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Re: st: arima, t-stat, test give inconsistent results


From   vwiggins@stata.com (Vince Wiggins, StataCorp)
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: arima, t-stat, test give inconsistent results
Date   Mon, 18 Sep 2006 12:27:15 -0500

Richard Boylan <rtboylan@gmail.com> asks,

> I get t-stats of zero for the MA coefficients, but if I do a joint
> test, they are highly significant.
>
> That does not seem right to me at all.
>
> I am inlcuding the regression output.

Here are Richard's estimation results and test.

 arima y, arima(2,1,2)

ARIMA regression

Sample:  2 to 233
    Number of obs      =       232
                                               Wald chi2(4)       =    645.52
Log likelihood = - 409.4094                    Prob > chi2        =    0.0000

------------------------------------------------------------------------------
            |                 OPG
        D.y |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
y           |
      _cons |   .7414199   .1078346     6.88   0.000     .5300681    .9527718
-------------+----------------------------------------------------------------
ARMA        |
         ar |
        L1. |   .2857905   .0498606     5.73   0.000     .1880654    .3835155
        L2. |  -.8643362   .0507385   -17.04   0.000    -.9637818   -.7648907
         ma |
        L1. |  -.2812952   182.0783    -0.00   0.999    -357.1481    356.5856
        L2. |   .9999997   1294.639     0.00   0.999    -2536.445    2538.445
-------------+----------------------------------------------------------------
     /sigma |    1.399279   905.7593     0.00   0.999    -1773.856    1776.655
------------------------------------------------------------------------------

. test  L.ma L2.ma

 ( 1)  [ARMA]L.ma = 0
 ( 2)  [ARMA]L2.ma = 0

          chi2(  2) =   77.40
        Prob > chi2 =    0.0000


Such results are not uncommon with ARIMA models.  The immediate cause is
simple -- the estimates of the parameters for L1.ma and L2.ma are highly
correlated.  So, individually they are insignificant, but jointly they are
significant.  Put another way L1.ma and L2.ma are nearly collinear.  You
cannot identify their effects individually, only jointly.

What that explanation does not address is why.  Looking at the nearly
identical size, but opposite value coefficients on L1.ar and L1.ma, I would
guess that Richard has what is called a "common factors" problem.  This means
that he has included AR and MA terms in his ARIMA model that do not belong.

With non-ARIMA models, adding useless regressors is not much of a problem.
With the AR and MA terms in ARIMA, however, such terms can be highly
collinear, particularly when an AR and MA term at the same lag are included
and neither belongs in the model.  

The classic example is including AR(1) and MA(1) terms to model what is
actually a white-noise process.  The two terms are virtually unidentified, but
can have large and often individually significant coefficients.  Yet, when
tested jointly, particularly when using an LR test, the two terms are
insignificant.  Richard's problem on its head.

The bottom line is -- do not include so called common factors in an ARIMA
model.

The issue of common factors is fairly well known in the literature, see for
example Box, Jenkins, and Riensel (1994) or various texts by Harvey, though it
is typically discussed only with respect to convergence of the estimators.

I have surmised that common factors is the reason why the autocorrelation (AC)
and partial autocorrelation (PAC) functions are so assiduously used to
determine the AR and MA order of ARIMA models, though the treatment I have
found in the literature has been sketchy at best.  Simply throwing extra terms
into an ARIMA model, with the idea of testing such terms to determine lag
order, leads to unidentified or nearly unidentified models.

I would welcome comments, references, or refutations related to this
conjecture about the universal use of AC and PAC.


-- Vince 
   vwiggins@stata.com


References

Box, G E P, G M Jenkins, & G C Reinsel (1994). Time Series Analysis:
	Forecasting & Control (3rd ed), Prentice Hall.

Harvey, A C (1993). Time Series Models (2nd ed). Cambridge University Press.

Harvey, A C (1989) Forecasting Structural Time Series Models and the Kalman
	Filter. Cambridge University Press.

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