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So, you’ve spent all this time building a regression model for your risk team. It’s for stress testing, it’s important, maybe even for regulatory reporting. You get a nice high R-squared, you feel good about it... but then you notice something weird. The coefficients, the numbers that tell you how important each input is, they seem to jump around when you use different slices of your data. And when you plot the errors, you see this weird, cone shape.
Suddenly, that model you were so confident in feels… a little shaky. This is a classic scenario you’ll see on the exam, and it’s all about diagnosing what’s wrong under the hood of your Ordinary Least Squares, or OLS, regression. Getting this right is the difference between a reliable model and one that could get you into a lot of trouble.
Think of the assumptions of OLS regression like the rules for building a sturdy house. You need a solid, level foundation. You need the walls to be properly spaced. You need to make sure you didn't forget a major support beam. If you violate these rules, the house might still be standing, but you really wouldn't want to be inside during a storm. Today, we’re going to be the home inspector for our models.
The three big violations you need to know are heteroskedasticity, multicollinearity, and omitted variable bias. They sound complicated, but the ideas are pretty straightforward.
Let's start with the first one: heteroskedasticity. Fun word. All it means is that the variance of your model’s errors isn't constant. Imagine your model is predicting stock prices. If it's consistently off by about a dollar for low-priced stocks, but it’s off by ten or twenty dollars for high-priced stocks… that’s heteroskedasticity. The errors get bigger as the variable gets bigger. On a chart, this is that classic ‘fan-shape’ or ‘cone-shape’ in your residual plot that the exam question might describe.
Now, what’s the consequence? This is a huge point. Heteroskedasticity does *not* make your coefficient estimates biased. Let me say that again. The coefficients are still unbiased and consistent. On average, they’re correct. The problem is, they are no longer *efficient*. The model isn't the Best Linear Unbiased Estimator, or BLUE, anymore. Your standard errors are wrong.
This means all your hypothesis tests—your t-stats, your p-values—are totally unreliable. You might think a variable is statistically significant when it's not, or vice versa. Your model is pointing in the right direction, but you have no idea how wide the road is.
How do you spot it? Well, you can look at the plots. But to be sure, you use a formal test like the Breusch-Pagan test or the White test. The exam will expect you to know those names. They basically check if your independent variables can predict the size of your errors. If they can, you’ve got a problem. The most common fix, and the one you should look for first, is to use robust standard errors, sometimes called White-corrected standard errors. It's like putting new shock absorbers on your model to handle the bumpy road. It adjusts the standard errors so your t-stats are valid again, without even changing the coefficients.
Okay, that’s problem number one. Let's move to the second one: multicollinearity.
This happens when two or more of your independent variables are highly correlated with each other. They're telling you the same story. Think about building a model to predict a company's sales, and you include both 'revenue in US dollars' and 'revenue in Euros'. They're almost the same piece of information, just expressed differently. Or the example from the source material: 'interest rate spread' and 'credit default swap basis'. They tend to move together.
So what happens? The model gets confused about who to give credit to. It can't figure out the individual impact of each variable. Your overall model might still be great at predicting—you might have a really high R-squared—but when you look at the individual coefficients for those correlated variables, they might be statistically insignificant. And worse, they can be really unstable. Add or remove a few data points, and the coefficients can swing wildly.
The big takeaway here is just like with heteroskedasticity: your OLS estimators are still unbiased and consistent. But they have huge standard errors, making them inefficient. You just can't trust the individual coefficients.
To find it, you can start by looking at a simple correlation matrix of your inputs. If you see high correlations, say above 0.8 or 0.9, you should be suspicious. But the real tool for the job is the Variance Inflation Factor, or VIF. The VIF for a variable tells you how much the variance of its coefficient is 'inflated' because of its relationship with the other variables. A common rule of thumb is that a VIF greater than 5 or 10 is a sign of trouble.
How do you fix it? The simplest way is often to just remove one of the highly correlated variables. If they're telling you the same thing, you might not need both. Another option is to combine them into a single, composite variable.
Now for the third, and arguably the most serious, problem: omitted variable bias. OVB.
This is the one that truly breaks your model. It happens when you leave out a variable that actually matters for the outcome, AND that variable is correlated with one of the variables you *did* include. This is like building the house but forgetting a major, load-bearing wall.
Unlike the other two issues, omitted variable bias makes your coefficient estimates biased and inconsistent. That means they are just plain wrong, and even with more and more data, they won't get any closer to the true value. Your model isn't just inefficient; it's actively misleading you.
How do you find it? This is less about a single statistical test and more about your brain. It's about economic theory, about domain knowledge. Does your model make sense? Is there something obvious you forgot? For example, if you're modeling housing prices based only on square footage, but you left out the neighborhood... well, neighborhood is probably correlated with square footage (bigger houses are in certain neighborhoods) and it definitely affects the price. Your square footage coefficient is now going to be biased because it’s trying to pick up some of the effect of the neighborhood it's correlated with.
The main remedy is pretty obvious: include the omitted variable! If you can't get the exact variable, maybe you can use a good proxy for it.
So let’s walk through how you'd tackle this on exam day. You're given a scenario. First, read it carefully for clues. Does it mention unstable coefficients? Fan-shaped residuals? A high R-squared but insignificant variables? These are your signposts.
Next, you'd think about a visual check. Plot the residuals. See any patterns? If you see that fan-shape, you're immediately thinking heteroskedasticity. You'd confirm with a Breusch-Pagan or White test, and your go-to fix is robust standard errors.
If the problem is more about wobbly coefficients or that high R-squared with low t-stats, your mind should go to multicollinearity. You'd want to check the VIFs. Let's say a portfolio manager sees the VIFs for 'economic growth' and 'inflation' are both over 10. They're highly correlated. The best first step is probably to remove one of them, assuming theory supports it.
But if you look at the model and think, "Wait, theory says X should be in here, and it's not," you should worry about omitted variable bias. This is the most severe problem because it creates bias. The fix is to add that variable back in.
The single most important thing to keep straight for the exam is the consequence of each violation. Let's say it one more time. Heteroskedasticity and multicollinearity make your estimates inefficient—you get large standard errors, invalid tests—but they do NOT cause bias. Omitted variable bias DOES cause bias. Your estimates are fundamentally wrong. That distinction is a favorite for exam writers.
Let's try a quick question. A risk analyst models credit spreads. She plots the residuals and sees that they get wider as the predicted spread increases. What's the issue, what test confirms it, and what's the fix? (Pause) Okay, that widening pattern is the classic fan-shape. The issue is heteroskedasticity. You'd confirm it with a Breusch-Pagan or White test. And the most common fix is to apply robust standard errors.
Here's another one. An analyst builds a model to predict returns. 'Earnings growth' and 'revenue growth' have a correlation of 0.95. The model's R-squared is high, but the t-stats for both those variables are tiny. What's the problem and what tool quantifies it? (Pause) That's a textbook case of multicollinearity. The tool to measure it is the Variance Inflation Factor, or VIF.
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So, to wrap it up: for heteroskedasticity, think 'fan-shaped errors' and fix with robust standard errors. For multicollinearity, think 'correlated inputs' and 'wobbly coefficients,' check the VIF, and fix by removing a variable. And for omitted variable bias, the most dangerous one, think 'missing variable,' which leads to biased estimates, and fix by adding it back in.
Mastering these diagnostics will make you so much more confident, both on the exam and in your day job. You'll be able to build models you can actually trust.
Alright, that's it for this one. Keep up the great work. I'll talk to you next time.