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Multiple Regression Applications. Lecture 15. Today’s plan. Relationship between R 2 and the F-test. Restricted least squares and testing for the imposition of a linear restriction in the model. ^. ^. R 2. We know. We can rewrite this as. Remember:

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today s plan
Today’s plan
  • Relationship between R2 and the F-test.
  • Restricted least squares and testing for the imposition of a linear restriction in the model
slide3
^

^

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
r 2 2
^

^

^

^

^

^

^

R2 (2)
  • We know from the sum of squares identity that
  • Dividing by the total sum of squares we get
r 2 3
R2 (3)
  • Thus we have

or

or

  • If we divide the denominator and numerator of the F-test by the total sum of squares:
f stat in terms of r 2
F-stat in terms of R2
  • Even if you’re not given the residual sum of squares, you can compute the F-statistic:
  • Recalling our LINEST (from L13.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
relationship between r 2 f
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 accept 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
restricted least squares
Restricted Least Squares
  • Imposing a linear restriction in a regression model and re-examining the relationship between R2 and the F-test.
  • 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

restricted least squares 2
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
restricted least squares 3
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
testing linear restrictions
Testing linear restrictions
  • We wish to test the linear restriction imposed in the Cobb-Douglas log-linear model:
  • Test for constant returns to scale, or the restriction:

H0:  +  = 1

  • We will use L14.xls to test this restriction - worked out in L15.xls
testing linear restrictions 2
Testing linear restrictions (2)
  • The unrestricted regression equation estimated from the data 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
testing linear restrictions 3
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?
  • First, we rewrite the restriction:  = 1- 
  • Our restricted model is:

(lnY - lnK) = a + a(lnL - lnK) + e

or

G = a +Z + e*

testing linear restrictions 4
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

testing linear restrictions 5
Testing linear restrictions (5)
  • On L15.xls we find a sample n = 32 and an estimated unrestricted model giving us the following information:
testing linear restrictions 7
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

testing linear restrictions 8
Testing linear restrictions (8)
  • The F table value at a 5% significance level is:

F0.05,1,29 = 4.17

    • Since F* > F0.05,1,29 we will reject the null hypothesis that there are constant 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)
testing linear restrictions 9
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:
testing linear restrictions 10
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(G|L,K)
  • We need our unrestricted model to have the dependent variable G, or:
  • Where G = (lnY - lnK)
  • We can test this because we know that  +  - 1 = 0.121 since  +  = 1
  • estimating this unrestricted model will give us the unrestricted R2
testing linear restrictions 11
Testing linear restrictions (11)
  • From L15.xls we have :

R2* = 0.871

R2 = 0.963

  • Our computed F-statistic will be
testing linear restrictions 12
Testing linear restrictions (12)
  • On L15.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 would accept the null given this F-statistic is very small
testing linear restrictions 13
Testing linear restrictions (13)
  • From this we can conclude that we have a model where there are increasing returns to scale.
  • We don’t know the true value, but we can reject the restriction that there are constant returns to scale.
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