Econometrics

1
Econometrics

• Lecture 2
• STOCHASTIC REGRESSORS, INSTRUMENTAL VARIABLES AND
  WEAK EXOGENEITY

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OLS SOME RESULTS CONCERNING UNBIASEDNESS AND
CONSISTENCY

• CASE 1 NON-STOCHASTIC REGRESSORS AND ALL
  ASSUMPTIONS OF THE NORMAL CLASSICAL LINEAR
  REGRESSION MODEL SATISFIED
• OLS estimator unbiased and fully efficient.
• CASE 2 STOCHASTIC REGRESSORS AND X INDEPENDENT
  OF u
• the OLS estimator of ? is unbiased. It is also
  true that the OLS estimator is efficient and
  consistent.
• CASE 3 STOCHASTIC REGRESSORS AND X and u are
  asymptotically uncorrelated
• The OLS estimator consistent but biased in finite
  sample
• CASE 4 STOCHASTIC REGRESSORS AND X and u are
  correlated even asymptotically
• In this case, and it follows that that , and
  so OLS is inconsistent in general.
• In this case OLS estimator cannot be saved need
  a different estimator

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Cases where the OLS estimator cannot be saved

• Simultaneous Equation Bias
• Given the following structural system
• The reduce form equations are
• Clearly X is correlated with the error term u

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Cases where the OLS estimator cannot be saved

• Errors in Variables
• Observe
• x x v
• where v is random measurement error
• True model Y x? u
• (x-v)? u
• x? (u - v?)
• Estimated model Y x? ?
• Estimates
• The last term will not tend to zero. OLS
  estimator inconsistent

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Cases where the OLS estimator cannot be saved

• (3) The model includes a lagged dependent
  variable and has a serially correlated
  disturbance
• Suppose we estimate
• Y ?1 ?2 Xt ?3Yt-1 ut
• in which
• ut ?ut-1 ?t
• with ? non-zero.
• By lagging the equation for Y by one period, it
  is clear that Yt-1 is correlated with ut
  irrespective of the sample size (given that ??0).

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Cases where the OLS estimator cannot be saved

• (4) The disturbance term in the equation we are
  interested in estimating is correlated with the
  disturbance term in the equation which determines
  one of the regressors in our model
• Given the following system
• In this case estimating the first equation will
  show correlation between regressor and error
  term

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Instrumental Variables Estimator

• Each of these cases can (but will not
  necessarily) lead to failure of the zero
  correlation assumption.
• An alternative estimator (the Instrumental
  Variables IV estimator) exists which is
  consistent even where the assumption of zero
  asymptotic correlation of regressors and
  disturbances is not satisfied.
• IV is consistent whether or not the regressors
  and disturbance are correlated. If they are not
  correlated, OLS will also be consistent..
• We cannot check directly the validity of an
  assumption that the regressors and disturbance
  are uncorrelated. However, a decision to use the
  IV estimator might be made on the basis of
• Prior information theory might tell us that the
  orthogonality assumption (meaning here no
  correlation between a regressor and the equation
  disturbance) about a particular variable is not
  likely to be satisfied.
• Empirical testing of the assumption of no
  correlation indirectly, though the use of a
  Wu-Hausman test.

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What Is an Instrumental Variable?

• In order for a variable, z, to serve as a valid
  instrument for x, the following must be true
• The instrument must be exogenous -
• Cov(z,u) 0
• The instrument must be correlated with the
  endogenous variable x
• Cov(z,x) ? 0

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More on Valid Instruments

• We have to use common sense and economic theory
  to decide if it makes sense to assume Cov(z,u)
  0
• Sargan Test
• We can test if Cov(z,x) ? 0 Just testing
• H0 p1 0 in x p0 p1z v
• Sometimes refer to this regression as the
  first-stage regression

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IV Estimation in the Simple Regression Case

• For y b0 b1x u, and given our assumptions
• Cov(z,y) b1Cov(z,x) Cov(z,u), so
• b1 Cov(z,y) / Cov(z,x)
• Then the IV estimator for b1 is

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IV versus OLS estimation

• Standard error in IV case differs from OLS only
  in the R2 from regressing x on z
• Since R2 lt 1, IV standard errors are larger
• However, IV is consistent, while OLS is
  inconsistent, when Cov(x,u) ? 0
• The stronger the correlation between z and x, the
  smaller the IV standard errors

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The Effect of Poor Instruments

• What if our assumption that Cov(z,u) 0 is
  false?
• The IV estimator will be inconsistent, too
• Can compare asymptotic bias in OLS and IV
• Prefer IV if Corr(z,u)/Corr(z,x) lt Corr(x,u)

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IV Estimation in the Multiple Regression Case

• IV estimation can be extended to the multiple
  regression case
• Call the model we are interested in estimating
  the structural model
• Our problem is that one or more of the variables
  are endogenous
• We need an instrument for each endogenous variable

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Multiple Regression IV

• Write the structural model as y1 b0 b1y2
  b2z1 u1, where y2 is endogenous and z1 is
  exogenous
• Let z2 be the instrument, so Cov(z2,u1) 0 and
• y2 p0 p1z1 p2z2 v2, where p2 ? 0
• This reduced form equation regresses the
  endogenous variable on all exogenous ones

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Two Stage Least Squares (2SLS)

• Its possible to have multiple instruments
• Consider the structural model, and let y2 p0
  p1z1 p2z2 p3z3 v2
• Here were assuming that both z2 and z3 are valid
  instruments they do not appear in the
  structural model and are uncorrelated with the
  structural error term, u1

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Best Instrument

• Could use either z2 or z3 as an instrument
• The best instrument is a linear combination of
  all of the exogenous variables, y2 p0 p1z1
  p2z2 p3z3
• We can estimate y2 by regressing y2 on z1, z2
  and z3 can call this the first stage
• if then substitute y2 for y2 in the structural
  model, get same coefficient as IV

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More on 2SLS

• While the coefficients are the same, the standard
  errors from doing 2SLS by hand are incorrect, so
  let Stata do it for you
• Method extends to multiple endogenous variables
  need to be sure that we have at least as many
  excluded exogenous variables (instruments) as
  there are endogenous variables in the structural
  equation

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Addressing Errors-in-Variables with IV Estimation

• Remember the classical errors-in-variables
  problem where we observe x1 instead of x1
• Where x1 x1 e1, and e1 is uncorrelated with
  x1 and x2
• If there is a z, such that Corr(z,u) 0 and
  Corr(z,x1) ? 0, then
• IV will remove the attenuation bias

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Testing for Endogeneity the Hausman Test

• Since OLS is preferred to IV if we do not have an
  endogeneity problem, then wed like to be able to
  test for endogeneity
• If we do not have endogeneity, both OLS and IV
  are consistent
• Idea of Hausman test is to see if the estimates
  from OLS and IV are different

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Testing for Endogeneity

• While its a good idea to see if IV and OLS have
  different implications, its easier to use a
  regression test for endogeneity
• If y2 is endogenous, then v2 (from the reduced
  form equation) and u1 from the structural model
  will be correlated
• The test is based on this observation

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Testing for Endogeneity (cont)

• Save the residuals from the first stage
• Include the residual in the structural equation
  (which of course has y2 in it)
• If the coefficient on the residual is
  statistically different from zero, reject the
  null of exogeneity
• If multiple endogenous variables, jointly test
  the residuals from each first stage