The true regression model is

𝑦
=
𝛽
0
+
𝛽
1
𝑥
1
+
𝛽
2
𝑥
2
+
𝑢
.

However, a researcher omitted
𝑥
2
and estimated the following model:

𝑦
=
𝛽
0
+
𝛽
1
𝑥
1
+
𝑒
.

The estimated slope parameter from the second regression equation can be written as

𝛽
^
1
=
𝛽
1
+
𝐶
𝑜
𝑣
(
𝑥
1
,
𝛽
2
𝑥
2
+
𝑢
)
𝑉
𝑎
𝑟
(
𝑥
1
)
.

This means that, when
𝐶
𝑜
𝑣
(
𝑥
1
,
𝛽
2
𝑥
2
+
𝑢
)
𝑉
𝑎
𝑟
(
𝑥
1
)

Question

The true regression model is

𝑦
=
𝛽
0
+
𝛽
1
𝑥
1
+
𝛽
2
𝑥
2
+
𝑢
.

However, a researcher omitted 
𝑥
2
 and estimated the following model:

𝑦
=
𝛽
0
+
𝛽
1
𝑥
1
+
𝑒
.

The estimated slope parameter from the second regression equation can be written as

𝛽
^
1
=
𝛽
1
+
𝐶
𝑜
𝑣
(
𝑥
1
,
𝛽
2
𝑥
2
+
𝑢
)
𝑉
𝑎
𝑟
(
𝑥
1
)
.

This means that, when 
𝐶
𝑜
𝑣
(
𝑥
1
,
𝛽
2
𝑥
2
+
𝑢
)
𝑉
𝑎
𝑟
(
𝑥
1
)
<
0
, the parameter estimate will have ...

BlackTom AI · Accepted Answer

Omitted-variable bias equals the covariance of the omitted part with the included regressor divided by the variance of the included regressor. Mathematically the bias in β1̂ is Bias(β1̂) = E[β1̂] − β1 = Cov(x1, β2 x2 + u) / Var(x1). 1. Relevant knowledge points and concepts - Omitted-variable bias occurs when a relevant regressor (x2) is left out and the omitted term (β2 x2 + u) is correlated with an included regressor (x1). - The sign of the bias is the sign of Cov(x1, β2 x2 + u)/Var(x1). Var(x1) is always positive, so the covariance sign determines bias direction. - “Downward bias” means the estimator tends to be smaller than the true parameter (β1̂ < β1 on average). 2. Step-by-step reasoning - Start from the provided expression: β1̂ = β1 + Cov(x1, β2 x2 + u)/Var(x1). - Take expectations (or interpret the deterministic bias term): E[β1̂] = β1 + Cov(x1, β2 x2 + u)/Var(x1). - Define Bias = E[β1̂] − β1 = Cov(x1, β2 x2 + u)/Var(x1). 3. Intermediate steps / calculation logic - Since Var(x1) > 0, the sign of Bias = sign[Cov(x1, β2 x2 + u)]. - If Cov(x1, β2 x2 + u) < 0, then Bias < 0, so E[β1̂] < β1. That is, the estimator is biased downward. 4. Why the correct answer is correct - A negative covariance means the omitted component tends to move opposite x1; adding that negative term shifts β1̂ downward relative to β1. Hence the estimate underestimates the true slope on average (downward bias). 5. Why other possible answers are incorrect - “Upward bias” would require Cov(x1, β2 x2 + u) > 0; that contradicts the given < 0 condition. - “Unbiased” would require the covariance to equal zero, which is not the case here. - “No bias” or “inconclusive” are incorrect because the sign of the covariance gives a definite negative bias when it is negative. 6. Conclusion Because Cov(x1, β2 x2 + u)/Var(x1) < 0 implies Bias(β1̂) < 0, the estimated slope will have a downward bias. This matches the provided correct answer: downward bias.

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