# Why does my fixed effects return symmetric predicted values?

I am running a fixed effects model. My data consists of multiple variables for two time points for about 180 countries. My aim is to assess the effect that variable iv2 has on dv1, while controlling for other variables such as iv1. There is something odd though and I have the strong suspicion it has something to do with the two time points.

## Question:

When I run the fixed effects regression, the predicted values as well as the residuals have the same absolute value, but the opposite sign. The outcome puzzles me and I cannot wrap my head around it. Obviously, the residuals now violate the regression assumption of independent errors (serial correlation). Why does the model show this behaviour? Does this compromise the analysis of the effectiv2 has on dv1?

I would appreciate a formal mathematical explanation, but an intuitive explanation as to why this behaviour occurs in a fixed effects model (and not in random effects, for example) will also do the trick.

Here is a reproducible example in R:

# load the data:
df <- structure(list(country = c(1, 1, 2, 2, 3, 3, 4, 4, 5, 5), year = c(2010,
2015, 2010, 2015, 2010, 2015, 2010, 2015, 2010, 2015), dv1 = c(28.61,
31.13, 38.87, 39.46, 68.42, 70.39, 79.36, 80.55, 70.14, 71.48
), iv1 = c(-20.68, 0, NA, NA, -19.41, -18.73, 24.98, 25.23, 21.23,
-21.06), iv2 = c(-4.23, NA, NA, NA, -4.92, -4.22, 9.19, 9.37,
4.15, -3.92)), .Names = c("country", "year", "dv1", "iv1", "iv2"
), row.names = c(2L, 3L, 5L, 6L, 8L, 9L, 11L, 12L, 14L, 15L),class ="data.frame")