node5 Regression STAT 510 - Applied Time Series Analysis

node5 Regression STAT 510 - Applied Time Series Analysis -...

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This is Google's cache of http://onlinecourses.science.psu.edu/stat510/node/5 . It is a snapshot of the page as it appeared on 9 Aug 2010 04:28:35 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 1: Introduction and Basics Regression Submitted by gfj100 on Mon, 03/22/2010 - 14:05 Review of Regression Time series is a single list of numbers, x 1 , x 2 , . .. x n . What is the structure of regression data? In time series, we are trying to find dependency between neighbors. What are we trying to do with regression? In regression we are trying figure out the relationship between a set of variables. ( x 1 , z 1 ,), ( x 2 , z 2 ) . .. ( x n , z n ) What is the model for regression? A model that determines the linear dependency between these two variables has a slope, an intercept and an error term: x i = β 0 + β 1 z i + ε i What other assumptions do we make? We assume that: 1. the mean of ε i is zero, 2. the variances of ε i are the same, var i ) = σ 2 (homoskedascity), 3. the errors in the independent variable are independent (ε i are independent of ε j ) , and 4. the errors are normal. We could extend this model to more than one z i , for example: x i = β 0 + β i z i 1 + β i z i 2 + . .. + β q z iq + ε i
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with all of the the same assumptions on the errors. What about regression in the context of a time series? The z i could be t . This would be a reasonable model for data growing in time. Is this model stationary? Stationary implies that it has a fixed mean and variance. Is this the case here? x t = β 0 + β 1 t + ε i Another thing we could do, if we had two time series, (GNP growth and unemployment, for instance), would be regression to compare the two. We could assume that the relationship between the two series is: x t = β 0 + β 1 z t + ε i , where x t and z t are both time series This may be a simple way to address the relationship between these two time series? In the simple case of regression, we have pairs of observations, and regression fits a line with the least amount of "error", i.e., so that we minimize the squared distance from the line to each point. Basically, we move the β 0 's and the β 1 's around until we minimize the square of this deviation (above). This is why regression is sometimes called least squares fit. In minimizing this distance, we make the assumptions listed above so that we can draw conclusions about other things. For instance, if we have a fit, how good is this fit? Can we find standard errors that will tell us if = 0. Does this geometric interpretation of least squares give us any information about this? No. However, if we assume the errors have μ = 0, variance is σ
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node5 Regression STAT 510 - Applied Time Series Analysis -...

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