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### Regression Extensions

W&W Chapter 14

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.

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

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

The Results

We can compare the two groups as follows:

For D=0, Yp = -6.4 –2.02(0) + .000179X

= -6.4 + .000179X

For D=1, 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.

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.

Summary

More generally, if D is a 0-1 dummy variable in a regression model,

Yp=a + b1D + b2X

Then the regression line where D=1 is parallel and b1 units higher than the line where D=0.

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

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 +

Estimating the model

Suppose we estimate our new model and obtain the following results:

Yp = -5.1 –1.4D1 – 3.6D2 + .0021X

For autocracies (D1=0, D2=0), Yp = -5.1 + .0021X

For anocracies (D1=1, D2=0), Yp = -6.5 + .0021X

For democracies (D1=0, D2=1), Yp = -8.7 + .0021X

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.

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?

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).

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.

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.

Example

Dependent Variables

1)Ideological Change =

Ideological PositioniCongress k+I - Ideological PositioniCongress k

2)Abstention Change =

Abstention RateiCongress k+I - Abstention Rate iCongress k

Expectation: If departing members change their voting patterns more and abstain more, then this constitutes evidence of shirking.

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?

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).

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).

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.

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) + 0.54*District Political Change + 0.016*Electoral Slack + 0.000025*Seniority

Y = 0.074 + 0.54*District Political Change + 0.016*Electoral Slack + 0.000025*Seniority

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) + 0.54*District Political Change + 0.016*Electoral Slack + 0.000025*Seniority

Y = 0.146 + 0.54*District Political Change + 0.016*Electoral Slack + 0.000025*Seniority

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.

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) + 0.19*District Political Change - 0.00082*Electoral Slack - 0.000080*Seniority

Y = 0.00077 + 0.19*District Political Change - 0.00082*Electoral Slack - 0.000080*Seniority

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) + 0.19*District Political Change - 0.00082*Electoral Slack - 0.000080*Seniority

Y = 0.26077 + 0.19*District Political Change - 0.00082*Electoral Slack - 0.000080*Seniority

This means that members that are leaving Congress have changes in abstention rates that are 26% higher than members staying in Congress.

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 +

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).

Interactive dummy variables

For D=0 (non-democracies), the model reduces to:

Y = + 2X +

For D=1 (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).

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 N[0, 2]

A violation of non-constant variance is called heteroskedasticity (errors are not the same across the range of X values).

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.

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.

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