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# Time Series: Stationarity and ARMA watch

1. Hey

Can anyone help me with a bit of time series.

I'm just struggling how to tackle times serise.

So given a time series, I understand the I firstly must try to remove any trend and seasonality, in order to make the series stationary. And with regards to stationary, does this simply mean that the series should not depend upon t? So if this is the case a stationary series is then random, much like white noise.

However this is where I am now getting confused. It says that ARMA models can only be used for time series which are stationary. But if stationary series are just random, with now dependance upon t, what would be the point in trying to model them? As in if I were to use an AR model, these kind of models are usually of the form

Xt = a1 Xt-1 + Et

but this clearly shows that Xt depends upon previous data. And a stationary time series is one that is not dependent upon time

Thanks
2. Not dependent on t is when the series is trend stationary (i.e. a series is now stationary because the the trend has been removed). You also have stationarity in terms of first differences.

A stationary series isn't just when you have a series that's white noise. Stationary is just when the first and second moments (mean, variance) do not change over time. You make a series stationary but you can still have certain AR or MA characteristics that can model it.

Furthermore, take a random walk model. It's still 'modelable' (not a real word) as it is identified by an equation, but obviously it doesn't tell you a lot about future estimations, other than the best prediction of next period's value is this period's value.
3. (Original post by little_wizard123)
Not dependent on t is when the series is trend stationary (i.e. a series is now stationary because the the trend has been removed). You also have stationarity in terms of first differences.

A stationary series isn't just when you have a series that's white noise. Stationary is just when the first and second moments (mean, variance) do not change over time. You make a series stationary but you can still have certain AR or MA characteristics that can model it.

Furthermore, take a random walk model. It's still 'modelable' (not a real word) as it is identified by an equation, but obviously it doesn't tell you a lot about future estimations, other than the best prediction of next period's value is this period's value.
So what should be the general approach?

Something like this:

Plot the time series. Identify and trend and seasonality.

Remove the seasonality and trend in an attempt to produce iid residuals

If the residuals are iid then I can stop and I have found a suitable model

If not then should I be reconsidering the way in which I tried to remove the trend and seasonality, or should I start to consider an ARMA model? If I need to consider an ARMA model should this be applied from scratch or with trend and seasonal components in place?

Thanks
4. And when would smoothing the data and using an exponentially weighted moving average be of use?
5. Yeah, I preferred to use the SACF and SPACF to work out the order of integration etc. That's pretty effective.

White noise is just a special case of a stationary series. I think that process is pretty spot on tbh. Although plotting the transformed series (the stationary one) and looking at the SACF and SPACF would tell you directly whether the model is white noise or can be better expressed as an ARMA model.
6. (Original post by singh246)
And when would smoothing the data and using an exponentially weighted moving average be of use?
Well I think you might 'smooth' if you have seasonality, for example lots of spending around christmas. For example, you couldn't predict January's expenditure from the last month's, but this is what an AR model is defined as! So the seasonality can be taken care of by smoothing.

Just had a quick look, and Wikipedia is pretty good for exponential smoothing.

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