文档介绍：Recitation Notes 4
Konrad Menzel
October 6, 2006
1 Grouped Data and Instrumental Variables
How does OLS work?
As usual, we start with our standard regression model
Yi = α+ Xiβ+ εi
where the Xis are scalars. Let’s just recall what OLS actually does
1 N
Yi(Xi − X¯ ) E [XY ] − E [X]E [Y ] E[XY ] − E[X]E[Y ]
βˆ= N i=1 = N N N N−→∞→
LS 1 N 2 2 2 2
¯ EN [X ] − EN [X] E[X ] − E[X]
N i=1 Xi(Xi − X)
where EN [·] is a shorthand for the sample mean. So what exactly in this line of reasoning tells us that
we have to estimate the different moments in that formula from the same data set? - the answer to this
question is that this can’t be seen in the formula because it’s not true. From what we know about the
probability limits (. we can interchange the limit operation with addition, multiplication, division ­
and actually any other continuous transformation), we could in principle replace each sample average in
that expression with anything that plims to the right value. This opens up a wide range of possibilities,
so we could .
• completely ignore problems with data which are missing at random (. independent of the values
of Xi and Yi), and just estimate the moments off plete data we have
• estimate the moments from entirely different data sources as long as we are confident that they
cover the same population and have the same sampling design.
• we could put together ”pseudo-panels” bining independent cross sections for different points
in time.
• use moments from other sources that are aggregated at a group level, . E[XiYi|group Gi], and
then apply the law of iterated expectation by summing over the marginal distribution P(group Gi)
which gives us back the unconditional expectation (this is sometimes a great option if we can’t
access microdata about a particular variable - mention Josh’s paper on the Vietnam lottery).
•...and much more along these lines
The whole trick is that we just h