Econ2206 Notes

Topics: Normal distribution, Estimator, Scientific method Pages: 292 (9594 words) Published: August 21, 2013
ECON2206
 Revision
 Notes
  W2
 –
 SIMPLE
 REGRESSION
 MODEL
 
  MOTIVATION
  Much
 of
 applied
 econometric
 analysis
 are
 interested
 in
 “explaining
 y
 in
 terms
 of
 x”
 and
  confront
 three
 issues:
  1) Since
 there
 is
 never
 an
 exact
 relationship
 between
 y
 and
 x,
 how
 do
 we
 account
 for
  the
 “other
 unobserved”
 variables?
  2) What
 is
 the
 function
 relationship
 between
 y
 and
 x?
  3) How
 do
 we
 invoke
 a
 ceteris
 paribus
 relationship
 or
 a
 causal
 effect
 between
 y
 an
 x?
 
 
  The
 simple
 linear
 regression
 model
 is:
  y
 =
 β0
 +
 β1x
 +
 u
 
 
  -­‐ “u”
 is
 the
 stochastic
 error
 or
 disturbance
 term
 and
 represents
 all
 those
 unobserved
  factors
 or
 other
 factors
 other
 than
 x.
 
  -­‐ If
 all
 other
 factors
 (u)
 is
 held
 fixed,
 so
 that
 change
 in
 u
 is
 zero,
 we
 can
 observe
 the
  function
 relationship
 between
 y
 and
 x.
  -­‐ If
 we
 take
 the
 expected
 value
 of
 the
 model,
 (Δu
 =
 0
 and
 Δβ0),
 then
 we
 can
 see
 that
 x
  has
 a
 linear
 effect
 on
 y.
 
 
  We
 will
 only
 get
 reliable
 estimates
 of
 β0
 and
 β1
 if
 we
 make
 restricting
 assumptions
 on
 u.
 As
  long
 as
 β0
 is
 included
 in
 the
 model,
 nothing
 is
 lost
 by
 making
 the
 assumption
 that
 the
  expected
 value
 of
 u
 in
 the
 population
 is
 zero;
 E(u)
 =
 0.
 
  ZCM
  Our
 crucial
 assumption
 is
 by
 defining
 the
 conditional
 distribution
 of
 u
 given
 any
 value
 of
 x.
  This
 crucial
 assumption
 is,
 the
 average
 value
 of
 u
 does
 not
 depend
 on
 the
 value
 of
 x.
 
 
  E(u|x)
 =
 E(u)
 =
 0
 
  This
 is
 the
 zero-­conditional
 mean
 assumption
 (ZCM)
  -­‐ The
 average
 value
 of
 the
 unobserved
 factors
 is
 the
 same
 across
 the
 population.
 
  -­‐ An
 important
 implication
 of
 ZCM
 is
 that
 u
 and
 x
 are
 uncorrelated.
 
 
  OLS
  Ordinary
 Least
 Squares
 (OLS)
 is
 a
 method
 for
 estimating
 the
 unknown
 parameters
 in
 a
  linear
 regression
 model.
 The
 estimates
 for
 β0
 and
 β1
 are
 found
 by
 minimizing
 the
 sum
 of
  squared
 residuals.
 That
 is,
 the
 distance
 between
 the
 observations
 in
 the
 sample
 and
 the
  responses
 predicted.
 
  -­‐ Fitted
 values
 and
 estimates
 are
 denoted
 by
 a
 HAT
  -­‐ The
 values
 predicted
 for
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