Econometrics · Lesson 7 of 18

Omitted variables, irrelevant variables and measurement error

Every regression is a bet that the model is right. This lesson is about what happens when it is not: a variable that matters is left out, one that does not matter is put in, a regressor is recorded with error, or a straight line is fitted to a curve. Each mistake damages the estimates in its own way, and the exam tests whether you know which way. The lab at the centre lets you break each assumption on purpose, with a truth you know, and see what it does.

← Econometrics moduleLesson 7 of 18 · about 55 minutes

A question to hold on to

A researcher estimates Model A: y = β₀ + β₁S + β₂A + ε and Model B: y = β₀ + β₁S + ε, where S is study hours and A is ability. β̂₁ = 7.1 in A but 2.1 in B (both significant), and β̂₂ = 1.9 (significant). This suggests that: (A) students with lower ability also spend fewer hours studying; (B) students with lower ability spend more time studying; (C) there is no way that students of even high ability can get more than 40 marks; (D) none of the above.

Delhi School of Economics 2015Solved at the end ↓

New here?What is the Ashanagar survey?The invented household survey behind this page's examples: 600 households, generated from a model whose true coefficients we publish.

The Ashanagar survey is invented. The statistics modules used one district of 60 households. Econometrics needs more households and more variables, so this is a new, larger survey of 600 households in the same imaginary district: income, spending, household size, whether the household is in the town or a village, the head’s schooling and earnings.

Because we wrote the data-generating process, we know the true answers, and every lab can show whether a regression recovers them. Household spending, in thousands of rupees a month, was generated as 1 + 0.45 × income + 0.9 × household size + 1.5 × town + an error with standard deviation 2.5. A real survey never tells you its true coefficients. This one does, so you can see how close an estimate comes.

It also holds a variable, the head’s ability, that a real survey could not observe. It is kept so that the lessons on omitted variables can show what happens when it is left out. Every figure here is illustrative, not official statistics.

Households: 600
Share in the town: 41%
Mean monthly income: ₹15.4 thousand
Mean monthly spending: ₹12.6 thousand
Mean schooling of the head: 8.2 years

How the module uses it →

The central result

Leaving a variable out

Suppose the true model is y = β₀ + β₁x₁ + β₂x₂ + u, but we do not have x₂, or forget it, and regress y on x₁ alone. Two things must hold for the damage to be real: x₂ affects y (β₂ ≠ 0), and x₂ is related to x₁. Then x₁ picks up the effect of x₂ that it travels with.

Omitted-variable bias

E (b_{1}^{s h or t}) = β_{1} + β_{2} δ, bias = β_{2} δ

δ is the slope of the regression of the omitted variable x₂ on the included one x₁. The bias is the effect of the missing variable times its relation to the included one.

	δ > 0 (x₁ and x₂ move together)	δ < 0 (x₁ and x₂ move oppositely)
β₂ > 0 (x₂ raises y)	Upward bias	Downward bias
β₂ < 0 (x₂ lowers y)	Downward bias	Upward bias

In the survey, schooling raises a head’s earnings (we wrote 1.3 per year) and so does ability (3 per unit), and abler heads study longer. Leave ability out and the schooling coefficient is 1.699; put it in and it is 1.298. The excess, 0.401, is the bias formula at work: 2.685 (the effect of ability) times 0.149 (how much ability rises with each year of schooling). The derivation shows that this is an exact identity in any sample, not an approximation.

Derive it

The omitted-variable bias formula

We want to show

If the true model is y = β₀ + β₁x₁ + β₂x₂ + u and we regress y on x₁ alone, the slope has expected value β₁ + β₂δ, where δ is the slope of the regression of x₂ on x₁.

Write the short regression's slope.
Leaving x₂ out, least squares gives the simple-regression slope:
$b_{1}^{s h or t} = \frac{\sum ( x _{1} - x ˉ _{1} ) ( y - y ˉ )}{\sum ( x _{1} - x ˉ _{1} ) ^{2}}$
Substitute the true model for y.
y − ȳ = β₁(x₁ − x̄₁) + β₂(x₂ − x̄₂) + (u − ū). Put it in the numerator and split it into three terms:
$b_{1}^{s h or t} = β_{1} + β_{2} \frac{\sum ( x _{1} - x ˉ _{1} ) ( x _{2} - x ˉ _{2} )}{\sum ( x _{1} - x ˉ _{1} ) ^{2}} + \frac{\sum ( x _{1} - x ˉ _{1} ) ( u - u ˉ )}{\sum ( x _{1} - x ˉ _{1} ) ^{2}}$
Recognise the middle term.
The fraction in the middle is the slope of the regression of x₂ on x₁, which we call δ. The last term has expected value zero when u is unrelated to x₁ (the first assumption of lesson 2). Taking expectations:
$E (b_{1}^{s h or t}) = β_{1} + β_{2} δ$
So the bias is β₂δ: the effect of the missing variable on y, times the relation between the missing variable and the included one.
The same thing as an exact identity in any sample.
No expectation is needed for the sample version: the short slope always equals the long slope plus the long coefficient on x₂ times δ̂. In the survey, regress the head’s earnings on schooling and town (short), and then also on ability (long); then regress ability on schooling and town (δ̂ is its schooling coefficient):
short slope 1.6988 = long slope 1.2979 + 2.6853 × 0.1493 = 1.6988.
Ability helps earnings (β₂ is positive) and is higher among the better schooled (δ is positive), so the bias is positive: the short regression gives schooling credit for ability’s work. The true schooling coefficient is 1.3.

Step 1 of 4

Four consequences worth knowing:

The bias does not shrink with more data. The estimator is inconsistent: it converges to β₁ + β₂δ, not to β₁.
The standard errors give no warning. The short regression’s t statistics and intervals are computed as if it were right. The lab shows a 95% interval that contains the truth far less than 95% of the time.
The sign is a product of two signs. Take earnings, schooling and parental income: parental income raises earnings and goes with more schooling, both signs positive, so leaving it out biases the schooling coefficient upward. In the opening question, the short and long slopes differ by 2.1 − 7.1 = −5 = β₂δ with β₂ = 1.9 > 0, so δ is negative.
If the short and long slopes on x₁ are exactly equal, the identity says β̂₂ × δ̂ = 0, so either β̂₂ = 0 or x₁ and x₂ are uncorrelated in the sample. It does not say Cov(x₁, y) = 0, and it does not say the omitted variable is irrelevant to y.

Watch out for

The mistakes examiners are fishing for

Thinking an omitted variable always causes bias. It does not if it is unrelated to the included regressors (δ = 0), however much it affects y. It then only lowers precision, by adding to the error.
Predicting the sign from β₂ alone. The sign needs both: what the missing variable does to y, and how it relates to the included one.

Interactive

The break-an-assumption lab

We know the truth here, because we write it: the coefficient on x₁ is 1. Each press of the button draws 1,500 new samples, fits least squares to each, and shows where the estimates land. Start with “Leave out a variable”: raise the effect of x₂ and its correlation with x₁ and watch the cloud of estimates slide away from 1 while the reported standard error stays calm. Then include x₂ and watch it come back. The other three tabs are the next three sections.

x2 affects y and is correlated with x1. Include it, or leave it out, and compare. True model: y = 1 × x1 + β₂ × x2 + noise (sd 2). The coefficient we want is the 1.

Effect of x2 on y, β₂1Correlation of x1 and x2, ρ0.6Sample size, n100

Average estimate1.594

Truth1

Bias0.594

Real spread (sd)0.226

Reported standard error0.217

95% interval holds the truth24%

5% test wrongly rejects the truth76%

Horizontal: the estimated coefficient on x1 in each of the 1,500 samples. Vertical: how many samples gave that value.

Estimates from 1,500 samplesTrue value, 1Average estimate

What it shows

With x2 left out the estimates centre on 1.594, not on 1. The formula β₁ + β₂ × ρ = 1 + 1 × 0.6 = 1.6 predicts it. The bias is the effect of the missing variable times its relation to x1. The reported standard error (0.217) matches the spread of the estimates (0.226), so it gives no warning: the interval is simply centred on the wrong number, and it contains the truth only 24% of the time.

The opposite mistake

Adding a variable that does nothing

Include a variable that has no effect on y (its true coefficient is zero) and nothing is biased. The estimates are still centred on the truth, because the model that includes it is still correct (its coefficient is just zero). The price is precision. If the irrelevant variable is correlated with x₁, it uses up some of the variation in x₁ that the regression could have used, and the variance of b₁ is multiplied by the variance inflation factor, 1 ÷ (1 − ρ²) in the two-variable case:

The cost of an irrelevant, correlated regressor

Var (b_{1}) = \frac{σ ^{2}}{S _{11} ( 1 - ρ ^{2} )}

The standard deviation of b₁ is multiplied by 1 ÷ √(1 − ρ²). With ρ = 0.6 that is 1.25, and with ρ = 0.9 it is 2.29. If ρ = 0 there is no cost.

Try it in the lab’s second tab. In the survey, household income is strongly related to schooling (correlation 0.77) and does not enter the earnings equation we wrote. Adding it to the earnings regression moves the schooling coefficient only from 1.298 to 1.253, and its own coefficient has t = 1.63, no evidence of an effect. But the standard error of the schooling coefficient rises from 0.035 to 0.0443.

So the trade is not symmetric. Leaving out a relevant variable gives bias that does not go away; including an irrelevant one gives variance that does go away with more data. That is why textbooks lean towards including a variable when in doubt, provided the sample is large. The price of too many regressors is felt in small samples, where it also costs degrees of freedom.

Pause and predict

You add an irrelevant variable that is uncorrelated with every other regressor. What happens to the variance of the other coefficients?

When the data are noisy

Measurement error

Where does a regressor come from? In practice from survey answers, rounded accounts, proxies. Suppose the recorded x is the true x* plus noise v, and v has nothing to do with x* or with u. Then the noise is in the regressor, and the result is attenuation: the slope shrinks towards zero by the reliability ratio λ = Var(x*) ÷ (Var(x*) + Var(v)), the share of the recorded variable’s variance that is true signal.

Derive it

Why measurement error in x pulls the slope towards zero

We want to show

If y = βx* + u but we only observe x = x* + v, with the error v unrelated to x* and u, then the slope of y on x converges to λβ, where λ = σ²_x* ÷ (σ²_x* + σ²_v) is between 0 and 1.

Rewrite the model in terms of what we observe.
Since x* = x − v:
$y = β x^{*} + u = β x + (u - β v)$
The error term of the regression we can run is w = u − βv.
That error is correlated with the regressor.
The observed x contains v, and so does w. Using that v is unrelated to x* and u:
$Cov (x, w) = Cov (x^{*} + v, u - β v) = - β σ_{v}^{2} \neq = 0$
The first assumption of lesson 2 fails, so least squares is no longer unbiased or consistent.
The slope converges to a shrunken value.
With a large sample, the slope of y on x settles at β + Cov(x, w) ÷ Var(x), and Var(x) = σ²_x* + σ²_v:
$plim b = β - \frac{β σ _{v}^{2}}{σ _{x^{*}}^{2} + σ _{v}^{2}} = β \frac{σ _{x^{*}}^{2}}{σ _{x^{*}}^{2} + σ _{v}^{2}} = λ β$
Put numbers on it.
If the true x has variance 1 and the error has standard deviation 0.5, λ = 1 ÷ (1 + 0.25) = 0.8, so a true slope of 1 is estimated as 0.8. With an error of standard deviation 1, λ = 1 ÷ 2 = 0.5, and half the effect disappears. Many simulated samples in the lab settle near those values, and the reported standard errors give no warning.
Two limits to remember. Error in y alone only adds noise to u: the slope is still unbiased, just less precise. And with several regressors the damage spreads: the other coefficients are also biased, in directions that depend on how the mismeasured variable relates to them.

Step 1 of 4

Here it is on the survey. Add random noise with standard deviation 2 to every head’s years of schooling (schooling’s own variance is 13.3, so λ = 13.3 ÷ (13.3 + 4) = 0.768). Regress earnings on schooling alone: the slope is 1.807 with the true schooling and 1.368 with the noisy version, a ratio of 0.757, close to λ. (Both are simple-regression slopes, so both carry the omitted-variable bias of chapter 1, and the ratio roughly isolates the attenuation.) With town and ability also in the regression, the damage is worse: the noisy slope is 0.876 against a truth of 1.3, because once the other variables are taken out, less of the variation in recorded schooling is true signal.

Where the error is matters:

Error in x: the estimate is biased towards zero, and the bias persists in large samples. The reported standard errors look fine, so the estimate is confidently too small.
Error in y (classical, unrelated to x): no bias. It just adds to the regression error, so standard errors are larger.
With several regressors, an error in one biases the others too, in directions that depend on how it is related to them.

The remedy is better data, or an instrument for the mismeasured variable (lesson 11).

Watch out for

The mistakes examiners are fishing for

Saying measurement error always biases estimates to zero. That is for classical error in a single regressor. Error in y does not bias, and with several regressors the direction can differ.
Concluding that a small estimated effect is probably true because it is precisely estimated. Attenuation shrinks the estimate but not the confidence in it, and more data makes the interval narrower around the wrong value. In the lab’s third tab, with an error of standard deviation 1, the 95% interval holds the truth only about 11% of the time at n = 100 and almost never at n = 300.

The wrong functional form

The wrong shape, and RESET

Suppose the true relation bends and we fit a straight line. This is an omitted-variable problem too: the missing term x² is related to x, so the line absorbs what it can. The line’s slope settles on a kind of average of the true effects over the data, and is wrong as a description of the effect at most particular values of x. The residuals are positive at both ends and negative in the middle (for a convex curve), so they are systematically patterned: if the data are ordered by x, the residuals look serially correlated (lesson 10). The fourth tab of the lab shows it: the average slope is right for the middle of the data and wrong at both ends, and predictions at the ends are off.

Ramsey’s RESET tests for this kind of error without having to guess its form. Fit the model, take the fitted values ŷ, then add ŷ² and ŷ³ to the regression and test their coefficients both equal to zero with an F test with 2 restrictions:

RESET

y = β_{0} + β_{1} x_{1} + \dots + γ_{1} y^^{2} + γ_{2} y^^{3} + v, H_{0} : γ_{1} = γ_{2} = 0, F \sim F (2, n - k - 2)

Powers of the fitted values stand in for omitted nonlinear terms of the regressors. The null is that the model is correctly specified.

On the World Bank data, the straight-line regression of life expectancy on GDP per head fails: F(2, 182) = 48.2, far above the 5% critical value of 3.05. Replace GDP by its log and the failure disappears: F = 0.3, p = 0.74. In the survey, spending is truly linear in income and RESET finds nothing: F = 0.38, p = 0.68.

Three things to say in an exam. Rejecting H₀ means misspecification: an omitted variable or a wrong functional form, not which. It is a general test and so has less power than a test aimed at a particular alternative. And a failure to reject does not prove the model is right: the lab’s fourth tab shows that with small samples and mild curvature RESET often misses it.

The comparison

Which mistake costs what

Mistake	Bias in the estimates	Variance and standard errors
Omit a relevant variable correlated with x₁	Biased, and inconsistent	Computed as if nothing were wrong, so intervals are centred on the wrong number
Omit a relevant variable uncorrelated with x₁	None for β₁ (the intercept may be biased)	The error variance is larger, so precision is lower
Include an irrelevant variable	None	Larger if it is correlated with the other regressors
Classical measurement error in x	Towards zero (attenuation), inconsistent	Look normal, so the estimate is confidently too small
Classical measurement error in y	None	Larger
Wrong functional form	Slopes biased as estimates of the effect at particular x, and predictions are off	RESET and residual plots often reveal it

What to say in an exam

A relevant omitted variable costs bias that more data cannot fix, and the standard errors will not warn you. An irrelevant included variable costs only variance. Measurement error in a regressor pulls its coefficient towards zero. A misspecified form is detected by RESET, which rejects without saying what is missing.

Solving the opening question

The opening question, solved

Delhi School of Economics 2015 · hard

β̂₁ = 7.1 in Model A (with ability) and 2.1 in Model B (without), and β̂₂ = 1.9. What does this suggest?

Apply the identity.
The short slope equals the long slope plus β̂₂ × δ̂: 2.1 = 7.1 + 1.9 × δ̂.
Solve for δ̂.
δ̂ = (2.1 − 7.1) ÷ 1.9 = −2.63, which is negative.
Read the sign.
δ̂ is the slope of ability on study hours. A negative value means students who study more hours have lower ability on average: lower-ability students spend more time studying. Leaving ability out (Model B) drags the study-hours coefficient down, because study hours stand in for low ability, which lowers marks.

Answer

(B) Students with lower ability spend more time studying. Option A has the sign of the relation the wrong way round, and C reads the coefficients as a ceiling on marks, which they say nothing about.

Revise

The lesson on one screen

Omitted-variable bias: E(b₁ short) = β₁ + β₂δ. Needs both β₂ ≠ 0 and δ ≠ 0. The sign is the product of the two signs.
Sample identity: Short slope = long slope + (long coefficient on x₂) × (slope of x₂ on x₁). Exact in any sample.
Consequences: Biased and inconsistent. Standard errors give no warning. More data does not help.
Irrelevant variable: No bias. The variance of b₁ rises by 1 ÷ (1 − ρ²) when it is correlated with x₁.
Bias against variance: Omitting costs bias that stays. Including costs variance that falls with n.
Measurement error in x: Attenuation: the slope tends to λβ with λ = Var(x*) ÷ [Var(x*) + Var(v)].
Measurement error in y: No bias. Larger variance.
RESET: Add ŷ² and ŷ³, F test with 2 restrictions. Rejection means misspecification, not which kind.

Practice

Check yourself

0 of 13 correct so far

Compute

The expected value of the short slope

The true model is y = β₀ + 2x₁ + 3x₂ + u. x₂ regressed on x₁ has a slope of 0.4. If y is regressed on x₁ alone, what is the expected value of the slope?

E(short slope)

Recovering the auxiliary slope

The coefficient of x₁ is 5.0 when y is regressed on x₁ alone, and 3.2 when x₂ is added. The coefficient of x₂ in the long regression is 0.9. Find the slope δ̂ of x₂ on x₁.

δ̂

The cost of an irrelevant variable

An irrelevant variable is correlated 0.8 with x₁. By what factor is the standard deviation of b₁ multiplied when it is included?

Factor

Attenuation

The true x has variance 4 and the measurement error has variance 1. The true slope is 0.5. What does the regression of y on the recorded x converge to?

Probability limit

Undoing attenuation

A regression on a mismeasured regressor gives a slope of 0.9. The reliability ratio of the regressor is known to be 0.75. What is the true slope?

True slope

RESET

A regression with 3 coefficients on 60 observations has residual sum of squares 120. Adding ŷ² and ŷ³ lowers it to 100. Find the RESET F statistic.

The sign and size of a bias

The omitted variable x₂ has a true coefficient of −2 and a slope of 0.5 when regressed on x₁. Find the bias in the short-regression slope on x₁.

Bias

A standard error after adding a correlated irrelevant variable

The standard error of b₁ is 0.20. An irrelevant variable correlated 0.6 with x₁ is added (ignore the change in degrees of freedom and in s). Find the new standard error.

New standard error

Quick questions

1Omitting a relevant variable that is correlated with an included regressor makes the OLS estimator of the included coefficient

2Including an irrelevant variable in a regression

3Classical measurement error in an explanatory variable biases its coefficient

4Classical measurement error in the dependent variable

5A rejection of the RESET test indicates that

Real questions

Past papers, by level

Foundation

Definitions and one-step calculations: a t statistic, an R², which assumption is which. Most CUET and JNU questions sit here.

1 questionPractise →

Standard

Restricted against unrestricted regressions, reading output, and statement questions, up to GATE and IIT JAM level.

2 questionsPractise →

Delhi School level

This topic’s hardest questions, from the DSE entrance, are worked through in Lesson 18: Omitted, irrelevant and mismeasured variables.

6 questionsLesson 18 →

All 3 past-paper questions for this lesson, easiest first: practise the full set →

These are questions from the TestStop bank. A free account answers 10 a day, each with its worked solution; All-Access opens every set with no daily limit.

The Delhi School’s harder questions on this material (the sign of a bias for complements, a linear cost function fitted to a quadratic, the variance of a short and a long regression) are in the hard-problems lesson at the end of the module. What to do about the bias from an endogenous regressor is in lesson 11, on instrumental variables.

← Lesson 6Logs, quadratics and units: reading a coefficient Lesson 8 →Multicollinearity

Omitted variables, irrelevant variables and measurement error

Leaving a variable out

The mistakes examiners are fishing for

The break-an-assumption lab

Adding a variable that does nothing

Measurement error

The mistakes examiners are fishing for

The wrong shape, and RESET

Which mistake costs what

The opening question, solved

The lesson on one screen

Check yourself

Compute

Quick questions

Past papers, by level

Foundation

Standard

Delhi School level

Read next

Omitted variables, irrelevant variables and measurement error

Leaving a variable out

The mistakes examiners are fishing for

The break-an-assumption lab

Adding a variable that does nothing

Measurement error

The mistakes examiners are fishing for

The wrong shape, and RESET

Which mistake costs what

The opening question, solved

The lesson on one screen

Check yourself

Compute

Quick questions

Past papers, by level

Foundation

Standard

Delhi School level

Read next