# How can the regression error term ever be correlated with the explanatory variables?

The first sentence of this wiki page claims that

In econometrics, an endogeneity problem occurs when an explanatory variable is correlated with the error term.

My question is that how can this ever happen? Isn't regression beta chosen such that the error term is orthogonal to the column space of the design matrix?

• The regression beta is chosen such that the residual is orthogonal to the column space of the design matrix. And this can give a horrible estimate of the true beta if the error term isn't orthogonal to the column space of the design matrix! (i.e. if your model doesn't satisfy the assumptions necessary to consistently estimate coefficients by regression). Commented Feb 22, 2017 at 1:34
• Orthogonality of the error term and the column space of the design matrix isn't a property of your estimation method (eg. ordinary least squares regression), it is a property of the model (eg. $y_i = a + b x_i + \epsilon_i$). Commented Feb 22, 2017 at 1:47
• I think your edit should be a new question because you seem to have substantially changed what you are asking for. You can always link back to this one. (I think you need to word it better, too - when you write "what would the effect be" then I'm not clear on the effect of what?) Note that asking a new question generally produces more attention which would be an advantage for you over editing an existing one. Commented Feb 3, 2018 at 13:46

You are conflating two types of "error" term. Wikipedia actually has an article devoted to this distinction between errors and residuals.

In an OLS regression, the residuals (your estimates of the error or disturbance term) $\hat \varepsilon$ are indeed guaranteed to be uncorrelated with the predictor variables, assuming the regression contains an intercept term.

But the "true" errors $\varepsilon$ may well be correlated with them, and this is what counts as endogeneity.

To keep things simple, consider the regression model (you might see this described as the underlying "data generating process" or "DGP", the theoretical model that we assume to generate the value of $y$):

$$y_i = \beta_1 + \beta_2 x_i + \varepsilon_i$$

There is no reason, in principle, why $x$ can't be correlated with $\varepsilon$ in our model, however much we would prefer it not to breach the standard OLS assumptions in this way. For example, it might be that $y$ depends on another variable that has been omitted from our model, and this has been incorporated into the disturbance term (the $\varepsilon$ is where we lump in all the things other than $x$ that affect $y$). If this omitted variable is also correlated with $x$, then $\varepsilon$ will in turn be correlated with $x$ and we have endogeneity (in particular, omitted-variable bias).

When you estimate your regression model on the available data, we get

$$y_i = \hat \beta_1 + \hat \beta_2 x_i + \hat \varepsilon_i$$

Because of the way OLS works*, the residuals $\hat \varepsilon$ will be uncorrelated with $x$. But that doesn't mean we have avoided endogeneity — it just means that we can't detect it by analysing the correlation between $\hat \varepsilon$ and $x$, which will be (up to numerical error) zero. And because the OLS assumptions have been breached, we are no longer guaranteed the nice properties, such as unbiasedness, we enjoy so much about OLS. Our estimate $\hat \beta_2$ will be biased.

$(*)$ The fact that $\hat \varepsilon$ is uncorrelated with $x$ follows immediately from the "normal equations" we use to choose our best estimates for the coefficients.

If you are not used to the matrix setting, and I stick to the bivariate model used in my example above, then the sum of squared residuals is $S(b_1, b_2) = \sum_{i=1}^n \varepsilon_i^2 = \sum_{i=1}^n (y_i-b_1 - b_2 x_i)^2$ and to find the optimal $b_1 = \hat \beta_1$ and $b_2 = \hat \beta_2$ that minimise this we find the normal equations, firstly the first-order condition for the estimated intercept:

$$\frac{\partial S}{\partial b_1} = \sum_{i=1}^n -2(y_i-b_1 - b_2 x_i) = -2 \sum_{i=1}^n \hat \varepsilon_i = 0$$

which shows that the sum (and hence mean) of the residuals is zero, so the formula for the covariance between $\hat \varepsilon$ and any variable $x$ then reduces to $\frac{1}{n-1} \sum_{i=1}^n x_i \hat \varepsilon_i$. We see this is zero by considering the first-order condition for the estimated slope, which is that

$$\frac{\partial S}{\partial b_2} = \sum_{i=1}^n -2 x_i (y_i-b_1 - b_2 x_i) = -2 \sum_{i=1}^n x_i \hat \varepsilon_i = 0$$

If you are used to working with matrices, we can generalise this to multiple regression by defining $S(b) = \varepsilon' \varepsilon = (y-Xb)'(y-Xb)$; the first-order condition to minimise $S(b)$ at optimal $b = \hat \beta$ is:

$$\frac{dS}{db}(\hat\beta) = \frac{d}{db}\bigg(y'y - b'X'y - y'Xb + b'X'Xb\bigg)\bigg|_{b=\hat\beta} = -2X'y + 2X'X\hat\beta = -2X'(y - X\hat\beta) = -2X'\hat \varepsilon = 0$$

This implies each row of $X'$, and hence each column of $X$, is orthogonal to $\hat \varepsilon$. Then if the design matrix $X$ has a column of ones (which happens if your model has an intercept term), we must have $\sum_{i=1}^n \hat \varepsilon_i = 0$ so the residuals have zero sum and zero mean. The covariance between $\hat \varepsilon$ and any variable $x$ is again $\frac{1}{n-1} \sum_{i=1}^n x_i \hat \varepsilon_i$ and for any variable $x$ included in our model we know this sum is zero, because $\hat \varepsilon$ is orthogonal to every column of the design matrix. Hence there is zero covariance, and zero correlation, between $\hat \varepsilon$ and any predictor variable $x$.

If you prefer a more geometric view of things, our desire that $\hat y$ lies as close as possible to $y$ in a Pythagorean kind of way, and the fact that $\hat y$ is constrained to the column space of the design matrix $X$, dictate that $\hat y$ should be the orthogonal projection of the observed $y$ onto that column space. Hence the vector of residuals $\hat \varepsilon = y - \hat y$ is orthogonal to every column of $X$, including the vector of ones $\mathbf{1_n}$ if an intercept term is included in the model. As before, this implies the sum of residuals is zero, whence the residual vector's orthogonality with the other columns of $X$ ensures it is uncorrelated with each of those predictors.

But nothing we have done here says anything about the true errors $\varepsilon$. Assuming there is an intercept term in our model, the residuals $\hat \varepsilon$ are only uncorrelated with $x$ as a mathematical consequence of the manner in which we chose to estimate regression coefficients $\hat \beta$. The way we selected our $\hat \beta$ affects our predicted values $\hat y$ and hence our residuals $\hat \varepsilon = y - \hat y$. If we choose $\hat \beta$ by OLS, we must solve the normal equations and these enforce that our estimated residuals $\hat \varepsilon$ are uncorrelated with $x$. Our choice of $\hat \beta$ affects $\hat y$ but not $\mathbb{E}(y)$ and hence imposes no conditions on the true errors $\varepsilon = y - \mathbb{E}(y)$. It would be a mistake to think that $\hat \varepsilon$ has somehow "inherited" its uncorrelatedness with $x$ from the OLS assumption that $\varepsilon$ should be uncorrelated with $x$. The uncorrelatedness arises from the normal equations.

• does your $y_i = \beta_1 + \beta_2 x_i + \varepsilon_i$ mean regression using population data? Or what does it mean precisely? Commented Feb 22, 2017 at 0:47
• @user1559897 Yes, some textbooks will call this the "population regression line" or PRL. It's the underlying theoretical model for the population; you may also see this called the "data generating process" in some sources. (I tend to be a bit careful about saying it is the "regression on the population"... if you have a finite population, e.g. 50 states of the USA, that you perform the regression on, then this isn't quite true. If you are actually running a population on some data in your software, you are really talking about the estimated version of the regression, with the "hats") Commented Feb 22, 2017 at 0:52
• I think i see what you are saying. If i understand you correctly, the error term in the model $y_i = \beta_1 + \beta_2 x_i + \varepsilon_i$ could have non-zero expectation as well because it is a theoretical generating process, not a ols regression. Commented Feb 22, 2017 at 0:54
• This is a great answer from statistical inference perspective. What do you think the effect would be if prediction accuracy is the primary concern? See the edit of the post. Commented Feb 3, 2018 at 4:35
• @denizenofthenorth When it comes to pure predictive accuracy, the prediction accuracy reigns supreme. You can have terrible violations of the usual assumptions, yet if the predictions, upon proper and thorough validation, are good enough to accomplish your task, go for it!
– Dave
Commented Oct 23, 2022 at 7:28

# Simple example:

• Let $x_{i,1}$ be the number of burgers I buy on visit $i$
• Let $x_{i,2}$ be the number of buns I buy.
• Let $b_1$ be the price of a burger
• Let $b_2$ be the price of a bun.
• Independent of my burger and bun purchases, let me spend a random amount $a + \epsilon_i$ where $a$ is a scalar and $\epsilon_i$ is a mean zero random variable. We have $\operatorname{E}[\epsilon_i | X] = 0$.
• Let $y_i$ be my spending on a trip to the grocery store.

The data generating process is:

$$y_i = a + b_1x_{i,1} + b_2x_{i,2} + \epsilon_i$$

If we ran that regression, we would get estimates $\hat{a}$, $\hat{b}_1$, and $\hat{b}_2$, and with enough data, they would converge on $a$, $b_1$, and $b_2$ respectively.

(Technical note: We need a little randomness so we don't buy exactly one bun for each burger we buy at every visit to the grocery store. If we did this, $x_1$ and $x_2$ would be collinear.)

### An example of omitted variable bias:

Now let's consider the model:

$$y_i = a + b_1x_{i,1} + u_i$$

Observe that $u_i = b_2x_{i,2} + \epsilon_i$. Hence \begin{align*} \operatorname{Cov}(x_{1}, u) &= \operatorname{Cov}(x_1,b_2x_2 + \epsilon )\\ &= b_2 \operatorname{Cov}(x_{1},x_2) + \operatorname{Cov}(x_{1},\epsilon) \\ &= b_2 \operatorname{Cov}(x_{1},x_2) \end{align*}

Is this zero? Almost certainly not! The purchase of burgers $x_1$ and the purchase of buns $x_2$ are almost certainly correlated! Hence $u$ and $x_1$ are correlated!

### What happens if you tried to run the regression?

If you tried to run:

$$y_i = \hat{a} + \hat{b}_1 x_{i,1} + \hat{u}_i$$

Your estimate $\hat{b}_1$ would almost certainly be a poor estimate of $b_1$ because the OLS regression estimates $\hat{a}, \hat{b}, \hat{u}$ would be constructed so that $\hat{u}$ and $x_1$ are uncorrelated in your sample. But the actual $u$ is correlated with $x_1$ in the population!

What would happen in practice if you did this? Your estimate $\hat{b}_1$ of the price of burgers would ALSO pickup the price of buns. Let's say every time you bought a \$1 burger you tended to buy a \$0.50 bun (but not all the time). Your estimate of the price of burgers might be \$1.40. You'd be picking up the burger channel and the bun channel in your estimate of the burger price. • I like your burger bun example. You explained the problem from the perspective of statistical inference, ie inferring the effect of burger on price. Just wondering what the effect would be if all I care about is prediction, i.e prediction MSE on a test dataset? The intuition is that it is not going to be as good, but is there any theory to make it more precise? (this introduced more bias, but less variance, so the overall effect is not apparent to me. ) Commented Feb 3, 2018 at 4:31 • @user1559897 If you just care about predicting spending, then predicting spending using the number of burgers and estimating$\hat{b}_1$as around \$1.40 might work pretty well. If you have enough data, using the number of burgers and buns would undoubtedly work better. In short samples, $L_1$ regularlization (LASSO) might send one of the coefficients $b_1$ or $b_2$ to zero. I think you're correctly recognizing that what you're doing in regression is estimating a conditional expectation function. My point is for that that function to capture causal effects, you need additional assumptions. Commented Feb 3, 2018 at 19:24

Suppose that we're building a regression of the weight of an animal on its height. Clearly, the weight of a dolphin would be measured differently (in different procedure and using different instruments) from the weight of an elephant or a snake. This means that the model errors will be dependent on the height, i.e. explanatory variable. They could be dependent in many different ways. For instance, maybe we tend to slightly overestimate the elephant weights and slightly underestimate the snake's, etc.

So, here we established that it is easy to end up with a situation when the errors are correlated with the explanatory variables. Now, if we ignore this and proceed to regression as usual, we'll notice that the regression residuals are not correlated with the design matrix. This is because, by design the regression forces the residuals to be uncorrelated. Note, also that residuals are not the errors, they're the estimates of errors. So, regardless of whether the errors themselves are correlated or not with the independent variables the error estimates (residuals) will be uncorrelated by the construction of the regression equation solution.