Missing intercept in fixed effects model output

Question

I am running a fixed effects model using plm in R and I'm struggling to understand the output.

1. Why is the estimate for the intercept term missing?
2. Why the R-Squared value is so low compared to the Adj. R-Squared?
3. Why the R-Squared value is so low compared to a random effects model I ran with R-Squared = 0.88?

> summary(fixed)
Oneway (individual) effect Within Model

Call:
plm(formula = Y ~ X, data = pdata, model = "within")

Unbalanced Panel: n = 271, T = 1-16, N = 1077

Residuals:
    Min.  1st Qu.   Median  3rd Qu.     Max. 
-10.4083  -1.5881   0.0000   1.6870   9.9549 

Coefficients:
  Estimate Std. Error t-value  Pr(>|t|)    
X 0.339422   0.023758  14.287 < 2.2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Total Sum of Squares:    10156
Residual Sum of Squares: 8101.5
R-Squared:      0.20226
Adj. R-Squared: -0.066294
F-statistic: 204.103 on 1 and 805 DF, p-value: < 2.22e-16
```

Answer 1

Please pay attention to the major differences between panel regression as implemented in the plm package and the usual lm or lmer functions. Quoting from the vignette with respect to the Within model that you have specified (emphasis added):

...the standard way of estimating fixed effects models with, say, group (time) effects entails transforming the data by subtracting the average over time (group) to every variable, which is usually termed time-demeaning... Panel data estimation requires to apply different transformations to raw series... the within transformation: $Q=I_{nT}−P$ returns a vector containing the values in deviation from the individual means. The Within function performs this operation.

If you're working with deviations from means as the Within model does, there is no intercept.

Panel data and time series can be very tricky. Intuition developed on standard linear regression can be misleading. Read and study very, very carefully. That much I know about panel data analysis.

I don't know enough about panel data analysis to answer your question about the adjusted $R^2$ in detail. The general idea is what's suggested in another answer: after appropriate corrections, there isn't any variance left to explain. What I don't understand is how the standard $R^2$ and its adjusted value are calculated with panel data regression.

It will be impossible to answer why your "random effects" model seemed to work better without more details about both the specific random effects model and the overall structure of your data.

Answer 2

I am unfamiliar with plm() but I can attempt an answer to questions 1 & 2. For 2), the adjusted R2 is the amount of variance explained relative to the simple mean of the data. Because your adjusted R2 is essentially zero, it suggests that the result of your formula has been to take the mean of the response variable Y. So I would expect that your effect estimate X=0.339422, is essentially the mean of Y. This answers your first question -- actually the intercept is not missing. The X=0.339422 is an intercept. What is missing is a slope.