Dummy Variables

1
Dummy Variables

• We have so far used Quantitative variables in
  regression
• Qualitative variables like gender, race,
  location, season etc also an important part of
  analysis
• Dummy variables are used to include such
  qualitative variables as explanatory variables in
  a regression model
• Logit, Probit, Tobit models include qualitative
  variable as dependent variable

2

• Dummy variable involves assigning 1 to
  observation of the chosen characteristic and 0
  for the rest.
• For gender, assign 1 for female observation and 0
  for male.
• For race(4), we need to create more than 1
  variable. 1st variable, 1 if African and 0 fo all
  other races.
• 2nd variable1 if White and 0 for all other
  races.
• 3rd variable1 if Asian and 0 for all other
  races
• 4rth variable1 if colored and 0 for all other
  races
• Important All 4 variables are not included in
  regression analysis, 4-13 are included. Omitted
  variable is the benchmark variable.
• Constant indicates the omitted benchmark variable
• Coefficients of the included variable is
  considered in relation to the constant

3
ANOVA Models

• Regression model with regressors that are
  exclusively qualitative in nature
• Sales explained only by seasonality
• If four seasons, include 3 dummy variable.
• Season excluded is benchmark is the constant
• Y234 35 D1 -21 D2 -67D3
• D1 Jan March D2Apr-June D3July-Sep
• D4Oct-Dec (benchmark variable)
• Jan-March season of highest sale
• July-Sep season of lowest sale
• Another way to do it is include ALL 4 dummies but
  exclude the constant. If not, it leads to perfect
  multicollinearity violating assumption of OLS

4
Interaction effects

• Dummy variables can be used to study interaction
  effect between various qualitative variables like
  race and gender.
• Yb1b2D1b3D2b4D3b5D4
• D1gender dummy D2-D4race dummies(b,w,i)
• Constantmissed gender and race dummy(female
  coloured)
• What is income of a male black? Introduced an
  interaction dummy D1D2
• Yb1b2D1b3D2b4D3b5D4b6D1D2
• b1b6income of male black

5
ANCOVA

• Regression model with both qualitative and
  quantitative model Incf(sex, education)
• Yb1 b2D1 b3X1
• Here the constant will differ for male female
• Constant b1 (D0, female)
• Constant b1b2 (D1, male)
• B3change in income for 1yr increase in education
• To understand what is the income of educated
  male, add an interactive term D1X1
• Yb1 b2D1 b3X1 b4 DX1
• If b4 is and significant it means educated male
  gets more than educated female. Educated male
  coefficient is (b3b4)

6
Other uses of dummy variable

• Check for structural breaks, better than chow
  test
• Deseasonalisation of data series
• Piece-piece linear regression (spline functions)

7
Multicollinearity

• Violation of the assumption of no linear
  relationship between the regressors
• B2 in multivariable reg is change in Y wrt to 1
  unit change in X1 holding other variables
  constant
• Important bcos isolating impact of X1 on Y is not
  possible if X1 is related to X2 included in the
  model
• In the event of perfect multicollinearity, slopes
  are indeterminate
• If relationship is not perfect even though there
  maybe some degree of linear relationship, slope
  can be estimated
• However the variances are extremely high in such
  cases and inference making is distorted
• t-value low and confidence intervals large
  because of high SE

8
Detection of multicollinearity

• High R-sq but insignificant t-values
• High pairwise and partial correlation between the
  regressors
• Auxiliary regression if the R-sq of a regression
  of the X-variables(Rasq) is higher than the R-sq
  of Yon X variables
• VIF (Variance Inflating Factor) 1/(1-Ra-sq)
• If VIFgt10, very high multicollinearity
• If VIF is close to 1, no multicollinearty
• TOL (Tolerance factor) 1/VIF
• TOL close to 0, high multicollinearity
• TOL close to 1, no multicollinearity

9
Remedial measures

• A priori information
• Combining cross-section and time series data
• Dropping variable and specification bias
• First difference or ratio transformation or in
  deviation from mean form
• Additional data multicollinearity is a SRF
  problem and usually caused by micronumerosity
  (sample sample)
• Factor analysis/ PCA