I am a beginner in Stata and I am having some trouble when undertaking regressions in Stata.

For some context, I am looking at the effect of technological progress on unemployment and I am using TFP growth as a proxy for technological progress. I am undertaking a time series regression using data from the UK from 1855-2016 and I am using the following model, including lagged values of TFP growth, real GDP growth, population growth and lagged variables of the dependent variable:

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Do i need to include a trend term in this?

I have done Dickey-Fuller tests to check stationarity and a Breusch-Godfrey test to check for autocorrelation - all variables are stationary and there is no autocorrelation. Do i need to undertake any more tests on this before analysing the results?

After this, I am undertaking a fixed effects panel regression analysis using data from 15 countries from 2000-2012 using the following model with country and year dummy variables:

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Is this the correct notation for a fixed effects model?

I have done a Hausman test for checking if fixed effects is preferable (it is), a modified Wald test to check for heteroskedasticity (it is present), a Woolridge test for autocorrelation (it is present) and a Pesaran's CD-test to check for cross sectional dependence (there is evidence of correlation across panel groups).

To solve all these problems I have clustered the standard errors by panelid to give more accurate results.

My question for this is regarding endogeneity - I am not sure how to test for this and I do not have an instrumental variable to use if there is endogeneity in my model. What should I do about this? Would I be able to say that endogeneity is a limitation for my analysis without testing or solving it?

I would really appreciate any help regarding these problems!

Thanks in advance!