Upload
cleopatra-reynolds
View
214
Download
1
Embed Size (px)
Citation preview
1
MF-852 Financial Econometrics
Lecture 9 Dummy Variables, Functional Form,
Trends, and Tests for Structural Change
Roy J. EpsteinFall 2003
2
Topics 0-1 Dummy Variables Linear Trend Transformations of Variables Tests for Structural Change
3
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?
4
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?
5
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.
6
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.
7
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.
8
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)
9
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)
10
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
11
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
12
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.
13
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.
14
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.
15
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.
16
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.
17
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.
18
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?
19
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.
20
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.
21
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.
22
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.