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321_09_slides15 - Introduction to Time Series Chapters 10...

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Introduction to Time Series 1 Chapters 10 + 11 + 12 Part II
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Trending Time Series Economic time series often have a trend Just because 2 series are trending together, we can’t assume that the relation is causal Often, both will be trending because of other unobserved factors We can control for the trend 2
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Trends a linear trend y t = α 0 + α 1 t + e t , t = 1, 2, … an exponential trend log( y t ) = α 0 + α 1 t + e t , t = 1, 2, α 1 =% growth/100 a quadratic trend y t = α 0 + α 1 t + α 2 t 2 + e t , t = 1, 3 y t t y t =e α1 t
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Detrending Adding a linear trend term to a regression is the same thing as using “detrended” series in a regression Detrending a series involves Regressing a variable in the model on t Use the residuals to form the detrended series By using the residual from the regression, the trend has been partialled out y t -det = y t α 0 - α 1 t 4
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Detrending An advantage to actually detrending the data (vs. adding a trend) involves the calculation of goodness of fit Time-series regressions tend to have very high R 2 , as the trend is well explained The R 2 from a regression on detrended data better reflects how well the x t ’s explain y t 5
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Example of Trend Modeling In a recent legal case, a downtown hotel claimed that it had suffered a loss of business due to what was considered an illegal action by others. In order to support its claim of lost business, the hotel had to predict what its level of business would have been in the absence of the alleged illegal action. In order to do this, experts testifying on behalf of the hotel used data collected before the period in question and fit a relationship between the hotel’s occupancy rate and overall occupancy rate in the city of Chicago. This relationship would then be used to predict occupancy rate during the period in question assuming no alleged illegal action. The regression equation is HX_occ = 16.1 + 0.716 Chi_occ Predictor Coef SE Coef T P Constant 16.136 8.519 1.89 0.069 Chi_occ 0.7161 0.1338 5.35 0.000 S = 7.506 R-Sq = 50.6% R-Sq(adj) = 48.8%
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Example of Trend Modeling Looks good but what about the residuals? 80 70 60 50 40 80 70 60 50 40 Chi_occ H X _ o c c R-Sq = 0.506 Y = 16.1357+ 0.716132X Regression Plot
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Example of Trend Modeling plot of the residuals… 30 20 10 2 1 0 -1 -2 Index SRES1 What’s wrong here?
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Example of Trend Modeling To take into account the downward trend in the hotel’s occupancy rate, we introduce a linear trend term in the model. t 2 t 1 0 t t Occ _ CH Occ _ HX ε β β β + + + = The regression equation is HX_occ = 26.7 + 0.695 Chi_occ - 0.596 Time Predictor Coef SE Coef T P Constant 26.694 6.419 4.16 0.000 Chi_occ 0.69524 0.09585 7.25 0.000 Time -0.5965 0.1134 -5.26 0.000 S = 5.372 R-Sq = 75.6% R-Sq(adj) = 73.8%
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Example of Trend Modeling The significant downward trend means that if you predict into the future period (beyond the sample period) in which the alleged damages occurred, the hotel occupancy rate is falling for reasons other than the allegation.
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