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

#### esprit88

##### New Member
Interaction between dummy IV and continuous moderator with log DV

Hello Community,

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.

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

The coefficient for Decline is -0.852***, which means that firms invest 85% less in R&D expenditures when Decline=1 (which relates to my first hypothesis)

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?.

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!

Cheers,
Tom

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#### JesperHP

##### TS Contributor
How do I now interpret the results of the interaction? When a firm is in decline (Decline=1), firms will invest 303% more in R&D expenditures?.
Consider the model

$$y_i = \beta_0 + \beta_1 d_1 + \beta_2 x_1 + \beta_3 x_1 d_1 + \epsilon_i$$

Now take the conditional expectation since that is what you are modelling and condition on $$d_1 = 1 , X_1=x_1$$:

Hence you get:

$$E [y_i \lvert d_1=1 , X_1=x_1 ] = \beta_0 + \beta_1 + \beta_2 x_1 + \beta_3 x_1$$

Compare this to expectation on $$d_1 = 0 , X_1=x_1$$:

$$E [y_i \lvert d_1=0 , X_1=x_1 ] = \beta_0 + \beta_2 x_1$$

So what is the difference? $$\beta_1 + \beta_3 x_1$$

So the difference depends on the levels of x_1 and $$\beta_3$$ describes how much larger or smaller an effect in the change in x_1 wil have on dependent variable when firm is in decline.

When a firm is in decline the effect of change in x_1 is $$\beta_2 + \beta_3$$

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#### esprit88

##### New Member
Thanks for the quick response, I forgot one important part to my assumed interpretation:

"When the firm is in decline (Decline=1), firms will invest 303% more in R&D expenditures with a one-percentage point increase in MOWN."

Yet, when looking at your second equation

Compare this to expectation on $$d_1 = 0 , X_1=x_1$$:

$$E [y_i \lvert d_1=0 , X_1=x_1 ] = \beta_0 + \beta_1 + \beta_2 x_1$$

So what is the difference? $$\beta_3 x_1$$
Should $$\beta_1$$ not be 0 as well since d1=0 times $$\beta_1$$ ?

Also, for me it is important to phrase the results. Could you maybe phrase your explanation in a way like I tried to do in my sentence? I am still not sure how to read the results. Further does your explanation imply that second question is wrong?

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%"

#### JesperHP

##### TS Contributor
Thanks for the quick response, I forgot one important part to my assumed interpretation:

"When the firm is in decline (Decline=1), firms will invest 303% more in R&D expenditures with a one-percentage point increase in MOWN."

Yet, when looking at your second equation

Should $$\beta_1$$ not be 0 as well since d1=0 times $$\beta_1$$ ?
Yes it should Im tired make several errors while writing but they should be edited now.

Also, for me it is important to phrase the results. Could you maybe phrase your explanation in a way like I tried to do in my sentence? I am still not sure how to read the results. Further does your explanation imply that second question is wrong?

#### JesperHP

##### TS Contributor
The effect of the dummy variable depends on the level of x_1 so evaluating the effect of whether or not a firm is in decline will depend on level of MOWN.

Hence this
The coefficient for Decline is -0.852***, which means that firms invest 85% less in R&D expenditures when Decline=1 (which relates to my first hypothesis)
is wrong unless x_1=0 which is uninteresting. To evaluate the effect of the dummy you can do several things:
1) Draw graphs in (x_1,y)-plane showing the two lines $$\beta_0 + \beta_2 x_1$$ which is the model for firms not in decline and the line $$( \beta_0 + \beta_1) + (\beta_2 + \beta_3)x_1$$ which is the model when firm is in decline.

The distance between these lines i direction of y is LEVEL effect. But as you see lines also have different slopes showing that changes in x_1 have different effect hence the sensitivity of y to changes in x_1 depends on whether or not the firm is in decline. If you want an interpretation of the lines you have to do a graph perhaps in excel.

#### JesperHP

##### TS Contributor
The second thing you can do is to evaluate the effect of the dumme at a chosen level of x1 fx the sample mean of x_1

$$(\beta_0 + \beta_1) + (\beta_2 + \beta_3) \bar x_1 - \big( \beta_0 + \beta_2 \bar x_1\big) = \beta_1 + \beta_3 \bar x_1$$

Which simply amounts to computing the distance between lines at the sample average ... the aforementioned LEVEL effect

So you have the number wrong.... but also the interpretation:
means that firms invest 85% less in R&D expenditures when Decline=1 (which relates to my first hypothesis)
no if computed at the sample average it is the AVERAGE firm that in AVERAGE invest ....(new number)...% less in R&D expenditures when Decline=1

alternatively you could calculate for firm with MOWN=25% , MOWN=50% MOWN=75% or simply show the whole line...graphs say a thousand words

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#### JesperHP

##### TS Contributor
Concerning the units your DV is logged and MOWN is in percentage so the change in DV resulting from a change in x_1 is a change measure in percentage point pr. percentage point. There is a special word for this?? Which is?

#### esprit88

##### New Member
Your answer comes surprising, especially as you remark that my interpretation of the IV and DV relationship is wrong. I followed Wooldridge, J. M. (2012). Introductory econometrics : a modern approach (5th ed.), p.234, and replicated his interpretation on dummy variables as IV on log(DV). In this condition, I only look at the relationship between IV Decline and DV and only as a next step start interpreting the effect of x_1 (MOWN).

#### JesperHP

##### TS Contributor
Very well I only have edition 4: So let me ask does the model on page 234 contain any interactionterm between dummy and quantitative IV just to check whether we are looking at the same thing?

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#### esprit88

##### New Member
The model described on p. 234 (Example 7.5, Log Hourly Wage Equation) does not contain any interaction between the dummy (Decline) and continuous moderator variable (MOWN). Are you implying that once you include the interaction to the regression, you cannot interpret the IV (Decline) as it is stated on p.234?

Attached you can find an overview of entire regression results table

In Model 2 (Self-Aspiration) I only regressed the IV (Decline), without any interaction effect. Here I got a coefficient -0.350*** for IV (Decline)

I assumed that since the results of each model is consistent, I only use the last Model 5 to explain my results.

#### JesperHP

##### TS Contributor
The model described on p. 234 (Example 7.5, Log Hourly Wage Equation) does not contain any interaction between the dummy (Decline) and continuous moderator variable (MOWN). Are you implying that once you include the interaction to the regression, you cannot interpret the IV (Decline) as it is stated on p.234?
No and Yes
No because I do not have the book in the exact same edition and in my book there is no table as the one you show so different editions differ and hence I cannot comment on whatever is written your book. But still on the other hand

Yes that is exactly the implication and the reasons are given in #5 and #6

#### esprit88

##### New Member
My mistake, the image of the table that I included are my regression results, not from Wooldridge. Anyway, so this means that for testing my first hypothesis, I need to refer to Model 2, bc no interaction effect is present.

In addition to my MOWN interaction, I also have a second one IOWN_TEN that interacts with Decline as well. They were all included in Model 5. Can I interpret both interaction in Model 5 or do I need to interpret them separately, speaking for MOWNxDecline use the coefficient of Model 3 and for IOWN_TENxDecline use the coefficients of Model 4.

#### JesperHP

##### TS Contributor
I do not remember you actually stating your "first hypothesis" although in #1 you mention that "first hypothesis" is related to the test of decline-dummy. So I cannot comment on
Anyway, so this means that for testing my first hypothesis, I need to refer to Model 2, bc no interaction effect is present.

In addition to my MOWN interaction, I also have a second one IOWN_TEN that interacts with Decline as well. They were all included in Model 5. Can I interpret both interaction in Model 5 or do I need to interpret them separately, speaking for MOWNxDecline use the coefficient of Model 3 and for IOWN_TENxDecline use the coefficients of Model 4.
You need to do modelselection based on method of model comparison. Which methods people use differ. Important factors in modelselection are 1) Theoretical arguments of what theoretically makes sense 2) Sattistical arguments related to nested models (F-testing as used in Woolridge see chapter 4 on testing multiple linear restrictions) and 3) Information criteria

Have you ever had any practise at these kind of procedures at your institution of education?
If you have just follow the procedure they preach

The point is to find a model "you think" is the correct model - then do interpretations and conclusions.

#### JesperHP

##### TS Contributor
I assumed that since the results of each model is consistent, I only use the last Model 5 to explain my results.
Given that all your variables seems significant you probably cant do any modelreduction. So this is probably the best way to go about matters. But then the model you suggested in #1 is really irrelevant.
I suggest you post a new thread where you state the relevant model your "first hypothesis" and then deal with problems one question at a time.