Understanding ARIMA results between stata12 and eviews7

[stata.diallo]

Hi,

I am new to time series forecasting (in fact, I've never done it before! so apologizes if I do something stupid... ) but am asked to take over a work by a colleague who used to estimate in eviews.

Here is his command in eviews and results.

ls d(life_expectancy_rwa) c ar(1) ma(1)

Dependent Variable: D(LIFE_EXPECTANCY_RWA)
Method: Least Squares
Date: 05/23/14 Time: 15:53
Sample (adjusted): 1962 2012
Included observations: 51 after adjustments
Convergence achieved after 65 iterations
MA Backcast: OFF (Roots of MA process too large)

Variable Coefficient Std. Error t-Statistic Prob.

C 0.162130 0.099824 1.624160 0.1109
AR(1) 0.927800 0.049352 18.79952 0.0000
MA(1) 1.362460 0.336141 4.053235 0.0002

R-squared 0.979104 Mean dependent var 0.410669
Adjusted R-squared 0.978233 S.D. dependent var 1.807632
S.E. of regression 0.266690 Akaike info criterion 0.251562
Sum squared resid 3.413928 Schwarz criterion 0.365199
Log likelihood -3.414832 Hannan-Quinn criter. 0.294986
F-statistic 1124.542 Durbin-Watson stat 0.210275
Prob(F-statistic) 0.000000

Inverted AR Roots .93
Inverted MA Roots -1.36
Estimated MA process is noninvertible


I tried to replicate this in stata, but failed to get the same results.

. arima D.life_expectancy_rwa, ar(1) ma(1)

(setting optimization to BHHH)
Iteration 0: log likelihood = -37.511483
Iteration 1: log likelihood = -27.47726
Iteration 2: log likelihood = -19.615086
Iteration 3: log likelihood = -17.964063
Iteration 4: log likelihood = -17.921561 (backed up)
(switching optimization to BFGS)
Iteration 5: log likelihood = -17.918088 (backed up)
Iteration 6: log likelihood = -15.461307
Iteration 7: log likelihood = -15.294291
Iteration 8: log likelihood = -15.273493
Iteration 9: log likelihood = -15.268665
Iteration 10: log likelihood = -15.267692
Iteration 11: log likelihood = -15.267676
Iteration 12: log likelihood = -15.267676

ARIMA regression

Sample: 1961 - 2012 Number of obs = 52
Wald chi2(1) = 573.25
Log likelihood = -15.26768 Prob > chi2 = 0.0000

------------------------------ -------------------------------------------------------
D. | OPG
life_expectancy_rwa | Coef. Std. Err. z P>|z| [95%
Conf. Interval]
--------------------+----------------------------------------------------------------
life_expectancy_rwa |
_cons | .4095347 1.426149 0.29 0.774
-2.385666 3.204736
--------------------+----------------------------------------------------------------
ARMA |
ar |
L1. | .9264644 .0386951 23.94 0.000
.8506233 1.002305
|
ma |
L1. | .9999997 . . .
. .
--------------------+----------------------------------------------------------------
/sigma | .3027849 .0237213 12.76 0.000
.256292 .3492779
-------------------------------------------------------------------------------------
Note: The test of the variance against zero is one sided, and the
two-sided confidence
interval is truncated at zero.



Thus, the results from forecast in Eviews and predict,y in Stata are quite different (increasing steadily in Eviews and reducing in Stata). What am I doing wrong?

Best regards.


--
Amadou B. DIALLO, PhD.
Senior Economist, AfDB.
[email protected]
+21671101789

[stata.diallo]

Hi,
This is the answer I received from StataCorp Tech Support.
Best regards.


Dear Amadou,

The reason for the differences is that Stata uses Kalman filter approach to estimate the -arima- models. For theoretical aspects see:

Hamilton, James "Time Series analysis".
Princeton University Press. 1994

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

There will be differences when the sample size is small, but as the sample size gets large the results converge to the same coefficient estimates.

You may confirm this by running the following example (using an artificial dataset generated with the user-written command -sim_arma-. Type, "findit sim_arma" to locate and install the command) in Stata:

clear
** Generate artificial dataset **
set seed 123
sim_arma y, nobs(50) ar(.92) ma(.75) spin(500)
rename _t time
tsset time

arima y , ar(1) ma(1)

And the output for the last command:

. arima y , ar(1) ma(1)

(setting optimization to BHHH)
Iteration 0: log likelihood = -72.584945
Iteration 1: log likelihood = -72.486973
Iteration 2: log likelihood = -72.482757
Iteration 3: log likelihood = -72.482314
Iteration 4: log likelihood = -72.481642
(switching optimization to BFGS)
Iteration 5: log likelihood = -72.481434
Iteration 6: log likelihood = -72.480168
Iteration 7: log likelihood = -72.480168

ARIMA regression

Sample: 1 - 50 Number of obs = 50
Wald chi2(2) = 131.62
Log likelihood = -72.48017 Prob > chi2 = 0.0000

----------------------------------------------------------------------------
| OPG
y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-----------+------------------------------------------------------------
-----------+----
y |
_cons | -3.181352 .9924067 -3.21 0.001 -5.126433 -1.23627
-----------+------------------------------------------------------------
-----------+----
ARMA |
ar |
L1. | .7544613 .0948148 7.96 0.000 .5686277 .9402948
|
ma |
L1. | .7223277 .1098259 6.58 0.000 .5070729 .9375825
-----------+------------------------------------------------------------
-----------+----
/sigma | 1.006188 .1368285 7.35 0.000 .7380088 1.274367
----------------------------------------------------------------------------
Note: The test of the variance against zero is one sided, and the two-sided
confidence interval is truncated at zero.


You may copy and paste the data in E-views and run the command:

ls Y c ar(1) ma(1)

To get:

Dependent Variable: Y
Method: Least Squares
Date: 05/28/14 Time: 16:24
Sample (adjusted): 2 50
Included observations: 49 after adjustments
Failure to improve SSR after 12 iterations
MA Backcast: 1

Variable Coefficient Std. Error t-Statistic Prob.

C -2.776981 0.784088 -3.541669 0.0009
AR(1) 0.708605 0.096439 7.347735 0.0000
MA(1) 0.999950 0.043335 23.07466 0.0000
. . .
. . .


You may now repeat the experiment with more observations (e.g. change
"nobs(50) to nobs(500)" and you will see that the coefficient estimates get much more closer.