Thank you.
Welcome, everybody, to today's lecture on estimation and inference.
I talked yesterday about possible violations of the standard assumptions, in particular
of the four Gauss-Malikov assumptions and the assumption of normality, normally distributed
errors, which is our assumption five.
Are there any questions?
It seems that this is not the case, so let me tell you what I will do today.
I will first talk about a more or less practical problem in econometrics, which is the selection
of the appropriate set of regressors in a regression.
This is not directly related to the problems of certain assumptions being violated, but
it is a typical and important problem in econometrics, as also the violation of assumptions is a
typical problem, so I found it appropriate to also talk about the selection of regressors
since we ended in the last lecture with a question, what exactly may be the right regressor
in the case of heterogeneity, or let's say proximate heterogeneity.
We talked about regression, where I was regressing stock prices on a constant, on a linear trend,
on two measures of total factor productivity, and then the problem is, of course, which
of these four regressors, let alone the constant, which of the three remaining regressors is
appropriate in the regression or is not appropriate in the regression.
In the context of multicollinearity, we can give a fairly clear answer, it is actually
quite easy to detect multicollinearity and find the appropriate means to combat multicollinearity,
but there are also problems associated with the procedure, which I have outlined yesterday,
because what I did was that I looked at the standard errors and the associated probability
in each of those regressions here, see them here, or see them here, the standard errors
not highlighted, but the probabilities highlighted, which of course relate to the standard errors
in the t-statistics, and I did not show you that these standard errors are actually the
correct standard errors, or that these are unbiased estimates of the correct standard
errors.
I just pretended that they are, but now I find it appropriate that you learn that this
is not necessarily the case.
This is why I talk about the selection of regressors, and you will see that later in
this lecture that finding the right standard errors is one of the problems associated with
the procedure, which I recommended for dealing with multicollinearity problems.
So far, for a short review of the last part of yesterday's lecture, let us now turn to
the problem, and this problem is that so far we have always pretended that we know the
true regressors.
So we have always pretended that we know what K is, how many regressors have to be in the
regression, and what the true regressors are.
But in reality, of course, we know neither.
We do not know what the appropriate K is, and we do not know what the true set of regressors
is, but we have to find out.
Now this gives rise to the question what happens if we specify the set of regressors incorrectly.
So let us study this question by looking at two competing models, model A and model B.
Model A is the more comprehensive model.
It specifies Y to be a function of regressors collected in a regressor matrix X, and other
regressors collected in a regressor matrix Z.
So X is multiplied by a parameter vector beta, which we want to estimate, Z is multiplied
by a parameter vector gamma, which we want to estimate, and then of course there are
errors U.
Model B is more parsimonious.
Model B specifies that Y is just a function of X beta plus eta.
So this is basically this kind of setup that we have been looking at so far with some regressor
matrix X.
And here we split the regressors in two different matrices and also split the parameter vectors.
Obviously, when we have this type of additional regressor, then the error term will in general
be a different error term from the error term in regression 42.
So I use different symbols here, but both of them are just errors.
U and eta are just errors with expectation zero, and we don't want to make that complicated.
So these errors can have a normal standard diagonal homoscedastic covariance matrix,
and they may even have a normal distribution.
So you may assume whatever you want about the errors here.
You'll see that nevertheless we run into trouble.
Now when we have these two competing models and we do not know which of the two models
is true, then there are basically two possibilities for errors we can make.
Either model A is true, the first comprehensive model is true, but we estimate erroneously
model B. Then we have what is called an omitted variables problem, because obviously in the
model which we estimate, model B, we omit certain variables which actually belong into
the model, namely the variables in a matrix Z. So this first case that model A is true
and we estimate model B is the omitted variables problem.
The same possible error is that model B is actually true, but erroneously we estimate
model A. If this is the case, then we specify regressors which actually do not belong to
the regression.
Namely, these regressors Z here, so we have a problem of superfluous regressors.
You'll see that the second problem is in some sense not as grave as the first problem, but
let me work out the details here so you can better understand why it is less grave, why
there are still problems also with the second problem.
We first turn to the omitted variables problem.
When we estimate model B, then obviously we get some estimator, beta hat B, the index
B here is of course for model B. This estimator is X prime X inverse times X prime Y because
we estimate the parsimonious model where we just have a regressor matrix X.
What is this estimator like?
We can replace the Y by the true model, but the true model must be taken from model A
because we assume that model A is true.
This is why we have the omitted variables problem.
The estimator beta hat B is actually X prime X inverse times X prime times X beta plus
Z gamma plus U.
Here things are different now from the standard setup.
As you know, we would not have this matrix Z gamma here, and then we would have nice
property that X prime X inverse, X prime X is just the identity matrix.
The two matrices cancel, one is the inverse of the other, so we just have beta there plus
then this linear combination of the error terms X prime X inverse, X prime U.
We've talked about this a lot, but now we also have this term Z gamma in the composition
of Y.
We can use this equation further and write it as X prime X inverse, X prime X beta.
That's the standard term here.
This is identity matrix.
These two matrices cancel, become just beta plus X prime X inverse, X prime Z gamma, which
is the new term, which will cause us some problems plus X prime X inverse, X prime U,
which is a term we also already know from our prior analysis.
Essentially, this is beta plus X prime X inverse, X prime Z gamma plus X prime X inverse, X
prime U.
We have made assumptions which dealt with this term here.
Either we assume the strict exogeneity of X and then the expectation of this term is
zero and the standard estimator, the standard please square estimator was combined.
This would not have us here, this assumption clearly, because even if the expectation of
this term here is zero, then we would still have this term X from X inverse, X prime Z
gamma and the expectation of this term is not zero very clearly.
The estimator would be biased even if we have strict exogeneity of X, even if Z is also
strictly exogenous.
Whatever we do, we always end up with the result that the estimate is biased.
This is what I've written here under assumptions A1 and A2.
We could also assume A3, A4, A5, it wouldn't matter.
We would have that the expectation of beta hat B is equal to beta plus X prime X inverse,
the X prime Z gamma so that in general, the estimate beta hat B is biased unless, but
this is a very specific condition, unless X and Z are exactly orthogonal to each other,
which would mean that X prime Z is equal to zero.
Under a specific circumstances, this can happen that X prime Z just cancel because the vectors
in X are exactly orthogonal to the vectors in Z, but that's a very rare case.
How about consistency?
Consistency can also not be established with the standard assumptions for the same reason.
You remember we had assumptions on consistency, which basically guaranteed that X prime X
inverse C or 1 over N X prime X inverse of that would converge in probability to a finite
and invertible matrix and X prime U divided by N would convert to zero, then this whole
term here would at least vanish asymptotically.
But even if we make these assumptions, the problem is not solved because we would be
left with this term here.
Again, we would see that 1 over N X prime X inverse C converges to a finite matrix,
a veritable finite matrix and X prime Z, the usual assumption would be that X prime Z over
N also converges to some finite matrix.
So we would have some finite matrix here which destroys the consistency property because the
limit when N goes to infinity, we would find that the estimator beta hat B has as its limit
beta plus the product of two finite matrices of which we know next to nothing, except that
they are not zero usually unless X prime Z becomes orthogonal to X becomes orthogonal
to Z, but that is, as I say, is a rare case.
So we can neither establish the unbiasedness property for this estimator beta hat B, nor
can we establish a consistency result.
Now, what else do we know?
We know that under the assumptions, the standard assumptions A1 to A4, the best linear unbiased
estimator for beta is the least squares estimator of model A because, as we have assumed, model
A is true in the case of omitted variables.
So what can we do?
We can write the estimator for model A, which I here call beta hat A, using some rules on
partitioned inverses as X prime X minus X prime Z, Z prime Z inverse C, Z prime X, and
the inverse of all of this times X prime minus X prime Z times Z prime Z inverse C times
Z prime Y.
If you know the rules on partitioned inverses, it will not be difficult for you to verify
that.
If you do not know them, it's not a big problem.
Just believe me that this here is the least squares estimator and therefore the blue for
beta when we just estimate model A. So we would get this formula here.
And essentially, you see that this can easily be true or that is quite likely that is true
because if the Z matrix were zero, well, then, of course, this expression, which I have written
down here, simplifies to the standard least squares estimator, which is the blue because
if the Zs are all zero, then this term here cancels and we are just left with X prime
X inverse C and this term here cancels.
So we're just left with X prime.
So we would have the standard least squares estimator in the case of Z equal to zero.
And if Z is not equal to zero, well, then the estimator makes this particular form here.
So what we see is that in general, the estimator beta hat A is different from the estimator
beta hat B because this is beta hat A and beta hat B. We have computed here and this
is obviously a very different expression.
Only in the special case where X prime Z is equal to zero, we would have the result that
beta hat A is equal to beta hat B. That means in this particular case, estimating the wrong
model B would still reside in the best linear and unbiased estimator for beta in the class
of all linear estimators.
But as I say, it's a special case and a very special case indeed.
So usually we would not find that this property here holds that X prime Z is equal to zero.
In all other cases, the beta hat B estimator would apparently not be the blue estimator
and quite likely it would be biased or actually, as we know, it is biased as long as X prime
Z is different from zero.
Now here is an exercise for you.
Define a so-called projector matrix P as P is equal to identity matrix minus Z, Z prime
Z inverse C, Z prime.
Those of you who are familiar with matrix algebra or who have worked through the slides
that I already discussed with you will immediately see this is a matrix which is idempotent.
Therefore, it is a projector matrix.
It's idempotent and symmetric, by the way.
So it belongs to those class of matrices which have eigenvalues just equal to one or to zero.
We talked about these, I think, in the last week.
I presented you the proof using this property here.
We don't need the property now.
I just defined the matrix this way and remind you that this is an idempotent matrix.
What you use this matrix for now is that you just pre-multiply the whole model A by this
projector matrix P. In this case, you would get some matrix PY, some vector PY on the
left-hand side of the equation.
I call this Y tilde.
On the right-hand side of the equation, you would get a matrix P times X, which I call
X tilde, and a matrix P times Z, which I call Z tilde, and, of course, the errors P times
U, I would call U tilde, that would be a new vector.
So essentially, what you do is you take this model here and multiply from the left-hand
side, all variables here with P, and then denote the model with tilde in the way in
which I have just described it.
So that's very easy.
Now, show that this pre-multiplycation of the A model with P filters out information,
which is unnecessary for the estimation of beta, so that the least squares estimator
of the transform model is still the blue for beta.
So it is equal to beta hat A. Good.
This completes the discussion of omitted variables, and you see that omitted variables cause serious
problems because we make it biased estimates, and the bias may be substantial.
So you might think that a good remedy for this problem is to start out with models
which include many, many variables, so that you make sure that you do not leave out any
variable which could possibly explain some share of the variance of the dependent variable.
If you do it this way, which is actually a possible way to proceed, then you have the
problem that perhaps you include more regressors in your model than you actually need to include
and are actually justified under the true model.
So you may easily run actually into the problem of superfluous regressors.
We look at this approach now by switching roles for model A and B. So we say that now
we assume that model B is true, and we estimate model A.
The first thing to note is, and perhaps it's easier to see this when we go back to the
two models, so we say that model B is true.
The first thing to note now is that you cannot say that model A is false.
Quite to the contrary, if model B is true, then model A is also true.
Because model A would be true with a value for gamma equal to zero.
So model B is actually nested in model A. Model A is true, and model B is true, and
we would estimate a true model, just with variables which have a coefficient estimate.
So in the population, the true coefficient for some of the variables which you include
in the regression is zero.
And you may say, so why not, then we estimate zero for these variables.
And indeed, some people argue that one can do it this way, in particular in VAR models.
This is a standard approach to just include all the variables and don't care whether they
have coefficients which are different from zero or whether they are equal to zero.
Because people argue, well, asymptotically, we will estimate the true coefficients anyway.
I would like to convince you that it is not quite as easy and that there are problems
associated with that.
OK, so we are here at the superfluous regressors problem, and I just noted that model A is
also true, namely with gamma equal to zero.
Which means that the estimator beta hat A is unbiased under just the assumptions A1
and A2.
And consistency can easily be established under the weaker assumptions A7 and A8, about
which we talked yesterday.
But the problem is that while beta hat A is unbiased, and in the case that assumption
A2 doesn't hold and is replaced by assumptions A7 and A8, still consistent, it is not the
case that beta hat A is variance, minimum.
So beta hat A would not be the blue, because we know the blue for this model is beta hat
B, and beta hat B is x prime x inverse c x prime y.
This is of course different from the model, from the estimator beta hat A, which as you
would have shown in the previous exercise that I just described, is equal to x tilde
prime x tilde inverse c times x tilde prime y tilde.
This estimator here differs from estimator beta hat B. Since we know that beta hat B
is the blue, we know that beta hat A is not the blue.
Maybe unbiased, but it is certainly not variance, minimum.
When beta hat A is known to be not variance, minimal, this means that the standard errors
which we estimate for beta hat A are greater than they need B.
So actually, estimating model A, if model B is true, implies that we have unnecessarily
high probabilities for type 2 errors for beta.
So there would be an unnecessarily high probability for accepting a null hypothesis, which is
in fact wrong.
This is of course something which we don't want to have in our econometric analysis.
Moreover, it also implies the possibilities of type 1 errors for gamma.
Because we would know the true gamma is 0, a type 1 error would mean we reject the hypothesis
that gamma is equal to 0 in favor of non-zero gamma, which would be a type 1 error.
And obviously, we don't want to have such type 1 errors.
So estimating model A, if in fact model B is true, implies higher probabilities for
type 2 errors and actually also higher probabilities for type 1 errors, because when estimating
beta hat B, the probability of the type 1 error for gamma is 0.
And here it would be equal to the significance level, so to alpha 2 to 0.5 or something else.
Coming up, misspecifications by either specifying too many regressors or specifying too few
regressors, so the orbital variables problem, misspecifications in general induce inefficient
estimation.
Inefficient estimation means we could do better, and more precisely, it means we have an unnecessarily
high variance of the estimator.
An exception in the case of the omitted variables problem is the case that x prime z is equal
to 0, so that the two sets of regressors are orthogonal to each other, but as I have said,
this is very rare.
If we specify too many regressors, so we have the superfluous regressors problem, then the
estimates maybe will still be unbiased or consistent under standard assumptions, but
the variance will be unnecessarily high.
Specifying too few regressors, which would be the omitted variables problem, yields in
general not only an unnecessarily high variance, but also biased and inconsistent estimates,
with the exception again of x prime z equal to 0.
Now in terms of estimation strategy, there are two possibilities.
One possibility is what is called general to specific modeling.
General to specific modeling means we start out with a very general model in which we
allow for all kinds of regressors, which we find interesting, arguing that if they don't
belong into this regression, if they are not a part of the true model, then we will, at
least asymptotically, since the estimates are consistent, we will estimate a zero coefficient
for the superfluous regressors.
So many people like this approach and say we do general to specific modeling, start
out with something which involves many, many regressors, and then we reduce the number
of regressors stepwise by eliminating the insignificant regressors.
Other people do it exactly the opposite way around.
They use a specific to general modeling strategy.
They start out with the most important regressors and then add, sequentially, other regressors,
retaining them if they are significant and picking them out if they are not significant.
So basically these people say, well, for instance, economic theory tells me there is a handful
of regressors which really should be included because theory knows that these regressors
explain the dependent variable.
So they start out with a set of most important regressors, for instance, derived from theory,
and then they look around and look for suggestions what else to include and include other variables
one by one, deleting them from the regression equation.
If the expanded regression equation does not produce a significant t-statistic for
this additional variable.
This problem is, or this way of proceeding, has the problem that the estimates will be
biased, right?
So it is not really clear whether we test the correct analysis.
How many people think that is better to use the general to specific method because their
estimates are at least consistent, if not even unbiased.
But as I have said, too many regressors, and this is the problem of the general to specific
approach, induce unacceptably high variance.
And if this is the case, then it may be that we have a regressor which actually belongs
into the true model, but it appears to be not significantly related to the dependent
variable.
So its coefficient seems to be not significantly different from zero because the standard error
for this regression coefficient, the estimated standard error is so big, bigger than necessary
actually that we think the variable is insignificant and then we throw it out of the general to
specific approach.
And then we are, of course, in trouble because once we have thrown out a variable which actually
belongs into the model, which is a part of the true model, then we have the omitted variables
problem.
And then all our subsequent estimates will be biased.
Good.
Just as a tiny remark, once again, hopefully we don't get tired of it.
The specific to general approach does not suffer from bias if additional regressors
are orthogonal to existing regressors.
Now how can we judge which set of regressors is appropriate?
Actually there is no clear cut statistical criterion or there are statistical criteria,
but each of these criteria has its disadvantages.
So none has really dominated usage.
Some people say, well, we just look at r bar squared.
So at the corrected measure of coefficient of determination, we talked about this r bar
square as an alternative to r squared.
And you recall that its definition is as I have written down here.
R bar squared, if you recall, has basically two elements in it which are important.
One element is that r bar squared decreases when the sum of squared residuals increases.
So we have u hat prime, u hat here.
You know that u hat prime, u hat is the sum of squared residuals.
So when the sum of squared residuals increases, then obviously since there's a negative sign
here, more is subtracted from 1, and therefore r bar squared is lower.
So increases in u hat prime, u hat, lead to a decrease of r bar squared.
The other issue is that we look at the number of regressors which we use, K. And for the
number of regressors we use K here, we see that r bar squared decreases with increasing
number of K. That was actually the main idea of constructing
r bar squared, if you recall.
So it looks like this is perhaps a way to find out which regressors we shall include
in our regression and which we shall not include in our regression.
But there is a but, like always, the problem is that r bar squared rises or falls with
an additional regressor, or the question whether r bar squared rises or falls with
an additional regressor depends on the set of regressors already in the regression.
So we do not get a unique statistic for one possible regressor which we examine where
we want to decide whether this regressor belongs into the regression or doesn't.
Whatever we do, it depends on the regressors we have already in the regression.
And if we use different approaches here, we will get different advice about whether
the new regressor shall be included in the regression or not.
So in that sense, r bar squared is not a very good measure of advice for the problem of
either permitted variables or superfluous variables.
Now as I have just emphasized in r bar squared, the fit of the model depends on 1 over n minus
k u hat prime u hat.
People have developed other criteria for judging the fit of a model.
And these run under the label information criteria.
I won't discuss the information criteria here in detail, I just mentioned the two most important
ones which are called Akaiike's information criterion, abbreviated AIC for Akaiike's information
criterion.
And the so-called Bayesian information criterion abbreviated BIC.
So we have AIC and BIC and actually we have a couple of other information criteria which
I do not introduce you to.
The Bayesian information criterion is more often actually called the Schwartz criterion
after it's a vendor.
So then it's usually abbreviated SC.
And as you can see from the formula here, the difference between AIC and BIC is more
a difference in detail than a difference in principle.
Because in both cases, the information criterion is such that it increases with u hat prime
u hat.
So the first term is always log over u hat prime u hat divided by n.
Same thing here, AIC and BIC are not different.
Now why actually do we have here the log of u hat prime u hat in the definition of the
information criterion?
This means as you will, as you easily see that when u hat prime u hat increases, so
when the sum of squared resilience increases, that then the information criterion increases.
Quite the opposite of what we have seen for R bar squared when which had the property
that when u hat prime u hat increases, then R bar squared decreased information criteria
increase.
Well, that's not a big problem.
You just have to bear in mind that the optimal AIC or the optimal BIC is always as low as
possible.
Whereas for R bar squared, we try to get high value for R bar squared for the information
criteria.
They are set up in such a way that we look for the lowest possible value of the information
criteria, either akaika or Bayesian.
So one may use one information criterion or one may use another one.
They are both equal in terms of the first term here, but they differ in terms of the
second term.
So for akaika's information criterion, you see that the second term is 2k over n.
So again, AIC increases with k, so gets worse when we have more regressors.
The same is actually true for BIC, which also increases with k.
So the value of BIC gets worse when k increases.
But in akaika, the k is normalized with respect just to n, and here it is normalized with
respect to log n over n.
So that's actually the only difference between AIC and BIC, the normalization of k.
And this results in the fact that BIC typically prefers the more parsimonious models than
AIC does.
So typically, AIC will take on its minimum in the specification, which involves rather
many regressors, whereas typically BIC will take on its minimum in the specification,
which involves rather few regressors.
So this is, of course, in some way dangerous, because many people first look at the regression
result, and then they decide which regression result they actually would like to publish.
And then they decide on the information criterion, which supports this regression result.
So for instance, they have a long regression, a regression with many regressors, and they
think this is a very nice regression, and they chose exactly what I would like to show.
So I use the Akaika information criterion because Akaika's criterion typically supports
long specifications with many regressors.
All people have the intention of showing that certain variables just don't matter, so they
are actually interested in short specifications, in specification where there are only fewer
regressors.
That may well be that people first estimate for any kind of specification or regression
models, and then they look at the BIC, and intentionally they choose the BIC knowing
that the BIC will advise them to use rather short regressions with few regressors.
So results would be biased by the a priori points of view of the researchers on what
their results should be, because the researchers would choose the information criterion, which
fits their basic views as best as possible.
So this is not good statistics.
You should not do this.
You should actually decide on which of the information criteria you would like to use
before you actually look at your data, before you run the first regression.
First fix and say, well, I will use AIC throughout, and then you look at the data and estimate
and apply the AIC, and then only the AIC and not the BIC.
Or conversely, you fix the BIC as your measure for the model before looking at the data,
and then you use the BIC consistently and do not look at the AIC.
That's actually the way to be used.
Or on the information criteria, like Akaiike and Bayesian, and there are two more actually,
I will give in my lecture on time series econometrics next year, but I won't go to the introductory
lecture.
Any questions so far?
It seems that this is not the case, so let us move on, and now I come back to violations
of our basic assumptions.
So we spoke already about the violation of assumption A2 and the violation of assumption
A5.
So violation of the assumption of strict exogeneity of the regressor matrix and violation of
the assumption that errors are normally distributed.
Let us now talk about the assumption A3, which postulates homoscedasticity and a diagonal
covariance matrix, which would be A4, for the shocks in this model.
So we have had those two assumptions.
They are actually quite related to each other.
A3 imposes homoscedasticity, so on the main diagonal of the covariance matrix, you always
find the same variance, and A4, as you know, postulates uncorrelatedness between shocks
of different observations.
So under A3 and A4, we would have that the covariance matrix of the U shocks is equal
to sigma square U, sigma square I, and now suppose that this is not true, that assumptions
A3 and A4 are violated.
And actually, the covariance matrix of U is just some matrix omega.
So we do not suppose any more that it's a diagonal matrix with all elements on the main
diagonal being equal, but rather we allow for any possible correlation between the disturbances
of U so that the covariance matrix of U would just be some matrix omega.
Of this matrix, we assume next to nothing except that it is symmetric and veritable
because any covariance matrix must always be symmetric and convertible just follows
from the fact that you do not have redundant shocks.
Now just as a result from linear algebra, if we have such a matrix which is symmetric
and convertible, we can decompose the matrix into two matrices P, where P prime P is exactly
equal to omega inverse.
This actually implies that if you take the inverse of the left-hand side and the inverse
of the right-hand side, it implies that omega is equal to P inverse E times P prime inverse
E.
During the setup, we may ask, how does actually the covariance matrix of U being equal to
omega affect the properties of the least squares estimator?
Well, that is rather easy to say if we concentrate on the first properties we have established,
namely on unbiasedness and variance minimality.
The first thing is that we easily see that already under assumptions A1 and A2, the least
squares estimator is still unbiased, namely the expectation of beta hat would be the expectation
of beta plus x prime x inverse E x prime U, same thing as that all the time, which is
equal to the expectation of beta times x prime x inverse E x prime.
And for U, we can, by the law of iterative expectations, replace E of U given x.
And we know that the expectation of U given x is equal to zero.
So the expectation of beta hat is equal to beta, even in the case where the covariance
matrix of the resilience, the covariance matrix of the sharks is different from sigma square
i.
Actually, this result of unbiasedness is true for any kind of distribution of the use, as
I have already repeatedly told you in this lecture.
However, the drawback of the general covariance matrix is that the ordinary least squares
estimator, which I will henceforth call the OLS, so ordinary least squares estimator,
is not any longer the blue for beta, even if assumptions a1 to a4, the standard cost
mark-up assumptions are satisfied, no, assumptions a1, a2 are satisfied, f3, f4 are of course
not satisfied, because we have this general covariance matrix omega.
So while the estimator is unbiased, even if the errors do not have this homoscedastic
structure expressed in assumptions a3 and a4, even though it is still unbiased, the estimator
is not any longer variance minimum.
Now try to get a feeling for this problem by doing the following exercise.
Suppose that equation 44 holds, this is the equation about the covariance matrix, right?
V of u is equal to some symmetric and vertical matrix omega.
If this is the case that assumption that equation 44 holds, and you now use standard econometric
software to estimate beta, then you will get beta hat as x prime x inverse x prime y, right?
The standard software just runs the usual OLS estimate.
The standard software then also gives you some output, which includes p values and t
statistics and of course, standard errors, estimated standard errors, and here you should
ask yourself, well, what are these estimates of the standard errors?
Actually, where do they come from?
And are these estimates reliable?
Basically, this is the question I have posed already at the beginning of this lecture,
when I was looking at these results here, and I was asking,
what do we actually know about those standard errors here?
Are they the right standard errors?
This problem comes up again, because what the commercial software does is
the commercial software just estimates the covariance matrix of beta hat
using the formula that we have derived in the standard Gauss Markov setup,
namely as sigma square u times x prime x inverse z.
So implicitly, the software always assumes that the covariance matrix of our residuals is equal
to sigma square u times i. And then if this assumption is correct, which the software
implicitly makes, then we get at least consistent estimates for the standard
errors of our coefficients, and we can use them for testing purposes.
Now, in this exercise, first thing you should do is compute just by hand the true covariance
matrix.
So my claim is the formula which commercial software uses to compute the standard errors
is inappropriate in the case of a general covariance matrix like v of u is equal to
sigma square u times omega. So one should not use this matrix here.
Now, is this a bug in the programming of a commercial software?
As I already told you, all commercial software will compute the standard error estimates,
just using this formula here. Even if test results indicate that this assumption which
underlies this formula here is violated, is not true, that the errors have some kind of correlation.
Still, standard software will always estimate the covariance matrix using this formula here.
Why is this so? Well, this is, sorry, this is part of the exercise. Question two.
Think about it. Why can commercial software not be programmed in such a way that actually
estimates omega? And then based on the estimate of omega computes the standard errors correctly.
And I give you a hint here, which was not yet a new set of slides I edited later.
When you want to answer the second question, then think about how many parameters do we need
to estimate matrix omega? That's a question. Yes, this is right. I will upload this set of slides,
but Steena was down this morning. At least I did not have access, so I couldn't upload it,
but I will do it as soon as possible. All right. Any other questions?
Good. Then let's move on. Another exercise would be, again, a computer-based exercise. So again,
take some programming language and now generate, again, matrix of regressors X. In previous
exercises, I have recommended to generate 100 by 3 matrix here. 100 by 1 matrix would already be
completely sufficient, right? So just generate some random matrix X of any kind of numbers.
And then again, go through a loop, let's say, a thousand times. So J, again, indicates the number
of the loop you are in. In each loop, you generate some vector of disturbances UJ, and then you
compute YJ as being equal to X plus UJ, which basically means that you have assumed that beta
is equal to 1, right? And then each time, estimate beta hat J. So the Jth estimate of the beta hat
vector as X prime X inverse C, X prime YJ. This beta hat J would just be a scalar. Save this
estimated beta hat J. So the best thing is to collect it in some vector in which you would then
get a thousand estimates of beta hat J for each of the J loops you run through. And also save the
standard error computed from using the standard formula sigma square U hat times X prime X inverse.
Now, after finishing the loop, then you compute the standard deviation of all the estimated beta
hat Js and compare this with the mean of all the standard errors. There's something missing here
with the mean of all standard errors of all standard errors. Sorry, what did I write there?
Yes. Yes, exactly. With the mean of all standard errors, that's what it meant to mean. So this of
all here is of all is just just just compared with the mean of all the standard errors because the
mean of all standard errors is actually what you would estimate as the variance, as the standard
deviation of the beta J, beta hat Js. And when you actually compute the standard deviation of all the
beta hat Js, you should get the same result. Now, that is standard. And you should actually get
that the standard deviation of all the beta hat Js here, approximately equals the mean of all the
standard errors. Next, redo this experiment, but modify it. So when you draw the disturbances UJ,
we had 100 by 1 random vector of disturbances UJ, which we used to construct the YJ estimates here.
When you draw those disturbances, do it in the following way. The first 50 disturbances
are just drawn as before. But for the remaining 50 disturbances, you increase the standard deviation
by a factor of 10. So you actually now don't have homoscedastic errors anymore, but you have
heteroscedasticity. The second 50 observations have a much higher variance than the first
observation. So think about what does the covariance matrix of the disturbances looks like in this case.
And then go through the loop and compare the true standard deviation of all the estimated
beta hat Js with the estimate of their standard errors, which the regression output would give you
when they estimate the standard error using the typical formula sigma hat square U times x prime
x inverse. And you should then find that the standard deviation of the estimated beta hat Js
is way larger, actually, than what you estimate when you use the standard formula here. So this
should convince you that applying this formula as commercial software does is giving a well,
greatly biased estimate, a greatly wrong estimate of the true standard deviation of the estimated
regression coefficients. Next thing we wanted to study is what actually would be the best
linear unbiased estimator, so the blue, for model 19 under the general covariance matrix,
which I specified an equation for before so that the covariance matrix of U is equal to
sigma square U times omega. Well, that is actually very easily done by just pre-multiplying the model
equation 19 by this matrix P, which was basically the square root of omega. Perhaps just to recall
again, no, this is already too far, we had written that omega inverse is equal to P prime
P. And sometimes loosely speaking, one says then that the P matrix is the square root of this
matrix omega inverse, right? Or the P inverse matrix is the square root of this matrix omega
because P prime P is equal to omega inverse, so this is like squaring matrix P just with a
transposition, and then you get omega inverse, or here the same thing for the inverse matrices.
Now we use this matrix P, which is the square root of the inverse of the omega matrix,
just a minute ago I said of the omega matrix, but it's the square root of the inverse of the
omega matrix, so that we get P y is equal to P x beta plus P u. Well, this P u are just denote by
U tilde. It is easy to see that the expectation of U tilde is the expectation of P times U,
is P times the expectation of U, and the expectation of U is zero by assumption one,
so U tilde is just an error term. Now as an exercise show that the variance of U tilde
is equal to I, the variance of U tilde is equal to the identity matrix. So when we transform the
model in this way by pre-multiplying with the square root of the omega inverse matrix,
then we would get a model which satisfies, because Markov assumptions one, two, four,
provided that assumption A2 was already holding for an x and u. So that seems to be a very easy
way out. We just need to take the square root of this omega matrix, pre-multiply the whole model
with it, and then we have a transform model, which we can estimate with the usual estimator,
with usual least squares estimator, and therefore we would be able to have a blue estimation of beta.
Actually, this is the way how we compute the blue for the new model where the covariance
matrix of U is equal to omega. That would be beta hat, and I call it GLS for generalized least
squares estimator, the generalization being that the covariance matrix is now not an identity
matrix, but it is a matrix omega, general matrix omega. Well, this matrix just applying
the least squares estimator to this transform model here would be x prime p prime p x inverse
times x prime p prime p y, and as we know that p prime p is equal to omega inverse,
and p prime p is equal to omega inverse also here. This estimator can be written as x prime
omega inverse x, the inverse of this matrix, times x prime omega inverse y.
Right? So this here is the generalized least squares estimator. You immediately see that it
is a generalization of the common ordinary least squares estimator, because in the special case
where omega is just equal to an identity matrix, of course then this omega inverse would also be
an identity matrix, and the estimator simplifies to the familiar formula x prime x inverse times
x prime y. So we would have the OLS estimator, but if omega is different from i, then this year would
be the blue. Now what you see in this formula 46 is that in this formula, the observations x are
being multiplied by omega inverse, both here and here, or here it is x and y which are being
multiplied by omega inverse, and here it is x prime and x which has been multiplied by omega inverse.
So this essentially means that the observations which we have on x and on y, they are weighted
by the inverse of the covariance matrix of u. Now I write here y, but I think I better ask you this
question now, and perhaps somebody knows the answer, otherwise I will give it to you if this
is perhaps too difficult to see. Why is it optimal to weigh the observations which are matrix x,
and in vector y, why is it optimal to weigh the observation with omega inverse? So with the
inverse of the covariance matrix of y, perhaps I, of u, excuse me, covariance matrix of u,
perhaps I should just explain why is this actually a way, where because again we could write a factor
one over n inside these parentheses here, and one over n here, and then essentially what would
happen is that we have certain weights here, basically covariancies expressed as a share of
all of the observations right here and there, so we can think of these as weights. Another question
is why do we weigh the observations with the inverse of the covariance matrix? Any thoughts
about that? Please raise your hand.
If there are no thoughts about it, then let me give you the answer. See, in the covariance matrix
of the disturbances, we would have a great element on the main diagonal, so we would have much
variance on the main diagonal for certain observations which are recorded with much error,
or where the relationship which we assume to be true, namely y is equal to x beta plus u,
is associated with a great error, so for those observations for which the relationship which
we want to estimate is rather shaky, rather unsure, rather unstable, associated with lots of error,
for those observations it is optimum to give little weight to these observations because we
use omega inverse here, so when the variance is high for certain observations, then we weigh
these information with the inverse of the variance, which means that we weigh them with a low weight.
It is easier to infer something about beta from observations where the variance of the u is small,
so weighing observations with the inverse of the variance of the u is optimal because it
filters out or puts more emphasis on those observations which we record with lower error,
with little error, with small errors, whereas it is also optimal to put little weight on observations
where we know that the error associated with these observations is great, and so this is why
it is optimal to weigh with the inverse of the covariance matrix.
This technique of weighing observations with the inverse of the covariance matrix of the error
is a very common technique in econometrics. For many, many estimators, you will see exactly this
principle that observations are weighed by the inverse of the covariance matrix, so many,
many estimators are actually set up exactly as this GLS estimator here, where the observations
are weighted by the inverse of the covariance matrix. Now, all of this here is a very nice
estimator. It's easy to derive. It's plausible that this is the estimator. By the way, it is
called the Aitken estimator after a guy called Aitken who developed it, and one might think,
well, why don't we just make use of this estimator in all suitable cases? The problem is, of course,
we just don't know omega, right? We don't observe omega. X and Y are data we have at our disposal,
but the omega we do not have at our disposal, and as you should find out in one of the previous
exercises, it is impossible to estimate the omega without further restrictions because there are
just too many parameters in omega to be estimated. We have more parameters in omega than we actually
have observations available, so we just cannot estimate all the parameters of the omega. The
exercise was about find out how many parameters there are in omega and see that there are more
of them than we have actually observations. So we need to move to specific forms of
omegas, of more general covariance matrices than just the identity matrix, and one possible
assumption would be that we have what is called heteroscalasticity. That is to say a violation
of assumption A3 now. So let us look at an omega matrix which is of the following form,
sigma 1 square on the main diagonal, then sigma 2 square, then sigma 3 square,
all the way down to sigma n square. All these sigma squares here are different,
or maybe different. At homoscalasticity, we would say all these sigma squares are the same,
that there's just one sigma square, so the covariance matrix is sigma square U times identity
matrix, but here we would have a diagonal matrix with n different variances for n different
shocks, which we have. Now again, first try to make your life simple by assuming that we know
those sigma parameters, which of course we don't. But if we do know them, then we could proceed as
follows. We could say omega inverse is of course just 1 over sigma 1 square, 1 over sigma 2 square,
and so forth, up to 1 over sigma n square. And so that's very easy. When you have a diagonal
matrix, then its inverse is just the same matrix with the diagonal elements inverted. So we would
have this as omega inverse. And therefore the P matrix would be 1 over sigma 1, so 1 over the
standard deviation rather than 1 over the variance of the first error, 1 over sigma 2, 1 over sigma
3, 1 over sigma n, and so forth. So that's also easily computed. Therefore, multiplying our model
with P would result in the transformed model 45 look like this. We would have in the vector of
dependent variables y1 being scaled down by sigma 1, y2 being scaled down by sigma 2,
and so forth, yn being scaled down by sigma n. So each of the dependent variables would just
be divided by the standard deviation of the respective shock, which is again this type of
weighing of observations with the inverse of its variance, which I have already talked about.
And the same thing would happen to the regressors. So each regressor in the first row of regressors
would be scaled down by sigma 1. And each regressor in the last row would be scaled down by sigma n.
So this would be the transformed model Py is equal to Px times beta plus then Pu. And Pu is
of course, analogously to the y vector, u1 over sigma 1, u2 over sigma 2, and then all the way
down to un over sigma n. The observations with large shocks receive lower weight in the estimation
here. That is the main idea. When the variance is great, when the standard deviation is great,
then these observations receive lower weight.
Therefore, this type of GLS estimator is also called weighted least squares.
It's also a nice idea. Actually, it's just a special case of the GLS estimator,
of the Aitken estimator. But the problem is that our initial assumption that we know all the sigma
squared terms here, sigma 1 squared, sigma n squared, this assumption is plainly not true.
So you might say, well, why don't we estimate them? But we cannot estimate them because we would
need to estimate then n plus k parameters. We need to estimate all those sigma parameters,
of which there are n, plus then the k parameters in the beta vector. So we would need to estimate
n plus k parameters, but we have only n observations. So impossible, right? Impossible
to estimate n plus k parameters with just n observations. We would have negative degrees of
freedom. So we just cannot estimate them this way. So what can we do then when we suspect,
or when we actually see from residual analysis that the homoskeleticity assumption in 3 is
violated, does not hold, there is no easy way out of that. Sometimes people try to model the
variances then. So one way here would be to assume that the variances of the shocks are
proportional to the values taken on by one of the regressors. Don't care which regressor,
actually one can also specify a linear combination of regressors. But let's say it is regressor x2.
So they would just make an assumption, hopefully based on some economic reasoning,
which would then say the variance of shock ui, given that regressor 2 takes on value xi2,
this variance is sigma squared u times xi2 squared. And this shall be true for all of the i's.
So if this is the case, then you see we have just one parameter which we have to estimate, because
for all the ui's, we would just have this unknown parameter sigma square u. We wouldn't have n
different sigmas, but we would just have one different sigma. And the idiosyncratic part
of the error variance is taken on by an observable quantity named by xi2.
So in this case, we would only have to estimate the sigma squared u, which means that we transform
the model to heteroskeleticity when we divide through by xi2. So for the
for the ith observation, we could write yi divided by xi2 is equal to xi prime times beta
divided by xi2 plus ui divided by xi2. And then it would be the case that the variance of ui
through are divided by ui2, given ui2 is equal to 1 over xi2 squared times the variance of ui
given xi2. Well, and this is then, of course, just sigma squared u. So we would have a covariance
matrix, which is almost get us homoscedastic after transforming the model to this form a 49 here.
So that the OLS is then the best linear unbiased estimator.
The related assumption which people sometimes make, or a related form of
heteroskeleticity might prove even a little easier. For instance, sometimes people assume
that the error ui is equal to some error eta i times xi2. Sometimes some regressor, which we
have one of the regressors doesn't need to be regressor number two, but any regressor or linear
combination or just any function of any regression regressor. So when there is this kind of
multiplicative relationship between some error, which is homoscedastic, eta i would seem to be
homoscedastic, but it is multiplied by xi2 so that ui is the product of eta i and xi2.
Ui as the product of eta i and xi2 is heteroscedastic. Then it would help already if we just
take the logarithms of the model, because the log of ui would then be equal to the log of xi2
plus the log of eta i and eta i is homoscedastic. So in this case, we could actually very easily
solve the issue of heteroscedastic errors by taking the log of the model provided that
the systematic part is also just multiplicatively related to the error terms and then estimate the
model in logarithms. So sometimes you may have good luck and you will find that when the model
estimates the levels of the variables, it is difficult because the error structure is not
homoscedastic that then taking logarithms relaxes things and you have a nice model
which seems to be homoscedastic. Now there is a third version which is actually the most
popular version today probably, because when we shy away from assumptions such as the ones that
I have just explained, there's also a way to estimate the heteroscedastic model using OLS.
When this is done, the covariance matrix turns out to be v of beta hat equal to x prime x
inverse e x prime omega x times x prime x inverse e. So this would be the covariance matrix if we
estimate a model which has its error covariance matrix, this omega matrix, and we just
estimate it by OLS. So then we get this here as covariance matrix. We can actually estimate
such a model by OLS because as we have seen, the estimate would at least be unbiased.
Now the problem that many people have thought is insurmountable for this covariance matrix here
is that we cannot estimate this n by n matrix omega because we don't have sufficiently many
observations to estimate all the parameters in omega. By the way, there are not n by n different
parameters in omega, but fewer, right? And this was part of the exercise. So don't think omega
has n by n parameters. That's not true or does have n by n parameters, but not n by n different
parameters. And we're just interested in the number of different parameters in matrix omega.
And then a guy named Halbert White, who unfortunately passed away a couple of years ago,
came along and said, well, actually we do not need to estimate omega to compute this
covariance matrix here. We just need to compute or estimate x prime omega x.
And that was actually a very simple idea. And nevertheless, it came rather late in the
development of econometrics because previously all people had thought, well,
in order to find this covariance matrix, we need to estimate omega. But White said, no,
that's not true. You just don't need to estimate omega. And we know we can't estimate omega from
n observations. We cannot estimate all the parameters of omega that we need to know,
but we can estimate x prime omega x. Because what kind of a matrix is x prime omega x?
See, the x matrix is the regressor matrix, which is of a dimension n by k. n is the number of
observations, which are many. k is the number of regressors, which typically are much less,
right? So if x is n by k, then x prime is k by n. So this product here, x prime times omega
times x is a k by k matrix. So k by k is a much smaller matrix with much less parameters
than an n by n matrix. And White said, hey, we just estimate x prime omega x. We do not estimate
the n by n matrix omega. We do not estimate this matrix here. But we estimate the k by k matrix x
prime omega x directly. And when we have estimated this matrix here, which is generally possible,
then we get this matrix here, the V of a beta hat matrix, the covariance matrix of a beta hat,
so the correct standard errors for a heteroscedastic model.
So how does this idea work? Well, when we have heteroscedasticity, so when the matrix,
the covariance matrix of the shocks is like here in equation 47, right? Diagonal matrix
with different variances on the main diagonal and zeros in the off diagonal elements.
If we have this heteroscedasticity assumption, then we know that 1 over n x prime omega x is
the same thing as 1 over n sigma i squared xi times xi prime, right? So just for all observations,
so for all n observations, some sigma i squared times xi times xi prime. Now you may say that
sigma i squared, we just don't know. So we cannot compute the sum here. So this is exactly
the problem that we don't know what the sigma i's are. However, under fairly general assumptions,
a consistent estimator for this mean here, and this is just a mean over many observations,
as you see, a consistent estimator for this expression would be 1 over n times the sum over
ui's hat squared xi xi prime. So what happens here is the xi xi prime, you just leave as they
are. We know what the xi xi primes are because they are from the regressor matrix, and you
replace the sigma i squared by ui hat squared. We know that usually we, in the homoscedastic model,
we estimate the sigma square u from the sum of the squared residuals, so from the u hat squared.
And exactly the same idea we use here by replacing the sigma i squared by the squared residuals
for each of the n variables. And thereby, we do not estimate each single sigma i squared,
but we just estimate, well, we estimate the sum here, right? We estimate this expression using
this here as an estimator. And that's easily computed, and Hobart-White showed that it is
a consistent estimate. So under the heteroscedasticity assumption 47, standard errors derived from the
covariance matrix estimate v hat of beta hat computed as x prime x inverse e times this weighted
sum here times x prime x inverse e are called White's heteroscedasticity consistent standard
errors. And you find them in virtually all computer output nowadays, right? It's easy
to compute them, and you can usually click an option that you do not want to have the standard
covariance matrix estimated, but you want to have White's heteroscedasticity consistent
covariance matrix estimate. Now, is there any risk in doing that? Not really, because suppose
that we were wrong in assuming that there is heteroscedasticity. So suppose that the omega
matrix actually is not heteroscedastic, but is homoscedastic. Well, in this case, all those sigma
i squares here would have the same value, which would essentially mean that we just estimate sigma
square u times 1 over n x i prime x i prime, right? So the special case of a homoscedastic
covariance matrix omega is completely included in White's setup. And what would happen if x prime
omega x simplifies just to x prime x because the omega matrix is just sigma square i times
an identity matrix? Well, then obviously we would have x prime x times x prime x inverse e.
So this whole term here would cancel, and we would estimate exactly the same covariance
matrix that we typically compute in the least squares set up under the assumption of homoscedasticity.
So we would just estimate sigma square u times x prime x inverse e as the estimate of the covariance
matrix of the beta of the beta heads. Okay, so this is the famous result, which Robert White
derived or published in 1980 in Econometrica. I give you the reference here, and that's a very
famous result. And it is very practical and implemented in basically all commercial software,
so you can easily use it if you feel that this is necessary. Any questions up to here?
No questions? Okay, exercises to verify what I have said here. You can go back to the previous
exercise, where I asked you to vary the variance of the disturbances for the later part of the
sample. So for observations 51 to 100, I said, well, make the variance that 10 times as big as
the variance of the disturbances for 1 to 50. Now suppose that you know that the variance for the
later part of the sample is 10 times the variance for the first part of the sample, then please
construct the generalized least squares estimate and estimate beta by OLS and by GLS in each loop.
And then compute the mean and the standard deviation of the beta heads for each of those
estimation methods. And then just look how they compare and which of these two estimates is better,
the OLS estimate and the GLS estimate. Obviously, you should find that the GLS estimate is
the estimate to be preferred. Still no questions?
Then let's not yet move on to autocorrelation because there's one more thing I want to tell you
about the heteroskeleticity consistent standard error. What White here has established is very,
very useful if the error terms are heteroscedastic. This is from 47 here. So if the omega matrix
is a diagonal matrix with different variances on the main diagonal, beware that you cannot apply
the white heteroskeleticity consistent estimate if the off-diagonal elements are not zero.
Actually, this is not really clearly spelled out in White's original publication. You see that this
is the case when you work through its proof, but it is not clearly spelled out. And I have seen
many cases where people thought they can use White's heteroskeleticity consistent errors
for just any matrix omega, also with non-zero off-diagonal elements.
That is not true, as you cannot do. So White's method has its limits. You need to have the
property that it is only on the main diagonal that you have varying variances, but for the
off-diagonal elements, you still need assumption A4. White just cures the violation of assumption A3.
He does not cure a violation of assumption A4, and that, as I say,
is not so really clearly spelled out in his econometric paper from 1980.
Good. No questions? There is a question.
Do we need to go through all the exercises on the slides to prepare better for the exams?
Yes. Well, see, when you ask me, will you be better prepared for the exams if you go through
all the exercises, the answer is yes. I mean, all those exercises are useful, and the more you do
of them, the better you will be prepared for the exam. If this is what you intended to ask,
then my answer is yes. If you intended to ask me if I will pose such programming exercises,
in the final exam, and somehow ask you to solve those programming exercises, then the answer is no.
In this case, you may say, well, you just don't do the programming exercises on,
not do as many of them as I give to you. You can do this because, as I say, this won't be a question
in the final exam that I let you program certain things that wouldn't even be possible. I would
like to ask you something like that, but it's not possible. But still, I think it is useful
because it greatly enhances your understanding if you work through those programming exercises.
If you're short of time, of course, you have to see which exercises you do and which exercises
you don't do. I must say, perhaps on a personal note, I'm somewhat unhappy with the design of
the master's program, and I communicated this also to Andreas Langer, to the program director,
in terms of this econometrics lecture, because this is clearly an inappropriate relationship we
have here between hours of lecture, four hours of lecture each week, and just one hour of
combinying lecture methods of econometric analysis, which is not even a tutorial, something like this.
So, as I already, I think, told you in the beginning of the lecture, Saeed Khordavardian
is absolutely not possible. It is absolutely impossible for him to cover everything which
I did in this lecture here in terms of some kind of exercises or a carbonite material or estimation
exercise or anything like this in this one hour of tutorial or a carbonite lecture. Actually,
it would probably be better if the master program were redesigned such that there are at most three
hours of lecture and then two hours of tutorial or a combinying lecture, or even the reverse,
just two hours of lecture and three hours of exercises. Because econometrics is very much
signed. We have to do lots on your own. We have to work through lots of material in order to grasp
what it is actually all about. So, it is quite time consuming. I'm aware of that and I would
really like to have a change in the design of the master program here, but that is difficult legally
because it has been set up in a certain way and there are many legal hurdles to change.
The program plus the program has a certain design, some symmetry between advanced macro,
advanced micro and advanced econometrics. It should actually be advanced econometrics,
but I'm still with the basic stuff here, which people don't really like to disturb.
So, indeed, this lecture requires an awful lot of independent work and in some way doing or
encouraging you or even forcing you to do this type of independent work, just sitting down with
your own pen and pencil and trying to understand what I did in the lecture, rewrite the proofs,
reproduce the proofs, understand the intricacies of econometrics is a lot of work.
But it is, of course, also what you need to be able to do when you want to do your independent
research sooner or later. And you have a research master here, a master which prepares you for
research and many of you probably think of going on with a PhD dissertation where you would
need to do everything just by yourself without any kind of explanation given to you by anybody,
unless you have specific problems, ask somebody, but usually you just sit there and have to read
papers, the scientific papers and have to get along with it and understand what's going on.
The master's programs help prepare you for that. And in some sense, of course, this lecture is a
preparation of that because you have to do lots already on your own. I do see the kind of knowledge
you bring with you from the bachelor's program and that is a big jump now to the master's level
and to this form of lecture. So as I say, I would actually prefer to split up the four hours of
lecture in this one semester into two semesters with two hours of lectures each and then
accompanying tutorials for econometrics. I think that would be more of a help. But this is not as
the problem is designed. I don't think it is likely it is going to change soon. Even though
Andreas Langer is also aware of the problem when I talked to him, that was quite clear that he
also understood the problem. You can talk to him yourself. It would probably be good if you talked
to him so that he also gets the feedback from students that they perhaps would also suggest
to distribute the econometrics class. Let's say that would be my suggestion evenly over two
semesters, rather than having all everything in the first semester. And you can see how he replies
to that. So that was a long answer to a short question, but perhaps it is necessary as
I think many of you may have had similar thoughts about the lecture. Since I answered it
in such a long way, there is no time anymore to talk about autocorrelation. So we will delay
this till next week, Monday, unless you have further questions now. If there are any questions,
then please, as always, raise your hand. Okay, it seems there are no questions.
Then I wish you a good rest of the week and see you again on Monday at noon.