# Comparing the regression coefficients of two time periods using unbalanced panel data and a fixed effects model

I'm currently trying to investigate how investors change their asset allocation due to liquidity risk and if their behaviour has changed since the financial crisis.

My data contains the asset allocation of about 200 investors over different time periods as well as a liquidity risk factor for each year and investor.

First I wanted to check if investors have higher allocations in "safe & liquid" assets with increasing liquidity risk by using a two way fixed effects model, where $Y_{i,t}$ refers to the allocation of investor i at year t in a liquid asset (e.g. bonds), and the lambdas to the investor and year fixed effects: $$Y_{i,t} = \beta_0 + \beta_1 \cdot Risk_{i,t} + \lambda_i + \lambda_t + u_{i,t}$$

In general I'm pretty happy with the results so far but now I would like to check if the 'behaviour' changed after the financial crisis, so basically I would like to know if there is a significant difference for $\beta_1$ before 2008 and after 2008.

My first idea was to add a time dummy variable that is 0 before 2008 and 1 after 2008 and check the significance of the interaction term: $$Y_{i,t} = \beta_0 + \beta_1 \cdot Risk_{i,t} + time_t + Risk_{i,t} \cdot time_t + \lambda_i + \lambda_t + u_{i,t}$$

However I encountered several problems, where I'm not sure how to handle them:

1. I think I need to exclude the time fixed effects to capture the differences of the time periods and avoid perfect collinearity between the time dummy and the year fixed effects. However, wouldn't that be equal to assuming that the data can be pooled for each of the time periods? How does this model compare to the first one where I use all time periods and the time fixed effect? For example if the first model has a positive coefficient and the second model shows a negativ coefficient for the time period before 2008 and a positive slope after 2008, would that prove that the positive correlation for the entire data is driven by the change in behaviour after 2008?

2. Is this method even sufficient for unbalanced panel datasets. I have much more observations after 2008.

3. For my first model I use two-way clustered standard errors to account for heteroscedasticity and autocorrelation, are they still appropriate if I remove the time fixed effects or should I only cluster the standard errors by investors in that case?
4. Is there a better approach? I personally would prefer to split the dataset and use the two way fixed effect model with each of the subsamples and then compare the slopes with some test, however, I wasn't able to find an appropriate one yet. I also thought about using a chow-test, but if I understand it correctly this would only be appropriate for time-series data?

Thank you in advance for your help! Also because I'm still learning how to handle panel data, I highly appreciate every additional information on when to use which type of standard error (robust, clustered etc.) or other infos that might help to improve the models.