The practical testing recipe remains almost the same. Suppose the null hypothesis is H zero: beta j equals beta j comma zero. Beta j is the unknown population coefficient, and beta j comma zero is the particular value asserted by the null, often zero. The alternative for the test on this slide is two-sided, so it allows beta j to be either below or above the null value.
First calculate t sub j. Its numerator is beta j hat minus beta j comma zero, the estimated distance from the null. Its denominator is the estimated standard error of beta j hat, which expresses that distance in units of estimated sampling uncertainty. A positive statistic means the estimate is above the null value, a negative statistic means it is below, and magnitude measures how many standard errors separate them. Although we continue to call this a t-statistic, under the large-sample argument its limiting reference distribution is standard normal.
For a significance level alpha, a two-sided test allocates alpha over both tails, alpha over two in the lower tail and alpha over two in the upper tail. The positive critical value is therefore the one-minus-alpha-over-two quantile of the chosen reference distribution. Taking the absolute value of t j folds the two tails together. Reject H zero when that absolute value exceeds the critical value. At alpha equal to zero point zero five with a standard-normal reference, for example, the cutoff is approximately one point nine six. Equivalently, reject when the two-sided p-value is below alpha.
The question at the bottom is exactly right: if asymptotic theory gives a standard normal limit, should the critical value come from normal zero one rather than a t distribution? In principle, yes. Exact finite-sample theory used a t critical value because sigma was estimated under normal errors; the large-sample theory yields a standard normal limit even after replacing unknown quantities consistently. In practice regression software commonly reports t-based values using n minus k minus one degrees of freedom. Flip to the next tab to see why that choice makes almost no difference once the residual degrees of freedom are large enough for the asymptotic approximation itself to be credible.
It turns out,
For a two-sided test of H_0:\beta_j=\beta_{j,0}, you can proceed much as before:
Calculate t_j=(\widehat{\beta}_j-\beta_{j,0})/\widehat{\operatorname{se}}(\widehat{\beta}_j).
Reject at significance level \alpha if |t_j| exceeds the 1-\alpha/2 critical value.
But,
Shouldn’t we use N(0,1) to find the critical value?
The code constructs a direct density comparison. Seq creates one thousand evenly spaced x values from negative three to positive three. D-norm evaluates the standard-normal density at those values. Each d-t call evaluates a t density at the same x values, with degrees of freedom set to two, ten, thirty, or fifty. Those degrees of freedom would correspond to n minus k minus one in the regression setting.
Data table combines the common x grid and the five density vectors. The quoted column names are the labels that will appear in the legend: N zero one for the standard normal and t with its relevant degrees of freedom for each t curve. The pipe passes that wide table to melt. Id variable equal to x keeps the horizontal coordinate fixed, while the five density columns are stacked into a variable column identifying the distribution and a value column holding its density.
G-g-plot uses this long table. Geom line maps x to the horizontal axis, value to the vertical axis, and variable to color, producing one colored curve per distribution. Scale color discrete removes the legend title by setting name to an empty string, and guide legend with n-row equal to two arranges the five entries across two rows. Theme lecture applies the deck style, and the last theme call puts the legend below the plot.
On the graph, the horizontal axis is a possible value of the standardized test statistic from negative three through three. The vertical axis is probability density. All curves are symmetric around zero and have total area one. The standard normal has the highest center among the displayed curves. At only two degrees of freedom, the t curve is lower and flatter near zero and allocates more probability to values far from zero. Those heavier tails extend beyond the plotted range as well. This is why a small-degrees-of-freedom t test uses a much larger critical value than a normal test.
As degrees of freedom increase, the uncertainty from estimating the scale becomes less important and the t distribution approaches standard normal. At ten degrees of freedom a difference remains visible. By thirty the curves are close, and by fifty they are difficult to separate by eye. The same convergence applies to critical values: for a five-percent two-sided test, the positive cutoff moves toward the standard-normal value of about one point nine six as the degrees of freedom grow.
This answers the previous tab’s practical question. Asymptotic theory formally calls for a standard-normal critical value. Software may continue to use the corresponding t value with n minus k minus one degrees of freedom. In a sample large enough to make the asymptotic normal approximation persuasive, those two reference distributions and their p-values are usually very similar. You can therefore use the p-values reported by standard regression software, while remembering that the real large-sample justification comes from asymptotic normality, not from exact normal errors.
The asymptotic theory calls for a standard-normal critical value. Because t_{n-k-1} approaches N(0,1) as its degrees of freedom grow, using the corresponding t critical value makes very little difference in a large sample.
End with the warning in red. Large-sample theory lets us remove the normality assumption, MLR dot 6, but it does not automatically validate the default OLS variance estimator when MLR dot 5 fails. Homoskedasticity means that the conditional variance of u given all regressors is the same constant sigma squared for every observation. Heteroskedasticity means that conditional variance changes with the regressors or across observations.
Under zero conditional mean and the other consistency conditions, heteroskedasticity does not by itself make the OLS coefficient estimates inconsistent. Beta hat can still converge to beta. What changes is the variance of that limiting distribution. The default homoskedastic OLS formula treats every observation as if it shared one common error variance. When the true variances differ, that is the wrong formula, and its estimated standard errors need not converge to the correct sampling standard deviations. The direction of the error is not fixed: default standard errors can be too small or too large depending on the pattern of heteroskedasticity.
This problem does not fade as n grows. More observations can make beta hat concentrate, but they also make an inconsistent variance formula converge more precisely to the wrong variance quantity. The resulting t-statistics, p-values, and confidence intervals can remain invalid. Asymptotic normality is useful only when the statistic is divided by a standard error that consistently estimates the appropriate asymptotic variance.
With independent observations, use heteroskedasticity-robust, or sandwich, standard errors. They keep the same OLS coefficient estimates but replace the homoskedastic variance formula with one that allows observation-specific squared residuals to contribute differently. If observations may be dependent within groups, such as repeated years for the same county or students within the same school, use standard errors clustered at the level where dependence can occur. Clustering allows arbitrary correlation within a cluster while relying on independence across clusters, so the clustering level must match the sampling and assignment structure.
Robust or clustered standard errors repair an uncertainty formula; they do not repair an endogenous regressor, omitted-variable bias, or a misspecified conditional mean. Connect this warning to the earlier inconsistency simulation. Endogeneity made the coefficient converge to one point two instead of one, and no standard-error choice could restore consistency. Heteroskedasticity can leave the coefficient centered correctly while making its default reported uncertainty wrong. In applied work you must address both questions: do the identifying assumptions make the coefficient target the right parameter, and does the variance estimator reflect the dependence and heteroskedasticity in the data?
The consistency of the default estimator of Var(\widehat{\beta}) does require the homoskedasticity assumption (MLR.5).
In other words, using the default variance estimator under heteroskedasticity remains a problem even when the sample size is large.
Use heteroskedasticity-robust standard errors with independent observations. When observations may be dependent within groups, use standard errors clustered at the appropriate level.