# [R] AR(1) with an error term arima.sim parameter question

Rolf Turner r.turner at auckland.ac.nz
Thu Dec 11 09:29:14 CET 2014

```On 11/12/14 20:09, Michael Selevan wrote:
> This makes sense, thank you for the thorough response!
>
> One follow up question though. Would your #2 option be the same as, say,
> not using the rand.gen at all and providing the following parameters
>
> y3 <- arima.sim(n=10, list(ar=0.8), innov=rnorm(10, sd=0.2))

No.  This will call rand.gen=rnorm() to generate innov.start, so
start.innov will be generated with a standard deviation of 1 rather
than 0.2.

>
> or even
>
> y4 <- arima.sim(n=10, list(ar=0.8), innov=rnorm(10, sd=0.2),
> innov.start=rnorm(10, sd=0.2))

Why didn't you try it?  It gives an error, saying start.innov is too
short.  It needs to be of length *28* according to the error message.
I got the argument name wrong (on the second attempt!) in my previous
posting.

y4 <- arima.sim(n=10, list(ar=0.8), innov=rnorm(10, sd=0.2),
start.innov=rnorm(28, sd=0.2))

should I think be the same as y2.  ***You*** try it and see!

(Set a seed prior to each calculation; that's what seeds are for!)

cheers,

Rolf Turner

<SNIP>

> On Wed, Dec 10, 2014 at 1:04 PM, Rolf Turner <r.turner at auckland.ac.nz
> <mailto:r.turner at auckland.ac.nz>> wrote:
>
>
>
>
>     On 10/12/14 20:21, Michael Selevan wrote:
>
>         Hello,
>
>         I am attempting to plot an AR(1) model with a standard deviation
>         and I am a
>         little confused as how to do that. I have been looking through the
>         interwebs and some documentation and I see that there is
>         potentially a few
>         different ways to do this.
>
>         First, simply using the documentation I came up with the command
>
>         arima.sim(n=10, list(ar=0.8), innov=rnorm(10, sd=0.2))
>
>         which would give me the standard deviation I want. Or I believe
>         that to be
>         the case. However, after some more searching and googling, I saw
>         an example
>         where someone used this as a means of adding the AR error term
>
>         error.model=function(n){rnorm(__n, sd=0.2)}
>
>         y = arima.sim(n=10, list(ar=0.8), innov=rnorm(10, sd=0.2), rand.gen=
>         error.model)
>         Now, I am a little confused by this. Would having the error term
>         in the
>         innov parameter as well as the rand.gen be redundant? What would
>         be the
>         expected differences between the two? Should only 1 be used?
>
>         Just looking for some clarification. Been searching and havent
>         found too
>         many examples that explicitly state how to add the error term to
>         an AR(1)
>         model.
>
>
>     It's a little bit subtle, but in a way that's not too important.
>
>     There is, in addition to "innov" a starting innovations vector
>     "start.innov" that is needed.  If either innov or start.innov is not
>     supplied their values get supplied by rand.gen().  So in your second
>     call to arima.sim() ***start.innov*** is being supplied by rand.gen()
>     (but ***innov*** will be taken to be equal to the argument supplied.
>
>     In your first call, where rand.gen() is not specified (and start.innov
>     is not specified), the supplied value of innov will be used and
>     start.innov will be produced by the *default* value of rand.gen()
>     which is rnorm(), you'll get rnorm(n.start,0,1).
>
>     Thus in your first call, the starting innovations will be done with
>     a different standard deviation than the other innovations.  Which is
>     probably not what you want.
>
>     Hence the second call is correct --- but it *is* kind of redundant
>     and confusing to supply "innov" as well as rand.gen().  The code
>     would be
>     clearer if "innov" were dispensed with and it was just left to
>     rand.gen() to do the work.
>
>     The following is not important, but it might be mystifying:  If you
>     leave out "innov" you will get a different result --- even if you
>     set a seed for the random number generators a priori.  E.g.:
>
>     # 1.
>     set.seed(42)
>     innov <- rnorm(10,0,0.2)
>     error.model=function(n){rnorm(__n, sd=0.2)}
>     y1 <- arima.sim(n=10, list(ar=0.8), innov=innov,
>                      rand.gen=error.model)
>
>     # 2.
>     set.seed(42)
>     error.model=function(n){rnorm(__n, sd=0.2)}
>     y2 <- arima.sim(n=10, list(ar=0.8),rand.gen=error.__model)
>
>     The vectors y1 and y2 are (surprisingly until you think carefully)
>     different.
>
>     This is because for y1, innov.start is generated *after* innov is
>     generated, whereas for y2 innov.start is generated *before* innov is
>     generated.  The first entry of innov for y1 will be the same as the
>     first entry of innov.start for y2.  So the sequence of innovations is
>     different.
>
>     Bottom line:  I would recommend *not* using the "innov" argument and
>     just specifying rand.gen() to get the standard deviations that you want.
>
>     HTH
>
>     cheers,
>
>     Rolf Turner
>
>     --
>     Rolf Turner
>     Technical Editor ANZJS
>
>
>
>
> --
> J. Michael Selevan

--
Rolf Turner
Technical Editor ANZJS

```