Omitted-variable bias

Key takeaways

Omitted-variable bias belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Omitted-variable bias to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Omitted-variable bias from memory before moving on to harder problems.

Reference excerpt

In statistics, omitted-variable bias (OVB) occurs when a statistical model leaves out one or more relevant variables. The bias results in the model attributing the effect of the missing variables to those that were included. More specifically, OVB is the bias that appears in the estimates of parameters in a regression analysis, when the assumed specification is incorrect in that it omits an independent variable that is a determinant of the dependent variable and correlated with one or more of the included independent variables.

In linear regression

Intuition Suppose the true cause-and-effect relationship is given by:

y = a + b x + c z + u {\displaystyle y=a+bx+cz+u}

with parameters a, b, c, dependent variable y, independent variables x and z, and error term u. We wish to know the effect of x itself upon y (that is, we wish to obtain an estimate of b). Two conditions must hold true for omitted-variable bias to exist in linear regression:

the omitted variable must be a determinant of the dependent variable (i.e., its true regression coefficient must not be zero); and the omitted variable must be correlated with an independent variable specified in the regression (i.e., cov(z,x) must not equal zero). Suppose we omit z from the regression, and suppose the relation between x and z is given by

z = d + f x + e {\displaystyle z=d+fx+e}

with parameters d, f and error term e. Substituting the second equation into the first gives

y = ( a + c d ) + ( b + c f ) x + ( u + c e ) . {\displaystyle y=(a+cd)+(b+cf)x+(u+ce).}

If a regression of y is conducted upon x only, this last equation is what is estimated, and the regression coefficient on x is actually an estimate of (b + cf ), giving not simply an estimate of the desired direct effect of x upon y (which is b), but rather of its sum with the indirect effect (the effect f of x on z times the effect c of z on y). Thus by omitting the variable z from the regression, we have estimated the total derivative of y with respect to x rather than its partial derivative with respect to x. These differ if both c and f are non-zero. The direction and extent of the bias are both contained in cf, since the effect sought is b but the regression estimates b+cf. The extent of the bias is the absolute value of cf, and the direction of bias is upward (toward a more positive or less negative value) if cf > 0 (if the direction of correlation between y and z is the same as that between x and z), and it is downward otherwise.

Detailed analysis As an example, consider a linear model of the form

y i = x i β + z i δ + u i , i = 1 , … , n {\displaystyle y_{i}=x_{i}\beta +z_{i}\delta +u_{i},\qquad i=1,\dots ,n}

where

xi is a 1 × p row vector of values of p independent variables observed at time i or for the i th study participant; β is a p × 1 column vector of unobservable parameters (the response coefficients of the dependent variable to each of the p independent variables in xi) to be estimated; zi is a scalar and is the value of another independent variable that is observed at time i or for the i th study participant; δ is a scalar and is an unobservable parameter (the response coefficient of the dependent variable to zi) to be estimated; ui is the unobservable error term occurring at time i or for the i th study participant; it is an unobserved realization of a random variable having expected value 0 (conditionally on xi and zi); yi is the observation of the dependent variable at time i or for the i th study participant. We collect the observations of all variables subscripted i = 1, ..., n, and stack them one below another, to obtain the matrix X and the vectors Y, Z, and U:

X = [ x 1 ⋮ x n ] ∈ R n × p , {\displaystyle X=\left[{\begin{array}{c}x_{1}\\\vdots \\x_{n}\end{array}}\right]\in \mathbb {R} ^{n\times p},}

and

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Omitted-variable bias

Start with the simplest possible case. Write down what Omitted-variable bias claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Omitted-variable bias before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Omitted-variable bias ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Omitted-variable bias

In research: Omitted-variable bias appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Omitted-variable bias in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Omitted-variable bias is common in secondary-school and first-year university syllabi. It links to neighbouring topics Experimental bias, Regression analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Omitted-variable bias outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Omitted-variable bias in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Omitted-variable bias means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Omitted-variable bias out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Omitted-variable bias in simple terms?

In statistics, omitted-variable bias (OVB) occurs when a statistical model leaves out one or more relevant variables. The bias results in the model attributing the effect of the missing variables to those that were included.

Why does Omitted-variable bias matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Omitted-variable bias?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Omitted-variable bias.