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node21 Focus on the AR Model (Auto Regressive) STAT 510 - Applied Time Series Analysis

Node21 Focus on the AR Model (Auto Regressive) STAT 510 - Applied Time Series Analysis

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This is Google's cache of http://onlinecourses.science.psu.edu/stat510/node/21 . It is a snapshot of the page as it appeared on 21 Jul 2010 08:54:24 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 Focus on the AR Model (Auto Regressive) Submitted by gfj100 on Sun, 03/28/2010 - 15:29 We already looked at one example of a stationary process, the moving average, MA . We know this is stationary process because we calculated the mean, variance and covariance function. In this section, we will focus on the AR model. We recall an AR ( p ) model: We will see later that this has zero mean. Can a stationary process have a non-zero mean, or does it have to have mean zero? Yes, but that mean cannot change through time. If the formula above has zero mean, does the following have mean α? No. Let's see why this is the case. If an AR ( p ) process had a mean μ then we could center it by subtracting the mean from all of the x t : Thus, the result ( x t - μ) has mean zero. If we rearrange this we get: The important point here is that the relationship between response variables and covariates is not as straightforward as in regression. The underlined part of the formula is α : α is not the mean, instead it has a relationship with all of the other parameters in the AR ( p ) including the
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mean. How can we deal with this in practice? What if we took a look at our data and it is floating around 100. How would we get rid of the 100? We could subtract the sample mean from the entire time series, which is essentially what we have doing above (except above we used the population value). So, in this way, it is easy to obtain effectively a zero mean AR. When we fit an AR ( p ) like this in R, it removes the mean automatically. One thing to note is that R is going to return a number of parameters along with the constant. It is important to know what it is returning. In practice, computer programs return different values (μ or α). What about the MA model? Is this the same problem. Take a look at the MA model: Is μ the mean? Yes, because there is no feedback. From this point on we will assume that the AR or MA that we discuss will have zero mean. The Backshift Operator Before we look at the models in more detail, we introduce the backshift operator , B . The backshift operator will allow us to write and analyze our models in a simpler way. If we apply B to x t we get: . If we apply B twice we get B 2 x t we get: We have already seen a similar operator, the difference operator , and now we can write this difference operator using back shift operator: How can we write an AR model using this backshift operator? Here is the AR model:
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Node21 Focus on the AR Model (Auto Regressive) STAT 510 - Applied Time Series Analysis

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