Regression Extensions

1
Regression Extensions

• WW Chapter 14

2
Introduction

• So far we have assumed that our independent
  variables are measured intervally. Today we will
  discuss how to interpret dummy variables in
  regression.
• Recall that a dummy variable takes on two
  possible values, 0 or 1.

3
The Regression Model

• Suppose we want to estimate the following model
• Y ? ?1D ?2X ?
• Y of militarized disputes a state gets
  involved in per year
• X amount of annual military spending
• D 1 if the state is democratic
• D 0 if the state is non-democratic

4
The Results

• We want to see if there is a difference between
  democratic and non-democratic regimes in terms of
  how many militarized disputes they get involved
  in.
• Suppose we estimate the following regression
  model
• Yp -6.4 2.02D .000179X

5
The Results

• We can compare the two groups as follows
• For D0, Yp -6.4 2.02(0) .000179X
• -6.4 .000179X
• For D1, Yp -6.4 2.02(1) .000179X
• -8.42 .000179X
• The coefficient for D (?1) is the change in Y
  that accompanies a unit change in D, which is
  either zero or one.

6
The Impact of a Dummy Variable

• We can see that the dummy variable changes the
  value of the Y-intercept (?). If we were to plot
  these two regression lines, they would be
  parallel lines with slope (.000179), and
  Y-intercepts 6.4 or 8.42.
• We can conclude that democracies get involved in
  about 2 fewer conflicts per year compared to
  non-democracies.

7
Summary

• More generally, if D is a 0-1 dummy variable in a
  regression model,
• Ypa b1D b2X
• Then the regression line where D1 is parallel
  and b1 units higher than the line where D0.

8
Multiple categories

• Suppose we want to expand our measure of
  democracy to three levels democracy, anocracy,
  and autocracy. We could create two dummy
  variables.
• D1 1 if anocracy, 0 otherwise
• D2 1 if democracy, 0 otherwise

9
Multiple categories

• We leave the third category (autocracy) out as a
  reference group. We always have one less dummy
  than there are categories because if we included
  all three, there would be perfect
  multicollinearity.
• Our new model is
• Y ? ?1D1 ?2D2 ?3X ?

10
Estimating the model

• Suppose we estimate our new model and obtain the
  following results
• Yp -5.1 1.4D1 3.6D2 .0021X
• For autocracies (D10, D20), Yp -5.1 .0021X
• For anocracies (D11, D20), Yp -6.5 .0021X
• For democracies (D10, D21), Yp -8.7 .0021X

11
Summary

• In this case we have three parallel regression
  lines. We can see that democracies are involved
  in 3 ½ fewer disputes than autocracies, and
  around 2 fewer disputes than anocracies. We can
  also see that anocracies get involved in 1.4
  fewer disputes than autocracies.

12
Example

• Severing the Electoral Connection
• Shirking in the Contemporary Congress
•  Lawrence S. Rothenberg and Mitchell S. Sanders
• American Journal of Political Science
•  
• Question If incumbents in Congress plan to
  retire or if they pursue higher office and face a
  distinct constituency, will they behave
  differently? In particular, will they have
  greater incentives to shirk?

13
Example

• Shirking Legislative behavior differs from what
  would be observed given perfect monitoring and
  effective punishment by constituents (i.e.,
  through elections).
•  
• Two Types of Shirking
• 1)   Ideological Shirking
• Members change their votes away from the
  ideological position of their district.
• 2)   Participatory shirking
• Members vote less frequently (casting fewer roll
  call votes).

14
Example

• Previous studies suggest that participatory
  shirking occurs, but that ideological shirking
  does not. Rothenberg and Sanders argue that this
  is largely a function of poorly specified
  measures of Congressional Shirking.
•  
• Research Design
• Compare a member's actions during the 4th
  quarter of one Congress with her actions in the
  4th quarter of the next Congress. Shortly before
  an election, those seeking reelection know that
  they will be judged by the electorate those not
  standing for reelection know that they are free
  to shirk.

15
Example

• Research Design
• They examine roll call votes taking place after
  July 1st of an election year in consecutive
  Congresses between 1991 and 1996. This produces
  366 cases from the 102nd Congress, 305 from the
  103rd Congress, and 327 from the 104th Congress.

16
Example

• Dependent Variables
• 1)   Ideological Change
• ?Ideological PositioniCongress kI - Ideological
  PositioniCongress k?
•  
• 2)   Abstention Change
• ?Abstention RateiCongress kI - Abstention Rate
  iCongress k?
•  
• Expectation If departing members change their
  voting patterns more and abstain more, then this
  constitutes evidence of shirking.

17
Example

• Independent Variables
• 1)   Retiring (Dummy Variable) equals 1 for
  individuals not running for reelection and not
  seeking other elected office (14.2 of their
  total 998 legislators) and 0 otherwise
• 2)   Pursuing Statewide Office (Dummy Variable)
  equals 1 for individuals leaving the House to
  seek statewide office (3.2 of the total) and 0
  otherwise
• 3)   Seniority years of prior service at the
  beginning of each Congress do senior members
  change their position less and vote more often
  than junior members?

18
Example

• Independent Variables
• 4)   Electoral Slack member's vote share
  (proportion of the two-party vote) in the prior
  election do electorally secure members have more
  liberty in voting?
• 5)   District Political Change absolute
  difference in the proportion of the two-party
  vote received by 1988 presidential candidate
  Michael Dukakis in the old and the new district
  (to reflect possible vote changes based on
  redistricting that occurred in 1992).

19
Example

•        There is evidence of member shirking,
  both in terms of ideological and participatory
  shirking. Retiring and Pursuing Statewide Office
  are significant and positive in both equations.
•        Impact on Ideological Change retiring
  members increase their ideological change by
  .039, while political aspirants increase their
  ideological change by .033 (the scale is -1 to 1
  for the ideology variable).

20
Example

•        Impact on Abstention Change Retiring
  members' abstention rates increase by .11 (11),
  which is substantial in an era where average
  abstention rates are near 5. Political
  aspirants' abstention rates increase by .15
  (15).
•        Small R2 values, especially for the
  Ideological Change model, suggest that there is
  substantial randomness associated with behavioral
  change.
•        Significant intercept in the Ideological
  Change equation indicates that legislator
  ideology is at least somewhat fluid for all
  members.

21
Comparing members running for reelection to those
leaving Congress

• 1)   For Ideological Change
•        Running for reelection (Retiring 0 and
  Pursuing Statewide Office 0)
•  
• Y 0.074 .039(0) 0.033(0) 0.54District
  Political Change 0.016Electoral Slack
  0.000025Seniority
• Y 0.074 0.54District Political Change
  0.016Electoral Slack 0.000025Seniority

22
Comparing members running for reelection to those
leaving Congress

•        Not running for reelection (Retiring 1
  and Pursuing Statewide Office 1)
•  
• Y 0.074 .039(1) 0.033(1) 0.54District
  Political Change 0.016Electoral Slack
  0.000025Seniority
• Y 0.146 0.54District Political Change
  0.016Electoral Slack 0.000025Seniority
•  
• This means that members that are leaving Congress
  are 7.2 more likely to change their ideological
  position (0.146 - 0.074) compared to members
  staying.

23
Comparing members running for reelection to those
leaving Congress

• 1)   For Abstention Change
•        Running for reelection (Retiring 0 and
  Pursuing Statewide Office 0)
•  
• Y 0.00077 0.11(0) 0.15(0) 0.19District
  Political Change - 0.00082Electoral Slack -
  0.000080Seniority
• Y 0.00077 0.19District Political Change -
  0.00082Electoral Slack - 0.000080Seniority

24
Comparing members running for reelection to those
leaving Congress

•        Not running for reelection (Retiring 1
  and Pursuing Statewide Office 1)
•  
• Y 0.00077 0.11(1) 0.15(1) 0.19District
  Political Change - 0.00082Electoral Slack -
  0.000080Seniority
• Y 0.26077 0.19District Political Change -
  0.00082Electoral Slack - 0.000080Seniority
•  This means that members that are leaving
  Congress have changes in abstention rates that
  are 26 higher than members staying in Congress.
  

25
Interactive dummy variables

• We can also interact our dummy variables with
  other independent variables if our theory implies
  such a relationship.
• For example, we might posit the following model
• Y ? ?1D ?2X ?3DX ?

26
Interactive dummy variables

• In our previous example, this would be warranted
  if we believed that military spending in
  democracies had a different effect on their
  propensity to use force than military spending in
  non-democracies.
• We might argue that democracies use increased
  military spending for defense, not for
  aggression, and thus that their rates of dispute
  involvement might vary (this is just a
  hypothetical argument).

27
Interactive dummy variables

• For D0 (non-democracies), the model reduces to
• Y ? ?2X ?
• For D1 (democracies), the model reduces to
• Y ? ?1(1) ?2X ?3(1)X ?
• Y (? ?1) (?2 ?3)X ?
• We can see that an interactive dummy changes both
  the intercept (by ?1) and the slope (by ?3).

28
Examining the Residuals of a Regression Model

• We make several assumptions about the error term
  in the regression model, ?.
• For example, we assume that the errors are
  normally distributed
• ?i ? N0, ?2
•  A violation of non-constant variance is called
  heteroskedasticity (errors are not the same
  across the range of X values).

29
Heteroskedasticity

• A simple way to detect heteroskedasticity is to
  plot the residuals against one or more of the
  independent variables.
• One common pattern that may indicate
  heteroskedasticity is a fanning out residual
  pattern.

30
Autocorrelation

• We also assume no auto-correlation, or that
  Cov?i, ?j 0 if i ? j. A typical violation
  occurs with time series data where the errors are
  related over time.
• To detect autocorrelation, it is useful to plot
  the residuals over time. We should observe a
  pattern in the residuals such that a high value
  is followed by another high value, and a low
  value is followed by another low value, etc.