Multiple regression applications
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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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Multiple Regression Applications

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Multiple regression applications

Multiple Regression Applications

Lecture 15


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


Multiple regression applications

^

^

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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