140_spline_breaks

# 140_spline_breaks - t but the intercept remains the same...

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ECON 140, Fall 2008 Alex Rothenberg Quick Note on Splines / Structural Breaks I made a mistake in my 8 AM section today. It wasn’t such a huge mistake, but I felt like it was instructive, so I’m sending out a note about it. When talking about structural breaks, I wrote down a model like this: P t = α + βt + ε t This is a *very* basic forecasting model. We want to predict P t using no other information than the date t . It’s probably not going to be a good model in practice, but let’s think a bit about it. Suppose we want to allow our model to change, so that before some date, t * , it has one slope while afterward it has another slope. One way to do this (formally) is to regress the following: P t = α + β ( t × { t < t * } ) + γ ( t × { t t * } ) + ε t This is the model I talked about in class. Note the following: E [ P t ] = α + βt for values of t < t * α + γt for values of t t * So, this model lets us have two different slopes for values of t above and below the break point
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Unformatted text preview: ( t * ), but the intercept remains the same. Because of this, it isn’t a true spline model. But it does allow us to free up the parameter of interest substantially. Another (possibly) better way is to do this regress the following: P t = α + βt + γ ([ t-t * ] × 1 { t ≥ t * } ) + ε t Now, consider the following: E [ P t ] = α + βt for values of t < t * α + βt + γ ( t-t * ) for values of t ≥ t * = α + βt for values of t < t * ( α-γt * ) + ( β + γ ) t for values of t ≥ t * So, this model allows us to have both diﬀerent slopes AND diﬀerent intercepts for values of t above and below the breakpoint. Notice that this model will be continuous as a function of t , whereas the model I presented in class will tend to jump discontinuously at t * . See section 5.3 in the book for more details. 1...
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