Hello
I am using the arima command to estimate an ARMA process. My query is about the the constant in the equation of the dependent variable "y". I generate teh data with the Stata code at the end of this post. For simplicity, I am generating an AR(1) process but it can be generalized to any ARMA. When estimating the AR(1) by ML, I obtain the output just below. In this output, the equation of the dependent variable "y" has a constant (26.78136). However, when I compute the sample mean with the command summarize, I do not obtain the same magnitude. What is 26.78136? This magnitude will be used by the command "predict" if you want to compute sample errors or predictions.
output:
**********************
. arima y in 1/199, ar(1)
(setting optimization to BHHH)
Iteration 0: log likelihood = -188.15232
Iteration 1: log likelihood = -188.14858
Iteration 2: log likelihood = -188.14848
Iteration 3: log likelihood = -188.14847
Iteration 4: log likelihood = -188.14847
ARIMA regression
Sample: 1 - 199 Number of obs = 199
Wald chi2(1) = 197.69
Log likelihood = -188.1485 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
| OPG
y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
y |
_cons | 26.78136 .1405286 190.58 0.000 26.50593 27.05679
-------------+----------------------------------------------------------------
ARMA |
ar |
L1. | .6852284 .0487352 14.06 0.000 .5897092 .7807476
-------------+----------------------------------------------------------------
/sigma | .6218523 .0335906 18.51 0.000 .5560159 .6876886
------------------------------------------------------------------------------
Note: The test of the variance against zero is one sided, and the two-sided
confidence interval is truncated at zero.
DO FILE:
***********************
*ARMA process
drop _all
set obs 200
gen time = _n
tsset time
*white noise
set seed 3342
gen nu_y1 = rnormal(0,0.7)
*AR(1) process
scalar dd = 8
scalar b1 = 0.7
gen y = dd/(1-b1) in 1
replace y = dd + b1*l.y + nu_y1 in 2/l
*Sample mean
summ y in 1/199
summ y in 2/199
*ML estimates
arima y in 1/199, ar(1)
I am using the arima command to estimate an ARMA process. My query is about the the constant in the equation of the dependent variable "y". I generate teh data with the Stata code at the end of this post. For simplicity, I am generating an AR(1) process but it can be generalized to any ARMA. When estimating the AR(1) by ML, I obtain the output just below. In this output, the equation of the dependent variable "y" has a constant (26.78136). However, when I compute the sample mean with the command summarize, I do not obtain the same magnitude. What is 26.78136? This magnitude will be used by the command "predict" if you want to compute sample errors or predictions.
output:
**********************
. arima y in 1/199, ar(1)
(setting optimization to BHHH)
Iteration 0: log likelihood = -188.15232
Iteration 1: log likelihood = -188.14858
Iteration 2: log likelihood = -188.14848
Iteration 3: log likelihood = -188.14847
Iteration 4: log likelihood = -188.14847
ARIMA regression
Sample: 1 - 199 Number of obs = 199
Wald chi2(1) = 197.69
Log likelihood = -188.1485 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
| OPG
y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
y |
_cons | 26.78136 .1405286 190.58 0.000 26.50593 27.05679
-------------+----------------------------------------------------------------
ARMA |
ar |
L1. | .6852284 .0487352 14.06 0.000 .5897092 .7807476
-------------+----------------------------------------------------------------
/sigma | .6218523 .0335906 18.51 0.000 .5560159 .6876886
------------------------------------------------------------------------------
Note: The test of the variance against zero is one sided, and the two-sided
confidence interval is truncated at zero.
DO FILE:
***********************
*ARMA process
drop _all
set obs 200
gen time = _n
tsset time
*white noise
set seed 3342
gen nu_y1 = rnormal(0,0.7)
*AR(1) process
scalar dd = 8
scalar b1 = 0.7
gen y = dd/(1-b1) in 1
replace y = dd + b1*l.y + nu_y1 in 2/l
*Sample mean
summ y in 1/199
summ y in 2/199
*ML estimates
arima y in 1/199, ar(1)
