Omitted Variable Bias

Most regression problems are noise problems, and noise problems have a cure: get more data. Omitted variable bias is not one of those. It is a problem of aim, not of steadiness. Your estimate converges beautifully — to the wrong number. A million observations will pin down that wrong number to four decimal places and give you a t-statistic of 40. This is why it is the first thing a good reviewer attacks and the last thing a proud researcher wants to hear.

Ice cream and drowning

Across a year, ice cream sales and drownings rise and fall together almost perfectly. Regress drownings on ice cream sales and you get a large, highly significant coefficient. Nobody believes ice cream drowns people. The obvious culprit is summer: hot weather sells ice cream and sends people into the water.

Now notice what the regression actually did. It had one variable to work with, and that variable happened to move with the true cause. So it handed the ice cream coefficient credit for everything summer did. The coefficient is not measuring ice cream. It is measuring ice cream plus a borrowed slice of summer, and the size of that slice is exactly what the formula below computes.

The trading version is less obvious and more expensive: a value signal that looks profitable because value stocks are small, a "market-neutral" book whose alpha is really a hidden short-volatility position, a factor that vanishes once you control for sector.

The formula, in three symbols

Suppose the truth is that two things drive the outcome $y$: the variable you measured, $x$, and a variable you left out, $z$.

$$y = \beta_1 x + \beta_2 z + \text{noise}.$$

In words: $\beta_1$ (beta-one) is the real effect of $x$, the number you are trying to learn. $\beta_2$ is the real effect of the omitted $z$. Now regress $y$ on $x$ alone. What you get back is not $\beta_1$ but

$$\hat\beta_1 \;\longrightarrow\; \beta_1 + \beta_2\,\delta .$$

In words: what you measure equals the true effect, plus the omitted variable's own effect multiplied by how strongly it moves with your variable. The third symbol, $\delta$ (delta), is the slope you would get from regressing the omitted $z$ on the included $x$ — a purely descriptive number saying "when $x$ goes up by one, $z$ tends to go up by $\delta$."

The extra term $\beta_2 \delta$ is the bias. It is a product, so it vanishes if either factor is zero.

Worked example one: the formula, verified by hand

Five observations, constructed so the truth is known exactly. The real model is $y = 1\cdot x + 2\cdot z$, with no noise at all.

$x$	1	2	3	4	5
$z$	2	1	4	3	5
$y$	5	4	11	10	15

So the true effect of $x$ is $\beta_1 = 1$. Now pretend you never recorded $z$.

Step 1 — the short regression of $y$ on $x$. Means are $\bar{x} = 3$, $\bar{y} = 45/5 = 9$. Cross-products: $(-2)(-4) + (-1)(-5) + (0)(2) + (1)(1) + (2)(6) = 8 + 5 + 0 + 1 + 12 = 26$. Squared $x$ deviations: $4+1+0+1+4 = 10$.

$$\hat\beta_1^{\,\text{short}} = \frac{26}{10} = 2.6 .$$

You would report 2.6 when the truth is 1.0. The estimate is 160% too large, and with no noise in the data it is not even slightly uncertain.

Step 2 — measure $\delta$, how $z$ tracks $x$. Mean $\bar{z} = 15/5 = 3$. Cross-products: $(-2)(-1) + (-1)(-2) + (0)(1) + (1)(0) + (2)(2) = 2 + 2 + 0 + 0 + 4 = 8$. Same denominator, 10:

$$\delta = \frac{8}{10} = 0.8 .$$

Step 3 — check the formula.

$$\beta_1 + \beta_2\,\delta = 1 + 2 \times 0.8 = 2.6 .$$

Exactly the 2.6 the regression produced. The bias is not an approximation or a tendency; it is a precise, computable quantity.

The scatter above is the relationship the bias depends on — the omitted variable against the one you kept. Slide the correlation toward zero and watch the fit line flatten: that flattening is $\delta$ going to zero, which is the only way to kill the bias without knowing $\beta_2$. It is also exactly what a randomised experiment buys you.

Worked example two: how much of your alpha is really size?

You regress fund returns on a value score and report the coefficient as the value premium. Suppose the truth is:

• value genuinely adds $\beta_1 = 1.5$ percentage points per unit of score;
• smallness adds $\beta_2 = 3.0$ points per unit, and you did not include it;
• value stocks skew small, so regressing smallness on value score gives $\delta = 0.6$.

Step 1 — compute the bias. $\beta_2 \delta = 3.0 \times 0.6 = 1.8$ points.

Step 2 — what you would report.

$$1.5 + 1.8 = 3.3 \text{ points.}$$

You publish a 3.3-point value premium. It is more than double the real one, and $1.8/3.3 = 55\%$ of your headline number is an unhedged bet on small caps. Hedge out size in live trading — as any risk system will insist you do — and most of the return disappears.

Step 3 — flip the sign to see the worst case. Suppose instead the omitted factor is a crowding drag with $\beta_2 = -3.0$, still with $\delta = 0.6$. Then the bias is $-1.8$ and you would report $1.5 - 1.8 = -0.3$: a genuinely profitable signal that appears to lose money. Omitted variables do not merely inflate; they can invert.

$\beta_2$ (omitted effect)	$\delta$ (how $z$ tracks $x$)	Bias	Effect on your estimate
positive	positive	positive	inflated
positive	negative	negative	understated, may flip
negative	positive	negative	understated, may flip
negative	negative	positive	inflated

What this means in practice

Every regression you present is an implicit claim that nothing important is missing. So make the claim explicit: name the two or three variables most likely to be lurking, and say why each fails one of the two conditions. "Sector matters for returns, but our signal is sector-neutral by construction, so $\delta = 0$" is a real defence. "We didn't have that data" is not.

Where you cannot measure the culprit, change the design rather than the regression. Fixed effects sweep out anything constant within a firm or a month. Matching compares like with like. Instrumental variables isolate the part of $x$ that could not have been touched by the omitted variable (Endogeneity and Instrumental Variables). Randomisation, where possible, forces $\delta = 0$ by construction and is the reason experiments outrank observational studies.

You can also bound your ignorance. The bias is a product of two numbers, so plausible ranges for each give a plausible range for the bias. If the estimate stays the same sign across every reasonable guess, your conclusion survives; if a modest $\delta$ wipes it out, you have learned that the result was never robust.

The formal statement of what breaks here is assumption two of the The Classical Linear Regression Assumptions, and the mechanics of partialling variables out are the The Frisch-Waugh-Lovell Theorem.