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Thread: Time Series ARIMA(p,d,q) - how to determine what p, d and q equal?

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    Time Series ARIMA(p,d,q) - how to determine what p, d and q equal?




    Hello,

    I (think) understand that in order to determine the parameter values of an ARIMA(p,d,q) model, I need to look at three things:

    1. PACF to determine the value of P
    2. The level of differencing to determine the value of D
    3. ACF to determine the value of Q (and if the process is stationary -> to then find the value of D)

    However, I have several questions about this process.

    1. Regarding the differencing, is it true that I should find out which level of differencing produces the lowest standard deviation. Thus, if second and third differencing produce higher standard deviations than first order differencing then my D value will equal 1. Is this correct?

    2. If my time series data shows some sort of linear trend (or the short term variation increases as time increases) is performing a log transformation a common approach to correct for this? And then at the end I would just need to "un-log" the data.

    3. Which PACF and ACF charts do I look at to determine the values for P and Q? Do I look at the PACF/ACF charts from the original data? Or do I look at the PACF/ACF charts from the differencing that I choose?

    4. What specifically from the PACF/ACF charts am I looking for in order to determine the values of P and Q, respectively? Do I simply just look for the highest lag that is significant? For example, what if I have an ACF chart that is slowly decreasing - what would my Q value be? What if I have a PACF chart that moves rapidly from positive to negative? If the ACF does NOT decay to 0, does that mean it's not stationary and therefore I need to difference the process?

    5. If I think the process is stationary, that means my D value = 0?

    Thanks to anyone who could help me out at all - I'm pretty much trying to teach myself Time Series analysis and unfortunately I feel it's a rather large and complex topic.
    Last edited by lancearmstrong1313; 08-17-2012 at 07:33 AM.

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    Re: Time Series ARIMA(p,d,q) - how to determine what p, d and q equal?

    Quote Originally Posted by lancearmstrong1313 View Post
    1. Regarding the differencing, is it true that I should find out which level of differencing produces the lowest standard deviation. Thus, if second and third differencing produce higher standard deviations than first order differencing then my D value will equal 1. Is this correct?
    Not always.
    Quote Originally Posted by lancearmstrong1313 View Post
    2. If my time series data shows some sort of linear trend (or the short term variation increases as time increases) is performing a log transformation a common approach to correct for this? And then at the end I would just need to "un-log" the data.
    Your explanation is not sufficient to take the log transformation. For a positive time series data, taking log may help to achive the normal assumption on residual.

    Quote Originally Posted by lancearmstrong1313 View Post
    3. Which PACF and ACF charts do I look at to determine the values for P and Q? Do I look at the PACF/ACF charts from the original data? Or do I look at the PACF/ACF charts from the differencing that I choose?
    First job is to make the time series stationary. So appropriate differencing comes first. In pure AR or MA looking at ACF/PACF help. Otherwise you need to do some iteration to get p & q.

    Quote Originally Posted by lancearmstrong1313 View Post

    4. What specifically from the PACF/ACF charts am I looking for in order to determine the values of P and Q, respectively? Do I simply just look for the highest lag that is significant? For example, what if I have an ACF chart that is slowly decreasing - what would my Q value be? What if I have a PACF chart that moves rapidly from positive to negative? If the ACF does NOT decay to 0, does that mean it's not stationary and therefore I need to difference the process?
    Refer q3. For a smaller lag of AR and MA one can form pattern. for larger lag of p & q ACF/PACF relationship is complex.




    5. If I think the process is stationary, that means my D value = 0?
    Correct.

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    Re: Time Series ARIMA(p,d,q) - how to determine what p, d and q equal?

    1. Regarding the differencing, is it true that I should find out which level of differencing produces the lowest standard deviation. Thus, if second and third differencing produce higher standard deviations than first order differencing then my D value will equal 1. Is this correct?
    You should use the differencing that makes sense given what your graphs show you and what the ACF/PACF suggest from the ideal types of ARIMA (that is certain ideal patterns have been identified that suggest various ARIMA parameters). An augmented Dickey Fuller test will also suggest whether the data is stationary or not. Be careful about over differencing a common mistake in ARIMA. It will distort your AR parameters if you do it (it eliminates AR that should be in the model). It is rare to have more than two levels of differencing in a model (including seasonal differencing).

    2. If my time series data shows some sort of linear trend (or the short term variation increases as time increases) is performing a log transformation a common approach to correct for this? And then at the end I would just need to "un-log" the data.
    Logging is used more often to deal with increasing (or decreasing) variability over time a violation of ARIMA assumptions. It also is used I believe to address seasonality although I am not sure of that. You don't have to transform the data back, but it is often done because it is harder to interpret the logged results.

    3. Which PACF and ACF charts do I look at to determine the values for P and Q? Do I look at the PACF/ACF charts from the original data? Or do I look at the PACF/ACF charts from the differencing that I choose?
    From the differencing. MA terms, and I think AR ones, are invalid when there is non-stationarity that is a trend of any kind.

    4. What specifically from the PACF/ACF charts am I looking for in order to determine the values of P and Q, respectively? Do I simply just look for the highest lag that is significant? For example, what if I have an ACF chart that is slowly decreasing - what would my Q value be? What if I have a PACF chart that moves rapidly from positive to negative? If the ACF does NOT decay to 0, does that mean it's not stationary and therefore I need to difference the process?
    There is no simple answer to that, it is the heart of the "art" of ARIMA. You are looking for patterns in the lags that are close to ideal types. Generally, the ACF shows patterns in the MA and the PACF for the AR. This is one starting point (Duke has a whole series of these).

    http://www.duke.edu/~rnau/411arim.htm



    5. If I think the process is stationary, that means my D value = 0?
    Yes although if you suspect seasonality, then you have to model that seperately. You should look at the Ljung Box statistic (SAS calls these white noise tests) to see if there is any remaining AC patterns which (if there are) means you have some parameter that is wrong.
    "Very few theories have been abandoned because they were found to be invalid on the basis of empirical evidence...." Spanos, 1995

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    Re: Time Series ARIMA(p,d,q) - how to determine what p, d and q equal?


    The responses you have received should be qualified to say that in the absence of time-varying parameters, error variance heterogenerity due to deterministic structural changes no pulses , no level shifts , no local time trends , no seasonal pulses what follows is true. In summary look for dominance between the ACF and the PACF. If the ACF dominates then the model might be an AR model, if the PACF dominates the model might be a MA model. Estimate and iterate to determine P,Q and any needed differencing. Review the tentative model errors for any needed pulse/level shifts/seasonal pulses/local time trends and add this structure to the model. Evaluate variance changes and parameter changes over time and evaluate linkages between the expected value and the variance of the errors for possible power transformations. If you need software to do this for you please go to http://www.autobox.com and pursue AUTOBOX.

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