Interpretting the intercept Fixed Effects or Random Effects model

[Ingrid de Boer]

I have a question about interpretting the intercept of a Fixed Effects analysis or a random effects analysis.

First of all let me explain the question I want to test. It is about the vicious cycle of corruption concerning three main causes (X1, X2, X3) which have an impact on Y (the level of corruption in a country). Corruption results in three main consequences of corruption (I name them A1, A2, A3). These three consequences eventually result in the three causes of corruption (X1, X2, X3). This is thus a vicious cycle. Which can be visiualized as I have done in the figure attached. I used a panel data set with a time range of 6 yrs.

Firstly I want to investige the impact of the causes of corruption on Y. To do so, I executed a Fixed Effect Analysis and a Random effects analysis, after that I used a Hausman test to concude which test is appropriate. I found that Fixed effect was appropriate.
From this test I got the following results (See attachment).

Providing a cons_ (intercept) of -96, which is according to me very strange. Since the mean of the Y (Corruption perceptions index, value between 0 and 100) over all countries is 43. So a positive value. I wonder if I executed the test right, and if this result is reason for concerns or not? I hope someone can help me out.

Later on I also want to test the impact of the consequences of corruption on the causes of corruption, so I need to be sure if I execute the right test.

Thanks in advance

[Clyde Schechter]

The constant term is meaningless in these models. That is because it is colinear with the fixed effects, and then an identifying restriction is imposed to break that colinearity. The value of the constant depends on which identifying condition is imposed. -xtreg, fe- does it one way. -regress ... i.Country2)- would do it differently. (-areg-, I believe, does it the same way as -xtreg, fe-.) In fact, in principle, you could force the constant term to be any number you chose by crafting an appropriate constraint to apply with -cnsreg ...i.Country2-. So the constant term, and the fixed effects themselves, are not interpretable at all.

[Ingrid de Boer]

Thank you for explaining it to me. If I understood it right, the coefficients and constant term are not interpretable at all. So the only outcome of the test is that I know which causes have an effect on Y? But I don't know what that effect is? For example I know that HDI2 is significant, so HDI has impact on Y, but we do not know the value for the cofficient since this is not interpretable?
Thank you for your help!

[Carlo Lazzaro]

Ingrid:
not quite.
The coefficient of the constant term has no (or trivial) interpretation, whereas the other coefficients do have interpetation.
As an aside, as per per you outcome you seem to have a sky-rocketing R^2 with 2 non significant coefficients. Are you sure that your regression model does not suffer from quasi-extreme multicollinearity?
Eventually: I might be wrong, but I fear that investigating the consequences of corruption on the causes of corruption (and if, as you stated beforehand, the causes of corruption generate the consequences of corruption) might bring about a reversal causation issue (i.e., a type of endogeneity).

[Ingrid de Boer]

Well, I checked for multicollinearity of the model (I hope I did it the right way, since I am not very familiar with Stata), and these are my results (see attachments).
Since I did the Fixed Effect Analysis only for determining the impact of the three causes of corruption on Y, I don't know if multicollinearity is already a very big problem at this stage.

Kind regards,
Ingrid

[Carlo Lazzaro]

Ingrid:
as -areg- does not support -estat vif-, you shoud also consider -estat vce, corr- after -areg-.
Anyway, from your results, it seems that -HDI2- and -GDPpercapita2- are highly correlated.
About the problem tha quasi-extremen multicollinearity can cause on your regression estimates, see the folowing Richard's teaching note: https://www3.nd.edu/~rwilliam/stats2/l11.pdf

[Ingrid de Boer]

Thank you Carlo,

I executed -estat vce-, which provides very different results (See attachment). When I interpret the results now, I think there is no multicollinearity?
About the constant (intercept), I have one more question. Since there is no intercept for Fixed Effects model, there is no β₀ in the model. But from Greene (2002), I understand that α_iaccounts for the unknown intercept for each entity (in this case "country"), where i is the number of entities (in this case 141 countries). Is the value for cons_ than the α_i in the formula? Or am I completely wrong here?

Y_it = β₁X₁+β₂X₂+β₃X₃+α_i+u_it

Thank you for sending me the link and your advice!

Kind regards,
Ingrid

[Carlo Lazzaro]

Ingrid:
you posted the outcome of the covariance matrix (-estat vce) instead of correlation (-estat vce, corr).
A technical note (-areg- entry, page 79, Stata .pdf manual) covers -_cons- meaning in this regression model.