Some Topics In Multivariate Regression

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Some Topics In Multivariate Regression
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Some Topics

1. Confidence intervals
2. Dummy Variables
3. One sided tests
4. Scale of data
5. Functional Form

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Woldridge refs

• Chapter 1
• Chapter 2.1, 2.2,2.3,2.5
• Chapter 3.1,3.2,3.3,3.4,3.5
• Chapter 4.1, 4.2, 4.5, 4.6
• Chapter 5

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1. Confidence Intervals (4.3)

• We can construct an interval within which the
  true value of the parameter lies
• We have seen that
• P(-1.96 t 1.96)0.95 for large N-K
• More generally

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• Interval b tc se(b) will contain b with (1-a)
  confidence.
• Where tc is critical value and is determined
  by the significance level (a) and the degrees of
  freedom (dfN-K)
• For the case where N-K is large (gt100) and a is
  5 then tc 1.96
• Same as the set of values of beta, which could
  not be rejected if they were null hypotheses
• The range of possible values consistent with the
  data
• A way of avoiding some of the ambiguity in the
  formulation of hypothesis tests
• Formally A procedure which will generate an
  interval containing the true value (1-a) times
  in repeated samples

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2. Dummy Variables

• Wooldridge Chapter 7.1-7.4
• Record classifications
• Dichotomous yes/no e.g. trial, gender etc
• Ordinal e.g. level of education
• OLS doesnt treat them differently
• Need to be careful about how coefficients are
  interpreted
• Illustrate with sex in the wage regression
• Use labour2006.dta and dummy.do
• Sex 1 iff female sex0 iff male

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• Our basic model is
• wagei ?1 ?2 educi ui
• This can be interpreted a predicting wages based
  on schooling i.e.
• Ewagei ?1 ?2 Eeduci
• Suspect that wages are systematically different
  between men and women
• wagei ?1 ?2 educi ?3 sexi ui

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• Now the prediction becomes
• Ewagei ?1 ?2 Eeduci ?3 if female
• Ewagei ?1 ?2 Eeduci if male
• Note that sex disappears when it is zero
• This translates into separate intercepts on the
  graph
• The extra for being a woman
• Testing if ?3 is significant is test of
  significant difference in wages between the two
  groups

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Interaction

• While the intercept could be different the slope
  could be also
• i.e. the degree of discrimination could be
  different between the two groups
• Different returns to education
• Model this by an interaction term
• wagei ?1 ?2 educi ?3 sexi
• ?4 educisexi ui

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• Now the prediction becomes
• Ewagei ?1 (?2 ?4 )Eeduci ?3 if
  woman
• Ewagei ?1 ?2 Eeduci if man
• Note that sex disappears when it is zero
• This translates into separate intercepts and
  slopes on the graph
• The extra for bringing a woman and an extra
  for each levl of education
• Testing if ?4 is significant is test of
  significant difference in return to education
  between the two groups

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3. One Tailed Test (4.2)

• Need to be careful about the interpretation of
  the null and alternative
• Think of example of gender discrimination
• State the Hypothesis we want to test
• H0 bsex gt 0 H1 bsex lt 0
• Calculate the test statistic assuming that H0 0
  true.
• t-8.33
• Reject null if tlt-critical value at chosen sig
  level
• Can reject null as -8.58lt-1.64

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Acceptance Region
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• Which you use is up to you. But
• Beware of translating directly from English
• Be aware of the implications
• Rule of thumb
• H1 what you expect e.g. guilt
• H0 what you fear e.g. innocent
• So the test procedure minimizes the prob of
  rejecting what you fear when it is true
• This notion works for a two sided test also

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• Redo the gender discrimination example from the
  other side
• State the Hypothesis we want to test
• H0 bsex lt 0 H1 bsex gt0
• Calculate the test statistic assuming that H0 0
  true.
• t-8.33
• Reject null if tgt critical value at chosen sig
  level
• Cannot reject null as -8.58lt1.64

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The difference between the two

• The first H0 bsex gt 0 H1 bsex lt 0
• 5 chance of rejecting null when it is correct
• i.e. of stating bsex lt 0 when in fact bsex gt 0
• i.e. of stating there is discrimination when in
  fact there is none
• The second H0 bsex lt 0 H1 bsex gt 0
• 5 chance of rejecting null when it is correct
• i.e. of stating bsex gt 0 when in fact bsex lt 0
• i.e. of stating there is no discrimination when
  in fact there is some

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4. Scale (2.4 6.1)

• The scale of the data may matter
• i.e. whether we measure consumption in or bn
  or even
• Basic model yi b1 b2 xi ei
• Change scale of xi xi xi/c
• Estimate yi b1 b2 xi ei
• b2 c.b2
• se(b2) c.se(b2)
• Slope coefficient and se change, all other
  statistics (t-stats, R2, F, etc.) unchanged.

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• Change scale of yi yi yi/c
• Estimate yi b1 b2 xi ei
• b2 b2 /c
• b1 b1 /c
• se(b2) se(b2)/c
• se(b1) se(b1)/c
• t-stats, R2, F unchanged
• Both X and Y rescaled yi yi/c, xi xi/c
• Estimate yi b1 b2 x ei
• If rescaled by same amount
• b1 b1 /c se(b1) se(b1)/c
• b2 and se(b2) unchanged
• t-stats, R2, F unchanged

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5. Functional Form (6.2)

• Four common functional forms
• Linear qt a ?pt ut
• Log-Log lnqt a ?lnpt ut
• Semilog qt a ?lnpt ut
• or lnqt a ?pt ut
• How to choose?
• Which fits the data best (cannot compare R2
  unless y is same)
• Which is most convenient (do we want elasticity,
  rate of return?)
• How trade-off two goals

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Elasticity and Marginal Effects