ARIMA and inconsistent results when differencing for stationarity

[Jen Ward]

Hi,

I am exploring the options for the -arima- command for an interrupted time series and noticed that different results are obtained depending on whether the differencing is done manually or within the -arima- command.

Using the sample data: "http://www.stata-press.com/data/r16/wpi1.dta", the two simple models below produce identical results:

* Model 1 - Differencing within arima
arima wpi, arima(1,1,4)


* Model 2 - Use already differenced variable in the model and set i = 0
generate d1_wpi = d1.wpi
arima d1_wpi, arima(1,0,4)

However, when I include a variable in the model to represent the introduction of an policy (for an interrupted time series) the two approaches estimate different coefficients for policy:

*** Create a policy variable - 1 from 1981 and 0 before then
g year = yofd(dofq(t))
g policy = 0
recode policy (0=1) if year>=1981

*Model 1
arima wpi policy, arima(1,1,4) // coefficient = 0.88 (p = 0.64)

*Model 2
arima d1_wpi policy, arima(1,0,4) //coefficient = 0.08 (p = 0.88)

Can anyone suggest why the models differ and how I should proceed?

[Andrew Musau]

If you specify a lower order of integration, you need to difference all variables involved. So

arima wpi policy, arima(1,1,4)

is equivalent to

tsset year
arima d.wpi d.policy, arima(1,0,4)

[Jen Ward]

Thanks Andrew Musau - so if I need to difference the outcome to render it stationary, all IVs will also need to be differenced, is that correct? Is this so that the impact of the introduction of the policy still matches its introduction (in my example quarter 1 of 1981)?

Also I am unsure if nocons should be specified once detrended - could you advise?

[Andrew Musau]

so if I need to difference the outcome to render it stationary, all IVs will also need to be differenced, is that correct?

The command allows you to specify only one order of integration. This allows consistency in the interpretation of coefficients. Do you have a case where the outcome has a unit root and the independent variable is stationary at levels?

Also I am unsure if nocons should be specified once detrended - could you advise?

If you difference a series only once, the assumption is that the original series has a constant average trend (e.g., a random walk). Therefore, you can include -nocons- after differencing if you want to allow a non-zero average trend in the series.

[Jen Ward]

Hi Andrew Musau , thanks again for your reply. I am very new to time series analysis and I must admit, having looked at online resources, that I had focused on exploring the outcome to check for stationarity as well as MA and AR but I had not considered the same should be done for the IVs. With binary IVs, which capture policy introduction as a step function, I am assuming that's also technically not stationary?

I have a question re. what you said:

If you difference a series only once, the assumption is that the original series has a constant average trend (e.g., a random walk). Therefore, you can include -nocons- after differencing if you want to allow a non-zero average trend in the series

Would you say including -nocons- is a good assumption and better estimate of the IV? is it similar to capturing a drift?

Could you point me to any resource?

[Andrew Musau]

My apologies, the statement should have read you include a constant if you want to assume that the series has a constant average trend. Here is a silly example. If we run arima with no autoregressive and moving average terms and no differencing, the reported constant is just the mean of the series.

webuse wpi1, clear
arima wpi, arima(0,0,0)
mean wpi

Res.:

. arima wpi, arima(0,0,0)

(setting optimization to BHHH)
Iteration 0:   log likelihood = -598.19748  
Iteration 1:   log likelihood = -598.19748  

ARIMA regression

Sample:  1960q1 - 1990q4                        Number of obs     =        124
                                                Wald chi2(.)      =          .
Log likelihood = -598.1975                      Prob > chi2       =          .

------------------------------------------------------------------------------
             |                 OPG
         wpi |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
wpi          |
       _cons |   62.77419   3.078078    20.39   0.000     56.74127    68.80712
-------------+----------------------------------------------------------------
      /sigma |   30.12136   4.769445     6.32   0.000     20.77342     39.4693
------------------------------------------------------------------------------
Note: The test of the variance against zero is one sided, and the two-sided
      confidence interval is truncated at zero.

.
. mean wpi

Mean estimation                   Number of obs   =        124

--------------------------------------------------------------
             |       Mean   Std. Err.     [95% Conf. Interval]
-------------+------------------------------------------------
         wpi |   62.77419   2.715952      57.39813    68.15025
--------------------------------------------------------------

.

If we difference once, the reported constant is the mean of the differenced series

webuse wpi1, clear
arima wpi, arima(0,1,0)
gen dwpi= d.wpi
mean dwpi

Res.:

. arima wpi, arima(0,1,0)

(setting optimization to BHHH)
Iteration 0:   log likelihood = -171.53628  
Iteration 1:   log likelihood = -171.53628  

ARIMA regression

Sample:  1960q2 - 1990q4                        Number of obs     =        123
                                                Wald chi2(.)      =          .
Log likelihood = -171.5363                      Prob > chi2       =          .

------------------------------------------------------------------------------
             |                 OPG
       D.wpi |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
wpi          |
       _cons |   .6951219   .1056506     6.58   0.000     .4880506    .9021933
-------------+----------------------------------------------------------------
      /sigma |   .9759591   .0618399    15.78   0.000      .854755    1.097163
------------------------------------------------------------------------------
Note: The test of the variance against zero is one sided, and the two-sided
      confidence interval is truncated at zero.

. gen dwpi= d.wpi
(1 missing value generated)

. mean dwpi

Mean estimation                   Number of obs   =        123

--------------------------------------------------------------
             |       Mean   Std. Err.     [95% Conf. Interval]
-------------+------------------------------------------------
        dwpi |   .6951219   .0883592      .5202061    .8700377
--------------------------------------------------------------

.

We difference a series once to make it stationary if the original series has a constant average trend, one example being a random walk. So without differencing, the presence of a constant allows for a non-zero mean in the series. If we difference once, it allows for a non-zero average trend in the series. Duke has some nice online notes on Arima models which you can find at the link below:

https://people.duke.edu/~rnau/411arim.htm

The relevant discussion involving differencing and the inclusion of a constant is under "Identifying the order of differencing in an ARIMA model". It is a while since I did anything with time-series models, so I have no favorite textbooks. You may find recommendations on the Duke site.

[Jen Ward]

Thanks for the links Andrew Musau

[Jen Ward]

Andrew Musau is there any Stata command which would allow me to include an AR and/or MA terms in a regression model? The 'introduction to time series' manual mentions -newey- but it seems to only be able to account for AR. I am interested in this as I would also be interested in modelling the trend (to establish whether linear or quadratic) and I don't think I can do this using -arima- .

[Andrew Musau]

arima can fit an ARIMAX model but I do not know of a command that can fit a regression model with ARMA errors, where the difference is described in the following blog post:

https://robjhyndman.com/hyndsight/arimax/

You can start a new thread, maybe someone knows of a command. If no luck, try R.