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How should I interpret interaction term in OLS?

Dummy: 0 is flat tax system, 1 is progressive one.

Cont: continuous variable ranges from -2.5 to 2.5.

Before interaction:

dummy | .0183619 .0166824 1.10 0.278

cont | .0616873 .0110176 5.60 0.000

After interaction:

dummy | -.0429928 .0201104 -2.14 0.039
cont | .0301204 .0118294 2.55 0.015
inter | .0784644 .0186501 4.21 0.000

What are possible explanations for the dummy turning negative?

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    $\begingroup$ What do the numbers mean? $\endgroup$ – Matthew Drury Nov 28 '16 at 4:25
  • $\begingroup$ Please, try ti improve formatting, it's hard to read. $\endgroup$ – utobi Nov 28 '16 at 6:14
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I guess the first column is the coefficient value, the second one is standard error, and thereafter are $t$ and $p$ values.

Before: dummy is close to 0. This means that flat/progressive tax system has little effect (no relationship to causality ;-) after controlling for the continuous variable (call it $x$).

After: dummy is -0.043 and significant. In case the dummy = 0, you just ignore the interaction (it is always zero) and you just look at the $x$ estimate. For observations where dummy is 1, you have to add the interaction term and the $x$ estimate, and you get $-0.043 + 0.078 + 0.030 = 0.065$. If $x = -1$ you get the tax system effect analogously $-0.043 - 0.078 - 0.030 = -0.151$.

The fact that insignificant dummy turns negative when you add interaction has rather simple intuition:

  • Before, you assume that both cases (dummy = 0 and dummy = 1) have a common slope ($x$ coefficient) but different intercepts

  • Afterwards, you allow both slopes and intercepts to differ. Obviously, changing slope will most likely change intercept too unless the slope line "turns around" $x=0$.

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You cannot interpret them without also looking at the intercept. It might be that dummy + intercept is still positive. But think about it this way: perhaps the two groups (flat vs progressive system) have widely different effects of the continuous variable. Think of the effect as a function over -2.5 to 2.5. Then the dummy shifts that line up and down while the cont coefficient is the slope parameter. One of the lines can either shift up or down relative to the other (the dummy coefficient) or its slope can change (the cont coefficient).

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