Time Dummies and Time Trend in the same equation Can we run this regression:
$Y_{it} = a + bX_{it} + c_2Time_2 + ... + c_TTime_T + ht + U_{it}$                     
$i = 1,2,..., N;$ $t = 1,2,3...,T$
Where
​


*

*$Time_T$ are time dummies 

*$t$ is the time trend.


In other words, can we run a regression with both time dummies and and a time trend? I suspect that we cannot, but am looking for a more formal explanation.
Thank you!
 A: The way you have written the model some parameter will be unidentified hence the answer is: NO.
To see this consider the model without covariates  $x_{it}$ without loss of generality
$$(1) \ \ y_{it} = a + \sum_{t=2}^T c_t T_t + h \cdot t + \epsilon_{it}$$
for $t=1,...,T$. 
For $t=1$ the term $th = h$ which we can add to the constant $a$ to get $b:=(a+h)$ hence the model can be rewritten
$$ y_{it} = b + \sum_{t=2}^T c_t T_t + h \cdot (t-1) + \epsilon_{it}$$
such that the time trend $h \cdot (t-1)$ is now $0$ for time period 1 for which the model includes no dummy. For the other periods the value of the time trend $h \cdot (t-1)$ is simply added to the dummy coefficient $c_t$ to get $\tilde c_t := c_t + h \cdot (t-1)$ allowing us to rewrite model
$$y_{it} = b + \sum_{t=2}^T \tilde c_t T_t  + \epsilon_{it}.$$
Because the model with time trend has been shown equivalent to a model without the timetrend the timetrend itself is not identified. 
You can easily simulate this in R to see what happens in a model where you simulate (1) and then tries to estimate it
set.seed(1)   
library(data.table)
T <- 5
N <- 1000
a <- 1
c <- c(0,abs(rnorm(T-1)))
h <- 2

id <- sort(rep(1:N,T))
time <- rep(1:T,N)

dt <- data.table(id=id,time=time)
setkey(dt,id,time)

dt[,y:=a+c[time]+h*time,by=id]

# pooled panel data regression
lm(y~as.factor(time)+time,data=dt)
a + h
c[2:T] + h*(2:T-1)

resulting in the output
Call:
lm(formula = y ~ as.factor(time) + time, data = dt)

Coefficients:
     (Intercept)  as.factor(time)2  as.factor(time)3  as.factor(time)4  
           3.000             2.626             4.184             6.836  
as.factor(time)5              time  
           9.595                NA  

> a + h
[1] 3
> c[2:T] + h*(2:T-1)
[1] 2.626454 4.183643 6.835629 9.595281

where the coeffient on time is not estimated and the constant term is $a+h$ and the time dummy coefficients are given by the formula $c(t) + h(t-1)$.
(note: I have simulated the model without error $\epsilon_{it}$ but the result are the same with error terms)
A: Yes you can and probably should. The trick is to properly identify a minimally sufficient model that may include events, level shifts, seasonal pulses AND ARIMA structure.
