THE CLASSICAL MODEL

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CHAPTER 4

• THE CLASSICAL MODEL

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Introduction
If the assumptions discussed in this chapter
hold, then OLS is considered the best estimator
available for regression models. When one or
more of these assumptions do not hold, other
estimation techniques sometimes may be better
than OLS. What do we do when one or more of the
assumptions is not met? The pros and cons of
alternative techniques must be weighed.
Sometimes we must make adjustments to OLS when a
particular assumption has been violated.
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4.1 The Classical Assumptions

• The regression model is linear in the
  coefficients is correctly specified and has an
  error term.
• The good properties of OLS estimators hold
  regardless of the functional form of the
  variables as long as the form of the equation to
  be estimated is linear in the coefficients. We
  also assume that the equation has been correctly
  specified and that the stochastic error term has
  been added.

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• The error term has a zero population mean.
• When the entire population of possible values for
  the stochastic error term is considered, the
  average of that population is zero.

• All explanatory variables are uncorrelated with
  the error term.
• The observed values of the independent variables
  are determined independently of the values of the
  dependent variable and the error term.

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• Observations of the error term are uncorrelated
  with each other (no serial correlation).
• If a systematic correlation exists between one
  observation of the error term and another, then
  it will become more difficult for OLS to get
  precise estimates of the coefficients of the
  explanatory variables. An increase in the error
  term in one time period does not show up in or
  affect in any way the error term in another time
  period.

• Serially correlated (autocorrelated)

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• The error term has a constant variance (no
  heteroskedasticity).
• The observations of the error term are assumed to
  be drawn continually from identical
  distributions. This type of violation makes
  precise estimation difficult, because a
  particular deviation form a mean can be called a
  statistically large or small deviation only when
  it is compared with the standard deviation of the
  distribution in question. OLS will generate
  imprecise estimates of the coefficients of the
  explanatory variables, and the relative
  importance of changes in Y will be very hard to
  judge.

Heteroskedasticity
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• No explanatory variable is a perfect linear
  function of other explanatory variables(no
  perfect multicollinearity).
• The relative movements of one explanatory
  variable will be matched exactly by the relative
  movements of the other even though the absolute
  size of the movements might differ. OLS will be
  incapable is distinguishing one variable from the
  other.

Collinearity
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7. The error term is normally distributed (this
assumption is optional but usually is invoked).
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4.4 The Gauss-Markov Theorem and the Properties
ofOLS Estimators
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OLS estimators can be shown to have the following
properties

• 1. They are unbiased. The OLS estimates of the
  coefficients are centered around the true
  population values of the parameters being
  estimated.

2. They are minimum variance. The distribution
of the coefficient estimates around the true
parameter values is as tightly or narrowly
distributed as possible for an unbiased
distribution. No other linear unbiased estimator
has a lower variance for each estimated
coefficient than OLS.
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3. They are consistent. As the sample size
approaches infinity, the estimates converge on
the true population parameters (i.e., the
variances get smaller, and each estimate
approaches the true value of the coefficient
being estimated).
4. They are normally distributed. Various
statistical tests based on the normal
distribution may be applied to these estimates
(chapter 5).
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4.5 Standard Econometric Notation
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End-of-Chapter Exercises 1, 3, 4, and 5