MF-852 Financial Econometrics

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MF-852 Financial Econometrics

• Lecture 9
• Dummy Variables, Functional Form, Trends, and
  Tests for Structural Change
• Roy J. Epstein
• Fall 2003

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Topics

• 0-1 Dummy Variables
• Linear Trend
• Transformations of Variables
• Tests for Structural Change

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Dummy Variables

• H0 often involves a change in a regression
  coefficient.
• Example Yi is cheese dogs consumed at party by
  ith person.
• Use regression to estimate mean number of cheese
  dogs eaten
• Yi ?0 ei
• Does the mean differ between men and women?

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Dummy Variables

• A dummy variable D has the value 0 or 1.
• 0 is for a baseline group
• 1 is for a contrast group.
• Suppose women are the baseline. Then Di 0 if
  the ith person is female, otherwise Di 1.
• What if men were the baseline?

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Dummy Variables

• H0 men eat same number of cheese dogs on average
• New regression is
• Yi ?0 ?1Di ei
• Female mean ?0 Male mean ?0 ?1
• Test H0 by testing significance of ?1.

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Dummy Variables

• Suppose 3 categories men, women, children. H0
  same mean for all.
• Define 2 dummies
• D1i 1 if woman, else D1i 0
• D2i 1 if child, else D2i 0
• Regression is
• Yi ?0 ?1D1i ?2D2i ei
• Effects ?0 ?0 ?1 ?0 ?2
• Test H0 with F test on ?1 and ?2.

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Functional Form

• We have specified a multiple regression as linear
  function
• Yi ?0 ?1X1i ?2X2i
• ?kXki ei
• But we have a LOT of flexibility in defining the
  variables.

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Transformations of Variables

• Examples
• Zi ln(Xi) Zi 1/Xi
• Zi Xi2
• Zi Xi Xi1 (first difference)
• Zi (Xi Xi1)/Xi1 ( change)
• Zi ln(Xi/Xi1) (compound g)

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More Examples of Valid Transformations

• Suppose Yi a0Xia1exp(ei) where a0 and a1 are
  coefficients.
• Take logs of both sides
• ln(Yi) ?0 a1ln(Xi) ei
• This is a linear regression model!
• ?0 ln(a0)

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Transformations in General

• We allow any term with 1 regression coefficient
  factored out in front.
• Yi ?0 ?1ln(X1i)X2i ?2X23i1
• But not
• Yi ?0 ?1ln(X1i)X2i?2X23i1

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Trend

• Trend the average increase (decrease) in Yi each
  period, after controlling for other factors.
• Only makes sense for time-series data.
• Define trend variable Ti i.
• T1 1, T2 2, etc.
• Yi ?0 ?1Ti ?2Xi ei

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Trend

• Interpretation Y changes on average by ?1 units
  each period, after controlling for X.
• Reflects net effect of omitted variables.
• Other trend models
• Ln(Yi) ?0 ?1Ti ?2Xi
• ?1 is average percent change in Y each period,
  after controls.

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Structural Change

• We assume that the model describes all of the
  data but this may not be accurate.
• The earlier example of a single mean for TV
  viewing for all populations (men, women,
  children) is simplest case where assumption might
  not be valid.

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Structural Change Testing, Generally

• H0 defines categories of interest in data, e.g.,
• Genders, age groups, geographic locations
  (cross-section data)
• Old vs. recent observations, special time periods
  (war, different regulatory regime) (time-series
  data).
• Define a dummy variable for each category other
  than the chosen baseline group.

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Structural Change Testing, Generally

• Include the dummy variables in the regression.
  This allows the different categories to have
  different intercepts.
• Equivalent to allowing different means.
• Yi ?0 ?1Di ?2Xi ei
• Test significance of dummies with t or F test, as
  appropriate.

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Structural Change Testing, Generally

• Next level of sophistication is to allow
  different categories to have different slopes for
  Xi.
• Create interaction term DiXi.
• Yi ?0 ?1Di ?2Xi ?3DiXi ei
• Test significance of ?1 and ?3 with F test.
• Can do this with categories gt 2.

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Structural Change Examples

• CAPM (time-series)
• (A)You estimate model to test if returns were
  significantly different during a subperiod in the
  data. This is an event study.
• (B)You estimate model with 20 weekly returns.
  Beta might have been different for the first 10
  weeks.

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Structural Change Examples

• Cross-section
• Model for prices charged by stores in different
  locations. Do stores have different prices after
  controlling for their costs? (from Staples-Office
  Depot merger)
• Baseball player salaries depend on years of
  experience and the square of experience. Does
  the players position also affect salary?

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Testing for Structural Change

• CAPM (A). Want to test if returns were higher in
  weeks 8-12. Define Di 0 if i lt 8 or i gt 12.
  Otherwise Di 1.
• Yi ?0 ?1Di ?2Xi ei
• Perform test of significance on ?1.

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Testing for Structural Change

• CAPM (B). Want to test if beta was different for
  weeks 1-10. Define Di 0 if i gt 10. Otherwise
  Di 0.
• Yi ?0 ?1Di ?2Xi ?3(DiXi) ei
• Perform F test on ?1 and ?3.

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Testing for Structural Change

• Store model. 50 stores in 3 different cities.
  Test if average markup is different across
  cities.
• Define D1i1 if in city 2, else 0.
• Define D2i1 if in city 3, else 0.
• Yi ?0 ?1D1i ?2D2i ?3Xi ei
• Perform F test on ?1 and ?2.

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Warning!

• Amount of data will limit how many structural
  changes you can test for.
• Model needs at least 5 data points per estimated
  coefficient (Epsteins rule of thumb).
• So you cant introduce lots of dummies
  indiscriminately.
• Slope changes are harder to measure than
  intercept changes.