Dummy Variables and Interactions

Dummy Variables What is the the relationship between the % of non-Swiss residents (IV) and discretionary social spending (DV) in Swiss municipalities?. reg def_social_head log_pctforeign if year==2005 Source | SS df MS Number of obs = F( 1, 1149) = 0.54 Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = def_social_h~d | Coef. Std. Err. t P>|t| [95% Conf. Interval] log_pctforeign | _cons | Data drawn from 6 different cantons (states).

Dummy Variables. tabstat def_social_head pctforeign, by(canton) Summary statistics: mean by categories of: canton canton | def_so~d pctfor~n | | | | | | Total |

Dummy Variables We can control for the fact that municipalities are drawn from different cantons by allowing the default expectation (intercept) for each canton to vary:. reg def_social_head log_pctforeign i.canton if year==2005 Source | SS df MS Number of obs = F( 6, 1144) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = def_social_h~d | Coef. Std. Err. t P>|t| [95% Conf. Interval] log_pctforeign | | canton | 2 | | | | | | _cons | Canton 1 is a ‘reference category’ – the intercept for canton 1 is “_cons”

Dummy Variables This is not the same as running separate regressions for each canton, because we still assume that the slope is identical for every subgroup. Social Spending = B0 + B1*Log_Pctforeign + B2*Canton1 + B3*Canton2 + B4*Canton3 … For municipalities in Canton 1: Social Spending = B0 + B1*Log_Pctforeign + B2*1 + B3*0+ B4*0 … Social Spending = (B0 + B2) + B1*Log_Pctforeign For municipalities in Canton 2: Social Spending = B0 + B1*Log_Pctforeign + B2*0 + B3*1+ B4*0 … Social Spending = (B0 + B3) + B1*Log_Pctforeign All we are doing is changing the starting value -- allowing the expectation when log_pctforeign = 0 to differ across cantons. B1 still describes the effect of Log_Pctforeign on Social Spending across the entire sample.

Dummy Variables What would happen if I added the following variables to the previous regression: A)Variable that measures whether a canton is German-speaking (1) or French- speaking (0) B)Variable that measures average GDP per capita in the canton. B)Variable that measures whether a municipality allows (1) or does not allow (0) immigrants to vote.

Dummy Variables and Interactions We can allow the relationship between log_pctforeign and def_social_head (the “slope”) to vary for each canton by using interactions: Social Spending = B0 + B1*Log_Pctforeign + B2*Canton1 + B3*Canton2 … + B4*Canton1*Log_Pctforeign + B5*Canton2*Log_Pctforeign … For municipalities in Canton 1: Social Spending = B0 + B1*Log_Pctforeign + B2*1 + B3*0+ B4*1*Log_Pctforeign + B5*0*Log_Pctforeign Social Spending = (B0 + B2) + (B1 + B4)Log_Pctforeign For municipalities in Canton 2: Social Spending = (B0 + B3) + (B1 + B5)Log_Pctforeign This is the same as estimating the relationship between social spending and log_pctforeign separately for each subgroup. We are assuming that the relationship differs in each canton.

Interactions Interactions should be justified by theory. For instance, we might reasonably assume that the relationship between % foreign and social expenditure would be different if municipalities allowed immigrants to vote. Voteright is a variable coded 1 if immigrants have voting rights in a municipality, 0 otherwise.. gen logpctforeignXvoteright = voteright * log_pctforeign. reg def_social_head log_pctforeign logpctforeignXvoteright voteright i.canton if year == def_social_head | Coef. Std. Err. t P>|t| [95% Conf. Interval] log_pctforeign | logpctforeignXvoteright | voteright | | canton | 2 | | | | | | | _cons |

Interactions Social Spending = B0 + B1*Log_Pctforeign + B2*Canton1 + B3*Canton2 + … B4*VotingRights + B5*Log_Pctforeign*VotingRights For non-immigrant voting municipalities in Canton 1: Social Spending = B0 + B1*Log_Pctforeign + B2*1 + B3*0+ +B4*0 + B5*Log_Pctforeign*0 Social Spending = (B0+B2) + B1*Log_Pctforeign For immigrant voting municipalities in Canton 1: Social Spending = B0 + B1*Log_Pctforeign + B2*1 + B3*0+ +B4*1 + B5*Log_Pctforeign*1 Social Spending = (B0+B2+B4) + (B1+B5)*Log_Pctforeign

Direct + Indirect Effects Direct Effect = Multivariate Regression Coefficient Indirect Effect = Bivariate – Multivariate