# Interpreting interaction between dummy IV and continuous moderator with log DV

I urgently need your help to read the results of my regression analysis for my master thesis, which I need to hand in next week.

My DV is the natural logarithm of R&D expenditures, IV is a dummy variable labelled Decline, where 1= Firm is in a decline and 0= Firm is not in decline. The moderator is managerial ownership measured in percentage (0-100% or 0-1.0), labelled MOWN.

My thesis has two goals: 1) Find the relationship between IV (Decline) and DV (R&D Expenditures, log) 2) Find the moderating effect of MOWN on the relationship between IV-DV

I regressed Decline, MOWN, and Decline X MOWN simultaneously and got following coefficients.

$log(RD) = 6.984 - 0.852Decline - 4.703Mown + 3.030MownDecline$

The coefficient for Decline is -0.852***, which means that firms invest 85% less in R&D expenditures when Decline=1 (correct?)

Coefficient for MOWN: -4.703* Coefficient for MOWN x Decline: 3.030

Contant: 6.984***

How do I interpret the results of the interaction now? When a firm is in decline (Decline=1), firms will invest 303% more in R&D expenditures with a one-percentage point increase in MOWN?.

Secondly, can I treat the MOWN variable as additional IV as well? Saying that MOWN, the moderator, has also a direct impact in the DV? Saying that "a one-percentage point increase in MOWN decreases DV by 470%"

Thank you a lot for your help. I really appreciate it!

• what do you mean exactly by IV ? Is it an "independent variable" or an "instrumental variable" ? Be clear when you ask a question. People have different background in statistics and they don't always have the same abbreviations. – PAC Aug 2 '13 at 11:36
• I meant independent variable, sorry for the confusion – Tom Aug 2 '13 at 11:44

The predicted value of your dependent variable can be found for any combination of Mown and Decline. When you have an interaction, looking at predicted values (and graphing them) is often a good way to see what is going on. You can also then exponentiate the predicted values.

You have log(RD)=6.984−0.852Decline−4.703Mown+3.030MownDecline

so you could make a table:

predicted log RD            Decline     Mown
6.984                          0         0
6.984 - 0.852                  1         0
6.984 - 4.703                  0         1
6.984 - 0.852 - 4.703 + 3.03   1         1


You could also make a graph with mown on the x axis, the predicted DV on the y axis, and one line for "decline = 0" and one for "decline = 1"

• Thank you Peter, but the IV (Decline) is a dummy variable.. when I put this on the x axis, there are only 2 values, how should I graph this? – Tom Aug 2 '13 at 11:43
• Sorry, I meant "mown" on the x axis, I corrected my post. – Peter Flom - Reinstate Monica Aug 2 '13 at 11:51
• Could you maybe help me with interpreting the positive coefficient of the interaction effect MownDecline? Does it mean that with increasing mown during decline, R&D will increase or perhaps that the impact of R&D expenditures during decline (-0.852) "less" ? – Tom Aug 2 '13 at 12:33
• It means that the effect of MOWN on the DV is different at different when DECLINE = 0 vs. 1; likewise, the effect of DECLINE is different at different levels of MOWN. The best way to see this is, as I said, to plug values into the formula and look at what happens, and to graph them. – Peter Flom - Reinstate Monica Aug 2 '13 at 14:37
1. Your interpretation of the effect of Decline is not correct : If Decline switches from 0 to 1, the % impact of Decline on R&D is 100[exp(-0.852) - 1]. You should read Dave Gile's post for the interpretation of a dummy variable in a log-linear model.

2. You should also be careful with your interpretation since I would suspect an omitted variable bias. If you have an unobserved bad shock for the firm, it may affect simultaneously the decline and the R&D expenditures. In that case, you would have to find a set of instrumental variables.

• Thanks, I know that you could compute a more exact estimate the way you suggested, but for simplicity I followed Wooldridge, J. M. (2012). Introductory econometrics : a modern approach (5th ed.), p.234, which state that taking the direct coefficient is also okay or is this a major mistake? – Tom Aug 2 '13 at 11:49
• My first point is always true and it is always wrong to interpret the coefficient directly as a percentage. My second point is also always true in the sens that you should always be careful to possible omitted variable bias. – PAC Aug 2 '13 at 12:04
• Ok I will change this immediately. One last question, is it possible to interpret the Decline to R&D expenditure relationship although I included the interaction effect Decline X Mown? One guy told me that once I include the interaction effect, the Decline coefficients has changed and I need to use the coefficient without the interaction effect. Thanks a lot! – Tom Aug 2 '13 at 13:58
• I think that ou should just look at your model and think about it. The interpretation of the Decline coefficient is obvious. – PAC Aug 2 '13 at 14:26