Assume that you have collected a sample of observations from over 100 households and their consumption and income patterns. Using these observations, you estimate the following regression
𝐶
𝑖
=
β
0
+
β
1
𝑌
𝑖
+
𝑢
𝑖
where
𝐶
𝑖
is consumption and
𝑌
𝑖
is disposable income. The estimate of
𝛽
0
will tell you:

Question

Assume that you have collected a sample of observations from over 100 households and their consumption and income patterns. Using these observations, you estimate the following regression 
𝐶
𝑖
=
β
0
+
β
1
𝑌
𝑖
+
𝑢
𝑖
 where 
𝐶
𝑖
 is consumption and 
𝑌
𝑖
 is disposable income. The estimate of 
𝛽
0
 will tell you:

BlackTom AI · Accepted Answer

β0 (the intercept) represents the expected level of consumption when disposable income Y is zero.

1) Relevant knowledge points and concepts
- In a linear regression C_i = β0 + β1 Y_i + u_i, β0 is the intercept and β1 is the slope. 
- Interpretation of β0: the conditional expectation of C when Y = 0, i.e., E[C | Y = 0] = β0, assuming the error has zero conditional mean (E[u | Y] = 0).
- Interpretation of β1: the marginal effect of income on consumption, i.e., how much consumption changes for a one-unit change in income.

2) Complete reasoning process step by step
Start from the model:
C_i = β0 + β1 Y_i + u_i.
Take the conditional expectation given Y_i = 0:
E[C_i | Y_i = 0] = E[β0 + β1·0 + u_i | Y_i = 0] = β0 + 0 + E[u_i | Y_i = 0].

3) Intermediate calculation steps
If the standard OLS assumption E[u_i | Y_i] = 0 holds, then E[u_i | Y_i = 0] = 0. Therefore:
E[C_i | Y_i = 0] = β0.

4) Why the correct answer is correct
This shows β0 equals the expected consumption level when disposable income equals zero. It is a level (intercept) interpretation, not a rate or share.

5) Why the other options are incorrect
- "The share of consumption that is independent of income changes." Incorrect because β0 is a level (dollar amount) of consumption at Y = 0, not a share or proportion of consumption unaffected by income.
- "The average disposable income across the sampled households." Incorrect because β0 pertains to consumption, not to the mean of Y.
- "The marginal change in consumption resulting from a one-unit increase in disposable income." Incorrect because that is the interpretation of β1 (the slope), not β0.

6) Conclusion
Therefore, β0 tells you the expected level of consumption when disposable income is zero. This matches the provided correct answer: "The expected level of consumption when disposable income is zero."

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