1 MF-852 Financial Econometrics Lecture 9 Dummy Variables, Functional Form, Trends, and Tests for...

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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.