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Thread: Stationarity and White noise

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    Stationarity and White noise




    Hi all,
    I'm just beginning to learn about time series analysis for my final year project at uni and I am confused about stationarity.
    A time series has to be stationary to be able to apply an ARMA model, however surely if a time series is stationary it is a white noise process and therefore cannot be predicted. Can anyone please point out what I am missing here?

    Thank you,
    Zeke

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    Re: Stationarity and White noise

    No, an AR(1) is stationary if the absolute value of "a" is less than 1.

    X_t = a*X_t-1 + epsilon_t

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    Re: Stationarity and White noise

    Ok, thanks but if X_t is stationary, it must have constant E(x_t)=mean and constant variance and therefore surely it must be white noise and no forecasting can be achieved because it is random data with no trend? Really struggling to understand here, I'm fairly new to stats and I'm trying to work out what basics I need to understand to apply time-series forecasting and possibly work out a model for a forecasting task. I'm supposed to be doing a literature review now looking at modelling methods for time-series'

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    Re: Stationarity and White noise

    Just because there is no long term trend unconditionally doesn't mean that you can't make a prediction given what occurred recently.

    For instance let's think of a different process. One that can only take the values 1 or -1. The next value in the sequence will remain the same with probability .99 and will switch with probability .01. Obviously without telling you anything about the starting position I could ask you what the probability of X_1000 being 1 is. Unconditionally you will probably say .5 right? But if I told you X_999 is +1 then you're going to say the probability that X_1000 is 1 will be .99.

    Time series can seem counter intuitive in that sometimes we talk about the unconditional properties of the sequence and sometimes we focus on the conditional properties of the sequence. It's the conditional properties that allow us to do better than just random guessing in some cases.
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    zeke512 (10-03-2017)

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    Re: Stationarity and White noise


    Ah I see, just because there is no long term trend in the data, the recent data points can still be used to predict the next data point with more accuracy than just the mean in the long term. Great thanks, feel better about this now

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