How does the interpretation change if I drop the linear terms?

[frederick lim]

Hi all,

I would like to know what is the difference in the interpretation between the two equations below, and generally, under what conditions can I drop the linear terms? Is there an issue related to identification if I use equation (1)?

weekly spending= time_fixed_effects + consumer_fixed_effects + B1*after + B2*treatment + B3*after*treatment + error (1)

weekly spending= time_fixed_effects + consumer_fixed_effects + B3*after*treatment + error (2)

The above examines the impact on weekly spending of the treatment groups compared to the control groups after a treatment unit subscribes to a service membership. Equation (1) includes the linear terms while equation (2) drops the linear terms. In this case, I think I will need to use equation (2) since including the time (in weeks) and consumer fixed effects, they will cancel off the effects of "after" and "treatment"? Please advise. Thank you.

Best,
Fred

[Clyde Schechter]

So, don't do either of those. Use factor notation (-help fvvarlist- if you don't know about this):

regress /*or other command*/ i.time i.consumer i.after##i.treatment

Assuming that after is constant at any given time, and each consumer always has the same value for treatment, then you are right that the "main effects" of after and treatment will be omitted due to colinearity. But you don't have to do this explicitly. If you use the command shown above, Stata will automatically omit whatever needs to be omitted.

A side benefit of doing it this way: if Stata doesn't omit something you think should be omitted, that will tell you that your data do not actually conform to your assumptions about constancy of after within time or constancy of treatment within consumer. Then you can go ahead and investigate why your data do not satisfy the constraints you expect and either fix the data or modify your expectations accordingly. If you just run the model with the interaction term but no "main effects" you will not find out if the data is wrong in this way. (Or, rather, you will only find out about it much later when you get some inexplicable results, or when somebody watching you present your results sees something wrong and calls you out on it. The earlier you discover any flaws in your data, the better. It is always best to write code that will fail in the presence of unexpected data and not just blunder on through it.)

[Phil Bromiley]

Clyde's code solves a problem Fredrick may not be aware of. It is seldom desirable to run interactions without including the main effects. Without the main effects, you're restricting the flexibility of the model in ways that may give you strange outcomes. Looking at b1 x1 + b2 x2 + b3 x1*x2 versus just b4 x1*x2, the second specification forces the influence of x2 to be 0 when x1 equals 0 while the first allows x2's effect to b2 when x1=0.. We seldom want the more restrictive specification.

By using ## instead of #, his regress statement automatically includes the main effects.

[frederick lim]

Thanks Clyde and Phil for the fast response. Appreciate it.

Cldye, in response to your suggest code, how would the original equation that I provided change? I'm assuming if I list down the first equation, which includes the linear terms, reviewers might question about the unidentification problem? Thanks.

Fred

[Clyde Schechter]

Well, you can just omit the B1*after + B2*treatment terms from equation 1, which leaves you with equation 2. But I would be sure to add an explanation that those terms are omitted only because they are colinear with the time and consumer fixed effects, respectively.

Of you could do it the other way. Show the full equation and then comment that due to the colinearity, in order to identify the model the after and treatment terms are omitted.

Either way is fine. I don't think there's any way, though, to just have an equation with no explanation. Because whichever way you write the equation, without further explanation, it looks wrong.

[frederick lim]

Thanks Clyde. In this case, is there's any change in the interpretation of B3, when comparing equation 2 and 1? Thanks.

[Clyde Schechter]

No, not in this case. In the general case, there would be an enormous difference--see Phil Bromiley's post above. But in this specific case, no.