Panel Cointegration, Moderating Effects and Multicollinearity I am running a panel fixed effects regression on 21 countries and 16 years. Its a secondary data taken from OECD website mostly. My model looks like this:
$$
\log{(GDP/Labor)}_{i} = \beta_0 +\beta_1 \log{(RnD)}_{i,t-1}+\beta_2 \log{(Human Capital)}_{i,t-1}+\beta_3 \log{(x_j*(RnD_j)}_{t-1}+\epsilon_t
$$
Where,
Rndi = Research and Development expenditure of country i
Rndj = Research and Development expenditure of country j
HC = Index of Human Capital
1). I have following problems with this model:
1) all variables have strong trend and they are expected to be non-stationary however, Im-Pesaran-Shin and Levin-Lin-Chu tests are giving contradictory results about non-stationarity so I cannot conclude about non-stationarity. In your opinion, which test is better for panel regressions? Also, is there any test I can do for cointegration in stata for panel regressions in addition to Westlund test? if I dont find evidence for cointeration, do I first difference the data? I lose ALOT of information and everything becomes insignificant if I do that.
2) One of the primary goals of my paper is to test for moderation effects of human capital on knowledge spillovers and productivity nexus. Problem is that there is a serious multicollinearity problem. VIF goes as high as 1000! How can I test for moderating effects other than using interactions? Moreover, in some cases I also find strong multicollinearity between my key variables and it really makes everything significant at < 0.001 with R2 less than 50.
3) How important is it to include year dummies in regressions? I lose all significance of my variables and things become very bad in terms of signs as well for my core variables when I use year dummies. 
I will be extremely grateful if you could please advice.
Thanks alot
Ali
 A: Question 1
If your outcome variable is integrated, you might consider using a single-equation generalized error correction model (GECM) as per Banerjee (1993) and De Boef (2001), as this model is agnostic to the stationarity of the predictors.
You might evaluate the stationarity of your outcome using:
$\log{(GDP/Labor)_{ti}} \sim \rho_{i}\log{(GDP/Labor)_{t-1i}} + \zeta_{ti} + \mu_{\rho_{i}}$,
where:
$\zeta_{ti}$ measures all disturbances to $\log{(GDP/Labor)_{ti}}$ in each time $t$ (assumed distributed normal), and
$\mu_{\rho_{i}}$ measures state-level variation in $\log{(GDP/Labor)_{ti}}$ (assumed distributed normal).
If $|\rho_{i}| \approx 1$, then you've got nearly integrated data, and the GECM, which also has the attractive properties of disentangling long-run effects, from both instantaneous change short term effects and from lagged short term effects.
The general form of the single equation GECM is:
$\Delta y_{t} = \beta_{0} + \beta_{c}\left[y_{t-1}-\left(\mathbf{X}_{t-1}\right)\right] + \mathbf{B}_{\Delta\mathbf{X}}\Delta\mathbf{X}_{t} + \mathbf{B}_{\mathbf{X}}\mathbf{X}_{t-1} + \varepsilon$,
where:
$\Delta$ is the first difference operator (e.g. $\Delta y_{t} = y_{t} - y_{t-1}$), and $\varepsilon$ may be decomposed into mixed effects (e.g. by including $\beta_{0i}$, for country-level random intercepts).
instantaneous short run effects are given by $\beta_{\Delta\mathbf{X}}$,
lagged short run effects are given by $\beta_{\mathbf{X}} - \beta_{c} - \beta_{\Delta\mathbf{X}}$, and
long run effects are given by $\left(\beta-{c}-\beta_{\mathbf{X}}\right)/\beta_{c}$.
This specification assumes a homogeneity of error correction processes. I haven't yet tried to derive a heterogeneous error correction specification... 
In Stata you can perform Hadri's test for unit-root in panel data on the residuals of such a model, to check them for stationarity.
Question 2
I do not know that I can say much useful here.
Question 3
The time dummies can be included in the GECM model, and presumably other dynamic times series models, often they are used as indicators of, for example, policies going into effect. I have done something similar, but used (time-varying) proportions (rather than 0/1 indicator variables) to represent the portion of the time period during which a policy was in effect (e.g. some policies go into effect January 1, some July 1, some December 21, etc.). On the other hand: you don't have tons of data, so I suppose it depends how many new variables you are adding.

References:
Banerjee, A., Dolado, J. J., Galbraith, J. W., and Hendry, D. F. (1993). Co-integration, error correction, and the econometric analysis of non-stationary data. Oxford University Press, USA.
De Boef, S. (2001). Modeling equilibrium relationships: Error correction models with strongly autoregressive data. Political Analysis, 9(1):78–94.
