Two definitions, and the whole lecture turns on the difference between them. Start with the blue callout. Homoskedasticity says the variance of the error, conditional on x, equals sigma squared. Here, u is the unobserved error, x represents the explanatory variables we condition on, variance measures the spread of u around its conditional mean, and sigma squared is one constant positive number. The blue phrase underneath is the key: wherever you look in the x distribution, the conditional spread of the errors is the same.
Now compare the red callout. Under heteroskedasticity, that conditional variance is f of x. The letter f does not impose a particular formula. It simply says the error variance can change with x. For example, income prediction errors may spread out as education or experience rises. Heteroskedasticity concerns the variance, not the conditional mean. We can still have expected u given x equal to zero while the size of typical positive and negative errors changes.
Read the three bullets at the bottom for why this matters now. Constant conditional variance is MLR five, and every standard error, t-test, F-test, p-value, and confidence interval in lecture 05 used it. Normality gave it to us there because independence of u from x means the entire conditional distribution of u, including its variance, does not change with x. This lecture removes that protection. We will keep asking two separate questions: whether OLS coefficients survive, and whether the usual testing machinery survives.
Read the difference: under homoskedasticity the spread of the error is the same number whatever x is. Under heteroskedasticity it is a function of x.
Look at the two panels. The horizontal axis is x, running from zero to one. The vertical axis is y, and each dot is one of one thousand simulated observations. Both panels were generated from the same conditional mean, one plus three times x, so the true intercept is one and the true slope is three. The blue line in each panel is the fitted OLS line through its cloud.
On the left, labeled Homoskedastic, the cloud has roughly the same vertical thickness at low, middle, and high values of x. The code generates those errors with a constant standard deviation of one point five, which is three times zero point five. On the right, labeled Heteroskedastic, the error standard deviation is three times x. Near x equals zero, the errors are therefore close to zero and the points hug the line. As x rises toward one, their standard deviation rises toward three, so the cloud opens into a fan. The upper and lower edges spread apart even though the middle stays on the same line.
That last comparison is the main lesson. Heteroskedasticity does not automatically tilt the fitted line or move the true coefficients, because the positive and negative errors can still average to zero at every x. It changes how much information observations at different x values carry, and therefore how uncertain the fitted line is. That is why the repair later in this lecture changes the estimated standard errors, not the coefficient estimates.
This slide states the problem we are going to solve. Suppose reality is heteroskedastic, so the conditional error variance changes with x, but you proceed as if the errors were homoskedastic. There are two consequences to examine, and you must keep them separate.
The first bullet asks about estimating beta j itself. Beta hat j is an OLS coefficient estimate, such as the estimated slope on x. We want to know whether heteroskedasticity changes its expected value, creates bias, or otherwise invalidates OLS as a way to estimate the conditional mean relationship. That is a question about the center of the estimator’s sampling distribution.
The second bullet asks about estimating the variance of beta hat j. That variance describes how beta hat j would move across repeated samples. Its square root is the standard error, which is the denominator of a t-statistic and therefore feeds directly into p-values and confidence intervals. This is a question about the spread of the sampling distribution.
People often hear that an OLS assumption has failed and conclude that everything is unusable. Here the two questions receive different answers. The coefficients remain centered correctly under the assumptions used for unbiasedness, but the default measure of their uncertainty does not. That distinction is why the solution later is to keep OLS and replace the variance estimator.
What are the consequences of assuming the error is homoskedastic when it is heteroskedastic in reality?
Take the first of the two central questions on its own. The screen asks whether OLS coefficient estimators remain unbiased when the error is heteroskedastic. Unbiased means that, over repeated random samples generated by the same process, the average value of beta hat j equals the true beta j.
Think back to the unbiasedness proof from lecture 01-3. In the simple regression case, beta hat one can be written as beta one plus a weighted sum of the errors. Conditional on the observed x values, the weights are fixed. If expected u given x is zero, then the expected value of that weighted error term is zero, leaving expected beta hat one given x equal to beta one. The multiple-regression argument has the same logic.
Now ask exactly where a constant error variance entered that proof. It did not. We needed the model to be linear in its parameters, random sampling, no perfect collinearity so the coefficient exists, and zero conditional mean so x is not systematically related to the error. Heteroskedasticity changes the size of the errors across x, but it need not change their conditional mean. Decide on that basis before you open the Answer tab.
Are OLS estimators unbiased when error is heteroskedastic?
Yes. The OLS coefficient estimator is still unbiased in the presence of heteroskedasticity, provided the assumptions that actually deliver unbiasedness, especially expected u given x equals zero, continue to hold. The sentence on screen gives the decisive reason: the proof of unbiasedness never used the homoskedasticity assumption.
Be precise about what this answer does and does not claim. It says heteroskedasticity by itself does not move the center of the sampling distribution of beta hat j away from beta j. It does not say every regression with heteroskedastic errors is automatically unbiased. If x is correlated with an omitted variable in u, zero conditional mean fails and OLS can still be biased. That is a different problem.
MLR five entered only when we derived the familiar variance formula for beta hat j. So keep every OLS coefficient estimate exactly as it is. The fitted values and residuals are also obtained in the ordinary way. The repair in this lecture concerns the second central question: how to estimate the variance and standard error of those unchanged coefficients. Flip forward with that distinction firmly in mind.
Yes. We do not need to use the homoskedasticity assumption to prove that the OLS estimator is unbiased.
This tab reviews the familiar variance formula under homoskedasticity. On the first line, the variance of beta hat j equals sigma squared divided by S S T x times one minus R j squared. Sigma squared is the common conditional variance of the error. S S T x is the total sum of squares for the explanatory variable x j, so more sample variation in x j gives more information and lowers the coefficient variance. R j squared comes from the auxiliary regression of x j on all the other explanatory variables. When x j is highly explained by those variables, one minus R j squared is small, the denominator shrinks, and multicollinearity raises the variance of beta hat j.
Sigma squared is unknown, so the second line replaces it with sigma hat squared. Look at its definition: add the squared OLS residuals u hat i from observation one through N, then divide by N minus k minus one. N is the sample size, k is the number of slope regressors, and the additional one accounts for the intercept. Those lost degrees of freedom correct for estimating k plus one coefficients before measuring the remaining error variation.
The blue phrase in the callout is the operational point. R and other software use this homoskedastic formula by default. The reported standard error is the square root of the resulting estimated variance. Every default t-statistic, p-value, and confidence interval then inherits this assumption. If the errors are actually heteroskedastic, sigma squared is not one common number, so this particular shortcut is no longer the correct variance formula.
We learned that when the homoskedasticity assumption holds, then,
Var(\widehat{\beta}_j) = \frac{\sigma^2}{SST_x(1-R^2_j)}
We used the following as the estimator of Var(\widehat{\beta}_j)
\frac{\widehat{\sigma}^2}{SST_x(1-R^2_j)} where \widehat{\sigma}^2 = \frac{\sum_{i=1}^{N} \widehat{u}_i^2}{N-k-1}
But, under heteroskedasticity,
Var(\widehat{\beta}_j) \ne \frac{\sigma^2}{SST_x(1-R^2_j)}
Under heteroskedasticity, the line above this question tells you that the true variance of beta hat j is not sigma squared divided by S S T x times one minus R j squared. There is no single sigma squared that describes every observation’s conditional error variance. The question now asks whether the estimator built from that formula might nevertheless work on average.
Read the expression from left to right. Var hat of beta hat j, subscript default, denotes the variance estimate produced by the conventional homoskedastic formula. The triple-bar definition says that this object is sigma hat squared divided by S S T x times one minus R j squared. The outer expectation asks for the average value of that default estimator across repeated samples. On the right is the actual sampling variance of beta hat j.
So the equality asks whether the default estimator is unbiased for the true coefficient variance when errors are heteroskedastic. This is not the same as asking whether beta hat j itself is unbiased. We already answered that question yes. Here we are testing whether the estimated spread around that correctly centered coefficient is itself correct. Answer before opening the next tab, because all of the hypothesis-testing consequences follow from this equality failing.
Is E[\widehat{Var(\widehat{\beta}_j)}_{default}] \equiv E\Big[\frac{\widehat{\sigma}^2}{SST_x(1-R^2_j)}\Big]=Var(\widehat{\beta}_j) under heteroskedasticity?
No. Once the conditional variance of u changes with x, the default homoskedastic estimator is not generally unbiased for the true variance of beta hat j. It treats every observation as though it had the same error variance, even though observations at different x values can contribute very different amounts of uncertainty to the estimated slope.
Do not accidentally transfer this answer to the coefficient itself. Beta hat j can remain unbiased because expected u given x is still zero, while the estimated variance of beta hat j is biased because the squared size of u changes with x. A correct center and an incorrectly measured spread can coexist.
Also notice what the one-word answer does not tell you. It does not say that the default variance is always too small or always too large. The direction depends on where the high-variance observations lie and on the leverage those observations have for estimating a particular coefficient. In the first simulation it will be too small and the test will over-reject, but that is one design, not a universal direction. What is universal is that you cannot trust the default standard error, t-statistic, p-value, or confidence interval under arbitrary heteroskedasticity. The Consequences tab now translates that failure into test behavior.
No.
So, what are the consequences of using \widehat{Var(\widehat{\beta}_j)}=\frac{\widehat{\sigma}^2}{SST_x(1-R^2_j)} under heteroskedasticity?
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Your hypothesis tests will generally have the wrong rejection rate.
The blue sentence above this question gives the consequence: hypothesis tests will generally have the wrong rejection rate. That follows as a chain. A biased variance estimate gives a wrong standard error after taking its square root. The standard error sits in the denominator of the t-statistic, so the t-statistic is mis-scaled. Its reference distribution and critical value no longer deliver the rejection probability you intended. P-values and confidence intervals are wrong for the same reason.
The question asks what people mean when they say hypothesis testing is biased. Be careful with the vocabulary. A test is a decision rule, not a coefficient estimate, so its bias is not described by an expected estimate being above or below a parameter. Instead, evaluate its behavior over repeated samples when the null hypothesis is true.
At a five percent significance level, the test is designed to reject a true null five percent of the time. If its actual rejection probability is higher, it over-rejects and produces too many false positives. If the probability is lower, it under-rejects and has an overly conservative false-positive rate. Think in those repeated-sampling terms, then open the Answer tab.
What does it mean to have hypothesis testing biased?
Biased testing here means that you reject a null hypothesis more often or less often than the chosen significance level intends. At a five percent level, the repeated-sampling promise is conditional: among hypothetical repetitions in which H zero is true, about five percent should end in rejection. Such a rejection is a Type I error.
If the standard error is underestimated, it makes the absolute t-statistic too large because the denominator is too small. More statistics cross the critical values, p-values become too small, confidence intervals become too narrow, and the test over-rejects. You then announce effects that are statistically significant more often than the nominal level permits. If the standard error is overestimated, the chain reverses: t-statistics are too small, p-values are too large, confidence intervals are too wide, and the test under-rejects.
The answer on screen therefore says roughly that testing bias is rejection too often or too rarely relative to your intention. It is not enough that an individual decision happened to be right or wrong. We judge the procedure by its long-run rejection rate. The next slide constructs data where the null is true in every repetition and measures that rate directly.
Roughly speaking, it means that you over-reject/under-reject the hypothesis than you intend to.