# Problem 2.2

### In the simple linear regression model *y = β*_{0} + β_{1} x + u, suppose that *E(u) ≠ 0*. Letting *α*_{0} = E(u), show that the model can always be rewritten with the same slope, but a new intercept and error, where the new error has a zero expected value.

_{0}+ β

_{1}x + u

_{0}= E(u)

Let

u = α_{0} + v

where

E(v) = 0

Now

y = (α_{0} + β_{0}) + β_{1} x + v

which has the same slope *β _{1}*, but intercept

*α*, and error

_{0}+ β_{0}*v*such that

*E(v) = 0*.

# Problem 2.4

### The data set BWGHT.RAW contains data on births to women in the United States. Two variables of interest are the dependent variable, infant birth weight in ounces (bwght), and an explanatory variable, average number of cigarettes the mother smoked per day during pregnancy (cigs). The following simple regression was estimated using data on n = 1,388 births:

bwght = 119.77 – 0.514 cigs

### (i) What is the predicted birth weight when *cigs = 0*? What about when *cigs = 20* (one pack per day)? Comment on the difference.

This model predicts a birth weight of 119.77 ounces with *cigs = 0*, and 109.49 ounces when *cigs = 20*. That is, a pack a day lowers birth weight by slightly more than ten ounces.

### (ii) Does this simple regression necessarily capture the causal relationship between the child’s birth weight and the mother’s smoking habits? Explain.

Birth weight is likely to be a function of many factors. Smoking may be one of those, or it may just be correlated other factors. That is, this relationship does not capture a causal relationship.

### (iii) To predict a birth weight of 125 ounces, what would cigs have to be? Comment.

For this model to predict a birth weight of 125 ounces, *cigs* would have to be about -10.18. A birth weight of 125 ounces, therefore, is probably in the residual for *cigs = 0*.

### (iv) The proportion of women in the sample who do not smoke while pregnant is about 0.85. Does this help reconcile your finding from part (iii)?

For 85% of the women in this study, smoking does not predict any of the variation in birth weight, supporting the notion that a birth weight of 125 ounces is in the residual for *cigs = 0*.

## References

Problems and data from Wooldridge *Introductory Econometrics: A Modern Approach, 4e*.