Studentized OLS Contrasts and Limiting t Laws

From an OLS Limit to a Studentized Contrast

An asymptotic distribution for an estimator is useful only if its scale can also be estimated. For a scalar contrast of OLS coefficients, the central question is whether the estimated standard error tracks the contrast’s sampling variability closely enough that dividing by it produces a standard normal limit. The answer is yes under regularity conditions and consistent covariance estimation. It does not, in general, mean that the resulting statistic has an exact Student t distribution at a finite sample size.

Consider observations \((Y_i,X_i)\), \(i=1,\ldots,n\), where \(X_i\) is a \(k\)-dimensional regressor vector and \(Y_i=X_i'\beta+u_i\). Assume the observations are IID, \(E[X_i u_i]=0\), \(Q=E[X_iX_i']\) is finite and positive definite, and the score \(X_i u_i\) has finite second moments. Write \(\Omega=E[u_i^2X_iX_i']\), and assume it is finite. These conditions give the usual asymptotic linear representation and normal limit for OLS; the contrast-specific variance must also be positive.

The matrix \(\Omega\) allows the conditional variance of \(u_i\) to depend on \(X_i\). Under homoskedasticity, \(E[u_i^2\mid X_i]=\sigma^2\), it simplifies to \(\Omega=\sigma^2Q\), so \(V_c=\sigma^2c'Q^{-1}c\). In general, the contrast’s variance depends on both the regressor distribution and the way error variability changes with the regressors.

Studentization Gives a Standard Normal Limit

Let \(\hat V_n\) estimate the covariance matrix of \(\hat\beta\), so its entries are on the \(1/n\) scale. For example, the heteroskedasticity-robust sandwich estimator is

where \(X\) is the matrix with rows \(X_i'\), and \(\hat u_i=Y_i-X_i'\hat\beta\). Suppose the OLS limit holds and \(n\hat V_n\xrightarrow{p}Q^{-1}\Omega Q^{-1}\). This covariance consistency is the key requirement: it ensures that the estimated variance for the particular contrast converges to its correct value.

Proof. Apply the continuous linear transformation \(c'\) to the assumed OLS limit. This gives \(\sqrt{n}(c'\hat\beta-c'\beta)\xrightarrow{d}N(0,V_c)\). Covariance consistency and continuity of the scalar map \(A\mapsto c'Ac\) imply \(nc'\hat V_nc\xrightarrow{p}V_c\). Since \(V_c>0\), taking square roots and reciprocals gives \(\sqrt{nc'\hat V_nc}\xrightarrow{p}\sqrt{V_c}\). Therefore, \[ \frac{c'\hat\beta-c'\beta}{\sqrt{c'\hat V_nc}} = \frac{\sqrt{n}(c'\hat\beta-c'\beta)}{\sqrt{nc'\hat V_nc}} \xrightarrow{d}\frac{Z}{\sqrt{V_c}}, \] where \(Z\sim N(0,V_c)\). The final ratio has distribution \(N(0,1)\), by Slutsky’s theorem. \(\square\)

When the Statistic Has an Exact t Distribution

The standard-normal limit just derived is an asymptotic statement. A different, stronger set of assumptions gives an exact finite-sample Student t law. Conditional on a fixed design matrix \(X\) of full column rank \(k\), assume \(u\mid X\sim N(0,\sigma^2 I_n)\), with \(\sigma^2>0\), and \(n>k\). Define the residual variance estimate \(s^2=\hat u'\hat u/(n-k)\), and use the conventional covariance estimate \(s^2(X'X)^{-1}\).

Proof. Given \(X\), the OLS estimation error is \((X'X)^{-1}X'u\). Consequently, \[ \frac{c'(\hat\beta-\beta)}{\sigma\sqrt{c'(X'X)^{-1}c}}\sim N(0,1). \] Let \(P_X=X(X'X)^{-1}X'\) be the projection onto the column space of \(X\), and let \(M_X=I_n-P_X\). The residual vector is \(M_Xu\). Because \(M_X\) is a symmetric idempotent matrix of rank \(n-k\), normality gives \(\hat u'\hat u/\sigma^2=u'M_Xu/\sigma^2\sim\chi^2_{n-k}\). The numerator is a linear function of the projection \(P_Xu\), while the residual sum of squares depends on \(M_Xu\). These projections are independent: their covariance is proportional to \(P_XM_X=0\), and jointly normal uncorrelated vectors are independent. Hence the standardized numerator is independent of the chi-square denominator. Dividing a standard normal by the square root of an independent \(\chi^2_{n-k}/(n-k)\) variable gives \(t_{n-k}\), as claimed. \(\square\)

Asymptotic t Statistics and Common Pitfalls

In applications, a studentized contrast is often called a t statistic because it is an estimate divided by an estimated standard error. That name alone does not determine its distribution. With a consistent heteroskedasticity-robust standard error, the statistic above converges to \(N(0,1)\), but it need not have any exact Student t distribution in finite samples. Even under homoskedasticity, exact finite-sample t inference relies on Gaussian errors and the conventional residual variance estimate; asymptotic normality requires less.

The distinction matters when selecting reference quantiles. Under the asymptotic result, a two-sided large-sample procedure uses standard-normal quantiles. Under the exact conditional result, the reference distribution is \(t_{n-k}\). As \(n-k\) increases, the \(t_{n-k}\) distribution approaches the standard normal, but that approximation does not make the finite-sample robust statistic exactly t-distributed.

• Pitfall: treating every estimate-over-standard-error ratio as exactly t. The incorrect reasoning is that estimating a standard error automatically creates a Student t distribution. The correction is that the exact t result requires the Gaussian homoskedastic projection argument; consistent standard-error estimation alone yields an asymptotic normal limit.
• Pitfall: using a covariance estimate with the wrong scale. The matrix \(\hat V_n\) estimates the covariance of \(\hat\beta\), which is order \(1/n\). The limit covariance is instead \(n\operatorname{Var}(\hat\beta)\), estimated by \(n\hat V_n\). Keeping these scales distinct prevents an erroneous factor of \(\sqrt n\).
• Pitfall: ignoring a zero contrast variance. If \(V_c=0\), the stated standardization is not justified. The theorem explicitly requires \(V_c>0\).

The practical point is precise: a scalar contrast needs both a valid OLS limit and a covariance estimate consistent for its variance. Once those conditions hold, studentization supplies a standard-normal asymptotic reference. Exact finite-sample t inference is a separate conclusion supported by stronger distributional assumptions.