Regression donkey here, refereeing paper and wanting to make sure of the interpretation of the interaction of two fixed effects. With panel data (say households i in state s at time t), the interaction of the state and time fixed effects
Interaction of fixed effects

\gamma_{s}*\lambda_{t} (is NOT the same as a statetime fixed effect\delta_{st} , right? The authors claim that the interaction term allows stateyear specific shocks, and that seems wrong to me. Am I right? Thanks! 
Interacting the FEs is not the same as putting country (say) FE and year FE separately. The interaction allows the time FE to have disperse effects on the countries. However, excluding the individual country FE and time FE is a series, as doing so does not allow for capturing permanent differences across countries (culture, institutions, etc), which could lead to a nontrivial case of omitted variable bias.

Grad student here: Could someone competent confirm/deny this assertion?
Interacting the FEs is not the same as putting country (say) FE and year FE separately. The interaction allows the time FE to have disperse effects on the countries. However, excluding the individual country FE and time FE is a series, as doing so does not allow for capturing permanent differences across countries (culture, institutions, etc), which could lead to a nontrivial case of omitted variable bias.

Grad student here: Could someone competent confirm/deny this assertion?
Interacting the FEs is not the same as putting country (say) FE and year FE separately. The interaction allows the time FE to have disperse effects on the countries. However, excluding the individual country FE and time FE is a series, as doing so does not allow for capturing permanent differences across countries (culture, institutions, etc), which could lead to a nontrivial case of omitted variable bias.
The first part is right, but the second part, the part following 'however', only matters of you are trying to recover the FE estimates themselves. If you only care about some other coefficients, then you will get the same result either way.
That is:
Reg y x i.state##i.year
And
Reg y x i.state#i.year
Will produce the same coefficient estimates for x.

And I just realized that this was 4 years ago so I'm not sure why I just directed this at OP.
Grad student: take out some data and try int. You'll get your answer.
OP, do you not have any data you can try this on to double check your intuition? Really straightforward.

clear
set seed 0214
set obs 1000
gen x=rnormal()
gen year=round(uniform()*15)+2000
egen country=cut(x), group(10)gen x2=rnormal()
gen y=50*rnormal()+x2qui {
reg y x2 i.country##i.yearscalar reg1_b=_b[x2]
reg y x2 i.country#i.year
scalar reg2_b=_b[x2]
egen cty_by_year=group(country year)
reg y x2 i.cty_by_year
scalar reg3_b=_b[x2]
}di "First reg x2 beta " reg1_b
di "Second reg x2 beta " reg2_b
di "Third reg x2 beta " reg3_b 
When you interact state and year dummies (i.e. when you include state, year, and state*year in the regresion, which by the way is the same as creating stateyear dummies and including them in the regression), you are assuming that the unobserved state level heterogenity varies over time. Also, you are assuming the time effect to vary by state. If you include state and year separately and no interaction, you are assuming that the unobserved state level heterogenity is constant over time.

Interacting state and year dummies (i.e. state, year, and state*year), would lead to a perfect fit (no variation and no standard errors). Suppose you have n=7 states and T=10 years for a sample size of nT=70. The interaction of state and year dummies creates 69 dummies (6 state dummies, 9 year dummies, 54 stateyear dummies). With constant and no other variables, you have 70 variables to estimate, one observation per parameter estimate. This doesn't make sense.
You should instead include state dummies and statespecific time trends only if the concern is statespecific effects. Does this make sense?
regress y x1 i.country##i.year % interaction of state and year dummies
regress y x1 c.year##i.country % state dummies and statespecific time trend