Re: st: -arima- vs. -sspace-

[Richard Gates <rgates@stata.com>]

Jorge Eduardo Pérez Pérez wrote:
>
> I have been fitting arima models using the state space representation
> and the -sspace- command instead of using -arima-. I am doing this
> because I need the filtered state estimates for other purposes. I have
> found that in some cases, -arima- achieves convergence while -sspace-
> does not, even using the same optimization technique. See an example
> below for an ARIMA(2,0,0). I have a couple of questions.
>
> 1. Why does this happen? I know that -arima- has its own
> implementation of the Kalman Filter coded in the Stata matrix
> language, while sspace is coded in mata and uses the mata optimization
> routines. Can anyone point out the differences between the two
> implementations? I would expect that a model that converges using
> -arima-, would also converge using the -sspace- routines that are much
> more up to date.
> 2. How do I get -sspace- to achieve the same results as -arima-?

Try scaling your data.  -sspace- estimates the variance and -arima- the
standard deviation.

When you have a nonconvergent problem, use the -iterate()- option
to terminate the optimization at a point where it looks like it is
having trouble and take a look at the estimates.



. do arima2

. clear

. * Input database
. input dlym

          dlym
  1. .0076563
  2. .0095813
  3. .007427
 ...
(omitted)

 44. -.0007679
 45. .000582
 46. .0012176
 47. .0005591
 48. -.0000923
 49. .0030711
 50. .0041749
 51. .0029995
 52. .0018442
 53. end

. * Declare time series
. gen t=_n

. tsset t
        time variable:  t, 1 to 52
                delta:  1 unit

. replace dlym = 1000*dlym
(52 real changes made)

. * Estimate arima
. arima dlym, arima(2,0,0) nocons

(setting optimization to BHHH)
Iteration 0:   log likelihood = -132.81204
Iteration 1:   log likelihood = -132.79176
Iteration 2:   log likelihood = -132.78632
Iteration 3:   log likelihood =  -132.7844
Iteration 4:   log likelihood = -132.78369
(switching optimization to BFGS)
Iteration 5:   log likelihood = -132.78337
Iteration 6:   log likelihood = -132.78305
Iteration 7:   log likelihood = -132.78304

ARIMA regression

Sample:  1 - 52                                 Number of obs      =        52
                                                Wald chi2(2)       =    193.17
Log likelihood =  -132.783                      Prob > chi2        =    0.0000

------------------------------------------------------------------------------
             |                 OPG
        dlym |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ARMA         |
          ar |
         L1. |   1.098012   .1706826     6.43   0.000     .7634798    1.432543
         L2. |  -.2922192   .1466112    -1.99   0.046    -.5795718   -.0048666
-------------+----------------------------------------------------------------
      /sigma |   3.066515   .1590157    19.28   0.000      2.75485     3.37818
------------------------------------------------------------------------------
Note: The test of the variance against zero is one sided, and the two-sided
    confidence interval is truncated at zero.

.
. *Estimate via sspace
. constraint 1 [dy]e.dy=1

. constraint 2 [dlym]dy=1

. constraint 3 [et]e.dy=1

. constraint 4 [ldy]l.dy=1

.
. sspace (dy l.dy l.ldy e.dy, state noconstant)  ///
>         (ldy l.dy, state noconstant )   ///
>         (et e.dy, state noconstant)     ///
>         (dlym dy, noconstant),          ///
>         covstate(diagonal)  constraints(1/4) tech(bhhh 4 nr) ///
>         iterate(15)
searching for initial values ...........
(setting technique to bhhh)
Iteration 0:   log likelihood = -133.26724
Iteration 1:   log likelihood = -133.08202
Iteration 2:   log likelihood = -132.96335
Iteration 3:   log likelihood = -132.89188
(switching technique to nr)
Iteration 4:   log likelihood = -132.84706
Iteration 5:   log likelihood = -132.78316
Iteration 6:   log likelihood = -132.78304
Iteration 7:   log likelihood = -132.78304
Refining estimates:
Iteration 0:   log likelihood = -132.78304
Iteration 1:   log likelihood = -132.78304

State-space model

Sample: 1 - 52                                    Number of obs   =         52
                                                  Wald chi2(2)    =     163.02
Log likelihood = -132.78304                       Prob > chi2     =     0.0000
 ( 1)  [dy]e.dy = 1
 ( 2)  [dlym]dy = 1
 ( 3)  [et]e.dy = 1
 ( 4)  [ldy]L.dy = 1
------------------------------------------------------------------------------
             |                 OIM
        dlym |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
dy           |
          dy |
         L1. |   1.098088   .1304214     8.42   0.000     .8424671     1.35371
             |
         ldy |
         L1. |  -.2922904   .1330764    -2.20   0.028    -.5531154   -.0314655
        e.dy |          1  (constrained)
-------------+----------------------------------------------------------------
ldy          |
          dy |
         L1. |          1  (constrained)
-------------+----------------------------------------------------------------
et           |
        e.dy |          1  (constrained)
-------------+----------------------------------------------------------------
dlym         |
          dy |          1  (constrained)
-------------+----------------------------------------------------------------
Variance     |
  dy         |   9.403747   1.845373     5.10   0.000     5.786882    13.02061
------------------------------------------------------------------------------
Note: Tests of variances against zero are one sided, and the two-sided confidence
    intervals are truncated at zero.

I hope this is what you need.

-Rich
rgates@stata.com
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