1. Multiple Regression Applications
Lecture 15
Lecture 15

2. 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
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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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4. R2 (2) We know from the sum of squares identity that ^
Dividing by the total sum of squares we get ^
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5. R2 (3)
Thus we have or or
If we divide the denominator and numerator of the F-test by the total sum of squares:
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6. 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
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7. 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
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8. 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
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9. 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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10. 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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11. 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
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12. 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
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13. Testing linear restrictions (3)
adding up the regression coefficients, we have: = 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*
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14. 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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15. Testing linear restrictions (5)
On L15.xls we find a sample n = 32 and an estimated unrestricted model giving us the following information:
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16. Testing linear restrictions (7)
The restricted model gives us the following information:
We can use this information to compute our F statistic:
F* = [( )/1]/(0.359/29) = 72.47
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17. 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)
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18. 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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19. 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 = since  +  = 1
estimating this unrestricted model will give us the unrestricted R2
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20. Testing linear restrictions (11)
From L15.xls we have :
R2* = 0.871
R2 = 0.963
Our computed F-statistic will be
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21. 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
The F table value at a 5% significance level is
The probability that we would accept the null given this F-statistic is very small
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22. 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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