Solutions to problem set 2
Problem 1.
(a) Zi is independent of (Yi, Xi, Wi)? Instrument relevance. Zi does not enter the population
regression for Xi. Z is not a valid instrument.
(b) Zi = Wi? The predicted X (X_hat) will be perfectly collinear with W. (Alternatively, the first
stage regression suffers from perfect multicollinearity).
(c) Wi = 1 for all i? W is perfectly collinear with the constant term.
(d) Zi = Xi? Z is not a valid instrument because it is correlated with the error term.
Problem 2.
(a) The error term is likely to include socio-economic status characteristics, such as family income,
which typically have a positive impact on GPA and are also likely to be positively correlated with
PC ownership.
(b) Parents in families with higher income are able to buy PC’s for their children: thus parental
income satisfies relevance. However, as argued in part 1, parental income is likely in itself to be
positive correlated with GPA: hence parental income does not satisfy validity. Consequently
parental income is not a good IV for PC.
(c) This is an example of a so-called natural experiment. Define a dummy variable, grant, which
is equal to one if the student received a grant and zero otherwise. Then, if grant was randomly
assigned (as stated in the question), it is uncorrelated with u: the determination of whether or not
a given student received a grant is independent of that student’s socio-economic characteristics
precisely because grant was randomly allocated. Thus grant is likely to satisfy validity.
Receiving the grant is however likely to make a student more likely to own a PC (of course, some
students who did not receive the grant might well still own computers): thus grant is likely to
satisfy relevance. So grant is likely to satisfy both of the requirements to be a good IV.
This does rely on the random allocation of the grant: if the university had preferentially awarded
grants to poorer students then the grant would typically be negatively correlated with u, because it
would be negatively correlated with parental income, and hence grant would not be a valid IV.
Problem 3.
(a) For the CEV assumptions to hold, we must be able to write tvhours = tvhours* + e0, where the
measurement error e0 has zero mean and is uncorrelated with tvhours* and each explanatory
variable in the equation. (Note that for OLS to consistently estimate the parameters we do not
need e0 to be uncorrelated with tvhours*.)
(b)The CEV assumptions are unlikely to hold in this example. For children who do not watch TV
at all, tvhours* = 0, and it is very likely that reported TV hours is zero. So if tvhours* = 0 then
e0 = 0 with high probability. If tvhours* > 0, the measurement error can be positive or negative,
but, since tvhours>=0, e0 must satisfy e0>=-tvhours*. So e0 and tvhours* are likely to be
correlated. As mentioned in part (a), because it is the dependent variable that is measured with
error, what is important is that e0 is uncorrelated with the explanatory variables. But this is unlikely
to be the case, because tvhours* depends directly on the explanatory variables. Or, we might argue
directly that more highly educated parents tend to underreport how much television their children
watch, which means e0 and the education variables are negatively correlated.
Problem 4.
Only (b), omitting an important variable, can cause bias, and this is true only when the omitted
variable is correlated with the included explanatory variables. The homoskedasticity assumption
played no role in showing that the OLS estimators are unbiased. (Homoskedasticity is only used
to obtain the standard variance formulas for the βj) Further, the degree of collinearity between the
explanatory variables in the sample, even if it is reflected in a correlation as high as .95, does not
affect the assumptions of the multiple linear regression. The problem only arises if there is a perfect
linear relationship among two or more explanatory variables.
