1 mf-852 financial econometrics lecture 9 dummy variables, functional form, trends, and tests for...
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![Page 1: 1 MF-852 Financial Econometrics Lecture 9 Dummy Variables, Functional Form, Trends, and Tests for Structural Change Roy J. Epstein Fall 2003](https://reader035.vdocuments.mx/reader035/viewer/2022062801/56649e4d5503460f94b43801/html5/thumbnails/1.jpg)
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MF-852 Financial Econometrics
Lecture 9 Dummy Variables, Functional Form,
Trends, and Tests for Structural Change
Roy J. EpsteinFall 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 – Xi–1 (first difference) Zi = (Xi – Xi–1)/Xi–1 (% change) Zi = ln(Xi/Xi–1) (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 + 1[ln(X1i)*X2i] – 2X2
3i–1 But not
Yi = 0 + 1ln(X1i)*X2i*2X23i–1
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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 > 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 player’s 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 < 8 or i > 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 > 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 D1i=1 if in city 2, else = 0. Define D2i=1 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 (Epstein’s rule of thumb). So you can’t introduce lots of
dummies indiscriminately. Slope changes are harder to
measure than intercept changes.