Computes normal Wald tests and confidence intervals for an ordinary least squares model using a heteroskedasticity-consistent covariance estimator.
Arguments
- object
An ordinary least squares model fitted by
stats::lm().- type
A character string specifying the HC estimator. The default is
"hcbeta".- alpha
Significance level. The confidence level is
1 - alpha.- null
Null values for the coefficient tests. Use a scalar to test all coefficients against the same value, or a numeric vector with one value per coefficient.
- ...
Method-specific constants passed to
vcov_hc(). Defaults are documented invcov_hc()and can be inspected withhc_methods().
Value
An object of class hcinfer containing the fitted HC covariance estimator,
coefficient tests, p-values, confidence intervals, diagnostics, and method
parameters.
Details
For each coefficient, hcinfer tests
$$H_0: \beta_j = \beta_j^{(0)}$$
against a two-sided alternative using the statistic
$$z_j = \frac{\hat\beta_j - \beta_j^{(0)}} {\sqrt{[\widehat{\Psi}_{HC}]_{jj}}}.$$
The reference distribution is the standard normal distribution. Confidence intervals are Wald intervals obtained by direct inversion of the test,
$$\hat\beta_j \pm z_{1 - \alpha / 2} \sqrt{[\widehat{\Psi}_{HC}]_{jj}}.$$
Bootstrap intervals and Student t quantiles are not used.
References
White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48(4), 817-838.
Cribari-Neto, F. (2004). Asymptotic inference under heteroskedasticity of unknown form. Computational Statistics and Data Analysis, 45(2), 215-233.
Examples
schools <- PublicSchools |>
dplyr::mutate(
income_scaled = income / 10000,
income_scaled_sq = income_scaled^2
)
fit <- lm(expenditure ~ income_scaled + income_scaled_sq, data = schools)
result <- hcinfer(fit, type = "hcbeta")
result
#>
#> ── 🔎 HCbeta robust inference ──────────────────────────────────────────────────
#> 📐 Model: `expenditure ~ income_scaled + income_scaled_sq`
#> Observations: 50 | Parameters: 3
#> 🥪 Robust covariance: HCbeta
#> Confidence level: 95.0% | Normal critical value: 1.9600
#> 💡 Use `summary()` for p-values, test results, confidence intervals, and
#> diagnostics.
summary(result)
#>
#> ── 🔎 HCbeta robust inference summary ──────────────────────────────────────────
#>
#> ── 📐 Model ──
#>
#> Formula: `expenditure ~ income_scaled + income_scaled_sq`
#> Observations: 50 | Parameters: 3 | Residual df: 47
#>
#> ── 🥪 Robust covariance ──
#>
#> Estimator: HCbeta
#> Confidence level: 95.0% | Normal critical value: 1.9600
#> Tests are two-sided normal Wald tests, one coefficient at a time.
#> Test results use alpha = 0.050.
#>
#> ── 🎯 Leverage diagnostics ──
#>
#> # A tibble: 6 × 2
#> statistic value
#> <chr> <chr>
#> 1 minimum 0.02669
#> 2 q1 0.03106
#> 3 median 0.03912
#> 4 mean 0.06
#> 5 q3 0.04962
#> 6 maximum 0.6508
#> Maximum leverage: observation 2 (index 2), value 0.6508
#> Average leverage: 0.0600
#> Concentration: 10.85 x average leverage
#>
#> ── ⚖️ Robust weights ──
#>
#> # A tibble: 6 × 2
#> statistic value
#> <chr> <chr>
#> 1 minimum 1.156
#> 2 q1 1.167
#> 3 median 1.187
#> 4 mean 1.276
#> 5 q3 1.212
#> 6 maximum 4.581
#> Maximum weight: observation 2 (index 2), value 4.5807
#> Median weight: 1.1869
#> Concentration: 3.86 x median weight
#>
#> ── ⚙️ Method parameters ──
#>
#> # A tibble: 12 × 3
#> parameter value role
#> <chr> <chr> <chr>
#> 1 c1 7 method constant
#> 2 c2 0.75 method constant
#> 3 lower 0.01 method constant
#> 4 upper 0.99 method constant
#> 5 mu_hat 0.94 estimated quantity
#> 6 s2_w 0.008504 estimated quantity
#> 7 phi_hat 5.632 estimated quantity
#> 8 a_hat 5.294 estimated quantity
#> 9 b_hat 0.3379 estimated quantity
#> 10 zeta 0.5 estimated quantity
#> 11 a_tilde 3.147 estimated quantity
#> 12 b_tilde 0.669 estimated quantity
#>
#>
#> ── 🔎 Coefficient tests ──
#>
#> # A tibble: 3 × 9
#> term estimate robust_se z p_value alpha test_result
#> <chr> <chr> <chr> <chr> <chr> <chr> <chr>
#> 1 (Intercept) 832.9 850.7 0.9791 0.328 0.050 ✅ do not reject H0
#> 2 income_scaled -1834 2309 -0.7945 0.427 0.050 ✅ do not reject H0
#> 3 income_scaled_sq 1587 1547 1.026 0.305 0.050 ✅ do not reject H0
#> ci ci_relation
#> <chr> <chr>
#> 1 [-834.3, 2500] includes null
#> 2 [-6359, 2691] includes null
#> 3 [-1446, 4620] includes null
#>
#>
#> ── 📏 Confidence intervals ──
#>
#> # A tibble: 3 × 4
#> term null_value interval interpretation
#> <chr> <chr> <chr> <chr>
#> 1 (Intercept) 0 [-834.3, 2500] includes null
#> 2 income_scaled 0 [-6359, 2691] includes null
#> 3 income_scaled_sq 0 [-1446, 4620] includes null
#> 💡 test_result is based on p_value < alpha. Do not reject H0 does not mean that
#> H0 is true.
confint(result)
#> # A tibble: 3 × 4
#> term conf_low conf_high level
#> <chr> <dbl> <dbl> <dbl>
#> 1 (Intercept) -834. 2500. 0.95
#> 2 income_scaled -6359. 2691. 0.95
#> 3 income_scaled_sq -1446. 4620. 0.95
hcinfer(fit, type = "hcbeta", c1 = 7, c2 = 0.75, lower = 0.01, upper = 0.99)
#>
#> ── 🔎 HCbeta robust inference ──────────────────────────────────────────────────
#> 📐 Model: `expenditure ~ income_scaled + income_scaled_sq`
#> Observations: 50 | Parameters: 3
#> 🥪 Robust covariance: HCbeta
#> Confidence level: 95.0% | Normal critical value: 1.9600
#> 💡 Use `summary()` for p-values, test results, confidence intervals, and
#> diagnostics.
hcinfer(fit, type = "hc5", k = 0.7)
#>
#> ── 🔎 HC5 robust inference ─────────────────────────────────────────────────────
#> 📐 Model: `expenditure ~ income_scaled + income_scaled_sq`
#> Observations: 50 | Parameters: 3
#> 🥪 Robust covariance: HC5
#> Confidence level: 95.0% | Normal critical value: 1.9600
#> 💡 Use `summary()` for p-values, test results, confidence intervals, and
#> diagnostics.
hcinfer(fit, type = "hc5m", k = 0.7, k1 = 1, k2 = 0, k3 = 1)
#>
#> ── 🔎 HC5m robust inference ────────────────────────────────────────────────────
#> 📐 Model: `expenditure ~ income_scaled + income_scaled_sq`
#> Observations: 50 | Parameters: 3
#> 🥪 Robust covariance: HC5m
#> Confidence level: 95.0% | Normal critical value: 1.9600
#> 💡 Use `summary()` for p-values, test results, confidence intervals, and
#> diagnostics.