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I have a question on interpreting coefficients in a regression model. To be specific, the regression model is as given below:

$$y=a+b_1x_1+b_2x_2+b_3Z+b_4Zx_1+b_5Zx_2$$

where

  y= continuous dependent variable
  x1= independent variable all values are positive (min=0)
  x2= independent variable all values are negative (max=0)
  Z = Normalized variable with values between (0,1)

Given this set up, how would we interpret

  a) positive and significant coefficient b1
  b) negative and significant coefficient b4
  c) What can we say about a joint test of b1+b4=0

x1 and x2 represent the same variable X (continous) and the goal is to allow for different slopes when X is positive and negative. Any help is much appreciated.

The question is motivated from the regression output in the following paper

Greve, Henrich R. "A behavioral theory of R&D expenditures and innovations: Evidence from shipbuilding." Academy of management journal 46.6 (2003): 685-702.

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    $\begingroup$ welcome to CV! if it is a homework please show your attempt and add self-study tag $\endgroup$ – Haitao Du Apr 11 '17 at 21:33
  • $\begingroup$ Hi. Actually no its not homework. I'm trying to understand the regression output from a published paper. $\endgroup$ – Anup Apr 11 '17 at 21:36
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    $\begingroup$ @Anup it would be very helpful if you could edit your post to add a full citation for the paper so others could look $\endgroup$ – shadowtalker Apr 11 '17 at 21:37
  • $\begingroup$ @ssdecontrol: I have added the reference. Thank you! $\endgroup$ – Anup Apr 11 '17 at 21:41
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Answering these in turn:

  1. This means:

    • when $Z = 0$, for every unit increase in $X_1$ above 0, $Y$ increases by $b_1$. So when $Z = 0$ and $X = 3$, if $b_1 = 2$ and $a = -1$, then $Y = 5$. If $X = 5$, then $Y = 9$.
    • we have sufficient evidence to reject the hypothesis that $b_1 = 0$, in favor of the hypothesis that $b_1 \neq 0$
  2. This means:

    • for every unit increase in $Z$, the effect of $X$ on $Y$, when $X > 0$, increases by $b_4$. From above, if $Z=0$, $X=3$, $b_1 = 2$, and $a = -1$, then $Y=5$. But if $Z=2$ and $b_4 = 0.5$, then $Y = 8$. So the effect of $X$ is "amplified" by the value of $b_4$. In particular, $Y$ can be expressed as a function of $X$ in the form $Y = (b_1 + b_4 Z) X$ when $X > 0$.
    • we have sufficient evidence to reject the hypothesis that $b_4 = 0$, in favor of the hypothesis that $b_4 \neq 0$
  3. A "joint test of $b_1 + b_4 = 0$" is the same as "a joint test of $b_1 = -b_4$". If we fail to reject this hypothesis, then every unit increase in $Z$ would offset $b_1$ by exactly 1, so an increase in $Z$ of 3 would reduce the effect of $X$ by 3. So our expression for $Y$ as a function of $X$ reduces to $Y = b_1 (1 - Z) X$.

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