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PROCEDURE FOR UNIVARIATE ARIMA ANALYSIS

VIEWS: 47 PAGES: 5

									                                                Prof. Steve De Lurgio

                    PROCEDURE FOR UNIVARIATE ARIMA ANALYSIS

Before doing analysis study the series of interest.
What is it?
What are some of the cause and effect relationships of the series?
What are some of the seasonal cause and effect relationships?
How is the series measured?
Is this one series or two or more series? That is, does the graph show one or
more series from a statistical standpoint?

Analyze plots of the series. Is the a lot of noise in the series? If so, then
this may be a very difficult series to analyze using ARIMA. Also, remember the
principle of parsimony. Complex ARIMA models are not expected, normally, there
are not more than one or two terms in the model unless there is more complex
seasonality. Simpler models forecast out of sample observations better than
complex, over-fitted models.


                            Identification

1) Plot Series, scatter diagrams, histograms, and standardized plots. Are
summary statistics consistent with approximately a normal distribution? If not,
consider alternative transformations.
2) Using a plot, confirm that expected patterns exist.
    Is there seasonality as expected? Why or why not?
    Is there variance stationarity? Why or why not?
    Is there level stationarity? Why or why not?
3) Look for outliers. Be cautious to ensure that seasonality is not the cause
of outliers. Don't mistakenly adjust seasonal values, thinking they are
outliers.
4) Investigate obvious outliers, document the outliers with full explanation?
Under all circumstances, keep an adjusted copy of your unadjusted data.
5) Is the series stationary in level and variance as evidenced by plots of the
series, autocorrelations, and partial autocorrelations.
6) If not stationary in variance, then transform the series as needed. Use
logs or power transformations as necessary.
7) If not stationary in level, then investigate the appropriate level of
differencing. Consider both seasonal and nonseasonal differencing. Be cautious
to not over-difference. When in doubt, fit an autoregressive model with the
appropriate φ value. φ should be close to 1.0, but not necessarily exactly one.
When in doubt about differencing, you can develop two models, one with AR(p) and
the other with appropriate differences.
8) When taking differences assure that the standard deviation goes down. After
taking differences repeat steps 1) through 7), particularly investigation of the
outliers created during the differencing process.
9)After taking differences, study auto and partial correlations for significant
patterns. These should provide a hint of the underlying stochastic process.
10)Be sure to use and confirm ARMA ACF and PACF patterns using charts of
profiles. Remember basic patterns and their behavior with positive and negative
φ and Θ parameters.
11)Avoid a shotgun approach where you put several parameters in the relationship
at the same time. You should build models iteratively.

                              Estimation

12) Estimate the model suggested from the autocorrelations or partial
autocorrelations.
13)If the estimation procedure has trouble converging on parameter estimates,
then increase the number of iterations, with 40 or 50 being the maximum.
However, investigate the cause of convergent problems, typically your model is
too complex or has redundant parameters.
14) Confirm that the nonlinear estimation procedure converged and terminated its
search as expected.
15) Request a correlation matrix between parameters. If the correlation between
parameter estimates is very high, then they may be redundant parameters. Try
dropping one at a time and both at a time. If there is no significant change in
Sum of Squared Errors, then they do not belong in the model. ì

Remember the concept of parsimony.

                             Diagnostics

16) Remember a good model should have the following characteristics:

      Statistically significant parameter estimates.

      Stationary and invertible parameter estimates.

       Low residual standard error compared to the standard deviation of the
original series.

       High coefficient of determination R2.
       Residuals which are white noise.
       No low-level or seasonal lag ACFs are more than 2Se from zero.
       Box-Pierce (also known as Q or Box-Llung) Statistic does not indicate
that there is a pattern in ACFs.
       No low-level or seasonal lag PACFs are more than 2Se from zero.
       Plots of residuals reveal that there are not outliers or nonstationarity.
If significant outliers exist, then your model may not be as efficient as it
might. Your analysis may be flawed

17) Forecasted values (not fitted values) are reasonable, particularly based
upon expert opinion.

      Forecast error variance profiles are reasonable.

18) Over fit a model to confirm that additional parameters are not necessary.

19) When necessary, generate several different alternative models of the series.

          Parsimony is the principle in choosing one model over the other, when
everything else is equal.

          Forecast error variance profiles can differ significantly from one
model to another. This is important in deciding to choose one model over
another. Choose that model that fits the data best, confirms expert opinion
about the underlying behavior of the series, and provides relatively accurate
forecast for actual data which was withheld during the fitting process.
However, the profile of forecast errors can be revealing about the
appropriateness of one model over another.

          A model with tight 95% confidence intervals on parameter estimates is
better than another model when everything else is equal.
                     Starting Over With a Fresh Perspective

When it is necessary to make sure that one has not overlooked an important
alternative or more appropriate model, then it is suggested that the analyst
consider alternative modeling strategies.

For example, if outliers were not adjusted properly, then an inappropriate model
may have been identified.

     CHECK FOR OUTLIERS.   EXPLAIN OUTLIERS.    ADJUST FOR OUTLIERS.

    Try different types of differences when this might yield alternative models.

    The second time you analyze a series goes much more quickly, possibly one
fifth of the time the first analysis took.

    Be sure to complete each of the above steps for the series which you have
chosen for analysis. Also, be sure to complete all the other requirements of
your minor report!

    When completing the steps above, you should record the characteristics of
important models, I have attached a suggested ARIMA log sheet. This should be
attached to you final report. I cannot answer your questions without analysis of
your log sheet. Religiously fill this sheet out. It will greatly reduce your
analysis time.

                                 GOOD LUCK!!!

                       BDS 545 FORECASTING
                                                     Prof. Steve DeLurgio

       ADDITIONAL COMMENTS ON THE STEPS OF ARIMA ANALYSIS

  Be sure to study your ARIMA Handout in its entirety in regards to the steps of
ARIMA analysis. To reinforce that presentation and to provide a slightly
different perspective on ARIMA modeling please find below an Overview of ARIMA.
While this overview addresses ARIMA modeling specifically, it is equally
applicable to any type of forecast modeling process.


                   OVERVIEW OF ARIMA ANALYSIS

                           IDENTIFICATION

PLOT THE SERIES AND ALL TRANSFORMATIONS

     Outlier Identification and Replacement - Don't Over Adjust, but be sure to
adjust extreme values.
     Speculate on the best model. State why the series trends, has seasonality,
walks randomly, or has cyclical variations. If you weren't using ARIMA
analysis, what type of model would you use.


PLOT ACFs, PACFs, and IACFs

     Is the series stationary?

     Take appropriate differences to achieve stationarity in mean
          and variance.
     Have you created any outliers in the process of achieving
          stationarity?

     Have outliers interfered with achieving stationarity or
          hidden the none stationarity?

     Have you over-differenced? You can tell if differences are
          needed by looking at plots of the series and by ACFs.



                           ESTIMATION

ITERATIVELY IDENTIFY, ESTIMATE, AND DIAGNOSE TENTATIVE MODELS

     Add parameters one at a time. Resist the urge to add several parameters at
one time. It is legitimate to do so, but be sure to go back and repeat the
analysis step by step, parameter by parameter to assure that you have not
converged on the incorrect model.

     Iteratively add and delete components of the model until no new
improvements are possible.

     Pay particular attention to the concept of parsimony. Track the level of
R-sq as additional parameters are added to the model. Are you modeling the
underlying process or its realization? Remember that an infinite order AR is a
first order MA, and vice versa.

     Are the parameter estimates statistically significant?


                            DIAGNOSES

     Are parameter estimates statistically significant?
     Are parameters uncorrelated? That is, are they redundant?

     Have you overfitted a model by modeling outliers?

     Are the parameter estimates stationary and invertible?

     Does the adjusted R-square increase when additional parameters were added?

     Use ACFs, PACFs, IACFs, plots of residuals, etc. to confirm the following:

     No significant correlations exist at low order lags.

     No significant correlations exist at or adjacent to seasonal lags.

     The Box-Pierce (chi-square) statistic is insignificant at about 20 to 30
lags. (Those of us using Forecast-Pro and RATS are forced to use the output of
RATS at only one lag. Those using SAS have a whole range of lag values to
analyze.)

     Are there extreme outliers in the residuals?   Then, adjust the causes of
these outliers.

     Compare the results of competing models.
     Remember that achieving white-noise is not the ultimate objective. Models
are defensible even if there are some high correlations, just so long as R-
square is relatively high with the final models. To better understand this,
consider the fact that the results of using Winters, Fourier Series Analysis, or
classical decomposition methods yield residuals (i.e. errors) that are extremely
autocorrelated.


                         THE BEST MODEL


     The best model will have desirable statistical and intuitive attributes.
It will:

     1) Have intuitive appeal. It will make sense from an intuitive and
theoretical standpoint. Not all of the parameters may seem logical or
intuitive, but differences and lags of the parameters should make sense.

     2) Yield multi-period ahead forecasts that are logical, theoretical sound,
and consistent with the past.

     3) Be parsimonious, simple, but effective.

     4) Have parameters that are statistically significant and uncorrelated.

     5) Yield the best intuitive forecasts and forecast error variance profiles.

     6) Be intuitive to managers.

     7) Clearly represent the patterns in demand that we expected.

     8) Yield out of sample forecasts that are defensible.


     Complicating the determination of the above is the fact that several
different models may yield similar attributes. In such cases, discard all but
one of the models if the models are really identical. If the model are telling
us slightly different things about the series, then we should use several models
if our goals is to achieve the highest accuracy of forecasts.


     IF YOU GET "NO" ANSWERS TO SOME OF THE ABOVE QUESTIONS, THEN IT IS TIME TO
REEVALUATE YOUR APPROACH, AND TRY AGAIN. IF YOU REACH A ROAD BLOCK IN YOUR
ANALYSIS, THEN YOU MAY HAVE TO STOP AND CONSULT ME.

DO NOT CONSULT ME UNLESS YOU HAVE A LOG OF YOUR ANALYSES, I WANT TO HELP BUT
CANNOT WITHOUT YOUR LOG COMPLETELY FILLED OUT!!! I HATE FILLING OUT LOGS, BUT
IT WORKS.

GOOD LUCK,

SUCCESSFUL ANALYSES IS FUN!

								
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