Simple Regressionintermediate

Variance of the OLS Estimators

Adding the homoskedasticity assumption SLR.5, $\mathrm{Var}(u \mid x) = \sigma^2$ (the error variance is constant across $x$ ), gives the sampling variance of the slope as $\mathrm{Var}(\hat{\beta}_1) = \frac{\sigma^2}{\sum (x_i - \bar{x})^2}$ . The variance shrinks with a smaller error variance $\sigma^2$ and with more spread in $x$ , and it inflates when $\sigma^2$ is large or $x$ barely varies. Because $\sigma^2$ is unknown, it is estimated by $\hat{\sigma}^2 = SSR / (n - 2)$ , whose square root $\hat{\sigma}$ is the standard error of the regression, and the reported standard error of $\hat{\beta}_1$ is the square root of the estimated variance.

Why it matters

How precise the slope is depends on three things, the noise in $y$ , the spread of $x$ , and the sample size. A noisier outcome (large $\sigma^2$ ) makes the slope harder to pin down. A wider spread of $x$ acts like a longer lever, anchoring the line more firmly, so more variation in $x$ lowers the variance. More data points in the denominator help too. Standard errors translate this variance into the everyday measure of how much you should trust the coefficient.

Formulas

Variance of the slope (under SLR.5)

\mathrm{Var}(\hat{\beta}_1) = \frac{\sigma^2}{\sum_{i=1}^{n} (x_i - \bar{x})^2}

Holds under SLR.1 to SLR.5. Falls as

\sigma^2

falls or as the total variation in

x

rises.

Estimated error variance

\hat{\sigma}^2 = \frac{SSR}{n - 2} = \frac{\sum \hat{u}_i^2}{n - 2}

Unbiased estimator of

\sigma^2

. The divisor

n - 2

corrects for the two parameters

\hat{\beta}_0

and

\hat{\beta}_1

already estimated. Its square root

\hat{\sigma}

is the standard error of the regression.

Standard error of the slope

\mathrm{se}(\hat{\beta}_1) = \frac{\hat{\sigma}}{\sqrt{\sum (x_i - \bar{x})^2}}

The square root of the estimated variance. This is the number Stata reports next to the coefficient.

Worked examples

Scenario

After `regress wage educ`, Stata reports a coefficient on `educ` of 0.54 with a standard error of 0.05, and a "Root MSE" of 3.4.

Solution

The standard error 0.05 is $\sqrt{\mathrm{Var}(\hat{\beta}_1)}$ , computed as $\hat{\sigma}$ divided by the square root of the total variation in `educ`. The Root MSE 3.4 is $\hat{\sigma}$ , the standard error of the regression, estimating the standard deviation of $u$ . If education had been less spread out in the sample, the denominator would shrink and the standard error would rise, making the slope less precise.

NoteStata labels

\hat{\sigma}

as "Root MSE" in the regression header.

Common mistakes

✗The standard error of the regression $\hat{\sigma}$ is the same as the standard error of $\hat{\beta}_1$ . $\hat{\sigma}$ estimates the spread of the errors around the line. The standard error of $\hat{\beta}_1$ measures the precision of the slope estimate and equals $\hat{\sigma}$ divided by $\sqrt{\sum (x_i - \bar{x})^2}$ .
✗The variance formula $\sigma^2 / \sum (x_i - \bar{x})^2$ holds without homoskedasticity. It relies on SLR.5, $\mathrm{Var}(u \mid x) = \sigma^2$ . Under heteroskedasticity this formula is wrong and robust standard errors are needed instead.
✗More variation in $x$ makes estimates worse. The opposite is true. Greater spread in $x$ enlarges the denominator $\sum (x_i - \bar{x})^2$ and lowers $\mathrm{Var}(\hat{\beta}_1)$ , so the slope is estimated more precisely.
✗Homoskedasticity is needed for OLS to be unbiased. Unbiasedness needs only SLR.1 to SLR.4. SLR.5 is required for the simple variance formula and for OLS to be the best linear unbiased estimator, not for unbiasedness itself.

Revision bullets

•SLR.5 homoskedasticity: $\mathrm{Var}(u \mid x) = \sigma^2$ , constant error variance
• $\mathrm{Var}(\hat{\beta}_1) = \frac{\sigma^2}{\sum (x_i - \bar{x})^2}$
•Variance falls with smaller $\sigma^2$ , more spread in $x$ , larger $n$
• $\hat{\sigma}^2 = SSR/(n-2)$ ; $\hat{\sigma}$ is the standard error of the regression (Root MSE)
• $\mathrm{se}(\hat{\beta}_1) = \hat{\sigma} / \sqrt{\sum (x_i - \bar{x})^2}$ , reported by Stata

Quick check

Under SLR.1 to SLR.5, the variance of the OLS slope decreases when:

What does the assumption of homoskedasticity, SLR.5, state?

Connected topics

E(u|x)=0 Unbiasedness Gauss-Markov CLM normality Heteroskedastic Robust SE

Sources

Wooldridge (2019), Ch. 2.5
Wooldridge, Jeffrey M. Introductory Econometrics: A Modern Approach. 7th ed. Cengage Learning, 2019. ISBN 978-1-337-55886-0.
Section 2.5 adds the homoskedasticity assumption SLR.5 and derives the sampling variance of the OLS estimators and the standard error of the regression.
Wooldridge (2019), §2.5 (Theorem 2.2)
Wooldridge, Jeffrey M. Introductory Econometrics: A Modern Approach. 7th ed. Cengage Learning, 2019.
Theorem 2.2 gives $\mathrm{Var}(\hat{\beta}_1) = \sigma^2 / \sum (x_i - \bar{x})^2$ and the unbiased estimator $\hat{\sigma}^2 = SSR/(n-2)$ .

How to cite this page

Dr. Phil's Quant Lab. (2026). Variance of the OLS Estimators. Derivatives Atlas. https://phucnguyenvan.com/concept/efm-ols-variance

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