Multiple Regression Applications

1
Multiple Regression Applications

• Lecture 14

2
Todays plan

• Relationship between R2 and the F-test.
• Restricted least squares and testing for the
  imposition of a linear restriction in the model

3
R2

• We know

• We can rewrite this as

• Remember
• If R2 1, the model explains all of the
  variation in Y
• If R2 0, the model explains none of the
  variation in Y

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R2 (2)

• We know from the sum of squares identity that

• Dividing by the total sum of squares we get

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R2 (3)

• Thus

or
or

• If we divide the denominator and numerator of the
  F-test by the total sum of squares

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F-stat in terms of R2

• The F-test for the joint hypothesis H0 b1 b2
  0 can be written in terms of R2

• Recalling our LINEST (from L12.xls) output, we
  can substitute R2 0.188
• We would reject the null at a 5 significance
  level and accept the null at the 1 significance
  level

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Relationship between R2 F

• When R2 0 there is no relationship between the
  Y and X variables
• This can be written as Y a
• In this instance, we cannot reject the null and F
  0
• When R2 1, all variation in Y is explained by
  the X variables
• The F statistic approaches infinity as the
  denominator would equal zero
• In this instance, we always reject the null

8
Restricted Least Squares

• Imposing a linear restriction in a regression
  model and re-examining the relationship between
  R2 and the F-test.
• Example of Cobb-Douglas production function
• In restricted least squares we want to test a
  restriction such as

Where our model is

• We can write ? 1 - ? and substitute it into the
  model equation so that
• (lnY - lnK) a a(lnL - lnK) e

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Restricted Least Squares (2)

• We can rewrite our equation as G a ?Z e
• Where G (lnY - lnK) and Z (lnL - lnK)

• The model with G as the dependent variable will
  be our restricted model
• the restricted model is the equation we will
  estimate under the assumption that the null
  hypothesis is true

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Restricted Least Squares (3)

• How do we test one model against another?
• We take the unrestricted and restricted forms and
  test them using an F-test

• The F statistic will be

• refers to the restricted model
• q is the number of constraints
• in this case the number of constraints 1 (?
  ? 1)
• n - k is the df of the unrestricted model

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Testing linear restrictions

• Estimation when imposing a linear restriction in
  the Cobb-Douglas log-linear model
• Test for constant returns to scale, or the
  restriction
• H0 ? ? 1
• We will use L13_1.xls to test this restriction -
  worked out in L14.xls

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Testing linear restrictions (2)

• The unrestricted regression equation estimated
  from the data (L13_1.xls) is

• Note the t-ratios for the coefficients
• ? 0.674/0.026 26.01
• ? 0.447/0.030 14.98
• compared to a t-value of around 2 for a 5
  significance level, both ? ? are very precisely
  determined coefficients

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Testing linear restrictions (3)

• adding up the regression coefficients, we
  have 0.674 0.447 1.121
• how do we test whether or not this sum is
  statistically different from 1?
• Note the restriction ? 1- ?
• Our restricted model is
• (lnY - lnK) a a(lnL - lnK) e
• or
• G a ?Z e

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Testing linear restrictions (4)

• The procedure for estimation is as follows
• 1. Estimate the unrestricted version of the model
• 2. Estimate the restricted version of the model
• 3. Collect for the unrestricted model
  and
• for the restricted model
• 4. Compute the F-test
• where q is the number of restrictions (in this
  case q 1) and (n-k) is the degrees of freedom
  for the unrestricted model

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Testing linear restrictions (5)

• On L14.xls our sample n 32 and an estimated
  unrestricted model provides the following
  information

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Testing linear restrictions (7)

• The restricted model gives us the following
  information

• We can use this information to compute our F
  statistic
• F (1.228 - 0.351)/1/(0.359/29) 72.47

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Testing linear restrictions (8)

• The F table value at a 5 significance level is
• F0.05,1,29 4.17
• Since F gt F0.05,1,29 we will reject the null
  hypothesis that there are constant returns to
  scale
• Note that the test rejects constant return. As a
  b gt 1, we might conclude there are increasing
  returns to scale.
• NOTE the dependent variables for the restricted
  and unrestricted models are different
• dependent variable in unrestricted version lnY
• dependent variable in restricted version
  (lnY-lnK)

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Testing linear restrictions (9)

• We can also use R2 to calculate the F-statistic
  by first dividing through by the total sum of
  squares

• Using our definition of R2 we can write

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Testing linear restrictions (10)

• NOTE we cannot simply use the R2 from the
  unrestricted model since it has a different
  dependent variable
• What we need to do is take the expectation
  E(GL,K)
• We need our unrestricted model to have the same
  dependent variable G, or

• Where G (lnY - lnK), Z (lnL - lnK)
• We can test this because we know that ? ? - 1
  0.121 since ? ? 1
• Estimating this unrestricted model will give us
  the unrestricted R2

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Testing linear restrictions (11)

• From L14.xls we have
• R2 0.871
• R2 0.963
• Our computed F-statistic will be

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Testing linear restrictions (12)

• On L14.xls we have 32 observations of output,
  employment, and capital
• The spreadsheet has regression output for the
  restricted and unrestricted models
• The R2 and sum of squares are in bold type
• F-tests on the restriction are on the bottom of
  the sheet
• We find that Excel gives us an F-statistic of
  72.4665
• The F table value at a 5 significance level is
  4.1830
• The probability that we could not reject the null
  given this F-statistic is very small

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Testing linear restrictions (13)

• From this we can conclude that we have a model
  where there are increasing returns to scale.
• We dont know the true value, but we can reject
  the restriction that there are constant returns
  to scale.