interaction opposite signs

[Faisal Abdullah]

I have the following regression
Leverage = @ + dummy + Cash + cashXdummy
Dummy is 1 for international firms and zero for non-international.


The results are puzzling
Cash gives significant positive effect
The variable (cash * dummy) is significant and negative.

Are the mentioned two opposite signs okay? What this means. I did the regression in several methods and still provide the same results. I also expand the smaple size and still have the same outcomes. Is this results fine or maybe I did something wrong? I learned that the multicoloniarty is not an issue of concern with interactions

many thanks

[Clyde Schechter]

There is nothing unusual or surprising about this. You need to understand the meanings of the coefficients in a model with interactions like yours. The coefficient of dummy is the expected effect on leverage of being an international firm conditional on cash = 0. The coefficient of Cash is the expected effect of an additional unit of cash among the non-international firms. The effect of an additional unit of cash among the national firms, if you want to find it, is given by the coefficient of cashXdummy plus the coefficient of cash. Since the interaction term is negative, this means that the impact of an extra unit of cash is smaller among international firms than it is among non-international firms.

[wanhaiyou]

There is nothing unusual or surprising about this. You need to understand the meanings of the coefficients in a model with interactions like yours. The coefficient of dummy is the expected effect on leverage of being an international firm conditional on cash = 0. The coefficient of Cash is the expected effect of an additional unit of cash among the non-international firms. The effect of an additional unit of cash among the national firms, if you want to find it, is given by the coefficient of cashXdummy plus the coefficient of cash. Since the interaction term is negative, this means that the impact of an extra unit of cash is smaller among international firms than it is among non-international firms.

Hi Clyde,
If we want to test the effect of cash on dependent variable, are we need to test the joint significant of cash and cash*dummy?
In addition, if one of these two variables (cash and cash*dummy) is not statistical significant and the other variable is significant,
then we want to conclude the effect of cash on dependent variable, how should we can do? If the joint test is significant,the coefficient of
cash is beta1 and the coefficent of cash*dummy is beta2, how can I calculate the marginal role of cash on dependent variable (Note:
I assume that cash*dummy is not significant).

Thanks very much for your help!

Best regards,
wanhaiyou

[Clyde Schechter]

So, when you run an interaction model, generally the first step in interpretation is to look at the results for the interaction term. If the interaction term is near zero in practical terms, not statistically significant, and if there is no compelling theoretical reason to believe that an interaction ought to exist, then I would simply revert to a model without any interaction term, and interpret that in the usual way.

If the interaction term is substantively or statistically significant, or if theory suggests that it is important anyway, then we have to interpret the model. In your case, we are dealing with a variable, cash, which I assume is a continuous variable, and a dummy, which I will take to be a 0/1 dichotomy.The most important thing to understand about an interaction model is that the question "what is the effect of cash on the outcome variable" has no answer. It is the very essence of an interaction model that no such single effect exists. The use of an interaction model implies that there is one effect of cash on the outcome when dummy = 0, and a different effect of cash on the outcome when dummy = 1. (If the effects are not different, then the interaction coefficient would be zero, or at least very close to zero, and you would revert to a regression without the interaction term, as pointed out above.)

The effect of a unit increase in cash on the outcome when dummy = 0 is given by the coefficient of cash. The effect of a unit increase in cash on the outcome when dummy = 1 is given by the coefficient of cash plus the coefficient of the interaction term. Both should be reported. While you can calculate these by hand, if you like, or using the -lincom- command, if you have run your model using factor variable notation, you can take advantage of the -margins- command to get them quickly and easily:

regress outcome c.cash##i.dummy

margins dummy, dydx(cash)

The output from the -margins- command will show you each of those two effects, along with standard errors, confidence intervals, and t-tests of statistical significance.

Note that it is also possible to issue the command

margins, dydx(cash)

which will also produce a result that one might call "the effect of cash on the outcome." But, one would be wrong in calling it that. That output is actually a kind of average effect of cash over both dummy = 0 and dummy = 1, but its value depends on the proportions of observations with each of those values for dummy, and it will not, in general, replicate in other samples. It does not represent the actual effect of cash on outcome for either value of dummy. And it represents an "average" or "marginal" effect of cash on outcome only if the proportions of dummy = 0 and dummy = 1 observations in future samples will always be the same as in the current one. So it is not really a useful statistic and I rarely, if ever, report it.

In interpreting statistical significance, it is important to note that interaction models work differently from models without interaction terms. The statistical significance of the cash coefficient does not tell you whether or not cash has a significant impact on outcome. No single statistic can tell you that in an interaction model: the effect of dummy differs, depending on the value of cash. The coefficient of dummy tells you whether the impact of cash on outcome is significantly different from zero when dummy = 0. The significance of the dummy variable does not tell you whether the dummy variable has an effect on the outcome. It tells you only whether the difference in outcome between dummy = 1 and dummy = 0 cases with cash = 0 is statistically significant. Now, I don't know for sure what your variable cash represents, but from its name alone, I would guess that cash = 0 is an unusual, if not an impossible, situation in your data. If that is the case, then the coefficient of the dummy variable has no practical importance to you. If you want to know the impact of the dummy variable at more meaningful values of cash, you can again use the -margins- command:

margins, dydx(dummy) at(cash=(1000 10000 100000))

Note: the numbers 1000, 10000, 100000 are just numbers I made up to illustrate the syntax. In practice, you would use actual meaningful, important values of cash.

[wanhaiyou]

So, when you run an interaction model, generally the first step in interpretation is to look at the results for the interaction term. If the interaction term is near zero in practical terms, not statistically significant, and if there is no compelling theoretical reason to believe that an interaction ought to exist, then I would simply revert to a model without any interaction term, and interpret that in the usual way.

If the interaction term is substantively or statistically significant, or if theory suggests that it is important anyway, then we have to interpret the model. In your case, we are dealing with a variable, cash, which I assume is a continuous variable, and a dummy, which I will take to be a 0/1 dichotomy.The most important thing to understand about an interaction model is that the question "what is the effect of cash on the outcome variable" has no answer. It is the very essence of an interaction model that no such single effect exists. The use of an interaction model implies that there is one effect of cash on the outcome when dummy = 0, and a different effect of cash on the outcome when dummy = 1. (If the effects are not different, then the interaction coefficient would be zero, or at least very close to zero, and you would revert to a regression without the interaction term, as pointed out above.)

The effect of a unit increase in cash on the outcome when dummy = 0 is given by the coefficient of cash. The effect of a unit increase in cash on the outcome when dummy = 1 is given by the coefficient of cash plus the coefficient of the interaction term. Both should be reported. While you can calculate these by hand, if you like, or using the -lincom- command, if you have run your model using factor variable notation, you can take advantage of the -margins- command to get them quickly and easily:

regress outcome c.cash##i.dummy

margins dummy, dydx(cash)

The output from the -margins- command will show you each of those two effects, along with standard errors, confidence intervals, and t-tests of statistical significance.

Note that it is also possible to issue the command

margins, dydx(cash)

which will also produce a result that one might call "the effect of cash on the outcome." But, one would be wrong in calling it that. That output is actually a kind of average effect of cash over both dummy = 0 and dummy = 1, but its value depends on the proportions of observations with each of those values for dummy, and it will not, in general, replicate in other samples. It does not represent the actual effect of cash on outcome for either value of dummy. And it represents an "average" or "marginal" effect of cash on outcome only if the proportions of dummy = 0 and dummy = 1 observations in future samples will always be the same as in the current one. So it is not really a useful statistic and I rarely, if ever, report it.

In interpreting statistical significance, it is important to note that interaction models work differently from models without interaction terms. The statistical significance of the cash coefficient does not tell you whether or not cash has a significant impact on outcome. No single statistic can tell you that in an interaction model: the effect of dummy differs, depending on the value of cash. The coefficient of dummy tells you whether the impact of cash on outcome is significantly different from zero when dummy = 0. The significance of the dummy variable does not tell you whether the dummy variable has an effect on the outcome. It tells you only whether the difference in outcome between dummy = 1 and dummy = 0 cases with cash = 0 is statistically significant. Now, I don't know for sure what your variable cash represents, but from its name alone, I would guess that cash = 0 is an unusual, if not an impossible, situation in your data. If that is the case, then the coefficient of the dummy variable has no practical importance to you. If you want to know the impact of the dummy variable at more meaningful values of cash, you can again use the -margins- command:

margins, dydx(dummy) at(cash=(1000 10000 100000))

Note: the numbers 1000, 10000, 100000 are just numbers I made up to illustrate the syntax. In practice, you would use actual meaningful, important values of cash.

Thanks very much for your valuable and detailed response.

Kindly regards,
wanhaiyou

[Faisal Abdullah]

Hi Clyde , thank you very much for your help. I feel more confident now

In Fact, when I did the interaction, the sign of the dummy variable changes from non significant to significantly positive.
dummy ( 1 for multinational and 0 for national) = positive and significant
Cash = negative and significant
cash interaction with dummy = negative and significant
regards

[Faisal Saleh]

Hi Clyde,
If I have two continuous variables that I want to regress their interaction against another continuous variable. For example, let's say that..
Y: firm performance
X1: proportion of independent directors on the board
X2: proportion of executive directors on the board

The results came as follows:
While X1's impact and X2's impact were both significantly negative, the interaction term (X1*X2) had a significant positive impact on Y, could you help me with the interpretation?

Please, note that the results of the separate regressions of X1 and X2 came as both having significantly negative impact. So, why the interaction term can have a positive impact?

[Clyde Schechter]

The interaction term doesn't "have an impact" at all. It is the difference between the impacts of X1 and X2, or, actually it is a rate of change of the difference between the impacts of X1 and X2.

When you run an interaction model, you are stipulating that there is no such thing as "the impact of X1" or "the impact of X2." Rather you are stipulating that there are infinitely many impacts of X1, one for each value of X2. And vice versa. So fi the coefficient of X1 and the coefficient of X2 are negative, all that means is that the marginal effect of X1 on the outcome is negative when X2 = 0, and the marginal effect of X2 on the outcome is negative when X1 = 0. At any particular value of X2, the marginal effect of X1 will be the coefficient of X1 plus the value of X2 * the coefficient of the interaction term. So the implication is that as X2 increases, the marginal effect of X1 increases. If X2 is sufficiently larger than 0, then the net result will be a positive marginal effect. I suggest that you use the -margins- and -marginsplot- commands to get a better picture of what your model is doing. Since X1 and X2 are both proportions, I assume they range between 0 and 1 and roughly span that range.

regress outcome c.X1##c.X2 // PERHAPS OTHER VARIABLES
margins, at(X1 = (0(0.1)1) X2 = (0(0.1)1))
marginsplot, name(predicted_outcomes, replace)
margins, dydx(X1) at (X2 = (0(0.1)1))
marginsplot, name(marginal_effect_of_x1, replace)
margins, dydx(X2) at(X1 = (0(0.1)1))
marginsplot, name(marginal_effect_of_x2, replace)

[Zeenat Murtaza]

Hi Clyde,

Can you please help with these results and advise if I am right?

Dependent variable:	Model 1	Model 2	Model 3 (fam-sample)
f.control	-0.043 ** (0.020)
(ΔCashHoldings)t	-0.072 * (0.042)	-0.095 ** (0.049)	-0.493 *** (0.199)
(ΔCashHoldings)t−1	-0.076 (0.051)	-0.070 (0.045)	0.003 (0.015)
F.control * (ΔCashHoldings) t	- 0.088* --(H1) (0.052)
F.control.1		-0.047 * (0.029)
F.control.2		-0.048 (0.032)
F. control.3		-0.039 (0.067)
F.1* (ΔCashHoldings)t		-0.071 (0.094)
f.2 * (ΔCashHoldings)t		-0.083 * --(H2) (0.046)
f.3 * (ΔCashHoldings)t		-0.003 (0.986)
Mature * (ΔCashHoldings)t			0.365 * --(H3) (0.229)

in model 1, change in cash coefficient is negative which means when there is reduction in cash the firms have allocated more cash to stabilize dependent variable amidst greater financing restrictions. f.control is also negative which means firms invest less in dependent variable but the interaction is positive. Does this mean that when change is cash is negative, fam control starts to invest more in dependent variable while when change in cash is positive, fam control invest less. sorry im not getting? the research question is do fam use more cash for stabilizing dependent variable compared to another dummy group? please help in interpretation. i shall be very grateful.

[Zeenat Murtaza]

I would be highly obliged if someone can please help me with this issue

[Clyde Schechter]

Sorry, but I am completely confused by #9. I don't know what any of the variables in the tables are, and in your text you discuss all sorts of things that don't have any obvious correspondence to any of the variables in the table. At one point you say "the interaction is positive" but all the interaction coefficients shown in the table are negative. So I really don't know what you have here. Please provide a clearer explanation. It would also be helpful if you show the exact Stata commands that were used to create these results.

[Zeenat Murtaza]

Dear Clyde,

I apologise for missing information: Please find below command and table:

asdoc xtdpdgmm L(0/2).rnd_ta rnd2 chcash_ta chcash_L1 i.fam_1 c.chcash_ta#i.fam_1, model(diff) collapse gmm(chcash_ta, lag(2 6)) gmm(rnd_L1 firm_control, lag(1 6)) gmm(chcash_ta, lag(1 1) diff model(level)) gmm(firm_control rnd_L1, lag(0 1) diff model(level)) two vce(r)

Generalized method of moments estimation
Fitting full model:
Step 1 f(b) = .00232579
Step 2 f(b) = .4161465
Group variable: isin_code Number of obs = 1548
Time variable: year Number of groups = 123
Moment conditions: linear = 69 Obs per group: min = 1
nonlinear = 0 avg = 12.58537
total = 69 max = 23
(Std. err. adjusted for 123 clusters in isin_code)

	WC-Robust
rnd_ta		Coefficient	std.	z	P>z		95% conf.
L1.		0.811	0.135	6.010	0.000	0.546	1.075
L2.		0.040	0.034	1.160	0.246	-0.027	0.107
rnd2		-0.349	0.161	-2.170	0.030	-0.664	-0.034
chcash_ta		- 0.096	0.084	1.150	0.024	-0.067	0.260
chcash_L1		-0.027	0.033	-0.810	0.419	-0.091	0.038
fam_1
20-40%		-0.005	0.027	-0.190	0.047	-0.058	0.047
40-60%		- 0.010	0.037	0.280	0.078	-0.062	0.082
>60%		0.013	0.061	0.210	0.834	-0.107	0.132
fam_1#c.chcash_ta
20-40%		-0.387	0.217	-1.790	0.074	-0.813	0.038
40-60%		0.274	0.121	-2.270	0.023	-0.511	-0.037
>60%		0.006	0.243	0.020	0.981	-0.470	0.482
_cons		0.034	0.047	0.730	0.465	-0.057	0.125

[Clyde Schechter]

OK. So fam_1 is a categorical variable with 4 levels. I'll guess that the (omitted) base level is 0-20%. The explanatory variable chcash_ta is a continuous variable.

So the estimated slope of the rnd_ta:chcash_ta relationship is -00.96 when fam_1 = 0-20%. When fam_1 = 20-40%, that slope is -0.096 - 0.387 = -0.483. When fam_1 = 40-60%, that slope is -0.096 - 0.010 = -0.1006. Finally when fam_1 = > 60%, the slope is -0.096 + 0.013 = -0.083. To also get the standard errors, confidence intervals and other statistics you can use the -lincom- command. (In this code I assume that fam_1 is coded 0, 1, 2, and 3. If other numbers were used, change the code accordingly.)
[/code]
forvalues i = 0/3 {
lincom chcash_ta + `i'.fam_1#c.chcash_ta
}
[/code]
Note: -xtdpdgmm- is a user-written command. I assume that it was written to work with postestimation commands such as -lincom-, but as I am not familiar with the command itself, I do not know if this is the case or not.

[Zeenat Murtaza]

Thank you so much Clyde for your valuable time and feedback. I a not getting how to interpret interaction terms. As chcash is negative and fam is negative, the interaction term is negative for both these variables. Does this mean if chcash is reduced, fam increases investment in dependent variable. Besides, if the coefficient of interaction term is positive for a similar negative-negative relationship what does it imply? it mean reduction in one decreases the effect of fam ?? Please advise.

[Clyde Schechter]

Again, I don't understand what you are asking. I don't know what "investment" has to do with anything in your model. Let me just put it in generic statistical terms.

We have a model y = b0 + b1 X + b2 i Z + b3 c.X#i.Z + other terms. Here, as in your situation, X is a continuous variable and Z is a categorical variable. For simplicity let's assume Z just takes on the values 0 and 1.

By using an interaction model, we are stipulating that there is no such thing as "the effect of X," nor "the effect of Z." Rather there are different effects of X and Z, depending on the value of the other variable. The marginal effect of X on y when Z = 0 is given by the coefficient of X, b1. The marginal effect of X on y when Z = 1 is given by b1 + b3. From the signs of b1 and b3 alone, you can't really say anything about the direction of the marginal effect of X when Z = 1, because it depends on the value of b1 + b3. If b1 is negative and b3 is positive, whether b1 + b3 is negative or positive depends on whether the magnitude of b3 is smaller or larger than the magnitude of b1. So just looking at the signs doesn't give you enough information. You have to actually calculate those effects and then look at the results to see which direction they go in.

I strongly suggest that you run those -lincom- commands and stop looking at the regression output itself. Just focus on the -lincom- outputs to see what the marginal effects of chcash_ta are at each level of fam_1. And if you want to know whether different values of fam_1 increase or decrease that marginal effect compared to some other value, just look at the corresponding results directly and see.