WHAT IF SOME OLS ASSUMPTIONS ARE NOT FULFILED

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WHAT IF SOME OLS ASSUMPTIONS ARE NOT FULFILED?
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QUIZ ONCE AGAIN ?

• What are the main OLS assumptions?
• On average right
• Linear
• Predicting variables and error term uncorrelated
• No serial correlation in error term
• Homoscedasticity
• Normality of error term

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QUIZ ONCE AGAIN ?

• Do we know the error term?
• Do we know the coefficients?
• How can we know whether all the assumptions are
  fulfilled
• On average right gt ???
• Linearity gt ???
• X and e uncorrelated gt ???
• e serially uncorrelated gt ???
• e homoscedastic gt ???

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FUNCTIONAL FORM

• We assumed a certain functional form
• What if in reality the relation exists, but has a
  different form?
• Any function can be approximated by the Taylor
  expansion. Consequently
• So what we need to test is that ?s are all zero
  in

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FUNCTIONAL FORM cont.

• This is easy, cause we already know how to do it
  ?
• But there could be a lot of these terms (time
  consuming)
• Ramsey came up with the idea, that if ?s are all
  zero, then fitted ys should have no correlation
  with actual ys.
• So instead of testing whether ?s are all zero,
  we test whether they differ from zero ?
• RESET TEST
• Do your model
• Find the fitted ys
• Run a model with your Xß and all the powers of
  fitted ys
• Test the hypothesis that coefficients by fitted
  ys are zero
• If you cannot reject the null (i.e. that they are
  all zero), you have a functional misspecification
  problem gt you cannot say if your bs are correct
  estimates of ßs.

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NORMALITY OF THE ERROR

• We typically assume that es have a standard
  normal distribution N(O,s2).
• This helps us to derive t and F distributions.
• Although we never know es, we get es, who
  should be consistent with es (so the same
  distribution)
• We can test if the distribution of es is far
  from normal
• Jarque-Berra
• Check out the skewness and kurtosis
• Compare it to the values for normal distribution
• The null says that they are alike, if you reject
  the null, you reject the normality in residuals
• Does that hurt?

Central Limit Theorem
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STABILITY OF THE PARAMETERS

• We typically assume that ßs do not depend on the
  size of Xs (in other words, relation between Xs
  and ys is stable)
• However, we can actually have many subsamples.
• Then what? Look at your dots, see if such thing
  occurs, and run Chow estimation (just as in LAB)

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NO AUTOCORRELATION(AT LEAST NOT SERIAL)

• We also assume that subsequent error terms are
  independent of each other.
• Assume that it does not hold, so that
• What then?
• Our estimators are still unbiased
• We can also show they are consistent
• But the problem occurs with the estimators of the
  variance of the estimators (which we need to test
  the significance hypothesis)

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NO AUTOCORRELATION(AT LEAST NOT SERIAL)

• Consequently, our estimators of the standard
  errors are incorrect
• We cannot trust our t-statistics any more!
• KEEP THAT IN MIND,
• WELL COME BACK TO IT IN A SECOND

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HOW DO WE GET AUTOCORRELATION?

• What we need in the error term is white noise

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HOW DO WE GET AUTOCORRELATION?

• Positive autocorrelation (rare changes of signs)

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HOW DO WE GET AUTOCORRELATION?

• Negative autocorrelation (frequent changes of
  signs)

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HOW DO WE GET AUTOCORRELATION?

• Model misspecification can give it to you for
  free ?

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NO HETEROSCEDASTICITY

• We also assume that error terms do not depend in
  size on the size of Xs
• Assume that it does not hold, so that
• What then?
• Our estimators are still unbiased
• We can also show they are consistent
• But the problem occurs with the estimators of the
  variance of the estimators (which we need to test
  the significance hypothesis)

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HOW DO WE GET HETEROSCEDASTICITY?

• What we need is error terms independent of SIZE
  of X.

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HOW DOES THE V MATRIX LOOK?

• We know that
• This holds for both heteroscedasticity and for
  autocorrelation.
• But arent there any differences?
• Heteroscedasticity is about the diagonal (values
  along the diagonal differ and should be always
  the same)
• Autocorrelation is about what happens outside the
  diagonal (they should be zero and they deviate
  from that)

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Testing for hetero

• Breusch-Pagan approach
• The alternative hypothesis assumes that
  s2is2f(zi), where f(.) is continuous
• Run your model yixißei
• Run the regression of e2 on any set of variables
  (x, y, whatever)
• Use the R2 from this regression (it has ?2
  distribution with p dof, where p is the no. of
  variables in the auxiliary regression)
• Test
• H0 no heteroscedasticity in this form (does not
  say NO heteroscedasticity in general!)
• H1 heteroscedasticity in the assumed form

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Testing for hetero

• White approach
• Heteroscedasticity occurs because some
  interrelations between xs are not accounted for
• Run your model yixißei
• Run the regression of e2 on all interactions of
  xs
• Use the R2 from this regression (it has ?2
  distribution with K(K1) dof, where K is the no.
  of interactions in the auxiliary regression)
• Test
• H0 no heteroscedasticity in this form (does not
  say NO heteroscedasticity in general! this form
  is rather general though ?)
• H1 heteroscedasticity in the assumed form

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Testing for auto

• Durbin-Watson approach
• The alternative hypothesis states, that there is
  autocorrelation of order 1 (two closest e are
  correlated)
• Run your model yixißei
• Get your e2
• Do the statistic on these e2

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Testing for auto

• Durbin-Watson approach continued
• If there is no (or weak) autocorrelation, the
  last term would be equal (or close) to 0, so the
  whole statistic would be 2.
• IF DW lt 2
• blue positive autocorrelation, green
  inconclusive
• no color no autocorrelation,
• IF DWgt2
• blue negative autocorrelation, green
  inconclusive, no color no autocorrelation
• YOU CAN USE IT ON SMALL SAMPLES EVEN

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Testing for auto

• Breush-Godfrey approach
• There is autocorrelation of order s (s closest e
  are correlated)
• Run your model yixißei
• Get your e2
• Run the auxiliary regression on lagged e in the
  form of etx??1et-1 ?2et-2 ?set-s
• Test the hypothesis that your ?s are zero
• The nice part is that TR2 (where T is the number
  of observations) of this auxiliary regression
  allows to test it as a combined hypothesis with
  ?2 distribution with s dof, where s is the no. of
  lags you take into account.
• MUCH NICER THAN DW, BUT REQUIRES BIG SAMPLES!

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CONCLUSIONS ABOUT AUTO AND HETERO ?

• What they both mean is that you can no longer
  trust the estimates of the standard errors
• You can still trust the estimators of your model,
  but you cannot test if they are non-zero (no
  valid hypothesis testing)
• If you have autocorrelation but a very big
  sample, you are asymptotically OK. so you need
  not to worry ?
• Big sample does not help for heteroscedasticity
  though ?
• In a small sample autocorrelation cannot be
  eliminated either
• What we have as a response is GENERALISED LEAST
  SQUARE estimator gt GLS same as OLS only helps to
  overcome the misestimation of standard errors.
• If no problems with auto and hetero, GLS less
  efficient than OLS (do not overuse it!)