Hi. I am using stata 16.1. I estimate a dynamic panel data model using the commando -xtreg, fe vce(robust)-
Background:
Specifically, I estimate an first-order autoregressive panel data model with time-invariant unobserved heterogeneity, using a Fixed-Effects (FE) estimator. I rely on T-asymptotics to eliminate the inconsistency in the FE estimators, caused by the presence of the lagged dependent variable. I have T=29 and varying N (I apply the model to different panel datasets, which have varying N. In some cases N<29, in other cases N>29). For several of reasons, I do not consider alternative estimators, e.g. in-sample instrumentable variables estimators by Anderson-Hsiao (1981) or GMM estimators by Arellano and Bond (1991) as relevant for/applicable to my data sample.

The model is formulated within an event-study framework (meaning that I include variables for each of the time intervals in the post-event window). I include only (functions of) one variable as the regressor, x_t: Its current value and lagged values. The structure is a little difficult to explain because I treat selected lags in composite variables to prevent an overfit of the model (I don't think this information is necessary for my question)

The ability to control for unobserved factors, that explain Y_it and are correlated with X_it, hinges on (I) the event study approach, of only considering short windows around the news announcement and (II) time-invariant unobserved factors.

Important for my question:
To validate that my model/data satisfy the assumptions sufficent/necessary for FE-estimators to be consistent (if we abstract for a moment from the inconsistency arising from the induced correlation between \alpha_{i} and \bar{\varepsilon}_{i}). I wish to assess the assumption of a "full rank of the matrix of demeaned explanatory variables". Particularly, I wish to assess the multicollinearity between the regressors in the model: the correlation between the contemporaneous x_t and its lagged values: x_(t-1), x_(t-2), x_(t-3).. x_(t-)

I run the commando -reg, y x_(t+1) +x_(t) x_(t-1) x_(t-2) x_(t-3) x_(t-4) x_(t-5) x_(t-6) i.(panel id)- (I note that the specification of variables is not entirely correct, as some of the lags denote "composite" variables)
And subsequently run the -estat vif-

I provide an example of the results I obtain for one data sample. The dependent variable is the euro-dollar continously compounded five-minute return and the x_t is the macro data release on business inventories. It takes on the "standardized unexpected component" in t and otherwise zero.

Code:
reg fiveminret fiveminretlag1 businessinventories_m5 businessinventories_0 businessinventories_5 businessinventories_10T15 businessinventories_20T30 businessinventories_35T45 businessinventories_50T70 i.news_id, robust cluster(news_id)
estat vif
Variable | VIF 1/VIF
-------------+----------------------
fiveminret~1 | 1.06 0.942444
busine~es_m5 | 1.07 0.930469
busines~es_0 | 1.07 0.932352
busines~es_5 | 1.07 0.932341
bus~es_10T15 | 1.15 0.871394
bus~es_20T30 | 1.22 0.822494
bus~es_35T45 | 1.22 0.820333
bus~es_50T70 | 1.37 0.731631
news_id |
2 | 2.05 0.488276
3 | 1.96 0.510915
4 | 2.04 0.489264
5 | 1.96 0.510838
6 | 2.17 0.461092
7 | 2.05 0.488546
8 | 1.96 0.510133
9 | 2.17 0.461749
10 | 2.16 0.462425
11 | 1.91 0.524322
12 | 1.96 0.509222
13 | 1.91 0.524632
14 | 2.04 0.489417
15 | 1.89 0.528895
16 | 1.96 0.510475
17 | 1.97 0.508800
18 | 1.89 0.529106
-------------+----------------------
Mean VIF | 1.73

Question 1:
To me, they appear highly correlated. Is this a major concern to my approach? My supervisor suggested me to follow this approach. At the same time, he does not want to answer my questions, when I believe "my" diagnostic tests results provide evidence against my aproach or is against the prevailing literature.
Question 2:
This might be a dump question, but I have a hard time rationalizing why a lagged variable is not perfectly correlated to its contemporaneous value? Obviously, my understanding is wrong (otherwise we would not be able to include the lagged independent variable in the regression models). Would it be possible for you to provide a theoretical explanation for why this is not the case that a variable and its lagged terms are not perfectly correlated?