Identifying assumptions of Difference-in-Difference versus Fixed Effects models I am reading the seventh edition of Introductory Econometrics: A Modern Approach by Jeffrey Wooldridge and I am a bit confused on the different identifying assumptions for difference-in-difference models and fixed effects models. The generalized model for diff-in-diff is of the following form:
$$
y_{igt} = \gamma_{g} + \lambda_{t} + \beta D_{gt} + \epsilon_{igt} \quad \quad \quad (1)
$$
where $i,g,t$ index individual, group and year respectively. Groups usually represents geographical units treated at the same time, but they can also represent, for example, low-wage vs high-wage workers. $D_{gt}$ is the binary treatment indicator. Now, for this model, Wooldridge argues that we need the well known parallel trends assumption which is usually showed by plotting the pre-treatment trends of each group. On the other hand, a generalized version of a fixed effects model follows this specification:
$$
y_{it} = \alpha_{i} + \lambda_{t} + \beta D_{it} + \epsilon_{it} \quad \quad \quad (2)
$$
That is, we have replaced the group/state fixed effects with individual fixed effects. From my understanding, the point estimate $\hat{\beta}$ does not change between DiD and FE. Instead, we change the inference (see this post). Wooldridge does not mention whether parallel trends plays any role in this type of model. Instead, he argues that $D_{it}$ should be randomized and not react to past shocks. This is consistent with the implementation of this same fixed effects models with state-specific time trends in the paper Woman's Suffrage, Political Responsiveness, and Child Survival in American History by Grant Miller (albeit the author calls this a diff-in-diff model I recognize it as a fixed effects one following Wooldridge). Miller argues that "only the timing of [treatment] is assumed to be exogenous". So, I guess my questions are the following:

*

*Should we care about parallel trends when trying to estimate (2), i.e a FE model?

*Should we care about exogenous treatment timing when trying to estimate (1), i.e a diff-in-diff model?

Thanks in advance.
 A: In reference to your first model you posit the following:

Groups usually represents geographical units treated at the same time, but they can also represent, for example, low-wage vs high-wage workers.

The $g$ groups may also have irregular (i.e., staggered) and/or intermittent (i.e., switching 'on' and 'off') exposure periods. A standardized posttreatment phase is not required. The pattern of the binary treatment variable can take on almost any pattern in the more generalized difference-in-differences (DiD) framework.
In reference to your second model you make this claim:

"...we have replaced the group/state fixed effects with individual fixed effects. From my understanding, the point estimate $\hat{\beta}$ does not change between DiD and FE.

Both models use fixed effects. The first model uses group fixed effects, represented by the parameter $\gamma_g$, while the second model uses individual fixed effects, represented by the parameter $\alpha_i$. The phrase "group" is a bit of a misnomer; the "group" fixed effects denote dummies for all $G - 1$ entities (i.e., cities, counties, states, countries, etc.). Suppose you observed individuals $i$ within states $s$. In this setting, $\gamma_s$ does not delineate a group or collection of treated states. Rather, it is estimating separate indicator variables for all states. It denotes, in accordance with my example, state fixed effects. The only difference between the two models is how we aggregate the data.
In some settings there is no natural unit $s$ where treatment is assigned. Instead, some individuals get treated at a particular point in time, and others do not. You could estimate the second model using the individual data and receive a similar estimate of $\beta$, provided you are observeing the same $i$ individuals over time. This is not a case involving repeated cross-sections data. Each $i$ must be observed before and after treatment.

Should we care about parallel trends when trying to estimate (2), i.e a FE model?

Absolutely.
The same DiD principles apply even in the presence of the individual data. If you observe outcomes of treated individuals before the exposure of interest, and we also observe other untreated individuals who experience the same trends over time, then we can still estimate the effect of treatment. The trends matter even at the lower level!
Again, this is a DiD equation at the individual level. DiD is a special case of fixed effects. The two models differ in aggregation. The "treatment" is well-defined for individuals, and so we can estimate the treatment effect at the $i$-level so long as we observe outcomes for all entities over the various time periods. For more information on this, review section 1.5 of these lecture notes.

Should we care about exogenous treatment timing when trying to estimate (1), i.e a diff-in-diff model?

Yes you should care about it. You must establish that your treatment is plausibly unconfounded. Some selection into treatment is allowed; this is why DiD is so popular. For example, treated states with certain characteristics may be much more likely to be selected to receive some treatment or exposure of interest. In mathematical terms, $E(y_0 | D_{st} = 1) \neq E(y_0 | D_{st} = 0)$. Again, you would hope most of the selection criteria involved time constant attributes specific to the state. Selection based upon past realizations of $y$ introduces bias.

In my cursory review of the referenced paper, it appears the authors do indicate that only the timing of suffrage laws (i.e., treatment) is assumed to be exogenous. They further claim that "state-specific differences that vary linearly over time are all purged from the estimate of $\beta$." They achieve this via the estimation of state-specific linear time trends (i.e., $\delta_s \times t$), which multiplies each unique state effect by a continuous linear time index. In my estimation, they view the differences in trends between states as more of a statistical problem to overcome (see Section III). I may not entirely agree this, but I should probably give the paper a full read before commenting any further.
In my opinion, I would start with investigating the stability of the trends before moving forward with your study. It is rare to find a DiD paper today that doesn't demonstrate, in some visual way, parallel trends before some exposure of interest.
A: I am answering this for posterity. Yes, we should care about parallel trends, as the key assumption that underlies the potential outcomes is that
$$
Y_{it}(D_{it}) = \alpha_i + \lambda_t + \tau_iD_{it}
$$
That is, the potential outcome of unit $i$ is an additive function of a unit-specific effect $\alpha_i$, a time effect $\lambda_t$ and a heterogeneous effect $\tau_{i}$ linked to the treatment $D_{it}$. Then, the individual treatment effect will be given by $Y_{it}(1)-Y_{it}(0) = \tau_i$. This model implies that units follow parallel trends, as absent the treatment, units would have different levels $\alpha_i$, but their changes would evolve in parallel. Concretely, we have that
$$
Y_{it}(0) - Y_{i,t-k} = \lambda_t - \lambda_{t-k}
$$
for all units $i$.
