negative values for mean squared errors in sae package for R

I have been using "sae" package for R to use small area estimations with spatial fay-herriot models (SFH). Using different distance matrices I occasionally obtained negative values of Mean Squared Errors (MSE).

The following link may reference a similar behavior:

scikit-learn cross validation, negative values with mean squared error

In any case here is a working example:

library(sae)

v1 <- c(0.000,0.089,0.081,0.082,0.058,0.075,0.062,0.043,0.000,0.037,0.065,0.056,
        0.046,0.055,0.034,0.043,0.043,0.027,0.013,0.011,0.036,0.029,0.017,0.081,
        0.000,0.093,0.081,0.062,0.077,0.066,0.046,0.000,0.036,0.063,0.054,0.044,
        0.053,0.033,0.041,0.041,0.026,0.012,0.010,0.035,0.028,0.016,0.073,0.091,
        0.000,0.080,0.066,0.085,0.070,0.048,0.000,0.036,0.062,0.053,0.043,0.053,
        0.032,0.041,0.041,0.025,0.012,0.010,0.034,0.028,0.016,0.071,0.076,0.077,
        0.000,0.053,0.083,0.065,0.043,0.000,0.039,0.071,0.059,0.047,0.057,0.035,
        0.044,0.044,0.027,0.013,0.011,0.037,0.030,0.017,0.060,0.070,0.075,0.065,
        0.000,0.070,0.084,0.076,0.000,0.032,0.053,0.051,0.041,0.065,0.041,0.039,
        0.055,0.023,0.011,0.009,0.030,0.031,0.019,0.065,0.074,0.083,0.084,0.060,
        0.000,0.076,0.050,0.000,0.037,0.066,0.056,0.045,0.055,0.034,0.042,0.042,
        0.026,0.013,0.010,0.035,0.029,0.017,0.056,0.067,0.072,0.069,0.077,0.079,
        0.000,0.065,0.000,0.033,0.057,0.054,0.044,0.071,0.040,0.041,0.055,0.024,
        0.011,0.009,0.032,0.030,0.017,0.051,0.060,0.063,0.062,0.084,0.067,0.079,
        0.000,0.000,0.030,0.052,0.049,0.041,0.063,0.051,0.038,0.067,0.024,0.011,
        0.009,0.031,0.040,0.027,0.015,0.018,0.019,0.026,0.004,0.022,0.013,0.000,
        0.000,0.064,0.036,0.045,0.057,0.030,0.051,0.057,0.032,0.077,0.097,0.110,
        0.070,0.066,0.089,0.024,0.029,0.029,0.041,0.009,0.035,0.021,0.000,0.018,
        0.000,0.059,0.071,0.098,0.045,0.050,0.077,0.028,0.082,0.045,0.040,0.099,
        0.048,0.054,0.054,0.059,0.059,0.072,0.039,0.065,0.051,0.031,0.000,0.050,
        0.000,0.077,0.060,0.069,0.043,0.056,0.051,0.034,0.016,0.013,0.045,0.036,
        0.021,0.042,0.047,0.047,0.059,0.033,0.053,0.045,0.024,0.000,0.058,0.079,
        0.000,0.075,0.067,0.054,0.071,0.048,0.040,0.018,0.015,0.055,0.045,0.026,
        0.028,0.033,0.033,0.046,0.015,0.040,0.030,0.005,0.000,0.094,0.068,0.089,
        0.000,0.053,0.053,0.099,0.033,0.062,0.027,0.022,0.083,0.049,0.037,0.046,
        0.050,0.051,0.060,0.055,0.056,0.070,0.046,0.000,0.043,0.071,0.069,0.053,
        0.000,0.051,0.050,0.074,0.031,0.014,0.012,0.041,0.037,0.020,0.018,0.023,
        0.023,0.035,0.023,0.029,0.033,0.032,0.000,0.053,0.053,0.071,0.065,0.061,
        0.000,0.080,0.089,0.045,0.016,0.035,0.066,0.095,0.057,0.024,0.030,0.030,
        0.043,0.012,0.036,0.027,0.002,0.000,0.075,0.063,0.086,0.102,0.050,0.070,
        0.000,0.039,0.063,0.028,0.023,0.094,0.066,0.038,0.038,0.042,0.042,0.052,
        0.050,0.047,0.058,0.058,0.000,0.035,0.062,0.060,0.046,0.086,0.078,0.049,
        0.000,0.030,0.011,0.021,0.042,0.057,0.035,0.018,0.022,0.022,0.031,0.005,
        0.027,0.016,0.000,0.039,0.091,0.045,0.057,0.076,0.037,0.051,0.075,0.031,
        0.000,0.069,0.063,0.095,0.052,0.076,0.016,0.019,0.019,0.027,0.004,0.023,
        0.014,0.000,0.076,0.070,0.038,0.048,0.062,0.031,0.045,0.062,0.027,0.085,
        0.000,0.104,0.078,0.061,0.090,0.014,0.018,0.018,0.025,0.004,0.021,0.013,
        0.000,0.084,0.063,0.035,0.043,0.056,0.028,0.058,0.056,0.039,0.076,0.099,
        0.000,0.070,0.076,0.105,0.020,0.024,0.024,0.035,0.005,0.029,0.017,0.000,
        0.022,0.096,0.050,0.064,0.085,0.040,0.058,0.091,0.035,0.084,0.050,0.045,
        0.000,0.056,0.069,0.005,0.011,0.011,0.026,0.000,0.019,0.010,0.010,0.016,
        0.058,0.047,0.068,0.070,0.041,0.114,0.090,0.068,0.052,0.039,0.067,0.074,
        0.000,0.103,0.006,0.010,0.010,0.019,0.000,0.014,0.007,0.006,0.057,0.071,
        0.032,0.045,0.061,0.026,0.070,0.061,0.042,0.086,0.082,0.106,0.091,0.097,
        0.000)
dmat <- data.frame(matrix(v1,byrow=TRUE,nrow=23))

y <- c(0.057,0.074,0.067,0.071,0.031,0.070,0.067,0.047,0.075,0.028,0.051,0.085,
       0.037,0.070,0.082,0.084,0.063,0.070,0.085,0.070,0.059,0.050,0.064)
x <- c(0.032,0.041,0.053,0.056,0.060,0.055,0.083,0.060,0.074,0.035,0.041,0.044,
       0.034,0.048,0.045,0.038,0.047,0.043,0.057,0.062,0.041,0.062,0.045)
vary <- c(0.00018,0.00014,0.00016,0.00003,0.00029,0.00015,0.00029,0.00039,
          0.00005,0.00008,0.00013,0.00017,0.00010,0.00027,0.00114,0.00051,
          0.00031,0.00002,0.00038,0.00024,0.00016,0.00019,0.00014)

fit1 <- mseSFH(y ~ x,vardir=vary,proxmat=dmat)
fit1$mse[fit1$mse < 0]

I'm not sure if this is the appropriate forum for the question.

Thanks in advance,

Joao

I'm pretty sure that this is due to bias correction that generally takes place when you have MSE. You can read about the formula for bias correction that is used in the references they provided in ?sae::meanSFH. In one of the articles, they provided a case study where the average MSE is negative. (I found this in Molina et al., 2009. They identify the bias correction in a few places, but it's very clear on pp. 452-453.)

You can visualize the errors and see how very close they are to zero.

data.frame(UnbiasedMSE = fit1$mse) %>% 
  ggplot(aes(x = seq_along(UnbiasedMSE), 
             y = UnbiasedMSE)) + 
  geom_line() + geom_point() + 
  scale_y_continuous(labels = scales::comma) + 
  theme_bw()