Continuous time-fixed effect in panel regression? I am doing regressions on panel data which basically have the following structure:
$$Y_{it} = X'_{it}\beta + P_{t} + e_{it}$$
$P_{t}$ indicates a counter for the period, so it takes the value 1 in period 1, 2 in period 2 etc. 
I know that in this case it just captures period specific shocks  etc. over all individuals $i$.
My question now, which is probably very stupid: Can I interpret $P$ (its coefficient) differently when it is coded in a more "meaningful" way? Say I want to capture the effect of earthquakes according to their strength, so that for example $P_1=0, P_2=7, P_3=2, P_4=0$ etc.
Can I interpret the coefficient of P in this case as the effect of earthquakes on $Y$? 
 A: The way you've written down the model you are making the assumption that the time trend is linear. A more flexible way of specifying the period specific shocks without such a strong assumption is to include time dummies:
$$y_{it} = X'_{it}\beta + \sum^T_{t=1}\delta_t d_t + \epsilon_{it}$$
To separate the earthquake shocks from the period specific shocks you could generate a variable $E_{it}$ which measures the strength of an earthquake that affects individual $i$ in period $t$ (or only $E_t$ is all individuals are affected the same way or if you don't know the peoples' location), and interact this variables with the time dummies.
$$y_{it} = X'_{it}\beta + \sum^T_{t=1}\delta_t d_t + \sum^T_{t=1}\gamma_t \left(d_t \cdot E_{t} \right) + \epsilon_{it}$$
The coefficients $\gamma_t$ would then capture the average effect of each earthquake ( if you have 4 earthquakes, then there will be four $\gamma$s) on the outcome $y_{it}$. Just having a linear trend $P_t$ absorbs too much as it picks up the random fluctuation in each year plus the earthquake effects whereas your other suggestion to recode $P_t$ as strength of each earthquake does the reverse, i.e. it picks up the average strength of the earthquakes (making again a linearity assumption which is certainly not true for earthquakes for which strength usually increases exponentially) but without further time controls it will also pick up the period specific effects.
