Multiple Regression Applications

Multiple Regression Applications Lecture 14

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 • 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

^ ^ ^ ^ ^ ^ ^ R2 (2) • We know from the sum of squares identity that • Dividing by the total sum of squares we get

R2 (3) • Thus 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 • 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

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

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

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 • 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

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

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*

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 L14.xls our sample n = 32 and an estimated unrestricted model provides 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 that the test rejects constant return. As a + b > 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)

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 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

Testing linear restrictions (11) • From L14.xls we have : R2* = 0.871 R2 = 0.963 • Our computed F-statistic will be

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

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.

Multiple Regression Applications