Random slope and cross-level interaction

[Sadrac Cenophat]

Dear All,
I am testing the curvilinear relationship between cInfComp (level-1 variable) and NPD. I also hypothesize that:

1. caccfin (level-1 variable) will flatten out the curvilinear relationship
2. cunav (level-2 variable) will steepen the curvilinear relationship.

Will my model be mis-specified if I do not include cInfComp and cInfComp2 in the model as random slopes? Please, find the Stata code below for further details.
Model 1:

melogit NPD c.caccfin##c.(cInfComp cInfComp2) c.cunav##c.(cInfComp cInfComp2) lnempl comp i.sector lnage foreign expo i.iso lnexptopman || _all:R.year || Country: cInfComp cInfComp2, difficult

Model 2:

melogit NPD c.caccfin##c.(cInfComp cInfComp2) c.cunav##c.(cInfComp cInfComp2) lnempl comp i.sector lnage foreign expo i.iso lnexptopman || _all:R.year || Country:, difficult

I should mention that the chi2 for model 1 and 2 are respectively chi2(4) = 5938.20, and chi2(2) = 5769.63. (both significant).

Thank you very much for your support.

[Clyde Schechter]

Your model would not be mis-specified in the sense of a model that is not a possible model for any data generating process. I might or might not be a mis-specification of the particular data generating process you are modeling.

The second model states that NPD is a quadratic function of cinfcomp, and that the location and width of the parabolic graph of that function depend on variables caccfin and cunav, but, at least given the values of caccfin and cunav, it is the same function regardless of which country the data come from.

The first model states that in addition to the parabolic graph's location and width depending on caccfin and cunav, they also differ idiosyncratically across countries.

So to choose between them you have to figure out which model is the better approximation to reality based on your understanding of what these variables are and how they are related to each other. I have no idea what these variable names even stand for, and even if I did, I suspect that these are not epidemiologic variables, so I would be out of my domain anyway. Hence, I can't advise you in any more specific way about this.

[Sadrac Cenophat]

Dear Clyde,
Many thanks for your support. I highly appreciate it. The study posits that informal competition (cInfComp: the degree to which a company is affected by uncompetitive practices of unregistered competitors) a) increases the likelihood of new product development (NPD) when informal competition is low, and b) decreases such likelihood as informal competition increases. "cunav" is a cultural variable that represents uncertainty avoidance (the extent to which members of certain societies/countries are comfortable with uncertainty). It is expected that the curvilinear relationship flattens out when uncertainty avoidance is high.
Following your remark regarding country idiosyncrasies, I suspect that the random-slope model might be more appropriate.
Once again thanks!
Sadrac