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Dummy Variables and Interactions

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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 = 1151 -------------+------------------------------ F( 1, 1149) = 0.54 Model | 155287.676 1 155287.676 Prob > F = 0.4631 Residual | 331138316 1149 288196.968 R-squared = 0.0005 -------------+------------------------------ Adj R-squared = -0.0004 Total | 331293604 1150 288081.395 Root MSE = 536.84 -------------------------------------------------------------------------------- def_social_h~d | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------------+---------------------------------------------------------------- log_pctforeign | 7.449417 10.14842 0.73 0.463 -12.4621 27.36093 _cons | 663.3708 27.61287 24.02 0.000 609.1935 717.5481 Data drawn from 6 different cantons (states).

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Dummy Variables. tabstat def_social_head pctforeign, by(canton) Summary statistics: mean by categories of: canton canton | def_so~d pctfor~n ---------+-------------------- 1 | 504.8596.135417 2 | 202.6653.2730006 3 | 474.0729.0919272 5 | 945.3242.0633038 6 | 377.7097.1101544 7 | 110.8237.0918817 ---------+-------------------- Total | 501.6756 1.689802 ------------------------------

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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 = 1151 -------------+------------------------------ F( 6, 1144) = 127.25 Model | 132602310 6 22100384.9 Prob > F = 0.0000 Residual | 198691294 1144 173681.201 R-squared = 0.4003 -------------+------------------------------ Adj R-squared = 0.3971 Total | 331293604 1150 288081.395 Root MSE = 416.75 -------------------------------------------------------------------------------- def_social_h~d | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------------+---------------------------------------------------------------- log_pctforeign | 220.8124 16.05465 13.75 0.000 189.3126 252.3123 | canton | 2 | -670.2634 67.5107 -9.93 0.000 -802.7221 -537.8047 3 | -94.19855 40.78477 -2.31 0.021 -174.2199 -14.17721 5 | 468.9531 35.77836 13.11 0.000 398.7545 539.1516 6 | -1193.182 86.82131 -13.74 0.000 -1363.529 -1022.835 7 | -468.4175 41.32189 -11.34 0.000 -549.4927 -387.3424 | _cons | 1180.505 40.99826 28.79 0.000 1100.065 1260.946 -------------------------------------------------------------------------------- Canton 1 is a ‘reference category’ – the intercept for canton 1 is “_cons”

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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.

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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.

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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.

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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 == 2005 ----------------------------------------------------------------------------------------- def_social_head | Coef. Std. Err. t P>|t| [95% Conf. Interval] ------------------------+---------------------------------------------------------------- log_pctforeign | 179.9446 19.08151 9.43 0.000 142.5076 217.3816 logpctforeignXvoteright | 85.12434 32.40203 2.63 0.009 21.55309 148.6956 voteright | 112.3438 417.7021 0.27 0.788 -707.1681 931.8557 | canton | 2 | -703.456 68.08389 -10.33 0.000 -837.0333 -569.8786 3 | -73.26651 41.22884 -1.78 0.076 -154.1556 7.62253 4 | -1033.193 60.82019 -16.99 0.000 -1152.519 -913.8663 5 | 361.7032 411.5004 0.88 0.380 -445.6411 1169.048 6 | -1079.953 424.194 -2.55 0.011 -1912.201 -247.7039 7 | -559.3247 409.6522 -1.37 0.172 -1363.043 244.3936 | _cons | 1160.733 414.167 2.80 0.005 348.1573 1973.31 -----------------------------------------------------------------------------------------

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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

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

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