What you see on this slide again is the conditional independence assumption, and I would like to
come back to this assumption which we have made in the last lecture, because I know it
is perhaps a little hard to understand when you see it formally in the way it is stated
here, saying that conditional on some observable variable xi, or need not actually be observable,
but in most cases it is, the conditional on some variable xi and the treatment di,
which is either one or zero, is independent of the potential outcomes. So conditional
on xi, it doesn't matter whether an individual is treated or is not treated for the question,
what kinds of potential outcome this individual has. Or to put it another way, conditional on
xi, the distribution of potential outcomes is the same in the group of treated and the group
of non-treated people. Now, as I say, I think that is perhaps a little hard to understand.
Currently, you know that this assumption is an assumption which shall be used or which we would
like to have satisfied if we do not have complete randomization like we discussed in the previous
section of the lecture. But in which sense this is equivalent to complete randomization will
perhaps become more apparent when I discussed this by presenting an example, which I will do
in this lecture. So this is point one of this lecture that I tried to explain better than I
did last time, the conditional independence assumption. And then I will move on to show
you what we can do when the conditional independence assumption does not hold, because
obviously there's still a strong assumption. And there are many cases, one of which we have already
discussed in the section, the review on basic econometrics, which is the case of missing
observation. So if we have unobserved variables, and we come back to this case, and then we actually
also have to deal with quite a bit of math today to see how we can handle this case and what kind
of consequences it has. If some assumption, some property, some variable is not observed, even
though this variable does defect the outcomes so that the CIA is violated. Okay, we discussed this
and went to some regression framework last lecture. And then I think we ended with this
slide here, which basically said that as long as we have randomization, everything is fine.
Because if we have randomization, then we know that the expectation of y1i given d i is equal
to one. So given treatment minus the expectation of y0i given that people are not treated is the
same thing as the ATT, the ATNT and the ATE. So there is no selection bias. And in terms of our
regression equation, which we discussed, which was this equation three here, we would then know
that we can estimate this equation here, which is a very simple equation in which we just regress
the observed outcomes on a constant and on the treatment variable. And that's it. We would know
that the estimate of delta is an unbiased estimate of the ATT, the ATNT and the ATT,
because all three of these effects are just the same. So everything is fine with randomization.
Sorry, wherever we give, that's where we are. All right. Now, let's move on. And perhaps first,
we should mention that regression three is appropriate to estimate the causal effect of
treatment on the observed outcomes provided, as I have just said, that there is random assignment
of the treatment. But now the question is, what do we do if there is no random assignment of d?
So again, think of the hospital example that only the sick people go to the hospital and
the healthy people do not. So very clearly, hospital treatment is not randomly assigned.
So suppose that there is some individual characteristic which makes it more or less
likely that some individual i receives a treatment. And for this individual characteristic, we will,
for the sake of simplicity, assume that this characteristic denoted wi is a binary variable.
So the characteristic just takes on two different values, one and zero, exactly in the same
way as d takes on just two variables, one, zero. So we could, for instance, imagine that the
characteristic is the sex of the person, either male or female, or it may be the ethnicity of
a person, either white or non-white. Or you may think of other types of zero-one variables, which
we measure in terms of a binary variable denoted here by wi. And this wi is observable.
So for all our individuals, let's now collect the wi binary variable in a column vector, which I
denote by w. So w is a column vector for all of the individuals, so individual one all the way up
to individual n. And this w then takes on the values zero and one, depending on what kind of
characteristic this individual has. So for instance, whether it is male or female.
Now here, the conditional independence assumptions comes into play, because it could be that outcomes
are potential outcomes, that potential outcomes are different for women than for men.
So maybe that plays a role what sex you are for a certain type of question, for instance, for
some type of health risk. Obviously, women have a higher health risk for breast cancer than
men have. So signing a treatment against breast cancer is something which would have different
potential outcomes for women than for men to give you a very trivial example. Now under the
conditional independence assumption, we would say that the potential outcomes y1i and y0i are
independent of the treatment provided that the treatment is conditioned on w. And what I want
to explain to you now is what this in practice means that treatment is conditioned on w.
So if the binary variable, the treatment is conditioned on w, then we have to distinguish
four different situations, whereas previously we just had two. Previously we just had two
situations, namely either the individual being treated or the individual being not treated.
So either di is one or di is zero. Now we have four situations, di is equal to one when wi
is equal to one and we have di is equal to one when wi is equal to zero, obviously.
And then of course the other two possibilities, there's no treatment and wi is one and there is
no treatment when wi is zero. So there are four combinations of the two binary variables. This is
actually why I took just a binary variable as wi because then I can limit my attention to four
cases, whereas obviously if wi could take on more values than just two, then we would have many more
cases than I have here. And it is easier to understand the whole thing when we just have
four cases. So the conditional independence assumptions now implies that for a given value
of wi, the di is independent of the potential outcomes or the potential outcomes are independent
of di. So there's mutual independence between the two outcomes on the one hand side and the di on
the other side. Or to put it again in terms of formula, we know that wi is equal to one. So this
would be the first case and the third case. Then we would say it is completely random whether a
person receives treatment or sorry receives treatment which would be up here or does not
receive treatment as it would be here. So given that person is say female, it is completely
random on whether the person receives treatment or not. And given that the person is male,
it is also completely random whether the person receives treatment or not. But it may be that,
for instance, you have many more cases where you administer some kind of medication against
breast cancer. If the person is a woman, then you have if the person is a man. So in this case,
the assignment to treatment may of course vary by much depending on the sex of the person that
either receives or does not receive the treatment. Obviously, probably there's not so much sense in
giving treatment to male persons unless perhaps one is interested in negative effects of
the medication. So for this, since men and women also have some things in common,
it may perhaps be also useful to study this with men, but I'm not an expert in this kind of
experience. All right. So essentially, the conditional independence assumption tells us
that we can think of the treatment variable being randomly assigned once we control for the
influence of the WI, once we distinguish the individuals by their WI status. We would say
that in each WI class, the treatment is randomly assigned. This is what the conditional independence
assumption means when it says that conditional on WI, the potential outcomes are independent of the
treatment. So what we might think of is then that we work with the regression specification of Form
9, that what would probably be the most natural way to proceed, that we say we regress the observed
outcomes now on a constant on the treatment variable and on the W variable, which is the
variable conditioning the treatment. And then we have an error term, of course. So I will try to
explain to you that this is not such a good idea. And it is important that you understand why this is
not such a good idea. So think about it properly now. Suppose we estimate this equation 9 here,
then the question is whether this properly controls for the influence of W. And to check this question,
the easiest way is that you look at the expected value of the observed variables conditioned on
what the D and W variables take on as values. So we would have the four cases that I just
discussed. The expectation of why a given treatment and given WI is equal to 1 is beta 1 plus delta
plus gamma, because all you have to do is an equation 9. You set D equal to 1, you set W equal
to 1. Expectation of U is, of course, always 0, so forget about it. So if both the variables are 1,
then the expectation is obviously beta 1 plus delta plus gamma, right? Beta 1 plus delta plus gamma.
If we take the expectation of a WI given treatment and WI is equal to 0, then by the same token,
the expectation is beta 1 plus delta, because W is 0, so gamma is not expected.
The expectation of why I give non-treatment WI being equal to 1 is beta 1 plus gamma.
And the expectation then with both variables equal to 0 is just beta 1. So that's fairly easy.
And it seems actually, well, this looks like something rather systematic with what we do here,
so where's the problem? The problem is here. We have on the previous slide four equations,
so four conditional expected values for just three parameters, which implies that the parameters
are over-determined. And to understand what this means that the parameters are over-determined,
look at parameter delta. Delta is the causal effect of treatment, so this is the key parameter
we are interested in. Now, we can always compute estimates of these expected,
of these conditional expectations here, right? So think of the conditional expectation here as
something which you approximate by taking just the average of the observations. So this would
mean take the average outcome variable for all those which received treatment and which are,
say, female. And here we take the average of all the outcomes for those which received treatment
and are male, and these are the two other cases. So we would have four equations, because we can
compute those averages, so these are data-driven numbers which we would have here. We would have
four equations in three variables, beta 1, delta, and gamma. So very clearly there's a problem with
solving the system of equations, because actually we are missing one parameter here. So for instance,
in this case, we could solve for delta in the following way. We could say delta is apparently
the conditional expectation of yi for the case treatment and wi is equal to zero,
minus expectation of yi for the case of non-treatment and wi equal to zero. Let's
check this on the previous page. So this was with two equations with wi equal to zero, so this
equation here and this equation here, we had to take the difference between treatment expectation
and the non-treatment expectation. So you see, beta 1 plus delta minus beta 1 is data. So that would
be one way to estimate delta by looking at the difference of these two conditional expectations
of the sample averages which match these two expectations here. Obviously, the second line
here is of less importance, but I've written down it anyway. We know that the expectation of yi given
treatment is of course the expectation of y1i, because this is the potential outcome in the case
of treatment. So knowing that there was treatment, we would have the expectation of y1i of the
potential output y1i, because this is actually what we observe. And conversely, here it is the
expectation of the potential output in the case of non-treatment, because given is non-treatment.
Well, the point is not the second line, this is more or less trivial. The point is that this first
line here need not be the way how we estimate delta. Because we could estimate delta also by
looking at the expectation of yi given treatment in the case wi is equal to one, minus the expectation
of why i give non-treatment and wi is equal to one. Because, go back to the system of equations
here, now we take the cases where wi is equal to one, so this case here and this case here,
and if we take this expectation minus this expectation, then you see this is beta one
plus delta plus gamma minus beta one plus gamma. So what we are left with is exactly the delta.
So we have actually two different ways here to estimate the delta, and we would come up with two
estimates delta, because in general these sample averages here and there on the difference would
be different from the difference of the sample averages in the case of wi being equal to one.
There is no reason why they should be the same. So we would come up with two different estimates
of delta, and we would not know which difference is actually true, perhaps both are wrong. So that
shows you that there is a problem actually with equation nine, because equation nine would not
allow us to arrive at a unique estimate of delta. Rather we would get two estimates of delta, and so
that wouldn't be very helpful. This is like if you look at two watches and they show different times,
then you don't know how late it is. So we know that it's hard to interpret the regression
coefficient now in nine unless we assume that the causal effect for wi equal to one is the same
as the causal effect for wi is equal to zero, which basically would mean that wi doesn't matter
for the potential outcomes. This is what the second lines in these equations tell you here.
So if it is true that there is no difference between this difference here and that difference
here, then of course it would basically mean that for the causal effect the wi doesn't matter.
That could by coincidence be the case, but in general it will not be the case. So that's not
really a solution. So we have the implicit assumption that the effect of di being equal to
one and wi being equal to one on the potential output equals the sum of the separate effects
of having either di is equal to one and wi is equal to zero and wi is equal to one with di being equal
to zero. So implicitly we assume that the effect of di being equal to one with wi equal to zero
plus the effect of wi being equal to one with di being equal to zero is the same as the effect of
di equal one and wi is equal to one. Now this assumption is not innocuous because it may be
the case that the effect of having both variables being equal to one simultaneously
is different from the sum of the effects of having either this variable equal to one or that variable
equal to one separately. So that's the main problem of equation nine that implicitly we make this
assumption that having both phenomena together is the same as the sum of the separate phenomena.
Since we have no reason to assume that this is actually the case we should consider
alternative specifications to equation nine and this type of alternative specification we would
get if we also allow for an interaction effect when di and wi both take on the value of one.
So when we know that the product of di times wi is equal to one right so the two things are
simultaneously equal to one if and only if the product is equal to one because if either of these
things were zero then obviously the product would be zero so di wi is equal to one only if both di
and wi are equal to one. So this is what we call an interaction effect there the fact that
vi is equal to one so that we have treatment interacts with the fact that wi takes on value
one and therefore equation the regression equation we think about now would be the observed outcomes
are explained by constant by treatment by the observable characteristic let's say
say female and male and by the interaction term between treatment and the the observable
characteristic so that we have a new coefficient beta four here and take this to be beta four
because implicitly i would say this here this delta is beta two and the gamma is beta three
even though i have given them different names but if you think of the whole parameter vector
as a vector of dimension four then obviously it makes sense to call the fourth component beta four.
So what we do now is that we collect all the multiplicative terms di wi in a new column thing
much the same way as before we have collected all the wi's in the column vector now i call this new
column vector z right so z is the product the interaction of d and w for all the objects of
observations we have so for all the new videos so we have d one w one here and di wi somewhere in
here and d and wn for the nth observation in here so that's a column vector of the product terms
and with this vector z we can then write equation 10 in vector notation as y is equal to beta one
jota plus delta d plus gamma w plus beta four z plus inheritor and if we now evaluate the conditional
expectations again for the four cases of treatment non-treatment and wi being one or zero then we see
that the expectation of yi in the case of both binary variables taking on value one is simply
beta one plus delta plus gamma plus beta four but if we switch off wi then the expectation is
only beta one plus delta because if we switch off wi so if this takes on variable value zero
then obviously here would be a zero for w and here would also be a zero because this is b times w
right so gamma and beta four would drop out and we are just left with beta one plus delta
however if we switch off the treatment and keep the wi at level one then the same thing occurs
just that the expectation would be beta one plus gamma because in this case we have switched off
the treatment here so this is zero the gamma continues to be gamma because w is equal to one
and this interaction term of course is also switched off
well in the case of non-treatment and wi being equal to zero obviously the value is beta one
so now with equation 11 we have four equations here and we have four parameters so the system
is well determined
which means that each parameter has a unique solution in this system of equations and you
can compute these solutions and i won't go through the computations here but it's obviously very easy
to solve for the coefficients here you can work through this at home so you would have a unique
representation of each of the coefficients here which means that you can estimate them actually
by simple averaging over the four classes of observations that we have depending on treatment
status and wi status now since we know that the di is given in each expression as either one or
zero so that in each expression either we have treatment or non-treatment we can then use this
representation which links the observed outcomes to the potential outcomes to replace the observed
outcomes in these equations here so there's always the observed outcome in here yi by the potential
outcomes which are implied by the treatment status so we can replace the yi here by y0i
right the yi here by y1i because here treatment status is one and so forth so we would have
the expectation of the potential outcomes in these expressions here
so we would find taking now the solution for delta so first line here is just this line
here it is the expectation of yi given treatment and wi is equal to zero minus expectation of
why i give non-treatment and same case wi equal to zero right so this is this line here
expectation of y1i apparently because treatment is one wi zero minus expectation of y0i because
treatment is zero still wi being equal to zero so this thing here we can now solve and we can say
that we know the expectation of y0i given di is equal to zero wi is equal to zero is the same
thing as the expectation of y0i given di is equal to one wi is equal to zero under the conditional
expectation assumption right under the assumption that conditional on wi being equal to zero
the di's the treatment status is actually as good as randomly assigned so the expectation of the
potential outcome y0i would not depend anymore on whether we have di equal to one or di is equal to
zero given that we know the w both cases refer to the wi equal to zero status so we can just exchange
the treatment status here and there so going from this line here we can come to this line here
or alternatively from this line here we can come to the third line here so you should read this
equation not as that i move from the second line to the third line which of course i can also do
but it is easier to understand when you read it as going from the first line to the third line
because from the first line here we would come to the third line by just replacing treatment status
here which means treatment by non-treatment again arguing well the conditioning variable wi
is the same in both cases so we know that treatment status is as good as randomly assigned
so it doesn't matter we have or the expectation of one y1i it doesn't matter whether we have
treatment or whether we have non-treatment the expectation is the same under the conditional
conditional independence assumption so in both cases we would have now the expectation of y1i
minus y0i for the same type of conditioning variables right di is one w is i is zero
same thing here or di is zero wi is zero same thing here so this means that this is the same
thing as the expectation of y1i minus y0i given that wi is equal to zero because in both lines
here we had wi is equal to zero which means that delta now we know how to interpret parameter delta
delta is the causal effect of the treatment conditional on wi being equal to zero so say
wi being male right i said one is female zero is male so in this case delta would be
the treatment effect for the male objects similarly we could explain what beta4 so we
could interpret beta4 so we would again start with the definition of beta4 so this first
equation here which involves these two differences right this is just this
expression here for beta4 okay so this expression here for beta4 we can again transform
by exchanging the treatment variables in this case i i have replaced the di equal to zero
here by di equal to one which i can do assuming that the cia holds because both is conditioned
on the same observable characteristic status so same six but for instance wi is equal to one
in this case here right and i do the same thing here moving from non-treatment to treatment for
the case of wi is equal to zero that's the same expectation again on y zero i here there
or y zero i here and there so we can write the whole thing as the expectation of the difference
between the potential outputs but given wi is equal to one right this would be here
exactly this is the difference of expectation of y1i minus expectation of y0i so this thing given wi
is equal to one minus the expectation of the difference of the potential outputs here given wi
is equal to zero which would be this difference here right between these two terms
so now we know the second term here however is the data right this we have computed on the
previous page the expectation of the causal effect given treatment status is male wi is equal to zero
this is delta so we know that beta4 is the same thing as well what is this here this is the
expectation of the causal effect given the treatment status is female right so same thing
actually as this term just in this term we have wi is equal to zero male and here we have wi is
equal to female is equal to one so it's female so we have here the difference of beta4 matters
to us the difference between the causal effects between female and male participants in a study
or in a treatment so beta4 is the difference between the causal effects conditional on wi
equal to one and wi equal to zero and remember that in the previous misspecified equation nine
where we had just three parameters uh we had i had said that implicitly we assumed that there
is no difference between the causal effects conditional on wi equal to one and wi equal
to zero because implicitly we have assumed that beta4 is equal to zero in equation nine
right when we go back to equation nine you see this here right there was no beta4 term here we
had this equation we were just regressed on treatment and on w right no beta4 no interaction
effect we knew this was not a good idea to do this because the system of equations is
over determined that two solutions for for delta so the basic problem is that we did not have
an interaction effect here which we later allowed for an equation 10 with a beta4 where beta9 was
of course a special case of equation 10 because this was the case where we assumed that beta4 is
equal to zero now we know what beta4 measures beta4 is the difference between the causal effects
conditional on either the individual being female or the individual being male so in equation nine
we had implicitly assumed that there is no difference on whether treatment applies no
difference in the causal effects to as to whether treatment applies to a woman or to a man
right but in general of course we should allow for such differences and therefore we have
interacted therefore we have allowed for the interaction effect
good this was the first important point i wanted to make in this lecture are there any questions
then please raise your hand if this is not the case then i will move on just with a brief
definition what we have just done is to get a feeling of what if what is meant by a saturated
regression model and here's the formal definition a regression model is called saturated if there
is exactly one parameter for each possible combination of the values of the explanatory
variables right so a regression model is called saturated if we have as many equations in terms
of conditional expectations of the outcome variable as we have parameters in the model
so very clearly model three which was the the first model we studied in the set of slides
y is equal to constant plus delta d plus error is a saturated model because there are just
two values the variable d can take on and so we have two conditional expectations and we have
two parameters beta one and delta therefore the model is saturated i have two conditional
expectations conditioning on the status of d on the treatment status and two parameters so model
three is a saturated model model 11 the model we have worked with last is also a saturated model
because there we have two explanatory variables d and w z being a derived
regressor as the product between d and w as you know so essentially there are two
statuses namely d is equal to one or zero and w is equal to one or zero so excuse me
there four different cases right as you know each variable taken on two values but four different
cases when we combine them so we have four different conditioning situations and we have
four parameters as we have worked out beta one delta gamma and beta four so model 11 is
saturated since we have as many parameters as there are combinations of d and w when two
variables are both binary whereas model nine was not saturated as you as i think you you know
understand we have just three parameters but we had four combinations of the possible outcomes of
d and w and therefore we were actually lacking one parameter so the model is not secure
so this seems like a really easy guide for us as to how how to specify a regression equation in
a causal setting you may say well not now we know how to do it the only thing we have to do is that
we have to provide always as many parameters as there are combinations of the explanatory
variables which define our conditional expectations so that seems easy
unfortunately it is not quite as easy simply for a problem of dimensionality
suppose we have three binary explanatory variables so we not only observe male female
but we also observe ethnicity race or however you want to call it white non-white right then with
three binary binary explanatory variables we want to have two to the three combinations so eight
different combinations which condition our expectation so this means that we need already
eight parameters now suppose that we have three explanatory variables with possible values minus
one zero and one each you know with gender now it is fashionable to say we just we don't have
two genders but we have at least three so male female and something which is either indeterminate
or split up again in different things let's just say it is male female and diverse
right and we have perhaps also three different ethnicities which we would like to consider so
for instance we have white and we have black and we have hispanic right so if that is the case that
we have three explanatory variables with three outcomes each we would already need 27 parameters
so in the saturated models the number of parameters can easily explode
meaning that if there are many different situations we have to take into account
among our individuals then the number of parameters can easily explode to such
dimensions that they cannot be usefully estimated anymore that happens already with three variables
and three outcomes you can imagine what will happen if you have more or less continuously
measured variables then obviously we would need infinitely many parameters if we take that
seriously so the saturated model is not always the solution to the problem of not having random
assignment because there may be just too many parameters which we would have to estimate given
a limited number of observations so for this reasons researchers often prefer to work with
models which are not saturated and that is as i have pointed out a problem because then the
parameters are over determined or involves an assumption that certain interaction effects are
actually zero so that beta four is equal to zero so mostly interaction effects are not exactly
zero but we can make of course assumptions that some interaction effects are small or negligible
so that doesn't play a big role if we have them included in our regression equation or not so
that is what this is what what researchers typically do when they work with models which
are not saturated that they make either explicitly or implicitly the assumption that certain
interaction effects are unimportant right this reduces the need for the parameters of the
interaction effect yeah so that's the way to handle this this problem if the curse of
dimensionality hits you i would strongly recommend that if you ever do such a type of work
then be aware of the problem and make explicit the assumption which interaction effects you
think are negligible because only then can you be sure that you identify the remaining
parameters in your model properly and only then can you also invite either other researchers
or do it yourself to test the explicit assumption that certain interaction effects are
negligible which may well be the case i mean it's not the case that always all interaction
effects matter greatly actually many may actually be just zero but you should be aware what you assume
what kind of interaction effects you assume to be zero and you should try to check as best as
possible whether this is possibly the case now first thing of course is that you should make sure
that or if the cia holds and if you think the cia does not hold then think about why it doesn't
hold and next think about what you may do about it all right so much for saturated model and
appropriate model specification if you want to use simple least squares procedure to estimate
the causal effect now we come to the next section and we will talk about the omitted
variable bias something we have already covered in the review of basic econometrics so much of
what i will do now will sound familiar to you but perhaps it is worth to to repeat the exercise and
i will point out some new aspects of omitted variable bias that we haven't studied so far
it is also something called the problem of unobserved heterogeneity which is basically
the same thing that people are just not equal along some dimension which we do not observe and
if we do not observe this dimension then obviously we omit some variable in our regression
so basically that's a section on violations of the cia that's a section on what can we do if
the conditional independence assumption which we have now discussed is is violated does not hold
in reality well that's actually a very usual case in applied burg there are many cases where we can
think of important variables which we cannot observe for instance if you want to explain
wages in some labor market study then one approach is typically that you relate wages to the type of
formal education the wage earners have received that's a variable which we can easily measure so
we can see what kind of school education or college education they have perhaps we even
know the grades they have received but you may also say that wages depend on some type of ability
of the birth and the ability beyond formal education is not usually observed so this would
be an omitted variable which has not been measured and which we can therefore not use to
explain the the endogenous variable wages therefore we would have the omitted variable bias
and what we want to do in this section here which as I already said at the beginning of the lecture
will be somewhat formal and mathematical and which we can handle in a certain way by trying
to get a feeling of what size this omitted variable bias actually has so that's actually
the important lesson you should learn in this section here that even in cases where we know
that we lack a certain variable that we omit a certain variable which is just not observed
then in these cases where we know that the coefficient which we estimate on the
observable variable is biased due to the omitted variable bias even in these cases we can try to
get a feeling of how big the bias is and therefore we can actually correct try to correct the bias
so this is what is the subject of this rather technical section here try to bear this in mind
if I lose you on my way through the math that is to follow
okay the important point which is sort of the clue which we use in order to assess the size
of the omitted variable bias is that the unobserved variables are often correlated
with the observed explanatory values actually as you will see if the omitted variables are not
correlated with the observed explanatory variables then we don't have a problem
right so that's fine but if they are correlated if the omitted variables are correlated then the
estimate of the observed variables of the coefficient of the observed variables will
be biased and that's a problem we compute what size this problem has and we will get
an idea on how we can correct this bias even though the one variable is still unobserved
so in the case that I just already gave you where we want to measure wages and explain them by
education and ability it is quite likely that ability is correlated with education because
students which are more able probably achieve higher degrees in education
there's probably not a one-to-one correlation but we can be pretty sure I would say in my common
sense simply that ability is positively correlated with education we may actually already have some
kind of idea how high this correlation is I mean everybody would agree the correlation is not
perfect so it's less than one obviously but it will be it will be positive and probably most
people would say that's a sizable correlation so you would probably not think that's a low correlation
like point two or point three or something like this more likely you would probably say it's a
correlation perhaps in the in the range of point seven so you may have an idea of how big the
correlation with the observed explanatory variable is either just intuitively like I'm
basically at it right now or perhaps because you have looked at other studies in other settings
with other data where ability was somehow measured and observed and somebody came up with an estimate
of saying well education and ability are correlated to a certain extent and the correlation is point
seven for instance right with a certain confidence of course but I mean that's intuitively
without having studied the matter I would say the correlation between ability and education is not
perfect but it is substantial and if I were asked to just give you any number of how high the
correlation is then I would probably say it's point seven it may also be point six or point eight or
something like this but I would probably opt for vote for a correlation higher than point five
anyway that's my subjective meaning of that now if we have a correlation between the observed
regressor and the unobserved variable which also explains the end of this variable then this
implies as we have already computed in the review of basic econometrics that we have a correlation
also between the observed regressor and the error term I will show this to you again in the next
slides but you may recall that we had this result already and that we had pointed out that this
violates the Gauss Markov assumption a2 so the exogeneity of the regressor matrix is not
given anymore it doesn't hold this assumption and it not only violates the Gauss Markov assumption
a2 strict exogeneity but it also violates the weaker assumptions which we had made to
guarantee the consistency of the estimate so to guarantee an asymptotically unbiased estimate
so these were assumptions a6.1 a6.2 and a6.3 that would also be violated if there is correlation
between the observed regressor and the unobserved omitted variable
now if a regressor is correlated with the error term of the regression we typically also speak
of an endogeneity problem which I have on the issue of econometrics these type of endogeneity
problems can arise for different reasons so in the review of basic econometrics we already
treated the errors in variables a problem which arises from say measurement terms currently we
do not really speak about measurement errors even though this is also a problem which is very
important in any kind of empirical research but we speak about an omitted variables biased due to
the fact that some important variable we cannot measure it's the same type of problem actually
but the cause for the problem is very different so these type of endogeneity problems are
endemic in the social sciences meaning they occur very very often they are perhaps more the rule
than the exception and there are various approaches to handling them which we will deal with in the
course of a lecture so one possibility is the so-called proxy variable approach another one
is the instrumented variable approach and then there's also the possibility of using panel data
regressions and as I said we will deal with them one by one
so now first to repeat what the problem is why is there an endogeneity problem if an unobserved
regressor correlates with an observed regressor so why actually does this translate into an
endogeneity problem where the observed regressor correlates with the error term question is so
what are the consequences of our exclusion of a relevant variable maybe involuntary this
exclusion simply because we don't measure the variable would like to include it but what we
cannot include it because it's not measured now let's first look at a true model so I call this
the true population model and this is why I use capital letters here so these are all vectors if
you like or actually they are random no sorry not vectors they are random variables right
which materialize which realize themselves say over time or in a cross-section infinitely many
times if you like so these are not yet the vectors which we observe but they are the random variables
which give rise to the observations which we make and which can collect them in but that's what I
mean by the true population model if you recall y is equal to a constant plus beta one times x one
plus beta two times x two plus an error term u and u as I say it's just a random variable so all
these are scanner random variables now suppose that this true model here gives rise to a number
of n observations so the n observations on y x one and x two we can measure obviously we do not
observe the use right this is the the error term but suppose we can measure the x one and the x two
and the y then we can order them in column vectors of size n by one so we would have column vectors
small y small x one small x two so this would be the setting where we observe both x one and x two
suppose y is wages x one is education x two's ability that would be the ideal case where we can
measure a variable which usually we cannot measure we will later consider the case where we cannot
measure variable x two now when we can make use of all available observations then the regressor
matrix which we have is x which consists of a vector of one's yuta plus vector x one and
vector x two so it's a matrix a regressor matrix with three columns and n rows
we will assume that this x has uh the usual property so in particular it satisfies the
Gauss Markov assumptions of course that's full column rank so we can just estimate the usual
regression model which i have written here in linear form or here in matrix form the way you
are used to advise equal to x beta plus u beta then of course it's just the parameter vector
with the components beta not for the constant beta one for x one and beta two for x two the
estimates i denote by hats as usual so beta hat is x prime x inverse the x prime y or x plus the
little penrose inverse times y so this is what we would like to estimate we would like to observe
column vector x two say ability but now um suppose that this is not the case suppose that we cannot
observe x two so we have to exclude it from the regression or which is the other possibility
we have not thought probably about the causal relationship and do not know or do not suspect
that x two also affects y so involuntarily we exclude x two from the regression equation
because we have not thought of the idea that x two also has an input whatever the reason is
we do not exclude we do not include x two now in the regression so in this case the regressor
matrix is x tilde and consists just of the vector of once and x one and the regression model we
estimate is then y is equal to beta zero times uta plus beta one times x one plus some new error term
nu i will denote this in matrix mutation as x tilde times b plus nu x tilde is of course the
shortened regressor matrix with a third column missing and i'll now write b rather than beta
because beta was the vector which consisted of beta zero beta one and beta two okay but b consists
just of beta zero and beta one so that's a short form of beta now what is the nu since we know that
the true model is actually beta zero uta plus beta one x one plus beta two times error plus error
we know that the nu is now beta two x two plus u so here plus beta two x two i think i just forgot
to mention the x two plus beta two x two plus u that would be the complete equation and this is
now included in the new error term nu so previously we had beta two x two plus u here since now we
do not include the x two in the regressor matrix automatically the term beta two x two becomes
part of the error term and this is why we give a different symbol to the error term now which is
nu right which is this thing here so the neglected random variable x two is now part of the
unobserved error term which makes sense because the error term captures everything which is
unobserved so in this case also the x two and also the coefficient beta two
now since the random variable nu or i call it here capital v that would be the random variable
all right here i call it small nu actually this is the realization and
um this here is the random variable
since the v contains uh the x two it follows now that the v is correlated with the regressor
matrix x tilde if x one is correlated with x two right so if the random variable x one
which explains why somehow is correlated with the random variable x two which also explains
why then we now know that the error term in the regression v is correlated with the regressor
matrix x tilde because we have the x two in here and the x one in here and x one and x two are
correlated so there would be correlation between v and x tilde and this violates the cosmarkov
assumption a two as i have pointed out and also violates um a 6.1 6.2 a 6.3
so we must expect that the OLS estimate of equation 15 is now biased so when we estimate
this equation 15 here we must expect that beta zero and beta one are estimated in the biased way
so the estimates won't make much sense or they are not really reliable when we have the omitted
variables problem denote these biased estimates by tildes so b tilde is equal to x tilde prime
x tilde inverse x tilde prime y in the usual way so that's the bupenro's inverse of x tilde times
y the corresponding residuals are nu tilde which is i minus x tilde x tilde plus times y
and you may recall that we called this matrix m usually so here i will denote it by m tilde
that's a projector matrix it is idempotent and actually symmetric i forgot to mention symmetric
here but it's not symmetric so we know that m tilde is the same thing as m tilde times m tilde
and you can easily verify this by just multiplying this out using the properties of the bupenro's
inverse so m tilde is is idempotent now let us check if the estimate b tilde is biased or unbiased
i mentioned here already since v and x tilde are correlated we would expect
that the estimates are biased so let us check if this is the case or let's check if b tilde is
unbiased what we do know is that the ols estimate the ols estimate beta hat is unbiased in the full
regression where we had assumed that we could observe x2 and where we included x2 as a regressor
in the regressor matrix so beta hat was fine i mean this satisfies the ols markov assumption
as we have assumed and the estimate is unbiased what we now want to do is that we relate the
estimate b tilde which we expect to be biased to the estimate beta hat and in particular we want to
relate the estimate we get for the first coefficient of for beta one beta one tilde
ooh beta one hat okay so this is the coefficient of the variable x1 which we observe
so ideally we would estimate that as beta one hat but now where we have the omitted variables
problem we estimate beta one tilde and what we would like to know is by how much differs the
estimate beta one tilde from beta one hat for this purpose it is helpful to consider an auxiliary
auxiliary regression of x2 and x1 and a concept now you may say well what what sense does make
to consider this auxiliary sorry this auxiliary regression because we do not know what x2 is we
don't observe it for us the omitted variable which we ignore so we cannot actually regress it on x1
no we don't we cannot do it in practice but we can do it in theory we can just write down
how the estimate would look like if we had knowledge of x2 so suppose we do this type
of regression because we know what x2 is so in this case we would regress x2 on x1
and of course on a constant and w would be an error term so we would run this regression x2
is constant gamma naught plus gamma one time times x1 this is the regressor and then there
would be an error term well actually the regressor matrix in this regression here is again x tilde
as you note because x tilde was the regression matrix in the regression with the omitted
whether the omitted variable was missing it consisted just of a constant and of x1
so this would be x tilde times a parameter vector gamma which consists of gamma zero and gamma one
and then there would be the error term so the least squares estimate of gamma would be
gamma tilde is equal to x tilde prime x tilde inverse x tilde prime x2 and that's of course
the peneros inverse of x tilde times x2 good now before i shock you with the next slide
let's still consider what we are doing here we have
um
the regressor matrix x which consists of yuta x1 and x2 and we know that the submatrix which consists
just of the first two columns yuta and x1 we have denoted by x tilde right so this is x tilde here
these two columns and the third column is x2 so basically what we have done is or what we may
do is that we write this x here as a partitioned matrix where it consists of two blocks the first
blocks block is x tilde and the second blocks block is x2 now you know that in computing the
ols estimate beta hat we have to compute the inverse of x tilde prime of x x prime x
so x we can write in this block form as x tilde and x2 and of course we can take the transpose
of this and write the block form here again then when you multiply out this in partitioned form
then you will actually arrive at a matrix which consists of four blocks the first block being
x2 the prime x tilde and the second block being x tilde prime x2 and the third block
in the second row being x2 prime x tilde and then x2 prime x2 right so you would have four blocks
and what we now would like to do is take the inverse of such a matrix here which consists
of four blocks and there's actually a theorem which tells us what this inverse looks like
and this is the theorem I give you here you may have seen that in a matrix algebra course but I
suppose most of you haven't it's just the result of matrix algebra and this even though it looks
complicated there is not much to understand about it just except that the formulas are true you can
actually easily verify that the formulas are true so what this theorem says that if a matrix is
partitioned into four blocks it can be inverted blockwise suppose we have a matrix of course must
be squared a square matrix otherwise the inverse wouldn't be defined the matrix is partitioned
into blocks a b c and d and we take the inverse of this matrix then the inverse is of the following
form here and I won't read it out to you the only thing you have to note is that down here
or actually in the other expressions too you see a matrix f or the inverse of f there was no f in
here this was just a b c d when f is defined here f is itself something somewhat complicated
expression and d minus c a inverse e b so f is related and actually made up of the four
blocks here and the inverse of this thing appears in here there's nothing to prove about that
actually you can just multiply out to convince yourself that this here is true it is complicated
I admit it but it's actually easy to verify just multiply this matrix here by the matrix a b c d
and you will see that the result is the identity matrix right well comes out really neatly and
easily so it's easy to convince yourself that a equation 17 is true by the way the situation
is symmetric because the inversion can also be computed in the following way a b c d inversi so
the same inversi is here can also be given this representation here in which we have again the
matrices b and c and so sorry b and c and d and then there is a new matrix g which was not in this
matrix here g is related to a b c d in this form it's a minus b b inverse c c and you can verify
again that this partitioned matrix here is the inverse of this matrix by just multiplying out
very easy actually essentially the difference between this representation and this representation
is of I mean the difference is zero so it would be the same matrix either you can compute it either
way to come to the same matrix but sometimes this representation is more useful and sometimes
this representation is more useful the difference basically being that this constructed here always
gives rise to sort of simple inverse here f inverse like d inverse here so it's just a matter
of taste or suitability in certain calculations whether you use representation 17 or 18 depending
on where you want to have the simple inverse either in the one one corner or in the two two
corner so here rather simple inverse and then the rest of the definitions would of course
be adjusted accordingly so that you get the right inverse there so as I say there's no economics
behind that this is just my matrix algebra the formula look complicated they will be somewhat
complicated and what follows but it's actually easy now we study our blue we know that if we have
both variables at our disposal x1 and x2 then we can estimate beta hat and this is the best linear
unbiased estimator because we have assumed that the Gauss Markov assumptions hold the blue is x
prime x inverse the x prime y we know that x as I have already pointed out is x tilde and x2
so we can partition matrix x as x tilde x2 and therefore x prime x is the same thing and as x
tilde prime x tilde x tilde prime x2 x2 prime x tilde and x2 prime x2 just multiply out blockwise
you'll easily find this result here so when we want to look at the beta hat here perhaps
just that you don't get lost on the way what we wanted to do is we wanted to relate the beta hat
to the b tilde estimate right I told you somewhere was it
oh yeah okay I told you we want to relate the b tilde for the incomplete model
to the beta hat and the b tilde was estimated from the x tilde matrices
so now you see we start here with the expression of beta hat and here are the x tildes here's
already x tilde prime x tilde and actually we have to take the inverse of this expression here so you
may expect there will be somewhere the inverse of x tilde prime x tilde which is part of the
expression of the b tilde so that's the way how we relate beta hat to b2 so all we have to do is
then use the partitioned inverse formula which I gave you on the previous slide namely in form
17 so the first of two forms use this to invert this matrix here in this case f is actually
scanner because f would be the inverse which comes in this two two place here and you see
this object here is already scalar because x2 is a vector so x2 prime times x2 is just the sum of
the squared components of the x2 vector so this is just a scalar it's a real number which is here
which implies that the f is also a scalar just a real number so it's not a complicated matrix it's
just a just a real number okay so the f is scalar and is actually equal to the sum of the squared
residuals of the auxiliary regression 16 which I had given to you where we regressed x2 on x1
and a constant so let's see that this is true f is equal to d minus ca inverse cb
well if you plug this in then you get d is x2 prime x2 minus and now you take a look at what
is ca inverse cb so you get this expression here you can reproduce that at home you see that here
we have x tilde x tilde prime x tilde inverse x tilde prime between x2 prime and x2 if here's
also x2 prime and x2 so this is the same thing as x2 prime minus as a x2 prime times m tilde
times x oh this should be x2 not xx okay this is x2 here so this is x2 prime m tilde x2
and well you know m tilde is either potent so we can also write this is the same thing as x2 prime
m tilde prime m tilde x2 what but x2 prime m tilde we had seen is the estimated error of
the auxiliary regression so this is w tilde prime w tilde and therefore this is the sum
of the squared residuals of the auxiliary regression 16 and it's just the real number
a positive real number
now this simplifies the one one element of the inverse because the one one element of the
inverse c is a inverse plus a inverse b f inverse is ca inverse which sounds complicated
but the f inverse is actually just one over well sum of squared residuals here and then you have
to compute what is a inverse b times ca inverse because you can pull the constant factor out here
and you see this is x tilde prime x tilde inverse x tilde prime x2 multiplied by its own transpose
i don't know if this is so knowing you here multiplied by their own transpose well and this
thing here looks like an ols estimator and in fact it is it is the gamma tilde again from the
auxiliary regression so this is the estimate of gamma in the auxiliary regression that's actually
why we used this x tilde or when we computed the auxiliary regression so that's gamma tilde times
gamma tilde prime and here we have x tilde prime x tilde inverse c plus well the squared product of
the estimated gamma tills and divided by the sum of squared residuals and we can do similar
computations for the one two and the two one element of the partitioned inverse so we can
compute this thing here and that thing here and you can reproduce that so obviously the two are
just the transposes of each other you can see this here right this thing is just the transpose of
this thing here because b and c are the transposes of each other so we get one over the sum of squared
residuals of the auxiliary regression times gamma tilde or gamma tilde prime both with a negative sign
yeah and now the rest here is just math you can now compute what beta hat is using the expressions
which we have arrived for the partitioned inverse c and i won't go through the details
here this is something for you to do at home but what you will arrive at is essentially that you
get beta hat is equal to beta tilde minus something in the first two components and it's equal to this
term here in the third component remember beta tilde was a vector with two rows
beta hat was a vector with three rows so this term here gives you the first two rows of beta hat
and this gives you the third row so this term here is actually beta hat not and beta hat one
right the first two components of beta hat and this here is beta hat two
so this is the result i was aiming at because this now relates the unbiased estimate beta hat
which we usually cannot compute because we don't know x2 to the estimate we can compute
and we will compute if we run the regression without the omitted variable without x2
this estimate is beta tilde so what we now know is that the beta hat is related to the beta tilde
but that beta hat differs from beta tilde by gamma tilde which are the coefficients
of the auxiliary regressions well times a scalar this thing here is a scalar right
residuals of the auxiliary regression times y dependent variable divided by the sum of squared
squares so this is just a scalar so we can write beta tilde not is equal to beta hat not plus
gamma tilde not times beta hat two and more importantly because this is just the constant
we're usually not so much interested in the constant and whether it's biased
i mean we can use this result but it's not so much the the term we are interested in
but here is what we are interested in beta one tilde so the estimate we actually get
is equal to the unbiased estimate of the first coefficient plus gamma one tilde times beta two
hat right y times beta two hat um well because um what happens here is that we have the beta
tilde minus gamma tilde times this thing but this thing is the same thing as this one here
as you can see and this one is beta two hat so for this term here we may also write beta two hat
okay so we know now in the second line of this thing here we have beta one tilde minus gamma
one tilde times beta two hat and this is exactly this equation here okay so now we know what the
bias of the beta one tilde will be because we can compute the expectation of beta one tilde very
easily from this equation here the expectation of beta one tilde is the expectation of beta one hat
which is of course beta one because beta one hat is unbiased plus the expectation of this term here
for which we know that the estimator beta one tilde is unbiased if and only if the expectation
of this product here is equal to zero
well will this be the case that the expectation of this product here is equal to zero
we can use now conditional expectations and the law of iterated expectations to compute this
expectation the expectation of gamma one tilde times beta two head is equal to the expectation
of a conditional expectation by the law of iterated expectations here i take the conditional
expectation of this product given x1 and x2. So given x1 and x2, the expectation of gamma 1 tilde
can be pulled out of the expectations operator. So this would be the expectation of gamma 1 times
the expectation of beta 2 hat given x1 and x2. But the expectation of beta 2 hat given x1 and x2
is just beta 2. So this is the expectation of gamma 1 beta 2, which are fixed parameters,
so it's equal to gamma 1 beta 2. So we know now that the expectation of gamma 1 tilde
times beta 2 hat is equal to gamma 1 beta 2. We know that beta 1 tilde is unbiased if this
expectation is equal to 0. So when would beta 1 tilde be unbiased? It would be unbiased if
either beta 2 is equal to 0 or gamma 1 is equal to 0. This is the conclusion of our math here,
beta 1 tilde is unbiased if and only if beta 2 is equal to 0 or gamma 1 is equal to 0.
But what does this mean? If beta 2 is equal to 0, well that would actually be trivial,
then this would mean in our example that the regressor x2 actually is not necessary.
x2 doesn't explain anything. Beta 2 was the coefficient of the unobserved variable x2.
So when we say beta 2 is equal to 0, then we say well x2 doesn't matter. Well obviously,
then we estimate the right model, the correct model, when we estimate y, when we regress y
just on beta 1. And then this thing here would be unbiased. So that's trivial that if beta 2
is equal to 0 that the estimate would then be unbiased. The important condition is that the
estimate can also be unbiased or is actually unbiased if gamma 1 is equal to 0. And now you
may not know anymore what gamma 1 was. So let me remind you, gamma 1 was the coefficient in this
auxiliary regression here where we related x2 to x1. Gamma 1 was the slope coefficient in this very
simple regression here where we regress x2 just on a constant and on x1. So we know this gamma 1
here is actually estimated as the covariance between x1 and x2 divided by the variance of
x1. Remember the very easy stuff we made in review of basic econometrics on how an OLS
estimator is constructed in the bivariate case, then the estimate of the slope parameter is just
the covariance of regressor and dependent variables divided by the variance of the regressor.
So the important thing is this gamma 1 here is actually equal to 0 if the correlation between
x1 and x2 is equal to 0. So if x2 is uncorrelated with x1 then gamma 1 would be equal to 0.
Gamma 1 equal to 0 is the condition for beta 1 tilde being unbiased.
So we conclude that if the variable x2 is uncorrelated with the regressor x1
then there is no problem in estimating without the omitted variable. Even though the omitted
variable does explain something, even though beta 2 is probably different from 0,
it doesn't matter that we omit the variable because the variable is orthogonal to x1 and
therefore we do not have a bias. But if there is a correlation then we will have a bias and the
bias will actually be gamma 1 times beta 2. So we may actually also assess the size of the bias
which we will do next time. Here is the bias. Gamma 1 beta 2 is called the omitted variables
bias and we may now get an idea of how big the omitted variables bias is because we may have an
idea of how big gamma 1 is, the correlation is. Possibly we may also have an idea of how big beta
2 is or at least we may know the sign of beta 2 or have an idea of what the sign would be. For
instance for ability the omitted variable we would suppose that beta 2 is positive. Suppose that gamma
1 is positive, perhaps 0.7 or something like this. So we would know that the bias in our estimate is
positive, that we estimate a coefficient when we neglect ability in our regression, we would
estimate a coefficient which is too high. So we would need to correct the bound. We must be aware
of that. This is how we will deal with this. We will discuss in practical examples on Thursday.
We will look at such examples as wages on ability at education, first education which we measure,
second on education which we do not measure, and also on the birth weight of babies given that
people smoke, have low income, and these kind of things. Supposing that certain variables we ignore,
we do not include in our regression. But that's it for today. We have derived the omitted
variables bias and we know it depends on the correlation between the observed
progressor and the unobserved progressor in this easy case. Are there questions now? Then please
raise your hand and next I will ask you if you want to go to interaction and ask them
in person if there are any questions or if we can answer them via the chat
here. Please raise your hand if you have any questions or comments.
Okay, I don't see anything. No questions, no comments. Probably this was hard stuff.
Please work through it until Thursday so that you understand it well. I know that was not an easy
lecture and it will become more easy in the next lecture by the way, so I could tell you a little
bit on that. But this one I know was a little harder. If you have questions or questions come
up when you prepare for next lecture, you're invited to pose them at the beginning of next lecture.
I will ask as usual if there are questions then please raise your hand there. But currently I
don't see any questions so I will stop the recording and stop this lecture. Say bye-bye
to you until Thursday afternoon.