## AR(1) Process first term

4

I'm trying to generate a very simple AR(1) Process with Mathematica using the ARProcess[] function. The process must have the following format:

$Y_t = \alpha + \beta Y_{t−1} + ϵ_t$

However, after reading the Mathematica help file and some posts about ARProcess[] at MMA.SE, I couldn't find a way to correctly generate the process. I seems that the ARProcess[] function cannot deal with the $\alpha$ term of the process.

For instance, consider the process described below:

$Y_t = 10 + .6Y_{t−1} + ϵ_t$ , where $ϵ_t$ is Normally distributed with mean $0$ and variance $4$ (i.e., $ϵ_t \sim N(0,2)$).

If I use ARProcess[{.6},4] I get

Mean[ARProcess[{.6}, 4][t]]
Variance[ARProcess[{.6}, 4][t]]


Out= 0

Out= 6.25

which are consistent with an AR(1) process of the form $Y_t = 0 + .6Y_{t−1} + ϵ_t$

I would like to be able to, though the ARProcess[] function, calculate the mean and the variance of the process $Y_t = 10 + .6Y_{t−1} + ϵ_t$ in order to get a Mean of $25$ and a Variance of $6.25$.

Is there any way to achieve this in Mathematica?

@Fred it does't work... what you've suggested works like an AR(2) Process of the form $Y_t=0+10 Y_{t-1} + .6 Y_{t-2} + \epsilon_t$, which is not stationary and, as such, the mean cannot be computed. – Rod – 2013-05-20T01:26:01.180

Sorry, I'm deleting the comment and will delete this one later. – Fred Kline – 2013-05-20T01:44:21.840

Don't worry @Fred... I think it's somehow a bug in Mathematica... – Rod – 2013-05-20T01:47:17.520

One other thought, is there a way to get the $10$ into the function? – Fred Kline – 2013-05-20T02:04:10.947

Ahah... That's exactly what I'm trying to figure out! I think it's not possible and, as such, it could be a bug in Mathematica... – Rod – 2013-05-20T02:05:22.503

3

The command ARProcess requires that the input term $e_t$ be a zero mean white noise with a specified variance. What you are trying to do is to have your input term be $10+e_t$ which is clearly not zero mean.

What you can do is to change variables. In your case, if you define a new process $z_t=y_t - 25$ then it will be the same as the $y_t$ process. To see this, consider the "new" problem:

$z_t = 0.6 z_{t-1} + \epsilon_t$

Substituting the definition of $z_t$ gives

$(y_t -25) = 0.6 (y_{t-1}-25) + \epsilon_t$

which, when rearranged is

$y_t = 0.6 y_{t-1} + 10 + \epsilon_t$.

which is the same as your orignal problem. This kind of thing is called "shifting the process". So the way to handle this is to work in the $z$ variable, get whatever answer you are looking for, and then at the very end shift back to $y$ via $y_t=z_t+25$.

Accordingly, you can use

varz = Variance[ARProcess[{.6}, 4][t]]


to get the variance of the $z$ process and

meanz = Mean[ARProcess[{.6}, 4][t]]


to get the mean of the $z$ process. Your original $y$ process has the same variance varz, and mean 25+meanz.

I explicitly wrote that $\epsilon_t \sim N(0,2)$, i.e., $\epsilon_t$ has $0$-mean (and not 10!). The AR(1) process has mean equal to $25$ and can accept an $\alpha$-term (in this case, 10) per definition. The problem is exactly that: the ARProcess[] doesn't accept this $\alpha$-term. BTW, the $\alpha$-term has no influence on the $\epsilon_t$ mean... – Rod – 2013-05-20T11:38:29.880

The above shows you how to use the Mathematica command ARProcess to act like an AR(1) process with nonzero mean. You cannot enter (as far as I know) a nonzero-mean input into ARProcess. But the point is: you don't need to. You can do the same thing by redefining your terms -- as shown above. There is no bug. – bill s – 2013-05-20T11:49:29.677

So what is the point in using Mean[ARProcess[]] if the result will be always zero? I mean, if the ARProcess[] function cannot deal with an $\alpha$-term, its much better to develop an AR function by yourself instead of using Mathematica's ARProcess[]... – Rod – 2013-05-20T12:25:19.603

1You said "You cannot enter (as far as I know) a nonzero-mean input into ARProcess"... so, this must be a bug, don't you think? Imagine two different processes: 1) $Y_t=10+Y_{t-1}+\epsilon_t$ and 2) $Y_t=0+Y_{t-1}+\epsilon_t$; they are clearly different but if you try to estimate both processes with EstimatedProcess[data, ARProcess] you will get exactly the same result, which is not consistent with one of the processes... so I'm getting wrong results if I use Mathematica-ARProcess... – Rod – 2013-05-20T13:16:43.683

I think it just doesn't work the way you expect. It seems to me that the most common procedure in any data analysis is to take out the mean. That's all you need to do in the case you cite and then it will give you sensible answers. I agree that it would be nicer if it allowed specification of the mean, but the documentation is pretty clear that it doesn't. – bill s – 2013-05-20T13:45:11.260

1I don't think the documentation is clear about removing the mean... the shift operator E in the Details uses 1 instead of an $a_0$-term. In the Applications part there is a single example (daily exchange rates of the euro to the dollar) where it removes the mean and then add it back for forecasting... I don't think an example is clear enough for defining the function... – Rod – 2013-05-20T14:09:19.463