.. slideconf::
   :slide_classes: appear

==============================================================================
Data Transformations
==============================================================================


Transformations
==============================================================================
Data often deviate from normality and exhibit characteristics
(skewness, kurtosis) that are difficult to model.

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- Transforming data using some functional form will often result in
  observations that are easier to model.

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- The most typical transformations are the natural logarithm and the
  square root.



Logarithmic Transformation
==============================================================================
Given independent and dependent variables, :math:`(x_t, y_t)`, the
natural logarithm transformation is appropriate under several
circumstances:

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- :math:`y_t` is strictly positive (the log of a negative number does
  not exist).

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- :math:`y_t` increases exponentially (faster than linearly) as
  :math:`x_t` increases.

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- The variance in :math:`y_t` appears to depend on its mean
  (heteroskedasticity).



Logarithmic Transformation
==============================================================================
Consider the relationship

.. math::

   y_t = \exp(\beta x_t)\exp(\epsilon_t),

where :math:`\epsilon_t \sim \mathcal{N}(0,\sigma)`.  

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- If :math:`\epsilon_t \sim \mathcal{N}(0,\sigma)`, then
  :math:`\exp(\epsilon_t) \sim \mathcal{LN}(0,\sigma)`.  

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- In this case,

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.. math::

   E\left[\exp(\epsilon_t)\right] = \exp(0.5\sigma^2) \\

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.. math::

   Var\left(\exp(\epsilon_t)\right) = \left(\exp(\sigma^2)-1\right)
   \exp(\sigma^2).



Logarithmic Transformation
==============================================================================
Thus,

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.. math::

   E[y_t] = \exp(\beta x_t) \exp(0.5 \sigma^2) \\

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.. math::

   Var(y_t) = \exp(2\beta x_t) \left(\exp(\sigma^2)-1\right) \exp(\sigma^2).

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- That is, :math:`E[y_t]` grows exponentially with :math:`x_t` and
  :math:`Var(y_t)` is heteroskedastic.



Logarithmic Transformation
==============================================================================
Taking the natural logarithm

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.. math::

   \log(y_t) = \beta x_t + \epsilon_t,

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- :math:`E\left[\log(y_t)\right]` grows linearly with :math:`x_t`.

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- :math:`Var\left(\log(y_t)\right)` is homoskedastic.




Logarithmic Transformation Example
==============================================================================
Given, :math:`\beta = 0.5` and :math:`\epsilon_t \sim
\mathcal{N}(0,0.15)`, the plot below depicts

.. math::

  y_t & = \exp(\beta x_t)\exp(\epsilon_t) \\
  \log(y_t) & = \beta x_t + \epsilon_t.

  .. ifslides::

  .. image:: /_static/Transform/logTransExample.png
     :width: 7.5in
     :align: center

.. ifnotslides::

  .. image:: /_static/Transform/logTransExample.png
     :width: 6in

.. ifnotslides::

  To create this plot, run the following script::

    setwd("~/Dropbox/Academics/Teaching/Econ114/S2013/RScripts/")
    library(quantmod)

    # Number of observations
    n = 500;

    # Parameters driving the true relationship
    beta = 0.5;
    sigma = 0.15

    # The independent variable
    x = seq(0.1,10,length=n)

    # Random shocks drawn for a normal
    eps = rnorm(n,0,sigma);

    # Create the independent variable based on specified model
    y = exp(beta*x)*exp(eps)

    # Plot levels of y and log of y
    pdf(file="logTransExample.pdf", height=6, width=10)
    par(mfrow=c(1,2))
    plot(x, y, type='l', main="(a)")
    plot(x, log(y), type='l', main="(b)")
    graphics.off()


Logarithmic Transformation Example
==============================================================================
Asset prices often display the characteristics that are suitable for a
logarithmic transformation.

  .. ifslides::

  .. image:: /_static/Transform/hogTransExample.png
     :width: 7.5in
     :align: center

.. ifnotslides::

  .. image:: /_static/Transform/hogTransExample.png
     :width: 6in

.. ifnotslides::

  To create this plot, run the following script::

    getSymbols("HOG", from="1993-01-01", to="2002-04-30")
    pdf(file="hogTransExample.pdf", height=6, width=10)
    par(mfrow=c(1,2))
    plot(HOG$HOG.Adj, ylab="HOG Prices", main="(a)")
    plot(log(HOG$HOG.Adj), ylab="Log HOG Prices", main="(b)")
    graphics.off()



Box-Cox Power Transformations
==============================================================================
Generally speaking, the set of transformations

.. math::

   y^{\alpha} = \begin{cases} \frac{y^{\alpha}-1}{\alpha} & \alpha
   \neq 0 \\ \log(y) & \alpha = 0, \end{cases}

Is known as the family of *Box-Cox power transformations*. 



Correcting Skewness and Heteroskedasticity
==============================================================================
Suppose a set of data observations, :math:`y_t`, appear to be right
skewed and have variance increasing with it's mean.  

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- A concave transformation with :math:`\alpha < 1` will reduce the
  skewness and stabilize the variance.

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- The smaller the value of :math:`\alpha`, the greater the effect of
  the transformation.

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- Selecting :math:`\alpha < 1` too small may result in left skewness
  or variance decreasing with the mean (or both).

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- The :math:`\alpha` that creates the most symmetric data may not be
  the best for stabilizing variance - there may be a tradeoff.




Box Cox Example
==============================================================================
  .. ifslides::

  .. image:: /_static/Transform/boxCoxExample1.png
     :width: 7.5in
     :align: center

.. ifnotslides::

  .. image:: /_static/Transform/boxCoxExample1.png
     :width: 6in

This plot was taken directly from Ruppert (2011).



Box Cox Example
==============================================================================
  .. ifslides::

  .. image:: /_static/Transform/boxCoxExample2.png
     :width: 7.5in
     :align: center

.. ifnotslides::

  .. image:: /_static/Transform/boxCoxExample2.png
     :width: 6in

This plot was taken directly from Ruppert (2011).




Geometry of Transformations
==============================================================================
Transformations can be beneficial because they stretch observations
apart in some regions and push them together in other regions.  

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- If data are right skewed, then a concave transformation will

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  - Stretch the distances between observations at the lower end of the
    distribution.

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  - Compress the distances between observations at the upper end of
    the distribution.
  
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- The degree of stretching and compressing will depend on the
  derivatives of the transformation function.




Geometry of Transformations
==============================================================================
For two values, :math:`x` and :math:`x'` close to each other, Taylor's
theorem says

.. math::

   |h(x) - h(x')| & \approx h'(x) |x-x'|.

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- :math:`h(x)` and :math:`h(x')` will be pushed apart where
  :math:`h'(x)` is large.

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- :math:`h(x)` and :math:`h(x')` will be pushed together where
  :math:`h'(x)` is small.

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- :math:`h'(x)` is a decreasing function of :math:`x` if :math:`h` is
  concave.
  



Geometry of Transformations
==============================================================================
  .. ifslides::

  .. image:: /_static/Transform/transGeometry1.png
     :width: 7.5in
     :align: center

.. ifnotslides::

  .. image:: /_static/Transform/transGeometry1.png
     :width: 6in

This plot was taken directly from Ruppert (2011).




Geometry of Transformations
==============================================================================
Similarly, if the variance of a data set increases with its mean, a
concave transformation will

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- Push more variable values closer together (for large values of the
  data).

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- Push less variable values further apart (for small values of the
  data).




Geometry of Transformations
==============================================================================
  .. ifslides::

  .. image:: /_static/Transform/transGeometry2.png
     :width: 7.5in
     :align: center

.. ifnotslides::

  .. image:: /_static/Transform/transGeometry2.png
     :width: 6in

This plot was taken directly from Ruppert (2011).