Homoscedasticity and heteroscedasticity

Consider the linear regression equation ${\displaystyle y_{i}=x_{i}\beta _{i}+\epsilon _{i},\ i=1,\ldots ,N,}$ where the dependent random variable ${\displaystyle y_{i}}$ equals the deterministic variable ${\displaystyle x_{i}}$ times coefficient ${\displaystyle \beta }$ plus a random disturbance term ${\displaystyle \epsilon _{i}}$ that has mean zero. The disturbances are homoscedastic if the variance of ${\displaystyle \epsilon _{i}}$ is a constant ${\displaystyle \sigma ^{2}}$; otherwise, they are heteroscedastic. In particular, the disturbances are heteroscedastic if the variance of ${\displaystyle \epsilon _{i}}$ depends on i or on the value of ${\displaystyle x_{i}}$. One way they might be heteroscedastic is if ${\displaystyle \sigma _{i}^{2}=x_{i}\sigma ^{2}}$ (an example of a scedastic function), so the variance is proportional to the value of x.

More generally, if the variance-covariance matrix of disturbance ${\displaystyle \epsilon _{i}}$ across i has a nonconstant diagonal, the disturbance is heteroscedastic.[8] The matrices below are covariances when there are just three observations across time. The disturbance in matrix A is homoscedastic; this is the simple case where OLS is the best linear unbiased estimator. The disturbances in matrices B and C are heteroscedastic. In matrix B, the variance is time-varying, increasing steadily across time; in matrix C, the variance depends on the value of x. The disturbance in matrix D is homoscedastic because the diagonal variances are constant, even though the off-diagonal covariances are non-zero and ordinary least squares is inefficient for a different reason: serial correlation.

${\displaystyle {\begin{aligned}A&=\sigma ^{2}{\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\\\end{bmatrix}}&B&=\sigma ^{2}{\begin{bmatrix}1&0&0\\0&2&0\\0&0&3\\\end{bmatrix}}&C&=\sigma ^{2}{\begin{bmatrix}x_{1}&0&0\\0&x_{2}&0\\0&0&x_{3}\\\end{bmatrix}}&D&=\sigma ^{2}{\begin{bmatrix}1&\rho &\rho ^{2}\\\rho &1&\rho \\\rho ^{2}&\rho &1\\\end{bmatrix}}\end{aligned}}}$

Heteroscedasticity often occurs when there is a large difference among the sizes of the observations.

• A classic example of heteroscedasticity is that of income versus expenditure on meals. As one's income increases, the variability of food consumption will increase. A poorer person will spend a rather constant amount by always eating inexpensive food; a wealthier person may occasionally buy inexpensive food and at other times eat expensive meals. Those with higher incomes display a greater variability of food consumption.
• Imagine you are watching a rocket take off nearby and measuring the distance it has travelled once each second. In the first couple of seconds your measurements may be accurate to the nearest centimeter, say. However, 5 minutes later as the rocket recedes into space, the accuracy of your measurements may only be good to 100 m, because of the increased distance, atmospheric distortion and a variety of other factors. The data you collect would exhibit heteroscedasticity.

One of the assumptions of the classical linear regression model is that there is no heteroscedasticity. Breaking this assumption means that the Gauss–Markov theorem does not apply, meaning that OLS estimators are not the Best Linear Unbiased Estimators (BLUE) and their variance is not the lowest of all other unbiased estimators. Heteroscedasticity does not cause ordinary least squares coefficient estimates to be biased, although it can cause ordinary least squares estimates of the variance (and, thus, standard errors) of the coefficients to be biased, possibly above or below the true of population variance. Thus, regression analysis using heteroscedastic data will still provide an unbiased estimate for the relationship between the predictor variable and the outcome, but standard errors and therefore inferences obtained from data analysis are suspect. Biased standard errors lead to biased inference, so results of hypothesis tests are possibly wrong. For example, if OLS is performed on a heteroscedastic data set, yielding biased standard error estimation, a researcher might fail to reject a null hypothesis at a given significance level, when that null hypothesis was actually uncharacteristic of the actual population (making a type II error).

Under certain assumptions, the OLS estimator has a normal asymptotic distribution when properly normalized and centered (even when the data does not come from a normal distribution). This result is used to justify using a normal distribution, or a chi square distribution (depending on how the test statistic is calculated), when conducting a hypothesis test. This holds even under heteroscedasticity. More precisely, the OLS estimator in the presence of heteroscedasticity is asymptotically normal, when properly normalized and centered, with a variance-covariance matrix that differs from the case of homoscedasticity. In 1980, White proposed a consistent estimator for the variance-covariance matrix of the asymptotic distribution of the OLS estimator.[2] This validates the use of hypothesis testing using OLS estimators and White's variance-covariance estimator under heteroscedasticity.

Heteroscedasticity is also a major practical issue encountered in ANOVA problems.[9] The F test can still be used in some circumstances.[10]

However, it has been said that students in econometrics should not overreact to heteroscedasticity.[3] One author wrote, "unequal error variance is worth correcting only when the problem is severe."[11] In addition, another word of caution was in the form, "heteroscedasticity has never been a reason to throw out an otherwise good model."[3][12] With the advent of heteroscedasticity-consistent standard errors allowing for inference without specifying the conditional second moment of error term, testing conditional homoscedasticity is not as important as in the past.

For any non-linear model (for instance Logit and Probit models), however, heteroscedasticity has more severe consequences: the maximum likelihood estimates (MLE) of the parameters will be biased, as well as inconsistent (unless the likelihood function is modified to correctly take into account the precise form of heteroscedasticity).[13] Yet, in the context of binary choice models (Logit or Probit), heteroscedasticity will only result in a positive scaling effect on the asymptotic mean of the misspecified MLE (i.e. the model that ignores heteroscedasticity).[14] As a result, the predictions which are based on the misspecified MLE will remain correct. In addition, the misspecified Probit and Logit MLE will be asymptotically normally distributed which allows performing the usual significance tests (with the appropriate variance-covariance matrix). However, regarding the general hypothesis testing, as pointed out by Greene, “simply computing a robust covariance matrix for an otherwise inconsistent estimator does not give it redemption. Consequently, the virtue of a robust covariance matrix in this setting is unclear.”[15]

There are five common corrections for heteroscedasticity. They are:

• View logarithmized data. Non-logarithmized series that are growing exponentially often appear to have increasing variability as the series rises over time. The variability in percentage terms may, however, be rather stable.
• Use a different specification for the model (different X variables, or perhaps non-linear transformations of the X variables).
• Apply a weighted least squares estimation method, in which OLS is applied to transformed or weighted values of X and Y. The weights vary over observations, usually depending on the changing error variances. In one variation the weights are directly related to the magnitude of the dependent variable, and this corresponds to least squares percentage regression.[16]
• Heteroscedasticity-consistent standard errors (HCSE), while still biased, improve upon OLS estimates.[2] HCSE is a consistent estimator of standard errors in regression models with heteroscedasticity. This method corrects for heteroscedasticity without altering the values of the coefficients. This method may be superior to regular OLS because if heteroscedasticity is present it corrects for it, however, if the data is homoscedastic, the standard errors are equivalent to conventional standard errors estimated by OLS. Several modifications of the White method of computing heteroscedasticity-consistent standard errors have been proposed as corrections with superior finite sample properties.
• Use MINQUE or even the customary estimators ${\textstyle s_{i}^{2}=(n_{i}-1)^{-1}\sum _{j}\left(y_{ij}-{\bar {y}}_{i}\right)^{2}}$ (for ${\displaystyle i=1,2,...,k}$ independent samples with ${\displaystyle j=1,2,...,n_{i}}$ observations each), whose efficiency losses are not substantial when the number of observations per sample is large (${\displaystyle n_{i}>5}$), especially for small number of independent samples.[17]

Absolute value of residuals for simulated first order heteroscedastic data

Residuals can be tested for homoscedasticity using the Breusch–Pagan test,[18] which performs an auxiliary regression of the squared residuals on the independent variables. From this auxiliary regression, the explained sum of squares is retained, divided by two, and then becomes the test statistic for a chi-squared distribution with the degrees of freedom equal to the number of independent variables.[19] The null hypothesis of this chi-squared test is homoscedasticity, and the alternative hypothesis would indicate heteroscedasticity. Since the Breusch–Pagan test is sensitive to departures from normality or small sample sizes, the Koenker–Bassett or 'generalized Breusch–Pagan' test is commonly used instead.[20] From the auxiliary regression, it retains the R-squared value which is then multiplied by the sample size, and then becomes the test statistic for a chi-squared distribution (and uses the same degrees of freedom). Although it is not necessary for the Koenker–Bassett test, the Breusch–Pagan test requires that the squared residuals also be divided by the residual sum of squares divided by the sample size.[20] Testing for groupwise heteroscedasticity can be done with the Goldfeld–Quandt test.[21]

Although tests for heteroscedasticity between groups can formally be considered as a special case of testing within regression models, some tests have structures specific to this case.

• Heterogeneity
• Spherical error

Most statistics textbooks will include at least some material on homoscedasticity and heteroscedasticity. Some examples are:

• Asteriou, Dimitros; Hall, Stephen G. (2011). Applied Econometrics (Second ed.). Palgrave MacMillan. pp. 109–147. ISBN 978-0-230-27182-1.
• Davidson, Russell; MacKinnon, James G. (1993). Estimation and Inference in Econometrics. New York: Oxford University Press. pp. 547–582. ISBN 978-0-19-506011-9.
• Dougherty, Christopher (2011). Introduction to Econometrics. New York: Oxford University Press. pp. 280–299. ISBN 978-0-19-956708-9.
• Gujarati, Damodar N.; Porter, Dawn C. (2009). Basic Econometrics (Fifth ed.). New York: McGraw-Hill Irwin. pp. 365–411. ISBN 978-0-07-337577-9.
• Kmenta, Jan (1986). Elements of Econometrics (Second ed.). New York: Macmillan. pp. 269–298. ISBN 978-0-02-365070-3.
• Maddala, G. S.; Lahiri, Kajal (2009). Introduction to Econometrics (Fourth ed.). New York: Wiley. pp. 211–238. ISBN 978-0-470-01512-4.

• Econometrics lecture (topic: heteroscedasticity) on YouTube by Mark Thoma