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Copy file name to clipboardExpand all lines: lectures/additive_functionals.md
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@@ -46,9 +46,9 @@ Asymptotic stationarity and ergodicity are key assumptions needed to make it pos
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Are there ways to model time series that have persistent growth that still enable statistical learning based on a law of large numbers for an asymptotically stationary and ergodic process?
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The answer provided by Hansen and Scheinkman {cite}`hansen2009long` is yes.
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The answer provided by Hansen {cite}`Hansen_2012_Eca` is yes.
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They described two classes of time series models that accommodate growth.
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He described two classes of time series models that accommodate growth.
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They are
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We describe how to construct, simulate, and interpret these components.
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More details about these concepts and algorithms can be found in Hansen and Sargent {cite}`hansen2008robustness`.
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More details about these concepts and algorithms can be found in Hansen {cite}`Hansen_2012_Eca`and Hansen and Sargent {cite}`Hans_Sarg_book`.
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Let's start with some imports:
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## A Particular Additive Functional
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{cite}`hansen2009long` describe a general class of additive functionals.
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{cite}`Hansen_2012_Eca` describes a general class of additive functionals.
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This lecture focuses on a subclass of these: a scalar process $\{y_t\}_{t=0}^\infty$ whose increments are driven by a Gaussian vector autoregression.
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When simulating we embed our variables into a bigger system.
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This system also constructs the components of the decompositions of $y_t$ and of $\exp(y_t)$ proposed by Hansen and Scheinkman {cite}`hansen2009long`.
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This system also constructs the components of the decompositions of $y_t$ and of $\exp(y_t)$ proposed by Hansen {cite}`Hansen_2012_Eca`.
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All of these objects are computed using the code below
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\end{aligned}
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$$
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Then the Hansen-Scheinkman {cite}`hansen2009long`, {cite}`Hans_Sarg_book` decomposition is
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Then the Hansen {cite}`Hansen_2012_Eca`, {cite}`Hans_Sarg_book` decomposition is
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