This is a useful post and subsequent discussion

When Can You Safely Ignore Multicollinearity?

http://statisticalhorizons.com/multicollinearity

have a look at especially point 2

No generally it is recommended to include both original variables inside the equation when interaction is included. You should consider another way to reduce multicollinearity.

In regression, here we suppose you have two predictor factors, one is X continuous , G categorical predictor, binary (multi is fine), we have 4 kinds of modeling

1. same interception, same slope BY G

the model is

y=a+bx+ e    e errors

2. different interactions same  slope; 

y=a+bx+r*G+e,  a is the interception for G=0, a+r is the interception for G=1

3. different interceptions  and different slopes

y=a+bx+r*G+w*X*G+e,  a is the interception for G=0, a+r is the interception for G=1

b is the slope for G=0, b+w is the slope for G=1

the above  three is commonly used. it is called hierarchical modeling, if high way interaction is included , all low way interaction and main effect should be included.

However there is another type: same interception different slopes

4. y=a+bx+w*X*G,  a is the interception for G=0 and G=1,  b is the slope for G=0, b+w is the slope for G=1

say for same case, if x=0, you know y=0, no mater what group you belong to.

Thant might be true, you can test if b=0 is true. In that case, you may use model 4

I use G as binary to explain, but G could be continuous or multi-categorical.

Model 4 is not popular, but it might be useful, depending on your problem.

@Miguel, The statement " set of variables that are highly autocorrelated" should be read as " set of variables that are highly correlated" it is multicollinerity. not "autocorrelated". For multicollinearity, you can use PCA (principal component analysis) or ridge regression, or transformation, see any applied linear regression book.

Yes, I mistyped and meant multicollinearity. I know that using *only* the interaction term is generally not preferred.

However ... in this case I think I can use it as a sort of "index" term. In this particular case one party (MAS) is hegemonic/dominant. There are a few municipalities were it runs unopposed (so ENP=1; ENOP=0), several where it runs opposed but by small fragmented parties (e.g. ENP=2.1; ENOP=5.2), and only a few places where it runs opposed by one or two serious opponents (e.g. ENP=2.3; ENOP=1.8).

So the "interaction" term actually spreads out the data nicely, so that it goes from zero (MAS is completely unopposed) to infinity (in reality about 12 or so) as support for MAS decreases AND it faces a fragmented opposition. The number is more moderate when MAS competes against a few parties.

I had planned to use ENOP and ENP only as control terms because I'm interested in the determinants of support for MAS. Clearly that support is mediated by ENP and ENOP, but all three terms are likely influenced by the other factors in my model (socioeconomic, demographic, etc.). But using all three terms (ENP, ENOP, and ENP*ENOP) created multicollinearity problems, which also washed out the effects of my variables of interest. Using the interaction effect only seems to resolve that problem, while still retaining the information about ENP and ENOP.

As you stated, it is fine, In fact, as you know, when you do the location transformation for predictor, there  is no  effect on the slope estimation. If two groups have different slopes, the two regression line is crossed. When you  move the initial point to the cross point, both groups have the same interception. If the cross point with X=0, you don't need do the transformation. If you put the Group main effect in, as I mentioned H0: b=0 will not be rejected. You may do this to verify, and put it in the discussion in your paper, because some referees may ask you such question. You need show them.

Yes another way of dealing with correlated variables is to add, multiply ... them. So what you do by only keeping the interaction term in the equation, is just this way of handling multicollinearity. If multiplication of these variables makes sense for the theory and interpretation, you are welcomed to do it. But let me remind that there are other ways of dealing with multicollinearity as well.

This is a useful post and subsequent discussion

When Can You Safely Ignore Multicollinearity?

http://statisticalhorizons.com/multicollinearity

have a look at especially point 2

I strongly suggest that you look at Sect 8.1 (on polynomial regression) of the book at the link below. Dr. Jones post is also correct. Best wishes. BTW since you are building an explanatory model rather than a predictive one you might find the paper by Shmueli to be of use.

http://www.amazon.com/Applied-Linear-Statistical-Models-Michael/dp/007310874X/ref=sr_1_1?s=books&ie=UTF8&qid=1458705125&sr=1-1&keywords=Kutner+Applied+linear+statistical

I think same as Yuanzhang Li  that PCR will be one of the best method, you alsow can use stepwise regression. More advanced but quite similar to PCR is PLS

I had considered stepwise regression, but when I do that (in Stata), most the VIF scores for the three multicolinear variables (ENP, ENOP, and ENP*ENOP) remain. And they are quite high. Or am I misunderstanding something in your suggestion?

I also don't think any kind of principal component analysis is useful, since ENP, ENOP, and ENP*ENOP all are based on the same underlying data (party vote shares). So there' no question that they tap into the same underlying dimension.

I think I'm coming to the conclusion that I will keep the "interaction" variable (ENP*ENOP) since I can also use it (in this particular case) as a degree of party dominance v. fractionalization model (since the major party I am interested in is a dominant party, so a zero on the interaction scale means the strongest dominance of that party and the higher values mean that party is least dominant in a highly fragmented field).

 If you decide to keep the interaction term cos is significant please also keep the individual terms in the interaction term even when not significant.

No, you need to retain them. In regression-style models, a model with a higher-order term (the interaction) is generally mis-specified if the component lower-order terms (the main effects) are not included.

Centering the individual variables before computing the product will usually reduce, but not eliminate, the correlations between the individual variables and the product term, which can help stabilize the estimation.

Is this true even if the interaction could be interpreted as something else? For example, in development economics, there's a simple indicator for Occupational Diversification by Tatu Vanhannen that is the multiplication of % urban population with % pop working in non-agriculture (as the number approaches 1, the country is highly urban/industrial). I don't think using that index requires using the component measures, does it?

Miguel, I apologize--I hadn't noticed there was a page one on this thread.

I think in some situations, and this may be one, you can make an argument for a product term as an index that is a meaningful predictor by itself. But it is not the same as the argument for using a product term to represent an interaction between two variables, which is the question you started with.

I think you're asking about switching from one interpretation to the other. The decision should be based on the theoretical meaning of the product term, which would ideally be based on a priori reasoning, not the result of doing it one way and running into trouble.

So, as I understand it, this is not now a question about dropping main effects from a model with an interaction model so much as a question about whether the product term stands alone with its own meaning, which is a question for topic experts, not methodologists. In that case, the individual effects could be construed as covariates, but you should also consider carefully their role in the hypothesis test, as the predictor of greatest interest, the product, can't be randomized, and of course they're correlated.

In sum, go back to the basics or your research question: What is the conceptual meaning of the product term, and if it is more an index, then what conceptual and statistical roles do the component variables play if you include them as covariates?

Pat

In my original model, all three terms were highly correlated with the dependent variable (also highly collinear). The R-square was nearly 0.9 so this was a strong model. The problem was that it eliminated the effects of many of the other variables of interest (my actual key independent variables).

Individually, both component variables and the composite/index/interaction term all work in the models, even though the R-square drops to between 0.5-0.7 (depending on specific models).

I'll have to be sure to specify this clearly in the discussion, as well as any notes or appendix.

Thanks!

A variable that is the product of two needs a good justification and interpretation guides.

You must be aware of the difficulty of interpreting the product of two variables without information on each one. For example, how to distinguish between PNE = 3, ENOP = 1 and ENP = 1, ENOP = 3? how you interpret the regression coefficient of this product in the model?

I would mean center ENP and ENOP. Then create an interaction between the centered ENP and ENOP. So, if you find the mean for ENP is 3.4 and the mean for ENOP is 4.6, then the centered ENP will be C_ENP = ENP - 3.4 and the centered ENOP will be C_ENOP=ENOP -4.6. Then make your interaction term C_ENOP*C_ENP. 

Hi. Miguel. centering variables only reduce multicollinearity issue not solve. When you decide to drop the main effect. You are basically arguing that their partial coefficients are near zero which imposes no loads to the Y of the interaction models. Another reason why you have collinearity issue is might be because of the multiplication of a dummy variable and a continuous variable, or you are multiplying a large number with another large number that generates an even greater number. This would definitely de-stabilize your models.

Centering is often not the way out.

Whether to drop the main effect and retain the interaction term is actually highly debated among researchers. the decision is totally arguable. Yet some reviewers or experts might hold different views against you. The situation depends.

Good luck.

I see this is old, but interesting.

I'd wonder if a different data set might give you a different result here. What about cross-validation? If you split the data set, would you get similar looking graphical residual analyses (with predicted y on the x-axis, and estimated residuals on the y-axis)?

You could compare different model results on the same graphical residual analysis scatterplot to compare fit, but remember to look for overfitting by splitting the data.

You could compare models with and without the main terms with that interaction term, but I think that if you do better without them, changing your sample might change that.

I'm not sure about a variable that is the product of two others, and in general would question keeping an interaction term without the main ones. That is why I wondered if it was a fluke of the sample you have that might not be repeatable.

Did you arrive at any conclusion, Miguel?

I would be interested in seeing what the argument is for dropping the relevant main effects (mentioned by Andrew Ma) when the interaction term is significant. To me, a significant interaction term makes the main effects interesting, whether large or small.

if you have a reason to keep the interaction term then you need to keep the main terms also. Does not make sense to keep interaction term without the interacting terms

The reason for avoiding collinearity is to increase the interpretability of the coefficients. This would hardly be the case if the relevant interaction term is significant.

Dr. Samuel Ogwu I've heard this said before. But sometimes the pval of the main effect is not significant, whist the interaction term is. In this case, wouldn't it make sense to drop the main effect?

If you are keeping the interaction term then also keep the main as it will make more sense to have the mains that are interacting in the model. This common in cross over validation.

Interaction= product of the interacting terms