Is least square dummy variable model better than random effects model?

Question

I have a panel dataset with one dependent and twelve independent variables. There are 50 individuals with data for 100 days. Theoretically, most of them should be significant. First, I checked for fixed effects using breusch and pagan lagrangian multiplier test. As I found the fixed effects to be significant, I performed the Huasman test. Finally, I chose random effects model. I am getting at most 4 variables significant using p-value for all models (pooled, fixed, and random). The problem is very low adjusted R-square(0.01 to 0.02). when I use least squares dummy variables model (LSDV) with all days as dummy variables, I find all the days to be significant apart from some explanatory variables and the adjusted square value becomes 0.90 (approx.). Since my days are 100, so there are 100 dummy variables. I also find R-square to be 0.90 when I use individuals' dummy variables (50) in the LSDV model. My questions are:

1. Is the difference between RE model and LSDV model R-squares due to the unobserved heterogeneity correlated with all of the regressors in all time periods?
2. Is there a limit to include dummy variables?
3. Is my LSDV model better than FE model or RE model?

Kindly also guide me to appropriate literature. The analysis is conducted in R using package plm.

Answer 1

Here is a start at an answer.

In your situation, where $i$ is an observational unit and $t$ is time and the model is $y_{it} = \beta X_{it} + \gamma_i + \epsilon_{it}$

and $\gamma_i$ is the unobserved heterogeneity, the LSDV should produce the same coefficients as the fixed effect (FE) estimator. However, the standard errors will be different. FE is better than LSDV when the number of individuals increases, but the number of time periods is fixed. LSDV is better in the opposite case, that is as the sample size increases, the number of observational units stays fixed (more or less).

My understanding of random effects (RE) is that if certain assumptions about the covariance structure hold, then RE is more efficient (smaller standard errors) than FE and LSDV. All three estimators will be consistent. However, FE and LSDV are agnostic to the covariance assumptions, so that when these assumptions are violated, then RE will be an inconsistent estimator, while FE and LSDV will remain consistent. This is the scenario you were testing for. In Mostly Harmless Econometrics, footnote 2 on page 223, Angrist and Pischke say that they they prefer FE (OLS) to RE (GLS) as "GLS requires stronger assumptions than OLS and the resulting efficiency gain is likely to be modest, while finite sample properties might be worse."

As far as the low adjusted $R^2$, it is not as big a concern as you might think if your goal is to estimate parameters (the $\beta$s). To see this, consider the five assumptions laid out in Wooldridge's Introductory Econometrics: A Modern Approach (or his grad text) for BLUE or the asymptotic analogue. The $R^2$ is absent from any of these assumptions for unbiased or consistent (and efficient) estimation.

However, if your goal is model fitting, then the model producing higher $R^2$ might be something to consider.

To throw in one extra bit of info, you have a long time period, 100, so you might consider looking at estimators that model time more seriously. One of these is the Arellano Bond estimator. See this blog post for a competing method together with useful references. A second method is multi-way clustering as in Cameron, Gelbach, and Miller. I suspect that an Arellano-Bond approach will be more appropriate in your situation.