Autoregression models

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Autoregressive models
Another useful model is autoregressive model.
Frequently, we find that the values of a series
of financial data at particular points in time
are highly correlated with the value which
precede and succeed them.
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Autoregressive models
Models with lagged variable
The creation of an autoregressive model generates
a new predictor variable by using the Y variable
lagged 1 or more periods.
Dependent variable is a function of itself at the
previous moment of period or time.
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The most often seen form of the equation is a
linear form
where yt the dependent variable values at the
moment t, yt-i (i 1, 2, ..., p) the dependent
variable values at the moment t-i, bo,
bi (i1,..., p) regression coefficient, p
autoregression rank, et disturbance term.
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A first-order autoregressive model is concerned
with only the correlation between consecutive
values in a series.
A second-order autoregressive model considers the
effect of relationship between consecutive values
in a series as well as the correlation between
values two periods apart.
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The selection of an appropriate autoregressive
model is not an easy task. Once a model is
selected and OLS method is used to obtain
estimates of the parameters, the next step would
be to eliminate those parameters which do not
contribute significantly.
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(The highest-order parameter does not contribute
to the prediction of Yt)
(The highest-order parameter is significantly
meaningful)
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using an alpha level of significance, the
decision rule is
or if
to reject H0 if
and not to reject H0 if
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Some helpful information
 
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If the null hypothesis is NOT rejected we may
conclude that the selected model contains too
many estimated parameters. The highest-order term
then be deleted an a new autoregressive model
would be obtained through least-squares
regression. A test of the hypothesis that the
new highest-order term is 0 would then be
repeated.
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This testing and modeling procedure continues
until we reject H0. When this occurs, we know
that our highest-order parameter is significant
and we are ready to use this model.
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Example 1
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We have to estimate the parameters of the
first-order autoregressive model
and then check if Beta1 is statistically
significant.
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Example 2
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Autogregressive Modeling

• Used for Forecasting
• Takes Advantage of Autocorrelation
• 1st order - correlation between consecutive
  values
• 2nd order - correlation between values 2
  periods apart
• Autoregressive Model for pth order

Random Error
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Autoregressive Modeling Steps

• 1. Choose p
• 2. Form a series of lag predictor variables
• Yi-1 , Yi-2 , Yi-p
• 3. Use Excel to run regression model using all p
  variables
• 4. Test significance of Bp
• If null hypothesis rejected, this model is
  selected
• If null hypothesis not rejected, decrease p by 1
  and repeat your calculations