Best econometric models to fit when there is no control group? I am trying to study the effects of a program, where everyone gets treated. I have 50% of the population in t and the rest of the 50% in t+1 period. I understand this gives me a slight opportunity of creating a control group for just one time period and run something like DID. Or I can run a Segmented TS analysis.

I was wondering if there is any better way?
If you dont want to used DID, a simple cross sectional analysis would do with the addition of a dummy variable to indicate whether the individual was treated in period $t+1$ or not (implying they were treated in period $t$

for example if you were trying to see if  test scores of students change depending on when they have their class on monday or tuesday. your data set would be:

$$
\begin{array}{c|lcr}
n &amp; \text{Test score} &amp; \text{IQ} &amp; \text{Class on monday} \\
\hline
1 &amp; 60 &amp; 80 &amp;0\\
2 &amp;80 &amp;65 &amp; 1 \\
3 &amp; 90&amp; 70 &amp; 0
\end{array}
$$

By the incorporation of dummies you can just use regression methods for cross sectional data.