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$$ y_{ist} = \alpha + \gamma T_{s} + \lambda d_{t} + \beta(T_{s} \cdot d_{t}) + \epsilon_{ist}, $$

where, for example, we may observe individual/entity $i$, in state $s$, at time period $t$. In this setting, $T_{s}$ indexes only those states exposed to treatment, 0 otherwise. The variable $d_{t}$ indexes periods after treatment in both treatment and control groups. Because $d_{t}$ is the same across all $s$, this model is used when treated states enter into the treatment condition at precisely the same time.

$$ y_{ist} = \gamma T_{s} + \lambda d_{t} + \beta(T_{s} \cdot d_{t}) + \epsilon_{ist}, $$

where, for example, we may observe individual/entity $i$, in state $s$, at time period $t$. In this setting, $T_{s}$ is a dummy equal to 1 in only those states exposed to treatment, 0 otherwise. The variable $d_{t}$ indexes periods after treatment in both treatment and control groups. Because $d_{t}$ is the same across all $s$, this model is used when treated states enter into the treatment condition at precisely the same time.

Your question, though, appears to be principally concerned with the inclusion of a single 'state-year' fixed effect using individual/firm level data. You specifically noted in your question that treatment is implemented at the $s$ level (as it typically does in a DD framework) and does not vary across individuals/firms within a state. To facilitate a better understanding of this, I simulated a three-level panel dataset in R with individual firms $i$. Below, this fake dataset is comprised of 2 firms embedded within 2 states observed over 3 years. Simple is better, sometimes. The last five variables (columns) show the 'state-year' effects (e.g., 'ny_19' = New York in 2019, 'ny_20' = New York in 2020, etc.). A ‘state-year’ fixed effect would absorb $D_{st}$ when that dummy only varies at the ‘state-year’ level. And it will not return the same estimate of $\beta$ if estimated with separate state and time effects. If we take out all the variation at the 'state-year' level, there may not be much left to explain with a 'state-year' treatment variable(s).

Your question, though, appears to be principally concerned with the inclusion of a single 'state-year' fixed effect using individual/firm level data. You specifically noted in your question that treatment is implemented at the $s$ level (as it typically does in a DD framework) and does not vary across individuals/firms within a state. To facilitate a better understanding of this, I simulated a three-level panel dataset in R with individual firms $i$. Below, this fake dataset is comprised of 2 firms embedded within 2 states observed over 3 years. Simple is better, sometimes. The last five variables (columns) show the 'state-year' effects (e.g., ny_19 = New York in 2019, ny_20 = New York in 2020, etc.). A ‘state-year’ fixed effect would absorb $D_{st}$ when that dummy only varies at the ‘state-year’ level. And it will not return the same estimate of $\beta$ if estimated with separate state and time effects. If we take out all the variation at the 'state-year' level, there may not be much left to explain with a 'state-year' treatment variable(s).

where $D_{st}$ is equal to unity for treated states during periods when treatment is in effect. $\gamma_{s}$ denotes state (unit) fixed effects; $\lambda$ denotes year (time) fixed effects. Note, these fixed effects replace $T_{s}$ and $d_{t}$, respectively, in the former equation. $D_{st}$ is the same as before $(T_{s} \cdot d_{t})$. Instead of doing this interaction manually, we code this dummy explicitly to reflect early/late adopter states, or possibly ones experiencing intermittent treatment exposure. It is for these reasons that researchers estimate the equation you referenced. Please review this post which details the coding of the treatment dummy. You can find further insights here.

where $D_{st}$ is equal to unity for treated states during periods when treatment is in effect. $\gamma_{s}$ denotes state (unit) fixed effects; $\lambda_{t}$ denotes year (time) fixed effects. Note, these fixed effects replace $T_{s}$ and $d_{t}$, respectively, in the former equation. $D_{st}$ is the same as before $(T_{s} \cdot d_{t})$. Instead of doing this interaction manually, we code this dummy explicitly to reflect early/late adopter states, or possibly ones experiencing intermittent treatment exposure. It is for these reasons that researchers estimate the equation you referenced. Please review this post which details the coding of the treatment dummy. You can find further insights here.