node26 SARMA SARIMA STAT 510 - Applied Time Series Analysis

Node26 SARMA SARIMA STAT 510 - Applied Time Series Analysis

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This is Google's cache of http://onlinecourses.science.psu.edu/stat510/node/26 . It is a snapshot of the page as it appeared on 31 Jul 2010 16:36:52 GMT. The current page could have changed in the meantime. Learn more Text-only version STAT 510 - Applied Time Series Analysis ANGEL Department of Statistics Eberly College of Science Home // Section 2: Time Domain Models SARMA / SARIMA Submitted by gfj100 on Sun, 03/28/2010 - 15:36 SARMA - Seasonal ARMA So far, we've not seen much seasonal data. The ARMA models that we've discussed do not allow for skipping lags. But, we may wish to have a model for monthly observations which depends on both the previous month and the month one year ago, for instance. SARIMA models will allow us to do that. So far we have discussed removing any trend, including seasonal trend. This doesn't mean that we have removed the dependency. We may have removed the mean, μ t , part of which may include periodic component. In some ways we are breaking the dependency down into recent things that have happened and long-range things that have happened. An example might be the weather. When you have a cold January it does depend on the December and November, but it can also depend on last January and the January before that because there are long-range weather trends as well as local (in time) weather trends. The same might be true of the economy. A pure SARMA means it only has seasonal components and is the first model that we want to discuss. We denote this by ARMA ( P , Q ) s . We can write the model using backshift operators in the following way: where S = the season. For instance, S = 12 with monthly data and S = 4 with quarterly data. If we write out the Φ part of this it would look like: The Θ piece is very similar.
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The first polynomial is the seasonal autoregressive operator and the second is the seasonal moving average operator. An ARMA (1,1) 4 would look like: This will give us: Is this a regular ARMA as well? (Think about setting certain things to 0.) This is essentially an ARMA model, except the lags between zero and four are omitted. What would the ACF and PACF look like for various seasonal ARMA models? Let's first look at a seasonal MA then a seasonal AR . Let's think about a monthly model and a seasonal MA , in other words an ARMA (0, 1) 12 . The model would be written as Is this model stationary?And, what would you expect of the ACF plot? Let's look at the auto covariance: When h = 0 you get something that corresponds almost exactly to a regular MA (1). What happens when h = 1? The autocovariance is zero, the terms will not overlap at all. What about for h = 2? No overlap. 3? Nothing. When will they line up again? At zero they lineup on top of each other, the next time is when h = 12. What ACF would you expect for this pure seasonal MA with S = 12?
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We should see one spike at 12.
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