Tests: The Linear Regression Model (Standard Errors of Estimates of Regression Coefficients)

Ivan Jacob Agaloos Pesigan

2020-12-31

# The Linear Regression Model: Standard Errors of Estimates of Regression Coefficients {#linreg-estimation-se-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

Standard Errors of Estimates of Regression Coefficients

vcovhatbetahat <- vcovhatbetahat(
  X = X,
  y = y
)
vcovhatbetahatbiased <- vcovhatbetahatbiased(
  X = X,
  y = y
)
result_sehatbetahat1 <- .sehatbetahat(
  vcovhatbetahat = vcovhatbetahat
)
result_sehatbetahat2 <- .sehatbetahat(
  X = X,
  y = y
)
result_sehatbetahat3 <- sehatbetahat(
  X = X,
  y = y
)

Biased Standard Errors of Estimates of Regression Coefficients

result_sehatbetahatbiased1 <- .sehatbetahatbiased(
  vcovhatbetahatbiased = vcovhatbetahatbiased
)
result_sehatbetahatbiased2 <- .sehatbetahatbiased(
  X = X,
  y = y
)
result_sehatbetahatbiased3 <- sehatbetahatbiased(
  X = X,
  y = y
)

Standard Errors of Scaled Estimates of Regression Coefficients (textbook)

slopeshat <- slopeshat(
  X = X,
  y = y
)
slopeshatprime <- slopeshatprime(
  X = X,
  y = y
)
sehatbetahat <- as.vector(
  sehatbetahat(
    X = X,
    y = y
  )
)
sehatslopeshat <- sehatbetahat[-1]
result_sehatslopeshatprimetb1 <- .sehatslopeshatprimetb(
  slopeshat = slopeshat,
  sehatslopeshat = sehatslopeshat,
  slopeshatprime = slopeshatprime
)
result_sehatslopeshatprimetb2 <- .sehatslopeshatprimetb(
  sehatslopeshat = sehatslopeshat,
  slopeshatprime = slopeshatprime,
  X = X,
  y = y
)
result_sehatslopeshatprimetb3 <- .sehatslopeshatprimetb(
  slopeshat = slopeshat,
  slopeshatprime = slopeshatprime,
  X = X,
  y = y
)
result_sehatslopeshatprimetb4 <- .sehatslopeshatprimetb(
  sehatslopeshat = sehatslopeshat,
  slopeshatprime = slopeshatprime,
  X = X,
  y = y
)
result_sehatslopeshatprimetb5 <- .sehatslopeshatprimetb(
  X = X,
  y = y
)
result_sehatslopeshatprimetb6 <- sehatslopeshatprimetb(
  X = X,
  y = y
)

lm() function

lmobj <- lm(
  wages ~ gender + race + union + education + experience,
  data = jeksterslabRdatarepo::wages
)
lm_se <- as.vector(sqrt(diag(vcov(lmobj))))

lm() function - scaled data

lmscaledobj <- lm(
  wages ~ gender + race + union + education + experience,
  data = as.data.frame(scale(jeksterslabRdatarepo::wages))
)
lmscaled_se <- as.vector(summary(lmscaledobj)[["coefficients"]][, "Std. Error"])
lmscaled_se <- lmscaled_se[-1]

result_sehatbetahat1 <- as.vector(result_sehatbetahat1)
result_sehatbetahat2 <- as.vector(result_sehatbetahat2)
result_sehatbetahat3 <- as.vector(result_sehatbetahat3)
result_sehatbetahatbiased1 <- as.vector(result_sehatbetahatbiased1)
result_sehatbetahatbiased2 <- as.vector(result_sehatbetahatbiased2)
result_sehatbetahatbiased3 <- as.vector(result_sehatbetahatbiased3)
context("Test linreg-estimation-vcov")
test_that("unbiased", {
  for (i in 1:length(lm_se)) {
    expect_equivalent(
      result_sehatbetahat1[i],
      lm_se[i]
    )
    expect_equivalent(
      result_sehatbetahat2[i],
      lm_se[i]
    )
    expect_equivalent(
      result_sehatbetahat3[i],
      lm_se[i]
    )
  }
})
#> Test passed 😸
test_that("biased", {
  for (i in 1:length(lm_se)) {
    expect_equivalent(
      round(result_sehatbetahatbiased1[i], digits = 1),
      round(lm_se[i], digits = 1)
    )
    expect_equivalent(
      round(result_sehatbetahatbiased2[i], digits = 1),
      round(lm_se[i], digits = 1)
    )
    expect_equivalent(
      round(result_sehatbetahatbiased3[i], digits = 1),
      round(lm_se[i], digits = 1)
    )
  }
})
#> Test passed 🌈

result_sehatslopeshatprimetb1 <- as.vector(result_sehatslopeshatprimetb1)
result_sehatslopeshatprimetb2 <- as.vector(result_sehatslopeshatprimetb2)
result_sehatslopeshatprimetb3 <- as.vector(result_sehatslopeshatprimetb3)
result_sehatslopeshatprimetb4 <- as.vector(result_sehatslopeshatprimetb4)
result_sehatslopeshatprimetb5 <- as.vector(result_sehatslopeshatprimetb5)
result_sehatslopeshatprimetb6 <- as.vector(result_sehatslopeshatprimetb6)
test_that("sehatslopeshatprimetb", {
  for (i in 1:length(lmscaled_se)) {
    expect_equivalent(
      result_sehatslopeshatprimetb1[i],
      lmscaled_se[i]
    )
    expect_equivalent(
      result_sehatslopeshatprimetb2[i],
      lmscaled_se[i]
    )
    expect_equivalent(
      result_sehatslopeshatprimetb3[i],
      lmscaled_se[i]
    )
    expect_equivalent(
      result_sehatslopeshatprimetb4[i],
      lmscaled_se[i]
    )
    expect_equivalent(
      result_sehatslopeshatprimetb5[i],
      lmscaled_se[i]
    )
    expect_equivalent(
      result_sehatslopeshatprimetb6[i],
      lmscaled_se[i]
    )
  }
})
#> Test passed 🥇