斯托克、沃森着《计量经济学》第六章

Chapter 6. Linear Regression with Multiple Regressors 6.1 Omitted Variable Bias（遗漏变量偏差）

OLS estimate of the Test Score/STR relation:

n

TestScore= 698.9 – 2.28×STR, R2 = .05, SER = 18.6

(10.4) (0.52)

Is this a credible estimate of the causal effect on test scores of a change in the student-teacher ratio?

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No: there are omitted confounding factors (family income; whether the students are native English speakers) that bias the OLS estimator: STR could be “picking up” the effect of these confounding factors.

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Omitted Variable Bias

The bias in the OLS estimator that occurs as a result of an omitted factor is called omitted variable bias. For omitted variable bias to occur, the omitted factor “Z” must be:

1.a determinant of Y; and

2.correlated with the regressor X.

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Both conditions must hold for the omission of Z to result in omitted variable bias.

Example #1: In the test score example:

1.English language ability (whether the student has

English as a second language) plausibly affects

standardized test scores: Z is a determinant of Y.

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2.Immigrant communities tend to be less affluent and thus

have smaller school budgets – and higher STR: Z is

correlated with X.

ˆβ is biased

•Accordingly,

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•What is the direction of this bias?

•What does common sense suggest?

•If common sense fails you, there is a formula…

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Example #2: Time of day of the test.

Time of day of the test could affect test scores, but is uncorrelated with the student-teacher ratio (STR).

Example #3: Parking lot space per pupil.

Schools with more teachers per pupil probably have more teacher parking space, but parking lot space has no direct

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effect on learning (learning takes place in the classroom, not parking lot).

Example #4: The Mozart Effect

A study published in Nature in 1993 suggested that listening to Mozart for 10-15 minutes could temporarily raise your IQ by 8 or 9 points.

A review of dozens of studies found that students who take

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optional music or arts courses in high school do in fact have higher English or math test scores than those who don’t.

Problem: The academically better students might have more time to take optional music courses or more interest in doing so, or those schools with a deeper music curriculum might just be better schools across the board.

Evidence from randomized controlled experiments: Many

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9 controlled experiments on the Mozart effect fail to show that listening to Mozart improves IQ or general test performance.

A formula for omitted variable bias : Recall the equation,

1ˆβ – β1 = 121()()n i i i n i i X X u X X ==−−∑∑ =

1

2

1

1n

i

i X

v n n s n =−⎛⎞⎜⎟⎝⎠

∑

where v i = (X i –X )u i ≅ (X i – μX )u i .