Using several cross-sectional regressions (for different years) vs. a single fixed-effects panel model

I have what I think is a relatively simple question about the merits of performing a single panel regression with fixed effects as opposed to individual yearly cross-sectional regressions with the same data.

Suppose I have an unbalanced panel with (up to) $n$ observations in $T$ years, with $T << n$. In a given year, each individual may receive either treatment 1 or treatment 2, or receive no treatment at all. (Note that the application of the treatments to individuals is not necessarily random). That is, I'm interested in $\beta_1$ and $\beta_2$ in the following model:

$$Y_{i,t} = \alpha_i + \beta_1 X^1_{i,t} + \beta_2X^2_{i,t} + \beta_3 W_{i,t} + \lambda_t + \epsilon_{i,t}$$

where $X^1_{i,t}$ and $X^2_{i,t}$ are indicators for whether treatment 1 or 2 (respectively) is applied to individual $i$ in year $t$, $W_{i,t}$ are controls, and $\alpha_i$ are individual fixed effects and $\lambda_t$ are yearly fixed effects.

If I run this panel regression, I find that $\hat{\beta}_1$ and $\hat{\beta}_2$ are not significantly different. However, if I run a series of cross-sectional regressions from the same panel for each year $t$ separately : $$Y_i = \beta_0 + \beta_{1,t} X^1_{i} + \beta_{2,t} X^2_{i} + \beta_{3,t} W_{i} + \epsilon_{i} \qquad \forall t \in T$$

I find that $\hat{\beta}_{1,t} \neq \hat{\beta}_{2,t}$ for each $t$, and sometimes by a large margin.

Wooldridge (2009, p.449) states "estimating two separate equations" is "sometimes...desirable," but does not elaborate on when this is the case. Can anyone provide a good explanation of the situations when running separate cross-sectional regressions for different years is more desirable (and produces better estimates) than a single panel data regression with individual fixed effects? My gut tells me that the former approach could lead to omitted variable bias, but what if the effects of the treatments change over time?

References:

Wooldridge, J.M. 2009. Introductory Econometrics: A Modern Approach. Edition 4e. South-Western, Mason, OH. p.449.

• (+1) How many years do you have in the panel and what is your research question? – Marquis de Carabas May 4 '16 at 21:08
• There are 6 years of data with about 3,000 observations (markets) per year. I'm interested in whether entry of one of two companies in a market leads to a change in price in that market. (The controls contain market characteristics and the presence of other competitors). – bluecanary May 4 '16 at 21:12
• In the actual model, I also have additional variables for entry in past or future periods (see Goolsbee and Syverson "How do incumbents respond to the threat of entry? Evidence from the major airlines." (QJE, 2008) for an similar model), but the question extends more generally. – bluecanary May 4 '16 at 21:17