Multiple Regression Applications Lecture 15 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 Lecture 15

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 Lecture 15

R2 (2) We know from the sum of squares identity that ^ Dividing by the total sum of squares we get ^ Lecture 15

R2 (3) Thus we have or or If we divide the denominator and numerator of the F-test by the total sum of squares: Lecture 15

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 Lecture 15

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 Lecture 15

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 Lecture 15

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 Lecture 15

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 Lecture 15

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 Lecture 15

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 Lecture 15

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* Lecture 15

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 Lecture 15

Testing linear restrictions (5) On L15.xls we find a sample n = 32 and an estimated unrestricted model giving us the following information: Lecture 15

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 Lecture 15

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) Lecture 15

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: Lecture 15

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 Lecture 15

Testing linear restrictions (11) From L15.xls we have : R2* = 0.871 R2 = 0.963 Our computed F-statistic will be Lecture 15

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 Lecture 15

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. Lecture 15