model and forecast conditional variances. The
variance of the dependent variable is modeled as
a function of past values of the dependent
variable and independent, or exogenous variables.
These models are widely used in financial time
series analysis.
3. Significance of ARCH and GARCH Models
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There are several reasons that you may want to model and
forecast volatility.
You may want to analyze the risk of holding an asset or
the value of an option;
Forecast confidence intervals may be time-varying, so
that more accurate intervals can be obtained by modeling
the variance of the errors;
More efficient estimators can be obtained if
heteroscedasticity in the errors is handled properly;
These models are also consistent with the volatility
clustering often seen in financial time series such as
returns data where large changes in returns are likely to be
followed by further large changes.
3. The ARCH Specification
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In developing an ARCH model, you will have to
consider two distinct specifications:one for the
conditional mean and one for the conditional
variance.
The conditional mean is the expected value of a
random variable when the expectation is
influenced (conditioned) by knowledge of other
random variables. The mean is usually a function
of these other variables. Similarly, the conditional
variance is the variance of a random variable
conditioned by knowledge of other random
variable.
3. An Example: The GARCH (1,1) Model
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In the standard GARCH (1,1) specification:
(1)
Yt  1   2 X 2   3 X 3   t
(2)
 t2   0  1 t21  1 t21
In fact, this is by far the most popular
specification.
3. An Example: The GARCH (1,1) Model, continued
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The mean equation given in (1) is written as a function of
exogenous variables with an error term. The objective of
modeling the conditional mean is to construct a series of
squared residuals  t2 from which to derive the conditional
variance.
From a time series of the squared residuals of the
conditional mean equation we develop the equation for
the conditional variance. Since t2 is the one-period ahead
forecast variance based on past information, it is called
the conditional variance. The conditional variance
equation specified in (2) is a function of three terms:
3. An Example: The GARCH (1,1) Model, continued
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The (1,1) in GARCH (1,1) refers to the presence
of a first-order GARCH term ( the first term in
parentheses) and a first-order ARCH term ( the
second term in parentheses).
An ordinary ARCH model is a special case of a
GARCH specification in which there are no
lagged forecast variances in the conditional
variance equation.
3. GARCH Specification
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In general, we could have any number of ARCH
terms and any number of GARCH terms. The
GARCH (p,q) model refers to the following
equation for  t2
 t2  0  1t21  ...  pt2 p  1 t21  ... q t2q
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The above model can be generalized even further
by including one or more exogenous or
predetermined variables as additional
determinants of the error variance
 t2  0  1 t21  1 t21   1 X t
3.The ARCH-M Model
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The X’s in equation (1) represents exogenous or
predetermined variables that are included in the
mean equation. If we introduce the conditional
variance into the mean equation, we get the
ARCH-in-Mean (ARCH-M).
Yt  1   2 X 2  3 X 3  ht2   t
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A variant of the ARCH-M specification uses the
conditional standard deviation in place of the
conditional variance.
3. The ARCH-M Model, continued
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The ARCH-M model is often used in financial
applications where the expected return on an asset
is related to the expected asset risk.
The estimated coefficient on the expected risk is a
measure of the risk-return tradeoff.
Note in the conditional mean equation the
variance is transformed into a conditional
standard deviation so that it is in the same unit of
measurement as the risk premium being modeled.