Review III: Failure of Assumptions

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Review III Failure of Assumptions

• Data Problems
• Misspecification
• Multicollinearity
• Heteroskedasticity

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Main Points from Last Class

• Yiß0,iß1X1,iß2X2,iß3X3,ißkXk,iu,i
• Causal relationship is more informative but also
  harder to estimate than correlation.
• Roughly speaking, identifying a causal
  relationship is equivalent to addressing the
  endogeneity (violation of assumption MLR 3).
• Another important assumption is random sampling.

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Omitted variables problem (Underspecification)

• Some variables that should be in the model are
  not included (sometimes called the omitted
  variable problem)
• If the omitted variables are correlated with any
  explanatory variable Xj, then Xj will be
  correlated with u. We call Xj endogenous. ßj will
  be inconsistent.
• If Xj is further correlated with other
  explanatory variables, then their coefficient
  estimates also become inconsistent even if they
  are exogenous.

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The Omitted Variables Bias
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How to address the omitted-variable bias?

• Add in the omitted variables
• Hardly possible unless we control the whole
  process.
• Random assignment of the independent variable of
  interest, conditional on observed control
  variables
• Occasionally possible
• The independent variable of interest is affected
  by some exogenous variables not in the system.
• Often possible. The focus of this course.

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Class quiz

• What are the possible omitted variables in the
  following cases?
• Regress wage on education
• Regress the prices of an apartment on its
  distance to the Central
• Regress GDP on capital and labor force

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Overspecification

• When redundant variables are added to the
  regression
• they do not affect the consistency of the
  estimates if they are not related to any of the
  variables in the regression. They only affect the
  precision of the estimates.
• Otherwise, the redundant variables may affect the
  estimates.
• Bad controls

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Bad control 1

• The control variable is an outcome of the
  independent variable of interest
• e.g. occupations in the wage model with education
  as the variable of interest

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Bad control 2

• Proxy variables that are the outcome of the
  independent variable of interest
• Note that a proxy variable is a variable that is
  not in the system (i.e. not directly affecting
  the dependent variable)

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A suggestion for choosing control variables

• Pay particular attention to control variables
  that are the outcome of the independent variables
  of interest.
• How to judge whether a variable is the outcome of
  another variable? Timing may be useful. Or use
  economic reasoning.

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Class quiz

• Think of some regressions you want to make to
  assess the relationship between some variables.
  Is there any problems as we have described above?

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Nonrandom Sampling (p. 310 or 326)

• Exogenous Sample Selection
• Sample selection is based on the observed
  independent variables.
• Estimates are not affected.
• Endogenous Sample Selection
• Sample selection is based on the dependent
  variables
• The error term is correlated with independent
  variables, violating assumption MLR. 3.
  Generally all estimates are inconsistent.
• yjaXjej
• If yigtc, then aXjejgtc, or aXjgtcej.
• Hence, Xi and ej are correlated.

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Nonrandom Sampling

• Stratified sampling
• The population is divided into nonoverlapping,
  exhaustive groups. Sampling frequency differs
  across groups.
• Whether estimates are affected depends on whether
  the stratification is decided by the dependent or
  independent variables.
• Homogeneity assumption across stratum?

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Measurement Errors (p. 302-309 or 318-325)

• If the measurement error is in the dependent
  variables and is unrelated to the independent
  variables, then the estimates are not affected
  (except for the intercept), but the variances of
  the estimates are larger.
• Measurement error in independent variables
• Classical Errors-in-variables (CEV) Assumption
  If the error is uncorrelated with the true
  variable, then the measured variable is
  correlated with the error, making it an
  endogeneity problem. The magnitude and direction
  of biases are hard to determine in reality.
  (Instrumental Variable approach can solve the
  engogeneity problem.)
• If the error is uncorrelated with the measured
  variable, the estimates are not affected, but
  their variances are.

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Outliers (p. 312-317)

• Outliers are observations that can significantly
  affect OLS estimates when they are added to the
  sample. These could happen because
• Typos
• Nature of some variables
• To address this problem, we may
• Manually pick and test the influence of outliers
• Using statistics package to pick outliers
• Use log specification that is less sensitive to
  outliers
• Use Least Absolute Deviations (LAD) estimators
  that put less weight on outliers.

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Functional Form Misspecification(pp. 289-295 or
304-310)

• If the adopted functional form is not the same as
  the true functional form, of course the estimates
  can be inconsistent.
• Is there really a true model and who knows what
  it is?
• Reduced-form approach Try different functional
  forms until it fits the data best.
• Structural-form approach Use the functional form
  suggested by economic theory (and under some
  assumptions).

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Multicollinearity

• Multicollinearity is not a violation of
  assumptions, so it does not affect the
  consistency of estimates.
• However, coefficient estimates become less
  precise (p.96-98)
• s2 Variance of the error term
• SSTj The sample variation in Xj
• Rj2 The linear relationships between Xj and
  other explanatory variables

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Heteroskedasticity

• Heteroskedasticity is present if the variance of
  u differ for different segments of the
  population, where the segments are determined by
  the different values of the Xs. (p. 54-55 and
  chapter 8). This is a violation of assumption
  MLR.5.
• Coefficient estimates are still consistent, but
  estimated variances of the estimates are
  inconsistent.

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Testing for Heteroskedasticity

• Breusch-Pagan test (BP test)
• u2d0d1X1d2X2dkXkerror
• Test H0 d1d2dk0
• An F test provided by most statistics package (a
  p-value is reported).
• Reject the null when p is small.
• White Test and its extensions
• Adding the squares and cross products of
  independent variables to the BP test.

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Heteroskedasticity-Robust Standard Error (p.260)

• When heteroskedasticity is present, the correct
  estimator of the variance of a coefficient is
• The heteroskedasticity-robust standard error (or
  Huber-White standard error) is reported by most
  statistics package.
• Note that this is a large-sample result, and it
  also requires some assumptions.