And just a look,
back, we have had the conditional independence sumption,
which basically sads says that it doesn't matter with a person is treated or not treated,
if we can condition the treatment status on some of service with veryble at ein.
So conditional on variable excited the treatment with then be independent of the potential outcomes,
and then this is as good as randomization.
That's what have eleverated on in the last lancer,
and obviously the question arrises,
what happens if this XI is not of served
If it is not observed,
then we can a condition on this verable,
but we would have some type of selection a bias.
Since we don't have something which is equivality to randomization.
Anyware. So this is why we then have dealt
with problems of unobservator variables. And we have computed the
error with some type of mathematical effort there.
And came to the conclusion that in a very simple set
up of a regression equasion with just two regressors,
as at just ted or the exposed that resposted to the
second question
If of those two veribles, one verible is not observed,
then the estimate of the reduced model,
which incorporates just the observable variable,
will in general be subject to and ormitted veribled bias.
And we have computed with our mathematical effort, the ormitted variables bias,
which in this tour, which in this case,
churns out to be the product of gamer one times beeter two.
Where beeter two is the true coefficient by
which the unupserved variable effects the dependent
Variable. So actually would like to estimate better two right
away and we would do so if we observed the unopserved variable,
better two with then measure the impact of the unopserved variable on the inductionals very ble.
This impact is doing there. I mean,
the me effect that we don't observe the veryable obviously,
does not enply that the very big doesn't
have an impact. So the impact better two is the there,
it is the hidden, so we are in the dependent variable.
Every wouldn't actually like to identify,
but we can't sometwee do not observely the regressor,
the unopserved veryble
So this beeter two is one factor, which etermined the
obmitted variable spies. The other factor is gamer one.
And then my one is essentially the correlation co-efficient,
between the observed variable and the enopserved very.
So between the observed regressor and the end upserved regressor. In the exilary regression,
which we had set up with the observed regressor,
would be regressed on a constant and on the enopserved regressor.
So this regression peroposition is gamer one and in this by
various setting as you know this would be the co-efficient of
correlation between very will ex-founded ex-two,
where ex-foundeds observed and ex-two
is adopserved
So the product of these two here, constitudes,
be omitted veribled spies, which means, that the estimate,
which we can expect for the influence of the observed verible,
differs from the true effect of the observed verible,
by this much, by this product you get
me when times be that too. We then there are two
special tases in which the omitted verible spies actually disenotics is,
so with this product here would be zero. One is the trivial case,
if be to two, equal to zero,
so if the unopserved verible doesn't impact on the independent verible,
then it action doesn't be long to the model in the first place,
So, bitte tu Zero obviously the ormitted verible spies, is Zero so.
Not including a veryble in the regression, which doesn't be long there,
doesn't cost any, any problem. Very clear. But if the beat and two is non Zero,
then the Gamer one is actually the decisive parameter here,
and then we see that, if Gamer one happened to be Zero,
that would be the second special as,
that is to say, if the unoperserved, variable,
does not core late with the observed redressure,
that we would also not have enomited variable spies,
but if we do have correlation between the observed and the outopserved variable,
then we would have,
not mitted verible spies,
provided better two is different from Zero
So that's be definition of the OVB, the ormitted variable spies.
And I just summarize this seear in two tables,
which blook on very much a like,
let's first look at the first table,
which gives you they ormitted variables biased of the estimator,
bieter one head.
So the estimator of the first, a cofficient in the
regression, the cofficient for the observed verever.
And you recall the estimator is random variable,
where's the ästimate we do one head
Is a real never. So for the ästimeter,
we know that,
if the correlation of the regressors ex one ex 2 in the population,
so the true correlation of the randomarilals ex one and ex 2,
if this correlation is great lathan zero.
And the influence of the second variable of ex 2 is great
10 zero 2, that we have a positive bias.
If the correlation is negative,
and be terd two is great properator than zero than biases negative,
and here is the converse,
peter two is negative and correlation is positive and the biases negative and And it is positive if
bothic correlations negative and the influence is negative.
Because this correlation here and there is essentially the true gamer,
but true gamer one, we had in the OVB formula.
So negative, times, negative gives than positive bias in positive times.
Positive gibt's positive bias and if the two peramitas have different
signed, then we have negative bias, ericane.
Now, the ormited variable spies for the estimator is not
necessarily the same thing as the ormitted variable spies for the
Because may happened that in those two variables, as random
variables, are positively core lated in the correlation,
it may happen as a core incident that sample we draw
is not core positively core lated with it other,
but raps, negatively, core lated.
And we may have the unfortunate event,
that we draw from this population,
here which has actually positive correlation,
but for some unlikely reason of for some reason and a case which throther unlikely,
the sandle we draw is such that the measure correlation
is actually
So, we have always to distinguish between the Estimator and the Estimator,
or between random variable and candle, which we drop.
Otherwise, of course, this science year of the bias are exactly the same.
So we know what the ormitted very blood with spice for
our estimate is by just looking at the cyple properties,
which is quite nice,
because we never know what the population, properties are,
we can just estimate the opulation properties by using center data.
But we never have knowledged of these things here,
we always always, we only always have knowledge of the Sabber moments.
And he will that know that the concrete estimate that we
got by estimating and incomplete model and optiming in ästimator be
to one head that this concrete estimate will be
biased and a positively or negatively, depending on what the sample correlation
between exponent exto is,
at of course depending and what we are soon for that
true cofficitions for the true cofficition beat at two,
so for the true influence,
with the second random verable has on
the endorgence
Not actually, we should always distinguish indotation between random variables and real numbers.
So we should thinguish between population moments and sample moments or itsons.
But this is somewhat come or some because then for very
similar concepts, we would have to have different simples.
Like for instance year, I used the same notation we
to one head both for the estimeter and for the estimet,
which is not really great.
I should have different simples there. I should per instance right here,
beter one head superscript n to make clear this is the
estimate computed from a sample of en observations for something
like that
I could do this, but this would be quite a bit of editional
notation, and would perhaps a not make it easier to
looks to reform you that, which give to you, even if
it very useful and the general finding is that in
text books and literature and almost everywhere, there is
no distinction innotation between random variables and there realization and a cemple,
really day is. Of course, sometimes there is,
and practical at the beginning of econometrics text,
which people distinctions between that, but enters sometime,
they will say, well, now please,
you have a hopefully understood,
what the difference between in estimeter and in estimet,
is we just use the same sible
For end of variables and procemples, for random variables and real numbers,
for estimators and estimators, whatever. We use the same simples,
because the fragilest are basically always analycas to each other.
And they have a different meaning, they have a different interpretation,
but they are so and analycas to each other that it
suffize is to just right them with one notation.
And you always have to know yourself when the interpretation as an estimator applies,
it when the ass to interpretation as an estimate applies.
And ancially this is what I will do in the future
to, I will not distinguish in not between
Random variables and real numbers. But rather,
I ask you hear that you should always be a where
of the diff difference and that you should train yourself in
thinking clearly in terms of eye the population or sapple relationships,
very offen formular, actually apply to booth interpretations.
But not always. So and you have to keep this
mind and as a say, train yourself,
thinking in these terms to perstand the distinction and it now
when which interpretation is perhaps a not emissable
Now, what we have done in all the smath here,
last clicture, with these things or was somebody road
kind of feel could follow, what I was doing.
I'd midd it is technical and little hard, the followy
have this been time on this. But very mind please,
we did this just for very simple, very simple model.
Namily a model with just two regressors. Ex one and
ex two and so two correfficions. And I was that one
regressors this not opsurble in the other one is of serviable,
that's actually the simples case you can choose for this setting
What I ask you to do in this exercise see or
where is it there? In this exercise, here,
that you do it for the generatings. Which is not
all that hard to do, when you have follow through,
what I have then in the sible taste.
So this exercise basically asks you to do exactly the same
thing as we have said in the specific example,
which are presented last vecture, but now what the general case.
So as say, well, there are two models,
model one is wise ex-one be one plus you.
So why is explain just
by X-Men
Or is just redressed on ex one with,
I am it a vactor be one.
A model two is a more comprehensive model,
which also a lot and influence of variables ex two,
with corefishing beater to. So,
professions are beater one beater two for ex one and ex two,
placed some error, of course,
here the air is could you, that it's could new.
So, this model here is ex bitter place new,
with a practitioned regressor matrix. So,
model one is just special case of modern two,
we are beater two,
would be equal to Zero and beater one,
would be equal to be one
Oder as you to do in this exercise,
is that you use the same partitioned inversion for you,
that I gave you last, we not last we class lecture.
And you say, to show that bitter one head,
so the eyestimate here, now not the estimate.
Yeah, the estimate this model, but our one head,
is equal to be one head, so the estimate model one,
minus ex one plus ex to bitter two head.
So this from your 2023 here is actually the complete
analog and to the formular we had Ah, hier.
Oh, ja. No, sorry.
Ähm, we had where? We had here. What
? We are we know. I think it's the same
thing. This formulay here return. So,
we don't want till there is equal to be the one head,
plus Gamer one till the beeter two at.
So the Gamer one till the hear is the same
thing as the exone plus exto here
And otherwise just this term has been moved to the other side.
But in principle, equation 203 year is sort of the multi-dimensional
version of the equation, we have to ramped last lecture.
And you can prove it by exactly the same means that we have used last lecture,
name the applied a formular for a promitional inversion.
So I ask you to do this exorcercercise here and prove
that this relationship holds authorit in the general cames, because
we will used this from in that in subsee connectors.
I give you a number of hims here a how to
solve this exorcercise
It's when that pretts you find it difficult, given
the difficulty of last lecture. But given the difficulty of
last dicture its also good that you go and to last
dicture again and projects see with the you have account this to,
what I was even there and practice,
where they in applied to very similar setting just with modern two expensory earables.
But as I say, we use,
we need this result 203 actually in letter dectors.
So, we should that you know that is true
No, let's turn to some examples, which proupes make
life on the leasier than it was on the two steak.
A first example is which the termination and I received or it the
question on this example at the beginning of this letter here,
to post wages are determined as follows.
So this is tart of the deprew model.
The wage is a constant plus beater one times education,
this beater two times ability plasties.
So this your completely distrapt this cribes the wages which people earn.
Lady pent on education and education and ability forget him of
the constant this is all yes, there, but doesn't really have any
kind of economic interpretation. It is an antior to supposed
that both beta one and beta to a positive.
So and sample mean beta one, should have the property
that the better your education is the hyper is your
lage. And the better your ability is the hyper is your lage.
So beta one and beta to a cquite naturally issued to be positive. No,
let is further is zoom that we can observe education,
the formal degree of education, somebody has a quite and school,
high school or University.
But we can not observe ability
More over that's a zoom, which I think is on
the eplacible that education and ability are positively correlated.
So the more able you are the higher will be on
everage at least for full population of for the sample,
the level of education that you get.
Now, when it could think of is,
since ability is not upserved that we estimate equation 25,
where wage is just regressed on education.
Oviously now, when we you regress wage just on education,
we have
new airterm new
Und it's got week. I call it we, bright.
So, new are to we, where is here the
air tom was you? And very clearly this new we
here is exactly equal to bieter two times ability plus you.
So the we captures everything, which is not captured by sorry,
by bieter not times youtale, does be the one times enducation.
Das ist we. Now from the analysis,
that we've made on the ormittled variable spies,
we know that the ästimate of bieter
one in equation 25 will overstate the true importants of elication
Because, we know that the correlation between ability at education is positive.
And we know that the correlation on ability is positive.
So this was the first case in my table.
It was this this case here. We have a positive buys.
I'm appositive buys, most likely also in the estimate.
Gut, so the estimate of Beta 1 in this model
here will overstate at through in pertence of education.
Now, that's a serious thing. Because itly bias is
positive it may make and a co-ficient,
which actually is insignificant
May make it look significant, because it sawn the large
and end actually tripy. Also, it may result in
a positive cofficient when the true confession the Zero.
So supposed that education actually doesn't really matter,
that it is just ability, which matters for you wages,
and the true cofficient on education is actually Zero.
So with a mid variables bias,
you make it the result,
that education seems to have a positive influence on waiters,
was in reality it as not
More over, it may resigned in a positive co-efficient when the true cofficient is negative. So it may actually term cofficients around,
which is parts on out,
such a great danger in our example here,
because it would be really supprising if education had a negative influence on your wages,
but in other settings, of course,
it maybe that you have a tiny negative true cofficient,
but new to a great or mitted very with bias,
you get a positive cofficient to prips even significantly positive.
And the same of true course
true for native bias
No, there is some of cavillary,
which you are should be a mind,
a positive bias is also card in upward bias,
quite naturally, a negative bias is also card downwork bias.
And if an estimate is closer to Zero,
in absolut value than closer is Zero,
then to be, let's say this way,
then estimate is closer to Zero that tood be,
then we card it biased toward Zero.
And this would be that case if the true cofficition and
absolute value is greater than the expectation of the estimate of
the estimate
Meint, in absolut Value. So, this can,
I that be the case that be to one is positive
and greater than the expectation of beta one till it.
Ord can be the case that be the one is negative.
And smaller than the expectation of your beat one till it,
in both cases, be to one till it,
would be call biased to
ord Zero
Do he is reword, example this stranguation has been estimited
by somebody, smoking doing premiency.
And the goal of the research here is to find out
if securite consumption, you're in premienced sea.
So securite smoking by pregnant women has edetry mental effect on
the both weight of new bonds.
Basically on the health of new boards,
because here this probably in the sumption that.
Babys which are really tiny and small are also this healthy
than they was things which have normal weight
But here we just look at the weight of the new boards and
this little table here, if you some data statistics,
so be weight is of cross the booth weight,
we have almost party in 100 observations here and the both weight mean of 1400
new boarm babys is 100 and 18 und 7. What actually,
I was not so sure what kind of measurement the use year,
because the certainly not cilogramm and in that the certainly not gramms.
So I supposed this something like American outses,
in which they measure the both weight of babys,
even those Have my doubts about own to see,
since the maximum birthweight, they observe, was 2071 ownses thats.
But if these were ownses, this would because to 9
kilo rams and I can't really imagine in 9 kilogram,
baby being born, even the some Americans are really fett.
Still, I mean, 9 kilogram that's poo, whole lot.
I wouldn't be like to be do women who carries that to baby.
Anyway, it we just take the unit as some unity. I also wondering a little bit at the spam here of this unit meaning that sort of the Biggest Baby,
the fettest Baby, was more than 10 times as heavy,
as this smallest Baby, so this must have been really tiny,
new borns here, at rilly fat Babys,
if this makes any sense, to me, but as proced does,
because the regression has been ästimated and the other data seem to be okay,
cigarity consumptions in security purday for a close to 1400 mothers,
where the mean is not so really interesting here,
because we know many mothers just don't smoke any securit,
so if you have a Instagram, you've probably,
see that many many mothers have a zero conception there,
and then some have probably substantially more than two securits.
The maximum was 50 secrets proday and of cost my minimum
is zero for the not
smoking mothers
And then there is the family income,
which ins measured in 1000 Dollars and per here orguestly.
209.000 Dollars is the mean,
which is reasonable ostimate of family income, every family income.
Maxi more more 65.000 Dollars pro year and well,
the minimal was just 5.000 Dollars for year.
So these are the digger, we have. And now we
supposed that the true model is actually that the birthweight depends on just two things,
which is a cross not true,
but let's the posit,
is like this the birthweight,
just depends on secrets consumption and on the family income
Und die Familie, die wir inkluden, because that that
people who have higher income, perhaps have better norrishment and if
they have better norrishment than they babys are also.
Better fan by they mother, I mean not
like receive more nutrians while they are in the mother's room.
And therefore the come out with a higher to oth weight everything als being equal. So,
let's a post, that this is the true model,
then if we had data on securite consumption and of family
and now we would just estimate the model as I have estimated at your, I ture somebody.
Else has estimated it is not my own. Output last
role is just be constant. This is not interesting.
The first role gibt's as the cofficient for the segarits,
which you is minus negative point for six, negative point for six.
This is a highly significant estimate as you can see,
because the standard earth is just zero point.
Nein, so the tease statistic is negative five.
Right on the peavalue is there.
So that's five
Family, income is once a significant,
not quite as significant as security consumption, by the way.
The confision is a point zero nine. Standard Arror is
points in a three. Teas statistic is three. So indeed the higher,
the income of family is, or of the mother,
it is the higher is the bothweight of the point on everage.
How ever, as you can see from this edition or autoport here,
we don't really explain and lot of the bothweight here,
because the are squared,
is just in the order of three percent
Hier ist die justed our are squeed mut much different.
Basically just three percent of the Variance and both waits,
we can explain my secret consumption and family income. And obviously,
I mean, there I'm many other things,
which you might but to plugin to such a regression,
if you want to explain both waits of bemies,
first thing I would plugin this actually the weight of the mother,
bright tom and breps the wait or the hight of the far that.
And these kind of physical characterists its are completely missing,
they would probably auto explain quite a bit year.
But anyway, it has been done this way and please
bear to mind the confession of figured consumption
is negative corner
Never negative found for six. So, negative influence of security consumptions on the
both weight a probably on the health of your base.
Now, there was a question. Oh, jetzt haben wir
die Hesse, googelt or known, I don't know.
I read to. Any talian baby holds the world,
record of new burns with 10 Kilogramm, back in
1955. So, the never could be right. Okay,
thank you very much than apparently, these are really outses.
I was that way that,
babys that have you would are are are are being board,
pure mothers, but trually,
I mean,
if they had 10 Kilogramm and it to itally you they
will be able to have 9 Kilogramm in the 9,
United States
Alright, thanks a lot for this. Gut. Genau. Surpose,
now we do not have the family income as a reducer.
I mean, this could have forness ins data protection reasons, right?
So the mothers come to haspitals and in the hospitals stay record
the bothweight and perhaps they also ask where had reasons for security consumptions and that's say mothers ponnestly these say a prule two-fully say,
oh many securits, they smode,
but we're have a hospitals are not a lot to ask with the family income is,
because this has a no health connection.
So the data set, you have compiled,
may just not being my just not contain the information on family income,
which is deployable, because we see that that when we have the family income,
than it terms out to be a significant progress,
also party, it is explains something.
But it could be the case that in a different study or setting,
you don't to have the family income, as a yet.
And they are for you just requests on secrets. So,
so posts, we estimate now this unders specified money,
because we know that the full model was actually the true
model with family income being significant
In this case, the estimate with the same set of data,
just pretending that we don't know the family in come.
The estimate for the cofficient of ciguretes is negative point five one.
And on the previous light it was negative point for six.
So it see that the absolute value of the cofficient assimate here is greater,
than the absolut value in the previous case. No,
if you didn't know economicicic theory in the way presented it to you,
then you would be confused and would ask where what now
is actually the influence of securite consumptions on worth way
Is it negative point five,
51 or is it negative point for 6s like in the previous regation,
because you would have around tour regressions and it would
have come up with two estimets and don't really know,
what is the true isding?
Now, we know, what an mitted varial spires is,
we can we pretty sure, what the better estimet,
is namely, of course,
the estimet in the regression, which included camily income.
So, let's look at this by running the Oxylitary regression.
Regress, the ormitted variable family income
On a constant and seared conception. Doing so,
results in a negative cofficient here, negative found for 5,
4, standard error ponsero 8,
soctive statistic is minus 6, so highly significant.
Therefore peer value is equal to zero. So we know
quite safety that securite consumption is negatively core lated with femily income.
Where income is low, people smoke wahr,
then we're I income is high
So, this means that in our derive relationship between the
estimated corificiant for secret consumption in the underspecified regression.
So in the regression with family income was missing,
that this estimate should actually be equal to beter webhed.
So the estimate of secrets in the poly specified model,
plus data one head,
which is this corificianteal,
in the oxemiory regreation
Times, the estimate of Vita 2, in the full
regression. So the estimate of the coalificient of family income
on both rights in the fully specified regution.
We have all forcal efficience now,
because we only pretented that family income both not observal.
So we can just plady in. Beeter one tild
it was, sorry. Now what do till do till
do this coalificient tear? And this estimate.
Negative point 5,
Negative Point 514 ist disco efficient be do want till them.
Be do one head was the confision,
we obtained for a tigered consumption, the fully specified regation,
this was a negative point for six, which we have here.
The delta one till the we have estimated hear,
this is negative point five for three, which I have here.
And the beat up two head was the co-efficient
on family income in the fully specified regression, which is
Zero-Point, Zero
So we can check with a our formular is correct,
just compute this here and you'll find that this,
what what is see over the break it, is negative point 5, 1,3.
So usentially it's exactly the same thing as hear the difference,
but with rehear and 4 here is just rounding, rounding error.
And so what we are very fight, somewhat experimentally now,
if you're mitted very into spies, has been correctly,
dirived by us in egonometric theory. And therefore,
we've also some no from our egonometric theory,
that this net estimate here is Verse, then, sorry now,
that this estimate here, I was of the rock slide,
that this estimate here for the underspecified regression,
where we just have securites as regressor,
is verse than this estimate here.
And actually, if we have just this regression,
because reverly income wasn't recorded,
then we can infer that the true cofficion will not be
quite as negative as this one,
is suming or knowing from other studies that the correlation,
but we in family income and securit consumption
is negative
We make to look a different studies with different data,
where people come up, saying well, consumption,
the collation between, family income and signate consumption,
is in the order of negative point 5, right?
And we made surpose that there is a positive income effect
of incount effect on birth weight.
So that we would be in a position to say that
this corvision here is certainly are to be in absolute terms.
So the true effect a certainly less strongly negative than
negative point 5,1 here
If we can do rive some ästimate of the beat
and to head from some othern study, which would,
I don't know,
have test it how fairly income is related to birthweight of turn it.
There we can actually also try to assess the obbitted verable spice in toddal.
And more is correct this corfficient estimate here by some
estimate of you omited very would spined.
But therefore office leaven would need to restructure to other studies,
which have permedy income at their posit,
of which certainly many exist because family incomes offen recorded
But it happened to be the case, we put ended that it happened
to be the case that was not recording are example here.
So that we had to estimate the underspecify. Okay,
here, it is now, the emitted variables a bias
is actually negative point searos hero for 7. Okay, any questions.
They let's come to an different approach are to a problem
nother approach of how to deal with the emitted variables variables
and this is the proxy verigable approach.
And their present this also
informed of example
So consider and way, equation that record nices,
that education experience and ability effect the wages.
So and this case the lack of wages. So again,
as oom that equation 206 here,
is the true model where the lack of wages with depend
on education on the experience and on ability with co-ficians beater
one beater two and beater three.
Clust constant which isn't just thing. Now,
to pose we have no masher of ability that the case,
we have already as cust.
The differents it this example for our discussion as a post
to do when we happy for being that now we also
have experienced in here
The true monated muffs three regressors, but one of these
those three regressors, we do not observe.
But we do opens are education and we do observe experience.
Or it. One way to middigate the or middle very
bespies is now the use other proxy very able.
And there's a proxy very able, of course,
we have to look for some very able which is closes as possible to ability.
So ability is out of serve,
but we may think of which very will could per head be available,
so observed, and highly core lated with ability,
of the same kind of informational content
There obviously one possible measure is the AQ-Score as a proxy very blood for ability.
But IQ measures intelligence and even there this disputed,
I'm be watch exactly is intelligence how do we defined in
the intelligence and does the IQ really measure intelligence in the
certain set that which we have defined.
All these are open questions,
but we do have an IQ-Score and the currently doesn't really
matter with and this is true measure of intelligence or not.
The question is where the ability so question hope to find,
but supposed we have defined somehow.
The question is where the ability is correlated with IQ scores,
which people have in test senters.
And probably most people with think when you,
do it a could we quite some degree of corration between IQ scores and ability.
So, we mays supposed that ability and IQ are positively
related for instance by means of an equation like equation
2027. Ability is it constant, plus data, three,
times IQ plus we three
I used data three here, rather than data one.
I mean, I've been delta zero here and this that I three.
So we may ask why, why don't three?
Because the state I three of the index three
relates to the index three here of the ability term,
which IQ is supposed to a praxi.
So that's why he's data three reven better one here.
Not the question is do we could unbiased estimetres of beta one and beta two,
if we used in the other Q and set of ability,
and if not do we at least get consistent estimetres
of beta one of beta two,
if we praxy ability by IT
So we'll steady this now in the example and also go little bit
here in do theiretical considerations, supposed we regress the lock of wages,
now on the three variables, we have valiable education experience and the IQ.
Then we estimate in the question like equation planet with some constant,
elf one education, plus LFA2 experience, plus LFA3,
IQ, plus some erotter.
I use elfers now rather than beaters in the previous equation,
because I don't know with a these cofficitions will be the
same as the co-officience in the equation which head ability here.
So they mean, will changed,
if we you have IQ rather than ability in the equation.
And there's on what we want to know is how the
elfers ever one ever two ever three here,
relate to the beater one, beater two, beater three indikation 206.
Sometimes the solution of the proxy verable is can't the plank
in solution to the obbitted variables are problem for the alys
reason that we just plagged in some other variable for the
variable,
verable which is music
The question switch arrise are are first,
how are the elpha coalificials and also the Aeroterm,
connected with the elpha coalitions, integration 208,
related to the beat a coalificials and the Aeroterm you, inequation 206.
So this one question we have to say.
And then also, which assumptions are needed for the pluginsolution
to provide consistent ästimetors of weeder one and inter two.
Or do we adjust this problem where the best way,
even though it looks a little complicated,
is that we first look at be concept
of a conditional expectation
Namely, the conditional expectation of ability, the unopserved variable,
conditioned on IQ, the observed praxi, that give.
And we may say, well, the expectation of ability
given IQ is perhaps, such a linear opproximission here.
Is such a linear expression, a constant plus deslta three
times IQ. So this thing here is something which is
at least in principle observable,
if we know what dead a zero and dead of three
If, it is true that the expectation of ability conditional on IQ is this,
then we would in furth that new three of we three,
the eritern of equation 208, no, sorry,
this was not the eritern of increation 2011,
but it was the air time of equation 2017,
yeah, mind,
it's one hear.
This is actually where they expected Valley,
the conditional expected Valley comes from, that this
airratter is just the difference with ween actual ability and the conditional
expectation of ability given IQ
So we decompose, the missing regressor,
now in two things namely and to ability or at namely actually,
in V3, and in the expectation of ability given IQ,
which we can right in this from year.
What we know is that, if we know played,
IQ in the regreation equation 208,
then the regression equation 208 accounts for the conditional
expectation of ability given IQ.
It doesn't not account for the
be three obviously
So this ding here is till something,
which is not observed that that's wides in eritern.
But we do account for this thing by plugging in the
IQ as a proxy progressor. So,
when we know look at the true model,
we can reright the true model and figure out,
what we actually ästimate, when we ex-estimate the proxy equation.
This year is the true model with ability and coffeesion we
don't three of ability, eriter view.
Now, we know that we can replace ability.
By the expression, conditional expectation of ability given IQ,
That's exactly the same thing as ability. The some of
these two terms. So we also know that the conditional
expectation of ability given IT is equal too that are not plus death three IT.
Oh, there is a you time is hing here.
As a you time is hing here, sorry.
They're not you top, plays data three IQ plus vithree.
Well, in that of course, we can collect terms.
So the you that term here, goes to the you
do, which we have you.
So actually art elfer not is equal two beta not,
plays beta three delta not
Peter Three, Delton Not, the Yota, which is missing here,
is part of the constant, goes into this clotificial tear.
So our other not is actually ego to beter not,
plus beter three deltons not. The beter one is
a parently equal two the other one. So,
there's no change of the beta of the conficial tear,
because eather one and beta one ästimate exactly the same thing.
And the certain assumptions which are not.
Same thing is true for beter two
and other two
And then I tou, the proxy regressure, regressure,
measures, the influence of beta three time state of three,
beta three, times Delta three,
this is what we ästimated by other foc.
And then we have a new are term e,
which is made up of you plus bitter three,
we three, because we have beta three,
be three year, we three was the are term from the occilary relationship.
So this is part of the are term.
So the are are are are are are term we ästimate
now in model 206, e,
is equal to you plus bitter three,
we three
So, this looks not so there, at actually,
because it seems we have a chance of saying that when we estimate elf one,
we actually astimate we to one the true cofficient.
And when we estimate elf at two,
we actually astimate beat to the true cofficient.
Quite pleasely when we estimate elf or 4,
we do not ästimate the true cofficient for ability.
But we estimate the best case actually,
beat a three times deal that through.
So we can not a cells year the elf or 4
with the beat
a three
But we can under some conditions, which I would spell out now.
As how she it and estimate on elfer one with estimates of weeder
one at ever two of weeder two and so far.
So, we can actually hope for consistant estimates of weeder one and weeder two.
Eve two essumptions in a particular are self-fight, sets fight,
next to the stand and assumptions of consumptions,
what you always make, one is
humpsume must be that the error term V3,
I for Observation I,
is uncore lated with education and experience,
for our edividual I or actually for
all lines
Because here in the airer term, e. We have we three.
And what we need to have as a property,
or what we need to sue,
but rechnet meets to be setist fight in real world, data.
So it needs to be setist fight as a property,
is that we three does not
courney with any of the regressors here. So it must
not courre late with education and it must not courneyed with experience.
For all persons eye which we upserve. And more over,
we need to have the property that UI this proped
of the airterm,
does not courre
late with IQ
So, sentially, emissionally, we had an equition which
had education and experience plus in possibility, possibility actually,
plus you. So we need the true model said,
this is constant plus be do one education,
plus be do two experience, plus be do through the ability,
plus you.
So alvis me you did not currently with experience and education.
We need not bother about any porrelation between you and experience
and education. But we do need tobother about correlation with
when you and IQ,
because IQ was not in the interial model
So we need to insure that there's no correlation to win to two,
because we know correlation with we in a regressor
and the air tome with lead to inconsistent estimation.
And the same thing holds hear, we three,
we do not need to assume that we three doesn't
colleague with IQ, because we three comes from an equation in which IQ
was or regressor, was an explaintory, random, earable.
So that was garanted by construction that we three doesn't correlate with IQ.
But we do have to ensure that we three doesn't for
a little with the regressors experience and education.
So this is why these to assumptions your actually now orgucizive,
for as tablishing, consistency of the asslimits ever one and ever two,
and if they are consistent that they would converge to the
true convision speeder one and be the two
So, that means, we do not,
we must not have contemporarinious at the ecorrelation between we through,
we the three and education and experience.
And this means that the variations of ability do not
depend on variations in education or in experience.
Once we have subtracted all the variations of ability explained
by IQ
Right. So that's purpose, so what difficult to see,
but be aware again of the fact,
that IQ is just capturing sort of the systematic port of ability.
Namily, the part which can we explained by IQ.
There for, when we say that should be no contemporaring
this correlation big ween we three end education et experience,
that we mean, essentially say,
that the variations of ability do not depend any more on
variations and education or experience,
after,
we have subtracted our variations of exability,
which are explain by IQ
Ja. On other words, pretts,
ability may very, and it is completely okay,
if Variations in ability, correlate with experience and education,
if these type of variations are captured by the IQ,
because it is a lot that regressors are core nated with it other,
education maybe
ex corrective with experience, experience maybe ecorrecting with IQ,
IQ maybe correlated with education. So all the variation in
ability which correlates with experience and education is harmless,
if it is captured by the Variations in IQ.
Then this does not effect the consistency of ourestimens.
But if they is some variation in ability,
which is not captured by the Variations in IQ.
But courleys with experience and education, then it will be
in we three. And then we would have a correlation
with ween the three and experience at education and this may
not be the case must not be the case,
if we want to
have consistent estimates
Oh, jetzt. So, what we essentially says that
we control for IQ as much as as IQ is a determined of ability?
And there these is humptions, what was dritten here,
I have just explited. And these assumptions the estimates for,
I've a not, and I've a three are also consisted.
Yven though, I've a three is not so valuable for us,
because in doesn't measure true effect of ability,
on wages
In our two have consistency in this oxilliary regation or that
say this plugin in redastion that's purpose of weather verting 40.
In our into have consistency in this plugin in regression,
when early need the absence of correlation between the regressors and the Aeroterms.
So we only need to have the absence of correlation with we regressor ex,
the regressors, pour observation 1 and the Aeroterm for Observation 1. This is in the something,
which we have oready mained in the review of basic econometrics,
in the fonds of dissumption
Meint? That's the necessary for consistency. If we want to goes dronger,
I want unbysness of the estimments,
then we actually need assumption 82 from the review,
big basic econometrics,
which was that the expectation of e-given ext as equal to zero.
And that actually emplies the epsets of correlation,
between regressors and error term for each peare in the several.
So we would have that the expectation of exi-i-prime eJ is
equal to zero for all eye and for all chain
Und den Matrix Notationing would right this as e of
expreme e is equal to zero.
So that's a much strunker, assumption. Well, see
now that already the weeker assumption here, what only holds
for all eye. And not for all eye and OJ,
that this assumption is already not you not yes.
And that there maybe actually many instances a which it is vionated.
So we will not bother much about unbased.
Estimation this is not so likely that we have unbased estimation in
this setting when very blue is missing and proxy by some
other variable. But rather will talk about consistent that investination
Okay. Is show you again,
how this works out in a impericular example,
which was published by Blackburn and Neumarkt in the quarterly general of genannt,
in 1992, where they actually ran
wage regression switch, which included some more regresses then just
the regresses, we talk about. But still,
I think, it is interesting to see how are regression
confessions behave, when ability is proxyed by IQ
So, I have you can see this well,
the coupy is not the best one act that I have to changes have
position of my screen little bit, to see it better.
What you have here as the dependent variable is a gain the lock of ways? No,
you see that the lot of ways is regressed on a number of variable,
some of which we know and so,
which we haven't that with it,
because they are until important for us.
So there is education and there is experience.
There's also. Ten year, that's merite,
status with the your merite
or not
Charakteristik of the Worker with they comes from the south.
I'm not from the south of they medited states. Urban Worker,
or not Urban Worker, Black Worker or not to Black Worker.
So these are additional individual characteristics, which I hadn't talk about
that obviously it make sense to includum and such an begrüßen.
And then, well, then the first regression stops.
So this is the regression in carnom,
one with the regression resets. Intercept is the constant,
this we are not interested in, we see
are squeed, is point to 5,
this doesn't change by much or the other two regressions.
So I want explain much about it.
I want to focus and what we see here
We see that that is a strahl positive effect of education on lord vages,
with a teast-statistic of more than 10. Eins, so highly significate.
Actually. And this would 10 us that education has
a return of 6'55555% in terms of majors.
So whatever the scale of education was that has been used here,
one point increase in the quality in your education,
would in prease your
wage by 6'55%.
So substitution
Experience, also has positive return not surprisingly, also highly significant,
a point zero one for with the much small our standard error.
So it's as this teastitistic, which is greater and for,
clearly significant. 10 year is also something,
which increases your wage, highly significant,
and then come the individual statistics,
which we don't want comment too much about,
a personal being married seems to earn higher wages than somebody was.
Not married, south has a negative influence urban has a
positive inpluence, black has a negative influence all these confisioned
astimets are positive,
are security intextive
Or it, but we don't have ability.
And so the office, when to head and said,
let us ask try to proxy ability by including the IQ.
Which we see here, and he see that the cofficient
point serious or a three six, is significant with this statistic of 3.6.
So has positive influence on wages, if you have higher IQ.
Now does this effect our cofficient effects else way,
what's quite interesting to see that they return to education
is reduced by that
So it seems to be the case,
that we had an ormitted variable spies here in the point service 65,
apparently, because the IQ correlates with education,
people who have a higher IQ, have a higher education,
that's not so surprising, therefore the omitted variable spies was positive,
this estimate is certainly a better estimate then the estimate,
which was of tained in columb one.
For experience some ever, there is
no change it on. The call ifician is completely unchanged
the standard ear is completely unchanged
Is this surprising, where probably not?
It is not to be expected that people with a higher
IQ have higher experience. Be with the lower IQ can
be long in the work in the work life or ready
happy in active, can have lots of experience actually.
So there is no reason to believe that there's a positive correlation
between IQ and experience. So that we would have head
to expect that the cofficient of experience doesn't changed by much.
Nordeth the cofficient on Tenger. Just to tiny little bit,
actually, in significant. Marriage, doesn't changed by much,
the cofficient is a centraly in same,
right, put 109 or point to oo
South changes it tiny little bit, but probably not even toos significant extent.
Perhaps they is some scorpionation, I don't know,
about people in the sound that not being as intelligent is people that they else wear,
but I do want to say that,
probably it just satistical artifact with we, and to see.
No, it say don't urban and not effect on black or just the tiny effect again.
And I better not world my moth shot,
that is certainly also not big, enough,
to qualify as shot of to significant decreens of die cofficition given die Standards,
die wir haben, because you see the edge sended areas,
the sended areas is point for.
So this is just about one standard area,
which the cofficition tans being reduced here, where as here,
you see that the standard area is point thereas are six or point thereas are seven,
so about the same,
range and the difference in the call efficient is almost two standard areas.
If you take the point series, there are six one
that's much greater decrease actually interns of call ificient,
as we have it for
south or problack
Oread. So, IT seens to capture some thing of
the observed of the unopserve verable ability. Now,
the office wint to head and said, well,
the press to should also impludent in term action term education times IQ,
because it could be that the return to education is higher
for people who have a higher IQ than it is for people with the noir IQ.
So, and that this reason not thing it made make
sense to haven interection perm in year
But this was not successful and that's why
want spend much time on this specification here,
you see first thing the energy action terms not significant.
Standard are this breator actually in the calificient.
The is, come on, lower thing one.
But IT than also because completely in significant. And education also
because completely in significant. So actually with including the interaction term,
you tick out everything which is related to ability ento education.
And this seems to edicate that this not a good assikly
What you mind think off those in this case and this is why
this customers cardam here too is. I mean,
you're never be able to accept this as definary side.
But what you my think of now is that you say will,
hey, it looks like of all those three veribles here,
it seems that the product term has the greatest tea Value.
But it's very week still, and less then one,
but it's much greater than to tea value here,
and the tea Value here.
So prohaps, we're should for the next atted,
kick out both IQ and education and just measure what the product
of education times IQ has is a as an influence on lot wages.
But that's additional research which the other seven done it which I
want do it you. But just as a suggestation,
if you have a result like this, we're everything sevenly,
is significant that perpsy should delite some of the regressions.
Okay. So this, I have said, now let's
go and see what we have learned from this exorcize.
It's seems that using proxy variables can lead to biased estimates,
if the unupserve
variable is coralated
Other explainatoring variables in the model,
because as I had pointed out,
sorry this was in the theoretical presentation,
what in the example, we spoke here,
because we know that is possible that the correlation,
a bit we the our term and the other explaintory very bit,
comes from the term we 3,
which we have introduced been to the regression by plugging in the IQ variable in place of unopserved ability.
So this would be a violation of and basic deception which
we have made and hear as an exercise please think
about it which previous assumption would be violated, if
we have this kind of bias by plugin in proxineriples.
There's applosed that in stit of equation 207 which have this
from here ability is related to the IQ, at just be IQ,
we have the true equation that ability is related to all three.
Observable brailables, education experience and IQ with conficience data one,
data to and data three
On that's also possible that education, actually increases your abilities,
quite possible that experience, expresses your abilities quite possible,
plossible and that your IQ is positively related to ability,
we had or he assume in equation
2010. Now of this world the case, then are equation
would have to be rearritant in the following for.
Nord wages is equite to a constant. I do rive this,
much at the same way, as a did the brief is example,
right you can very fine at home, that this is true,
plus now, elve one would be equal to beter
one plus beter three data one, because we would
have this influence of education, which takes the from data one
in the plugin solution. So but plugging in the IQ variable,
we would get the influence of beta three times data one education to the recression.
So when we regress on education, then actually this part
would be captured by their education regressor.
And civil learnly, beta three that two would compliment
the beta two impact of experience on lock waitest.
And elford two would also capture this editional
inference influence here
And I you would of course have the influence peter three
down to three, which is elf of 4 as before.
And then we would have the new ever term, e,
which consists of you and be three, be three.
So and this case it would follow that the plym of
elf of one is not be down anymore.
So the estimate is now not consistent anymore for the coffeesion
beed are one we are actually interested in.
Because we have the additional term back better three, Delta Man.
And similarly the plym of elf of elf of two,
head would not be equal to be to anymore,
because we have this additional Trump better three delta two
So, since it is probably the case that beta three is positive.
And that data one is positive. So the influence of
education on a ability is also, it's also positive,
we can in further what the direction of the ormitted
veribles bias is in these cases,
you just tape to compute with the bias is,
I that would be beta three time
stead one or them times that two.
And then we would know where here the ormitted to verible
spies on education
is positive
And same thing here, we also make the claim that
experience is corrected with ability, positive way.
So, there is then still positive biased for the returns
to education bieter one. But it is typically small,
then when we or mid IQ upgether. So,
we would probably go into the correct direction,
because some of the ability influence,
which had been projected on education,
would now be to captured by the intelligent agence of
the people
But some other part of the ability namey, the part of ability,
which is do to education with sudy captured by education and
therefore you don't have a positive omitted very own spirits.
Or it, let's talk about measurement eers in the dependent
veryble as the next source of endorginate tights.
It's very good that you enrept me with the question,
it's seems because eculated ask to if you
have questions
Shouldn't the new corfficience be in perent, it is bitter
one plus and bitter to dill their one on slide 59. I
mean, check if I mean, it's take there.
Oh yes. Ah, sorry, sory.
Yes, thank you rematch. Of course,
he should be perenthesies and he should also be perenthesies.
Right. This is not sufficient to have it had
the underlining break it seele. Now you're computery right of
course.
Yes, sorry
Okay, Mission and Eric in the Dependent Verible.
And we know excited talk in the next to sections
of delecture on two forms of missionment areas,
one will would be that we measure the dependent variable incorrectly
and the other will be that we measure the independent variable incorrectly.
So supposed that first we just measure the dependent variable with
error and the independent variables emission to with out herror.
So for instance, what we would like to explain is
why star, for example, anual family savings,
And what we have is not why start,
but we have why as an observible mensure of my start,
which are ever maybe someone to falls measure of myster.
So france that could be reported annual savings. And it's
quite likely that reported savings are not the true savings,
there maybe you tax reasons for instance,
why some people don't report all of the savings.
And so, that's a quite common setting actually that the
reported data, may have a bias.
Maybe maybe under reported
I would actually here in natural discuss the case of under reporting,
I just would scustber case of incorrect reporting,
because under reporting would basically introduce a new component of the constant,
which we are not interested in here,
our intent is carapy not to measure the degree of under
reporting that that we leave to the officials in text authority.
But we just want to know what the effect of miss reporting is.
So a true model, let's say, is why star,
is equal two constant plus peter on X1,
plus plus plus, peter case,
a plus view,
that we can drast right and as expeter plus you in metrics,
nation and we as zoom that because market of acceptions,
hold for the sequation
So, if we had wise star,
the oLS-Stimate would be, the best linieur unvised, estimator,
and of crossfit would be consistent because it's already unvised.
Unfortunately, we have this measurement error. And let's is so,
the metalman error even not is, well,
it's by definition, it's why miners why, star.
So, we can reright this definition of the
measurement error here as wise star, is equite
to why miners e, that. And then plagg in
for wise star in the ligarsion equation 30.
So, that we get wise star is expeter
So, that's strong, right? This would be why
now, this would be why? Sorry,
this star here is role, because what I do is
that I have the equation why star is ex bitta plus you.
And now I replace why star by why why minus you not
and bring the you not over to the other side.
So that I would have why with out the star is
equal to expeeter plus you plus you not.
And that I could right as ex bitta plus you
idifine we to be the some of you and innot
Is I have self, when I discussed the previous page,
I will is zoom that the expectation of the measurement error is zero.
So I will not a zoom that this under reporting of savings for instance
and which case the prettation of the measurement error would be negative.
That is a said, is not a critical assumption,
it just simplipized are discussion here, because if the expectation
of you not would be non-zero than we would just move
this expectation to the constant up their predation. Question is,
how is the relationship it we in the error term you not and the explainatory variables?
Because We know that you is not qualitted with ex,
since we have made the goes mark of sumptions.
But now we have the arturn V and the new component in V is you not,
and we do not ness surn now,
that you not is uncoordinated with ex.
You may find this surprising that I post this question,
because we have said, we measure the access with out error.
And you not is just the measurement error for the wine.
So why should there be an coordination at all?
But clearly, we have to big sure that that is
no coordination and between you not and the explainatory availables
Sondern, let is make a few is standard assumptions,
which seem to ensure, that there is no problem.
For instance let is is zoom that assumptions a six point,
two holds. So that the measurement are e not eye,
is a uncorrelated with the explainatory variables for all observations eye.
So that we have eas they the expectation of exi-ipran in not eye,
is equer to the product of the two expectations and therefore it is zero.
Well, then, we know,
when Assumption as six point,
two holds,
the own
ass estimeters
are consisted
No to please that if you not eye and UI are coalited,
are uncorrorflated, then we also know that the Variance of V,
which is this strong heard,
is equal to the some of the Variances of UI and eat not eye.
And therefore it just greater then the Variance of UI.
So the measurement area in the dependent variable results in the larger
are area Variance, then when no missurement areas occurse.
And when we have a larger a resilient variance
and I have supposed it it here,
as zooming that the measurement area is
uncorrelated with
UI, that we have a larger variance of the OLS-Estimator,
therefore larger standard ears, therefore it is no likely,
that are regressor is insignificant.
Now, we do have a problem, if this acception XA6 point,
two is not holding. So if it doesn't not hold,
than there are two cases to consider,
one of huch, I want discussed by two much,
which is the case that the core variance,
between e-not at the hold
regressor matrix ex is different from Zero,
well, this emplies that OLS is bytesd
If it is true,
that the co-Variance of e-not i XI is different from Zero.
But the co-Variance with when Elon I and ExJ for Jay
not being ego to I is, speaker to Zero.
Then OLS is inconsisted. So, so post that every educated,
people know there savings better than people with little educations.
So better education, but a educated people are better informed
about what they save, then people with little education,
which perhaps had a little disorder
in the finances
This case there maybe correlation between ex one, which is education,
and you not if why is self-reported saying, struggling actual sakes.
Here a would constructed case,
where people do not miss reports to to takes reasons in the sense of rich people,
reporting lower savings than the actually have, but heard may be,
that we have a correlation, which comes strong effect,
that the well educated people no pretty well,
what they have safe.
So there reporting error
is close to Zero
As the less educated people perhaps don't know,
so much how much savings they have or props dits they have that is native samings.
So the Arrow in self-reported savings, maybe less maybe dreater,
then for the educated people.
The diractial of such a bias is this than not clear.
So we would have biased and inconsistent destinments,
but which direction, this would have,
we would not really know,
because we would need to know what the correlation is between
the measurement error,
and the
regressor education
The Conclusion, therefore is that the measurement are in the dependent verible,
does not effect unvised this,
and all the consistency of the Estimates,
provided that the measurement area is uncoordinated with the explainatory variables,
but that the measurement are increases in this case,
we uncertainty of the Estimates.
How ever,
it is not necessarily the case that the measurement area is uncoordinated
with the regressors as the example has shown of better educated
people Knowing more about their savings that less educated people in this case,
we might even have a correlation,
but we in the regressor matrix ex and the new are turn,
which also includes the measurement are.
Orange, would now come to with the section on measurement
are in explanetory variable, but this I would only start if
there is no deseire to go to inter action,
so let me please ask, would you like to go
into interractive mode now and ask questions of comment on anything
Or other questions, which you have which you would like
to put into the chat, then do it's now.
Ist es das anybody want to meet in and interactive mode
than please wave to me? Ah, okay, a
paradinot. Or it, then let's sweet to the last 5
minutes on measurement errors in explanatory vaable.
So this case we look at a variable ex,
which I con ex star,
which is the
only explaintory variable
And supposed that business model here setisfized, the goes mark of
assumptions A1 and A2. So Ex-Star is strictly exagentist with respect
to you or you strictly exagentist with respect to ex star.
So it now there is something coming in in the chat.
Jetzt haben wir diese Again asking on the exense.
But I think I answer this question already last week or
even last thirst they, but once again,
I do not really want to posit take home final like You you get
such final in mycro and mycro exams, because I cannot
control that you do communicate with each other on questions,
which have exact mathematical solutions.
So it would be very easy for one of you to just fotograph is
stallution and sended around to anybody else in taking to take a homix.
And so that wouldn't be a series, exact, actually.
I would have course not allow to for you to communicate with each other,
but I could not control that is would not be than anyway.
And those students to respect that communication is disalaut,
would be in a deterrimental cause is copyar to people
who with ignor that it is disalat.
There would wouldn't be a fair example. So I hope
that I will be able to give you or ritten example,
but that depends on with that I that it will be
possible on much too to have their ritten example
for us morn group or what currently is actually my into
criticipation that it will not be allowed on much too to have this example.
And then this case, I have ass what it
is possible to post-point the example and have it sometime youing
the summer
Semester. Morlis, like we did it last year when it
that's for a post, with their postbound do to the lockdown.
I would give you orhythm exam in the summers semester if that is possible,
if the president tegries, that that's any postbot.
And I would know have know example then on marts two,
in probably also not second, it down on on the second time,
takes of the magics.
If that example, if postpolement, it is possible.
If postpolement, postpolement, is is not possible,
at the rhythm example is not possible in marge
I frankly don't yet know, what to do and how
to do you affair exam, where cheating is not possible?
I would have to think about that situation countly, I hope that,
I we give the promission to conspond the example, if
it kind of be head askedged on much too.
Now comes the question could you letter is not,
which state you will made your final decision, well,
I would help actually that I will get the decision on
the was comment on a possible pospolement as,
rather sue and would say prepse in the week art most two weeks time.
But can'tly it is not even completely clear that the exame cannot be
held on march two. The president has issued a statement,
which says that exams, britten exams, are for been,
for exams until to februarary 15 or forctino, something like this.
So even for exams, britten exams,
and this second have of february,
it is carn'tly at least not completely impossible that they take place,
as esgedged,
and a particular forc for smoderoups
I am a little skeptical that this will be possible.
But let's seen. As you know, infection rates probably
go down in Germany, so probably get some relaxation as
for as exact are concert. I will of course,
inform you as soon is possible, but for the time be,
I would recommend that you crepare for this example here,
taken to be on march too.
And because it could be that it would take place on march to,
so just march to,
so just learn in the way that you usually would have the exense,
this is not deterrimental to you,
because this actually the regular course
of things
And if the exam is handly us point,
than it's probably better or you that,
you will be from a head of the exam of course that.
Adoesn't tick place or takesplace at the different quite to time.
And you have if not time for the exams,
the take homepans which you get in micro and it from. But currently,
I haven't received to decision on my request for cosponent,
it so I can't tell you as a say, I think,
when we probably, when we can something like this,
and I will be able to tell you.
About it, but it's not in my hand when the
decision comes acquireds in the head of the University,
present to
There is the nether question, if the late with this exam,
be along with all other examps of the summer semester now,
it would not be a long with the other examps of the summer semester at the end of this summer semester,
but I would kite to truth the exam date,
let's say, in the middle of the summer semester.
Of course, this again,
depends on infection rates in Germany,
and wherether we still have restrictions and what kind of restrictions,
we have, you know,
we will just have digital learning in the next semester,
but I hope that props in may middle of may or something like this,
we would be able to have at least ritten,
it's ain,
and if this is possible,
that I would schedule the exam in the middle of the summer semester,
and not apply end
way of other example
Okay, any other questions. Gut,
wenn there's not point point in going on with Avision at
Eric and Explaintory verable, because time is you step.
So I think you very much for your attention which you
a good in the evening and see you then next peak
to stay