Heteroscedasticity means unequal scatter. In regression analysis, we talk about heteroscedasticity in the context of the residuals or error term. Specifically, heteroscedasticity is a systematic change in the spread of the residuals over the range of measured values. Heteroscedasticity is a problem because ordinary least squares (OLS) regression assumes that all residuals are drawn from a population that has a constant variance (homoscedasticity).

To satisfy the regression assumptions and be able to trust the results, the residuals should have a constant variance. In this blog post, I show you how to identify heteroscedasticity, explain what produces it, the problems it causes, and work through an example to show you several solutions.

## How to Identify Heteroscedasticity with Residual Plots

Let’s start with how you detect heteroscedasticity because that is easy.

In my post about checking the residual plots, I explain the importance of verifying the OLS linear regression assumptions. You want these plots to display random residuals (no patterns) that are uncorrelated and uniform. Generally speaking, if you see patterns in the residuals, your model has a problem, and you might not be able to trust the results.

Heteroscedasticity produces a distinctive fan or cone shape in residual plots. To check for heteroscedasticity, you need to assess the residuals by fitted value plots specifically. Typically, the telltale pattern for heteroscedasticity is that as the fitted values increases, the variance of the residuals also increases.

You can see an example of this cone shaped pattern in the residuals by fitted value plot below. Note how the vertical range of the residuals increases as the fitted values increases. Later in this post, we’ll return to the model that produces this plot when we try to fix the problem and produce homoscedasticity.

## What Causes Heteroscedasticity?

Heteroscedasticity, also spelled heteroskedasticity, occurs more often in datasets that have a large range between the largest and smallest observed values. While there are numerous reasons why heteroscedasticity can exist, a common explanation is that the error variance changes proportionally with a factor. This factor might be a variable in the model.

In some cases, the variance increases proportionally with this factor but remains constant as a percentage. For instance, a 10% change in a number such as 100 is much smaller than a 10% change in a large number such as 100,000. In this scenario, you expect to see larger residuals associated with higher values. That’s why you need to be careful when working with wide ranges of values!

Because large ranges are associated with this problem, some types of models are more prone to heteroscedasticity.

### Heteroscedasticity in cross-sectional studies

Cross-sectional studies often have very small and large values and, thus, are more likely to have heteroscedasticity. For example, a cross-sectional study that involves the United States can have very low values for Delaware and very high values for California. Similarly, cross-sectional studies of incomes can have a range that extends from poverty to billionaires.

### Heteroscedasticity in time-series models

A time-series model can have heteroscedasticity if the dependent variable changes significantly from the beginning to the end of the series. For example, if we model the sales of DVD players from their first sales in 2000 to the present, the number of units sold will be vastly different. Additionally, if you’re modeling time series data and measurement error changes over time, heteroscedasticity can be present because regression analysis includes measurement error in the error term. For example, if measurement error decreases over time as better methods are introduced, you’d expect the error variance to diminish over time as well.

### Example of heteroscedasticity

Let’s take a look at a classic example of heteroscedasticity. If you model household consumption based on income, you’ll find that the variability in consumption increases as income increases. Lower income households are less variable in absolute terms because they need to focus on necessities and there is less room for different spending habits. Higher income households can purchase a wide variety of luxury items, or not, which results in a broader spread of spending habits.

### Pure versus impure heteroscedasticity

You can categorize heteroscedasticity into two general types.

- Pure heteroscedasticity refers to cases where you specify the correct model and yet you observe non-constant variance in the residual plots.
- Impure heteroscedasticity refers to cases where you incorrectly specify the model, and that causes the non-constant variance. When you leave an important variable out of a model, the omitted effect is absorbed into the error term. If the effect of the omitted variable varies throughout the observed range of data, it can produce the telltale signs of heteroscedasticity in the residual plots.

When you observe heteroscedasticity in the residual plots, it is important to determine whether you have pure or impure heteroscedasticity because the solutions are different. If you have the impure form, you need to identify the important variable(s) that have been left out of the model and refit the model with those variables. For the remainder of this blog post, I talk about the pure form of heteroscedasticity.

**Related post**: How to Specify the Correct Regression Model

The causes for heteroscedasticity vary widely by subject-area. If you detect heteroscedasticity in your model, you’ll need to use your expertise to understand why it occurs. Often, the key is to identify the proportional factor that is associated with the changing variance.

## What Problems Does Heteroscedasticity Cause?

As I mentioned earlier, linear regression assumes that the spread of the residuals is constant across the plot. Anytime that you violate an assumption, there is a chance that you can’t trust the statistical results.

Why fix this problem? There are two big reasons why you want homoscedasticity:

- While heteroscedasticity does not cause bias in the coefficient estimates, it does make them less precise. Lower precision increases the likelihood that the coefficient estimates are further from the correct population value.
- Heteroscedasticity tends to produce p-values that are smaller than they should be. This effect occurs because heteroscedasticity increases the variance of the coefficient estimates but the OLS procedure does not detect this increase. Consequently, OLS calculates the t-values and F-values using an underestimated amount of variance. This problem can lead you to conclude that a model term is statistically significant when it is actually not significant.

**Related post**: How to Interpret Regression Coefficients and P-values

If you see the characteristic fan shape in your residual plots, what should you do? Read on!

## How to Fix Heteroscedasticity

If you can figure out the reason for the heteroscedasticity, you might be able to correct it and improve your model. I’ll show you three common approaches for turning heteroscedasticity into homoscedasticity.

To illustrate how these solutions work, we’ll use an example cross-sectional study to model the number of automobile accidents by the population of towns and cities. These data are fictional, but they correctly illustrate the problem and how to resolve it. You can download the CSV data file to try it yourself: Heteroscedasticity. We’ll use Accident as the dependent variable and Population for the independent variable.

Imagine that we just fit the model and produced the residual plots. Typically, you see heteroscedasticity in the residuals by fitted values plot. So, when we see the plot shown earlier in this post, we know that we have a problem.

Cross-sectional studies have a larger risk of residuals with non-constant variance because of the larger disparity between the largest and smallest values. For our study, imagine the huge range of populations from towns to the major cities!

Generally speaking, you should identify the source of the non-constant variance to resolve the problem. A good place to start is a variable that has a large range.

We’ve detected heteroscedasticity, now what can we do about it? There are various methods for resolving this issue. I’ll cover three methods that I list in my order of preference. My preference is based on minimizing the amount of data manipulation. You might need to try several approaches to see which one works best. These methods are appropriate for pure heteroscedasticity but are not necessarily valid for the impure form.

### Redefining the variables

If your model is a cross-sectional model that includes large differences between the sizes of the observations, you can find different ways to specify the model that reduces the impact of the size differential. To do this, change the model from using the raw measure to using rates and per capita values. Of course, this type of model answers a slightly different kind of question. You’ll need to determine whether this approach is suitable for both your data and what you need to learn.

I prefer this method when it is appropriate because it involves the least amount of tinkering with the original data. You adjust only the specific variables that need to be changed in a manner that often makes sense. Indeed, this practice forces you to think about different ways to specify your model, which frequently improves it beyond just removing heteroscedasticity.

For our original model, we were using population to predict the number of accidents. If you think about it, it isn’t surprising that larger cities have more accidents. That’s not particularly enlightening.

However, we can change the model so that we use population to predict the accident *rate*. This approach discounts the impact of scale and gets to the underlying behavior. Let’s try this with our example data set. I’ll use Accident Rate as the dependent variable and Population as the independent variable. The residual plot is below.

The residuals by fitted value plot looks better. If it weren’t for a few pesky values in the very high range, it would be useable. If this approach had produced homoscedasticity, I would stick with this solution and not use the following methods.

### Weighted regression

Weighted regression is a method that assigns each data point a weight based on the variance of its fitted value. The idea is to give small weights to observations associated with higher variances to shrink their squared residuals. Weighted regression minimizes the sum of the weighted squared residuals. When you use the correct weights, heteroscedasticity is replaced by homoscedasticity.

I prefer this approach somewhat less than redefining the variables. For one thing, weighted regression involves more data manipulation because it applies the weights to all variables. It’s also less intuitive. And, if you skip straight to this, you might miss the opportunity to specify a more meaningful model by redefining the variables.

For our data, we know that higher populations are associated with higher variances. Consequently, we need to assign lower weights to observations of large populations. Finding the theoretically correct weight can be difficult. However, when you can identify a variable that is associated with the changing variance, a common approach is to use the inverse of that variable as the weight. In our case, the Weight column in the dataset equals 1 / Population.

I’ll go back to using Accidents as the dependent variable and Population as the independent variable. However, I’ll tell the software to perform weighted regression and apply the column of weights. The residual plot is below. For weighted regression, it is important to assess the standardized residuals because only that type of residual will show us that weighted regression fixed the heteroscedasticity.

This residual plot looks great! The variance of the residuals is constant across the full range of fitted values. Homoscedasticity!

### Transform the dependent variable

I always save transforming the data for the last resort because it involves the most manipulation. It also makes interpreting the results very difficult because the units of your data are gone. The idea is that you transform your original data into different values that produce good looking residuals. If nothing else works, try a transformation to produce homoscedasticity.

I’ll refit the original model but use a Box-Cox transformation on the dependent variable.

As you can see, the data transformation didn’t produce homoscedasticity in this dataset. That’s good because I didn’t want to use this approach anyway! We’ll stick with the weighted regression model.

Keep in mind that there are many different reasons for heteroscedasticity. Identifying the cause and resolving the problem in order to produce homoscedasticity can require extensive subject-area knowledge. In most cases, remedial actions for severe heteroscedasticity are necessary. However, if your primary goal is to predict the total amount of the dependent variable rather than estimating the specific effects of the independent variables, you might not need to correct non-constant variance.

If you’re learning regression and like the approach I use in my blog, check out my eBook!

Jim Knaub says

Well, let me say that I have learned from Statistics by Jim (Frost), and I highly recommend it. These question and answer participations are a nice additional feature. – Jim K

Jim Knaub says

Richard Hart –

Perhaps I could interject something here.

Your project is not entirely clear to me, but yes, the main feature missed by ignoring heteroscedasticity is the large influence on the estimated variance of the prediction error, which with weighted least squares (WLS) regression will be larger with larger predictions. The OLS predicted values themselves are unbiased, but could be substantially different in practice from the WLS predictions when sample sizes are small.

Here is something I have sent to other people. Please note the spreadsheet, at the end, for finding a better value for the coefficient of heteroscedasticity than 0, which would indicate OLS:

=========================

Regarding heteroscedasticity:

If you might be interested, the following is with regard to the nature of heteroscedasticity, and consideration of its magnitude, for various linear regressions, including polynomials, which might be further extendable:

https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression

Please particularly note

https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity

and

https://www.researchgate.net/publication/333642828_Estimating_the_Coefficient_of_Heteroscedasticity

A spreadsheet tool for estimating or considering a default value for the coefficient of heteroscedasticity, developed for linear regression, is found here (with references):

https://www.researchgate.net/publication/333659087_Tool_for_estimating_coefficient_of_heteroscedasticityxlsx

That leads to the regression weight expression which can be entered as “w” into SAS PROC REG. I assume this is similar for other statistical software. Note that OLS regression is a special case of WLS (weighted least squares) regression, where the coefficient of heteroscedasticity is zero and weights are all equal.

See Brewer, K.R.W.(2002), Combined survey sampling inference: Weighing Basu’s elephants, Arnold: London and Oxford University Press, especially pages 111, and 87, 130, 137, 142, and 203.

========================

I am not certain that your problem is related to predicting totals from a sample, with a previous census as regressor data, but the following is for that case and some cases a little more complicated:

https://www.researchgate.net/publication/319914742_Quasi-Cutoff_Sampling_and_the_Classical_Ratio_Estimator_-_Application_to_Establishment_Surveys_for_Official_Statistics_at_the_US_Energy_Information_Administration_-_Historical_Development

(There I often used a coefficient of heteroscedasticity of 0.5, as described in the links above, which appeared robust to data quality issues often occurring with smaller establishments. Otherwise a larger value would usually have been an improvement.)

I hope this is helpful for you, but as I said, I’m not really certain that you are looking at the same thing. Maybe it is somewhat related though.

Cheers – Jim Knaub

Jim Frost says

Thanks for your thoughtful, detailed reply, Jim Knaub!

Richard, Jim K. is definitely more knowledgeable in this topic than I am! It looks like he gave you some top-notch advice!

Richard Hart says

Hi Jim – What a helpful article. We’re hoping you might be a life-saver for our project. We have a a linear equation that describes a industrial facility’s energy use based on the prior year of operation. It shows heteroscedasticity at the low end of production. One of our reviewers doesn’t like this. We’ve struggled to find a better model but we can’t.

At the bottom of the article you state, “if your primary goal is to predict the total amount of the dependent variable rather than estimating the specific effects of the independent variables, you might not need to correct non-constant variance.” So we’re wondering if we really need to fix this issue because what we’re ultimately measuring is the cumulative sum of differences between the modeled consumption and the actual consumption.

What do you think?

phabdallah says

Hi,

Thank you so much for your great expanation. Is there Heteroscedasticity in the graphs in the following links:

1- [link removed]

2- [link removed]

The second graph is for same data but after excluding patients aged over 85 years.

I am student doing analysis on sample size (n= 168614).

Thank you for your help.

Jim Frost says

Hi,

I don’t believe so. In the first one with the residuals by fitted values, it does taper off at the high end. But, that might be a consequence of there simply being fewer observations at the very high end, which means you don’t the full spread of values. You’re less likely to get the relatively less common observations that fall further away.

The 2nd model appears to be like the first model but with a truncated range on the residuals by fitted values. Cut off at the +2 standardized fitted value. I don’t know if it’s exactly the same data or not but it looks like it. I don’t evidence of heteroscedasticity there either. I just reread your comment more closely and noticed that they are the same data with patients aged 85+ removed. Interestingly that lops off your standardized fitted values at +2 very precisely. I wonder if including age related info in the model might improve the model? The predictions appear to be related to age.

However, both models appear to have a negative trend in the residuals by fitted values plot. You might need to address that.

Best of luck with your analysis!

Jim Knaub says

Understood. Interesting, thanks. I’m having some trouble following back on my phone to see what was said about the BMI data. You do still have heteroscedasticity when it takes fewer points in the y direction on the right side of the scatterplot to cover the same range as on the left. I estimated the coefficient of heteroscedasticity from the data and got about 0.6. I like including regression weight w, written in terms of the coefficient of heteroscedasticity, which should not matter much for predictions here, but could for the estimated variance of the prediction error for each predicted-y.

Jim Knaub says

Nice interpretation and good point Jim. According to page 111 of Brewer, K.R.W. (2002), Combined survey sampling inference: Weighing Basu’s elephants, Arnold: London and Oxford University Press, 1/x^2 would be the extreme (not accounting for influences such as omitted variables and data quality issues).

Jim Frost says

Thanks, Jim K. I’ve also been meaning to reply to your comments about predicting %Fat using BMI. I think there are several reasons why that model produces homoscedasticity.

The dependent variable is a rate (percent). That’s often a great way to recode variables to reduce heteroscedasticity. If I had used an absolute value, such as total fat in grams, the potential for heteroscedasticity increases because there would be a larger range from the smallest to largest values. Additionally, if the study had included a wider range of individuals (e.g., age), that would’ve produced an even wider range of values. It appears like using percentage and a narrowly defined population probably helped produce heteroscedasticity.

I mention in the post that recoding variables from absolute values to rates is my preferred method for fixing heteroscedasticity. I also think expressing variables as rates in these cases are often more meaningful than the absolute measure. Of course, that will vary by subject-area and specific application.

Marissa Albers says

Jim! Thank you! Very helpful and intuitive article.

Jim Frost says

Thank you, Marissa!

James R. Knaub, Jr (Jim) says

Howdy Jim Frost –

Something to consider:

Besides the analysis by Ken Brewer, it makes sense that a predicted value of 10,000,000 would have a larger variance of the prediction error than there would be for a predicted value of 10.

So if heteroscedasticity is the natural default, I’ve been looking at reasons why many regression scatterplots look homoscedastic. One interesting category is plots that look homoscedastic but really are heteroscedastic. An example is the nonlinear plot you have for predicting %Fat from BMI. We are interested in the vertical distribution of the points. It looks like the vertical range is a little greater on the lower end of the applicable BMI range, and thus likely the lower predicted y, i.e. y*, range. However, because the vertical point density at lower y* is greater, the variance of the prediction error is greater for larger y*. That is, points on the right side of the graph are spread out more in the vertical direction. When I measured it, the estimated coefficient of heteroscedasticity proved to be in the 0.5 to 1.0 range in my paper I last noted – closer to 0.5.

BMI is a very good (nonlinear) predictor for %Fat, at least for the range of BMI given. But for other models, we don’t always have such good predictors. But here we do have a straightforward model, apparently good data, and heteroscedasticity in the expected range for coefficient of heteroscedasticity.

Cheers – Jim Knaub

James R. Knaub, Jr (Jim) says

Joe, Jim F., et.al. –

From work done by Ken Brewer, discussed in https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity, it seems apparent that the best size measure which should be raised to the coefficient of heteroscedasticity to obtain the nonrandom factor of the estimated residuals is a good preliminary prediction of y. I’m working on background now to discuss how heteroscedasticity should be “ordinary,” and we only approximate homoscedasticity when the model has (a) omitted variable(s) of substantial importance, or maybe some other model issue, and/or data quality issues. So if one independent variable seems to have a lot to do with heteroscedasticity, it might often be a good single regressor. However, I think all the regressors should be used together for the size measure.

Cheers –

Jim K.

Jim Frost says

Hi Jim K.,

Thanks for sharing your article! I’ll need to take some time to digest it!

Joe C. says

You state,

“However, when you can identify a variable that is associated with the changing variance, a common approach is to use the inverse of that variable as the weight. In our case, the Weight column in the dataset equals 1 / Population.”

Since you’re squaring the error before summing, shouldn’t the weight be 1/Population^2 ?

Jim Frost says

Hi Joe,

It’s true that you square the differences and then sum for the SSE or variance, but that’s a separate matter than what weighted least squares (WLS) regression is doing. WLS uses the weight to shrink the variances of the residuals. The best weight depends on the relationship between the increase in fitted value and the increase in the residuals’ variances. That’s not necessarily going to be 1/X^2.

Waseem Jan says

can you please explain how to eliminate heteroscedasticity from panel data regression?

Jim Frost says

Hi Waseem, I’m pretty sure you’d use the same techniques that I discuss in this post. I’m not familiar with any techniques unique to panel data.

Patrik Silva says

Hi Jim, I am here again!

I would like to ask you a question that’s cooking my neurons!

I have a set of independent variables (IV) that I am assuming that they will affect my dependent variable (DV). However, when I use them in step-wise regression, they are reveling some consistence with my previous hypothesis and in fact they are statistically significant and reasonable R squared. Therefore, In this case do i need to check the residuals normality and homoscedasticity?

For now, I just want to prove that a particular variable (IV) is associate with my DV. If I am not interested to now how much, should I stop only by analyzing the R squared and P value? In this case of linear regression, which value should I talk about (R, R squared or Adjust R squared)?

Please, try to answer this question by explaining a bit about the differences between predictions, estimations and inferences?

Thank you in advance!

Sulaiman Inuwa Muhammad says

Thank you Jim for the wonderful explanation

Jim Knaub says

Fidel/Jim Frost –

I do not know what is available on R and other software, but one thing available on SAS is SAS PROC REG, for WLS. There you enter the regression weight as “w.”

Also, in SAS PROC REG, the estimated residuals can be printed, but unless it has changed, I noted long ago that it wasn’t really printing estimated residuals under the heading “residuals,” but actually the random factors of the estimated residuals (after accounting for regression weight).

Have a nice weekend.

Jim Knaub

Jim Frost says

Thanks, Jim Knaub! As usual, I appreciate your contributions!

fidel says

Hi, Jim. Would you please explain how to tell the software to perform the weighted regression? Should we go to Regression – Weight Estimation or Regression – Linear – WLS Weight?

Because I just went to Regression – Weight Estimation, and there was no option to Save the Residuals

Thank you, Jim

Great articles indeed you’ve posted. You do explain everything intuitively.

Jim Frost says

Hi Fidel, thanks so much for your kind words!

How to do this depends on the specific software that you’re using. The type of weighted regression that I refer to in this post is weighted least squares (WLS). So, I think that’s your best choice if you’re trying to do what I describe.

Jim Knaub says

PS – I have just worked with heteroscedasticity common to establishment surveys (finite population, cross-sectional surveys) that are used for production of official statistics.

https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity

https://www.researchgate.net/publication/324706010_Nonessential_Heteroscedasticity

https://www.researchgate.net/publication/319914742_Quasi-Cutoff_Sampling_and_the_Classical_Ratio_Estimator_-_Application_to_Establishment_Surveys_for_Official_Statistics_at_the_US_Energy_Information_Administration_-_Historical_Development

Jim Knaub says

Jim Frost – I am not sure what Frank means. Do you think he is saying that the ‘errors’ are not independent, so he needs to use the full GLS (generalized least squares) approach, not just dealing with heteroscedasticity? –

Frank – Don’t know, but you may find software that deals with GLS. Pretty sure SAS has that, maybe R and others. Whatever you do, you can test your model performance by saving out some data you have and checking to see if you would predict it well … besides the usual graphical residual analysis. – Maybe Jim Frost, the owner of this site, or someone else might have a better idea of what you need.

Cheers – Jim Knaub

Frank Sauerbier says

Hi Jim, thank you for this helpful information.

Could you gave me a hint how to deal with heteroscedasticity if the dependent var is somehow related to variance itself. In detail I investigate numbers (that are the standard deviation of a measure) and their dependence on the error rate (false positives). The residual plot is like a fan. But I think it could be intrinsic that variance increases with error rate.

Thanks in advance!

Frank

Jim Knaub says

Sani –

Probably the most used measure of heteroscedasticity is determined by the Iterated Reweighted Least Squares (IRLS) method. The algorithm is well-described in Carroll and Ruppert(1988), Transformation and Weighting in Regression, Chapman & Hall, Ltd. London, UK.

You could also see the following:

https://www.researchgate.net/publication/263809034_Alternative_to_the_Iterated_Reweighted_Least_Squares_Method_-_Apparent_Heteroscedasticity_and_Linear_Regression_Model_Sampling

and

https://www.researchgate.net/publication/263032446_Weighting_in_Regression_for_Use_in_Survey_Methodology (Note that the use of “w” in the notation on page 2 was not a good choice when avoiding confusion with the regression weight, of which this is a part. Sorry.)

I’d say that “pure” heteroscedasticity is ‘a feature, not a bug.’

You can use WLS and estimate variances of prediction errors and prediction intervals, keeping the heteroscedasticity in the error structure where it naturally occurs, with no transformation.

Cheers – J. Knaub

Jim Frost says

Thank you for sharing the excellent information!

Hiral Godhania says

Hello sir,

All of your articles are very useful .Sir could you please explain again how to detect heteroscedasticity and what are the implications of that.

Jim Frost says

Hi Hiral, this very post provides all of the information that you seek. It describes how to use residual plots to detect heteroscedasticity, the problems it causes, and possible solutions. You’re looking in the right place for the information you need!

Sani says

Good day Jim,

Please, can you help me with any method that can measure (not test) heteroscedasticity in a data set?

Jim Frost says

Hi Sani,

There are tests for heteroscedasticity, but I’m not overly familiar with them. I always go by the residuals by fits plot myself because the pattern is very distinct. I’ve heard of the White test and the Breusch-Pagan test. You can look into those if you need to.

Sorry, I just noticed that you’re looking for a measure rather than a test. Hmmm, these tests presumably need to measure and use a test statistic, maybe that is what you’re looking for? I’m not sure.

Jim

Senghort Kheang says

Dear guys

Do you have book 📚, article related with machine learning?

I just start learn but Prof. give me do research with topic: Real estate data visualization in Phnom Penh.

Can anyone help me?

Nick says

Really nice post. Thanks. Perhaps you could also mention using robust standard errors, Huber-White etc. This involves no data transformation but trades efficiency for robustness.