Tests: The Linear Regression Model (Residual Variance)

Ivan Jacob Agaloos Pesigan

2020-12-31

# The Linear Regression Model: Residual Variance {#linreg-estimation-sigma2hatepsilonhat-example}

Data

See jeksterslabRdatarepo::wages.matrix() for the data set used in this example.

X <- jeksterslabRdatarepo::wages.matrix[["X"]]
# age is removed
X <- X[, -ncol(X)]
y <- jeksterslabRdatarepo::wages.matrix[["y"]]
head(X)
#>      constant gender race union education experience
#> [1,]        1      1    0     0        12         20
#> [2,]        1      0    0     0         9          9
#> [3,]        1      0    0     0        16         15
#> [4,]        1      0    1     1        14         38
#> [5,]        1      1    1     0        16         19
#> [6,]        1      1    0     0        12          4
head(y)
#>      wages
#> [1,] 11.55
#> [2,]  5.00
#> [3,] 12.00
#> [4,]  7.00
#> [5,] 21.15
#> [6,]  6.92

Residual Variance

n <- nrow(X)
k <- ncol(X)
betahat <- betahat(
  X = X,
  y = y
)
RSS <- RSS(
  X = X,
  y = y
)
result_sigma2hatepsilonhat1 <- .sigma2hatepsilonhat(
  RSS = RSS,
  n = n,
  k = k
)
result_sigma2hatepsilonhat2 <- .sigma2hatepsilonhat(
  n = n,
  k = k,
  X = X,
  y = y
)
result_sigma2hatepsilonhat3 <- sigma2hatepsilonhat(
  X = X,
  y = y
)

Residual Variance (Biased)

result_sigma2hatepsilonhatbiased1 <- .sigma2hatepsilonhatbiased(
  RSS = RSS,
  n = n
)
result_sigma2hatepsilonhatbiased2 <- .sigma2hatepsilonhatbiased(
  n = n,
  X = X,
  y = y
)
result_sigma2hatepsilonhatbiased3 <- sigma2hatepsilonhatbiased(
  X = X,
  y = y
)

lm() function

lmobj <- lm(
  wages ~ gender + race + union + education + experience,
  data = jeksterslabRdatarepo::wages
)
lm_sigma2hatepsilonhat <- summary(lmobj)$sigma^2
lm_anova <- anova(lmobj)
lm_RSS <- lm_anova["Residuals", "Sum Sq"]
lm_sigma2hatepsilonhatbiased <- lm_RSS / n

sigma2hatepsilonhat <- c(
  result_sigma2hatepsilonhat1, result_sigma2hatepsilonhat2, result_sigma2hatepsilonhat3
)
sigma2hatepsilonhatbiased <- c(
  result_sigma2hatepsilonhatbiased1, result_sigma2hatepsilonhatbiased2, result_sigma2hatepsilonhatbiased3
)
context("Test linreg-estimation-sigma2hatepsilonhat.")
test_that("sigma2hatepsilonhat", {
  for (i in seq_along(sigma2hatepsilonhat)) {
    expect_equivalent(
      lm_sigma2hatepsilonhat,
      sigma2hatepsilonhat[i]
    )
  }
})
#> Test passed 🎊
test_that("sigma2hatepsilonhatbiased", {
  for (i in seq_along(sigma2hatepsilonhatbiased)) {
    expect_equivalent(
      lm_sigma2hatepsilonhatbiased,
      sigma2hatepsilonhatbiased[i]
    )
  }
})
#> Test passed 😀