Extract coefficient test results

Extracts the normal Wald test results from an hcinfer() object. If the requested significance level differs from the one used to create the object, only the reject column is recomputed. The test statistics and p-values are not affected by alpha and are never recomputed.

Usage

tests(object, ...)

# S3 method for class 'hcinfer'
tests(object, parm, alpha = object$alpha, ...)

Arguments

object: An object returned by hcinfer().
...: Unused. Passing named arguments raises an error.
parm: Optional coefficient names or integer positions to select a subset of coefficients. When omitted, all coefficients are returned.
alpha: Significance level used to compute the reject column. Must be strictly between 0 and 1. Defaults to the level stored in object. Changing alpha updates only the reject column; all other columns remain identical to the stored values.

Value

A tibble with one row per selected coefficient and the following columns:

term: Coefficient name.
estimate: OLS estimate $\hat\beta_j$.
null_value: Null hypothesis value $\beta_j^{(0)}$.
std_error: Robust standard error $\sqrt{[\widehat{\Psi}_{HC}]_{jj}}$.
z_value: Normal Wald statistic $z_j$.
p_value: Two-sided p-value $2\,\Phi(-|z_j|)$.
alpha: Significance level used for the reject column.
reject: Logical. TRUE when p_value < alpha.

Details

For each coefficient, the stored test is

$$H_0: \beta_j = \beta_j^{(0)}$$

against a two-sided alternative. The test statistic is

$$z_j = \frac{\hat\beta_j - \beta_j^{(0)}} {\sqrt{[\widehat{\Psi}_{HC}]_{jj}}},$$

and the p-value is $2\,\Phi(-|z_j|)$, where $\Phi$ is the standard normal distribution function. The null value $\beta_j^{(0)}$ is the one stored in the object, set when hcinfer() was called.

To test against a different null value, rerun hcinfer() with the desired null argument.

References

White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48(4), 817-838. doi:10.2307/1912934

Hinkley, D. V. (1977). Jackknifing in unbalanced situations. Technometrics, 19(3), 285-292. doi:10.1080/00401706.1977.10489550

Horn, S. D., Horn, R. A., and Duncan, D. B. (1975). Estimating heteroscedastic variances in linear models. Journal of the American Statistical Association, 70(350), 380-385. doi:10.1080/01621459.1975.10479877

MacKinnon, J. G. and White, H. (1985). Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties. Journal of Econometrics, 29(3), 305-325. doi:10.1016/0304-4076(85)90158-7

Davidson, R. and MacKinnon, J. G. (1993). Estimation and Inference in Econometrics. Oxford University Press.

Cribari-Neto, F. (2004). Asymptotic inference under heteroskedasticity of unknown form. Computational Statistics and Data Analysis, 45(2), 215-233. doi:10.1016/S0167-9473(02)00366-3

Cribari-Neto, F. and da Silva, W. B. (2011). A new heteroskedasticity consistent covariance matrix estimator for the linear regression model. AStA Advances in Statistical Analysis, 95(2), 129-146. doi:10.1007/s10182-010-0141-2

Cribari-Neto, F., Souza, T. C., and Vasconcellos, K. L. P. (2007). Inference under heteroskedasticity and leveraged data. Communications in Statistics - Theory and Methods, 36(10), 1877-1888. doi:10.1080/03610920601126589

Li, S., Zhang, N., Zhang, X., and Wang, G. (2016). A new heteroskedasticity-consistent covariance matrix estimator and inference under heteroskedasticity. Journal of Statistical Computation and Simulation, 87(1), 198-210. doi:10.1080/00949655.2016.1198906

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)

tests(result)
#> # A tibble: 3 × 8
#>   term             estimate null_value std_error z_value p_value alpha reject
#>   <chr>               <dbl>      <dbl>     <dbl>   <dbl>   <dbl> <dbl> <lgl> 
#> 1 (Intercept)          833.          0      851.   0.979   0.328  0.05 FALSE 
#> 2 income_scaled      -1834.          0     2309.  -0.794   0.427  0.05 FALSE 
#> 3 income_scaled_sq    1587.          0     1547.   1.03    0.305  0.05 FALSE 
tests(result, parm = "income_scaled_sq")
#> # A tibble: 1 × 8
#>   term             estimate null_value std_error z_value p_value alpha reject
#>   <chr>               <dbl>      <dbl>     <dbl>   <dbl>   <dbl> <dbl> <lgl> 
#> 1 income_scaled_sq    1587.          0     1547.    1.03   0.305  0.05 FALSE 
tests(result, alpha = 0.10)
#> # A tibble: 3 × 8
#>   term             estimate null_value std_error z_value p_value alpha reject
#>   <chr>               <dbl>      <dbl>     <dbl>   <dbl>   <dbl> <dbl> <lgl> 
#> 1 (Intercept)          833.          0      851.   0.979   0.328   0.1 FALSE 
#> 2 income_scaled      -1834.          0     2309.  -0.794   0.427   0.1 FALSE 
#> 3 income_scaled_sq    1587.          0     1547.   1.03    0.305   0.1 FALSE