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### Multiple Regression Applications

Lecture 15

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

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 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 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

- 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)

- 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)

- 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

- 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)

- 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)

- 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)

- 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)

- On L15.xls we find a sample n = 32 and an estimated unrestricted model giving us the following information:

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)

- 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)

- 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)

- 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)

- From L15.xls we have :

R2* = 0.871

R2 = 0.963

- Our computed F-statistic will be

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)

- 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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