Econometric Methods 1

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Econometric Methods 1

• Lecture 5 Stochastic regressors and IV estimation

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The non-stochastic assumption

• Suppose conducting trials of a pesticide effects
  on harvest
• Randomly choose which fields you apply pesticide
  and how much
• Collect data harvests (y), dose of pesticide is
  X variable helping to explain the size of the
  harvest from a field.
• Since the process of spraying field was
  undertaken by the experimenter, a statistician
  could then treat the dose as fixed.

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The non-stochastic assumption

• Now think of walking into randomly sampled
  households
• Ask them how many boxes of breakfast cereal they
  keep, how many pints of milk in refrigerator and
  how many people in the house, and particularly
  how many children in the house.
• All of these quantities are things you, the
  investigator, didnt know before you performed
  the interview and all could be correlated with
  holdings of breakfast cereal.
• For all of them can calculate a mean and a
  standard deviation, which allow you to say
  something about the relevant population. In
  other words we might treat them all as
  realisations of random variables.

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Estimation with stochastic regressors

• Consider modelling cereal holding per household
  with these variables.
• Run a regression of cereal holding with number of
  children as an explanatory variable.
• Would it be OK to treat the number of children as
  if it were a control variable like the pesticide
  in the field trial example?
• (1)
• Assumption we require is the independence of
  nchild and e, or, that the conditional
  expectation of e given nchild equals zero.
• With independence (1) can be estimated by OLS,
  and all desirable properties (unbiasedness) can
  be asserted.

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Estimation with stochastic regressors

• Recall OLS estimator with non-stochastic x in
  deviation form is
• b
• And E(b) ß because E(Sxu)
  0
• If x is stochastic we can no longer decompose
  E(Sxu) so conveniently. Taking expectations
• E(b) ß ESxu/ Sx2 where both u and x are
  r.v.s.
• Unfortunately, ESxu/ Sx2 ? ESx/ Sx2 x E(u)
  0 unless x and u are independent.
• So OLS will be biased unless x and u are
  independent

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Independence can be violated in a number of ways

• Simultaneity bias (reverse causality)
• Suppose having lots of breakfast cereal increases
  fertility (Unlikely!!!)
• Then 2nd model with nchild as dependent variable
  and cereal as explanatory variable. Then nchild
  contains e!
• Covariance of nchild and e, cannot be zero

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Independence can be violated in a number of ways

• Measurement error in X
• Suppose that the true number of children is
  nchild
• But we measure this with error nchild nchild
  ?
• Where ? is a random error with mean zero E(v)0
  and it is uncorrelated with the true nchild
• True model is
• But we estimate
• X is now correlated with the error term
• Cov(nchild,v) E(nchild.v) E(nchild.v)E(v2)
  

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Measurement error in X

• It can be shown (Wooldridge p322) that
• Where is the variance of the true X
  variable
• So bias in b is dependent upon the ratio
  Var(x)/Var(x)
• This is sometimes called the signal to noise
  ratio
• It is always less than 1
• So b is biased towards zero (attenuation bias)
• If variance of x large relative to measurement
  error then bias is small

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Independence can be violated in a number of ways

• Omitted variables
• These can be observed or unobserved
• The true model is Y b1 b2 X2 b3 X3 u
• But we omit X3 then
• E(b2)
• This equals ß2 only if ß3 0 or there is zero
  correlation between X2 and X3

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Consequences of stochastic X

• Suppose that there is positive correlation
  between x and u (reflecting lack of
  independence), then the OLS estimate of the slope
  is biased upwards, see formula.
• Can we assume independence?
• We need to be reasonably convinced there are
• (a) No feedbacks/simultaneity
• (b) No (serious) measurement error
• (c) No omitted variables likely to be correlated
  with x.
• Independence requires all of the observations on
  x are uncorrelated with all of the error terms
• Perhaps plausible that my error term is
  uncorrelated with other households number of
  children

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How about in time series?

• Much less plausible that independence holds
• Last years inflation error is uncorrelated with
  this years unemployment?
• It does not apply in model with lagged dependent
  variable
• yt a bxt gyt-1 ut
• Since one of the x variables (yt-1 ) is
  correlated with ut-1

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Asymptotic properties

• Recall consistency. If an estimator is
  consistent, its sampling distribution 'homes in'
  on the true value of the parameter as the sample
  size increases.
• limit P(b - b lt e) 1 for arbitrarily
  small e plim b b
• n ? ?
• Sufficient (not necessary) conditions for
  consistency are that the bias and sampling
  variance of estimator both go to zero.
• Example consider 1/n as an estimator of m.
  Biased (bias 1/n) in small samples and sampling
  variance s2 /n.
• Both of these go to zero as n goes to infinity,
  so it is a consistent estimator.
• Hence consistency might be a nice property to
  have in the absence of unbiasedness (at least if
  you've got a large sample).

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Properties of plim

• Plim has some convenient properties. In
  particular, if b1 is a consistent estimator of b1
  and b2 is a consistent estimator of b2 (i.e. plim
  b1 b1 etc.) then
• f(b1 ) is a consistent estimator of f(b1 ) for
  any function f. E.g. log b1 is a consistent
  estimator of log b1.
• f1 (b1 )/f2 (b2 ) is a consistent estimator of f1
  (b1 )/f2 (b2 ) for functions f1 and f2. This
  implies that b1 /b2 is consistent for b1 /b2, b1
  ? b2 is consistent for b1 ?b2 etc.
• These apply irrespective of whether b1 and b2 are
  independent or not.

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Asymptotic properties of regression coefficients

• Using the plim properties we can now write
• plim b b
• Note we have divided top and bottom by n to
  obtain sensible probability limits.
• The second term on the RHS will be 0 as long as
• (i) plim(Sxu)/n 0 and (ii) plim(Sx2)/n exists
  and is non-zero.
• The first part of this is satisfied if E(Sxu) 0
  
• (the contemporaneously uncorrelated case)
• This is a weaker condition than independence,
  since only xt and ut need to be uncorrelated (not
  all past and future values). This allows the
  Lagged Dependent Variable model.

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Instrumental Variables Estimation

• When x and u contemporaneously correlated need an
  alternative method of estimation.
• Instrumental variables (IV) is a general method
  which always yields consistent estimates, in
  particular when x and u are correlated
• The principle is simple - replace each of the
  problem x variables with an instrument z
  which is highly correlated with x but is
  uncorrelated with u.
• Two properties instrument relevance (i.e.
  important determinant of x) and instrument
  exogeneity that is not correlated with u. A
  valid instrument must be both.

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Instrumental Variables Estimation

• Intuition for relevance seems pretty
  straightforward.
• Exogeneity means instrument has no independent
  place in the regression only affects y through
  x.
• Instrument need not consist of a single variable,
  several could be used to instrument a single x.
• We start with the simpler case with one
  instrument for each problem variable.
• If we call the n ? k matrix of instruments W
  then the IV estimator of ß is given by
• bIV (WX)-1 Wy

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Instrumental Variables Estimation

• IV is effectively a two-stage procedure. Take
  case of one x and one instrument w. First do OLS
  regression of x gw e.
• Then Now let
• Now do the OLS regression
• IV is the two stages collapsed into one.
• In using only using that variation in x due
  to w
• Have purged it of variation that is correlated
  with u

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Variance of IV estimates

• IV estimates have higher variances than OLS
• The variance of the IV estimator is given by
• V(bIV) s2 (WX)-1 WW(XW)-1
• (for the single instrument per regressor case).
• s2 is estimated by (y Xb) (y Xb)/(n-k).

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Testing instrument validity

• Testing for instrument relevance is
  straightforward
• Estimate first stage regression to see if w
  affects x
• Rule of thumb (F-stat gt 10)
• Checking instrument exogeneity much harder and
  impossible when have just one w for each x
• We have
• Want to test w is uncorrelated with u cov(w,u)0
• But which us do we use here
• IV residuals not
  appropriate since may be inconsistent if w not
  exogenous instrument
• OLS residuals also likely to be inconsistent

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Testing instrument validity

• If these are more instruments than endogenous xs
  then we can test exogeneity, using the
  over-identifying restrictions test. See Stock
  and Watson p354. The intuition of it is that
  because there are more instuments than endogenous
  xs, then we can test if the instruments
  significantly enter the equation additionally to
  their role as instruments.
• Since cannot test exogeneity of instruments we
  always need a convincing story
• Appeal to economic theory or introspection