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Imagine the following regression model:

$\text{Abnormal Returns} = b0 + b1*SENT + b2*SIZE + b3*SENT*SIZE + e$

SENT is a standardized variable. SIZE is equal to 1 for "uncertain" firms, and 0 for certain firms.

I am examining whether sentiment affects abnormal returns more or less when firms are classified as uncertain or certain. Now, I have run the regression model and I understand how to interpret the results when ALL coefficients are significant and it is as follows:

(1) when there is no sentiment (SENT=0), the abnormal returns increases with b0 for firms classified as certain (SIZE=0, SENT*SIZE=0).

(2) when there is No sentiment (SENT=0), the abnormal returns increases with b0+b2 for firms classified as uncertain (SIZE=1, SENT*SIZE=0).

(3) when sentiment is high/low (SENT= 1 SD above mean or 1 SD below mean), the abnormal returns increases/decreases with b1 for firms classified as certain (SIZE=0, SENT*SIZE=0).

(4) when sentiment is high/low, the abnormal returns increases/decreases with b1+b2+b3 for firms classified as uncertain (SIZE=1, SENT*SIZE=1).

I have some doubts on how and if to interpret the results if only some coefficients are significant. Thus my questions are:

  1. Should I interpret the results if only b1(SENT) is significant? How do I interpret this?

  2. Should I interpret the results if only b2(SIZE) is significant? How do I interpret this?

  3. If only the interaction effect is significant, how do I interpret this?

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Even if the effects are not statistically significant, you should still include them in your interpretation of the model, with the caveat that the effect was not significant.

As for interaction effects, it's typically a rule of thumb that one shouldn't include interactions without main effects (although as with all rules of thumbs, there are exceptions to the rule).

So lets suppose you find only $\beta_3$ is significant in your results. I would still report all the estimated effects, but also note that we had only found enough evidence to conclude that SENT has a positive effect with firms of SIZE = 1.

In terms of interpreting insignificant results, let's use the example that $\hat \beta_1 = 1$, but is insignificant: "We observed that in our sample, the average abnormal returns from firms of SENT = 1 with SIZE = 0 was 1 higher than the average of abnormal returns from firms with SENT = 0 and SIZE = 0. However, we did not observe enough evidence to conclude that this was a real systematic difference between these two groups, rather than the possibility of random chance".

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  • $\begingroup$ I believe I understand you. What about if only SENT is insignificant but SIZE is significant and SENT*SIZE is significant? How should I interpret the coefficients? Say b1=0.001, b2=0.005 and b3=-0.003. Would you say that when sentiment is high (SENT=positive) the average abnormal returns increases with 0.005+-0.003 or with 0.001+0.005+-0.003? As the first coefficient is insignificant, I doubt whether to include this at all as the "total effect" for uncertain firms when sentiment is high. $\endgroup$ – Ras Jun 12 '15 at 18:12
  • $\begingroup$ You should definitely include $\beta_1$ (the estimated main effect of SENT), even though it is not statistically significant, when reporting the estimated value for when SIZE = 1 and SENT = x. $\endgroup$ – Cliff AB Jun 12 '15 at 18:20

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