Multiple Regression Models

Multiple Regression Models Lecture 13

Today’s plan • Functional form (what model do we estimate?) • Testing the explanatory power of the model • Adjustment to R2

Reading coefficients • With a bi-variate model we could easily determine how a change in X affects Y • With a multivariate model you must hold X1 constant to determine the effect of a change in X2 on Y • For this reason we call the slope coefficients in a multivariate regression the partial regression coefficients

Functional form • To return to arguments about functional form. How do we know what model to estimate? • Our example on earnings and years of education has some economic theory in its foundation - but basically an ‘ad-hoc’ specification. We know we want to test the relationship between earnings and years of schooling. • An example based on economic theory: the Cobb-Douglas production function Y = ALK • May want to test for constant returns to scale  +  = 1

Functional form (2) • We can transform the Cobb-Douglas equation into a form we can estimate by taking logs: ln Y = ln A +  ln L +  ln K • This is called log linear form since all the variables are in logs • The model is now linear in parameters so we can use least squares to estimate it • The log linear form gives us estimated coefficients that are elasticities: the estimates of  and  give us the elasticities of labor and capital with respect to output

Example with longitudinal data • L13-1.xls on the webpage. It contains information on companies in the UK private sector. Data from DATASTREAM; for US: COMPUSTAT • Note that this is a longitudinal (panel) data set - we are analyzing the same agents (the companies) over time • I have calculated the true output elasticity with respect to labor for a 100% change in labor and the true output elasticity with respect to labor for a 10% change in labor • Note that the larger the increase in the independent variable, the further the approximation is from the coefficient

Example with longitudinal data (2) • If we want to calculate the true change, we need to calculate: • If we want to estimate the Cobb-Douglas production function, we use the partial slope coefficients • We can calculate the partial slope coefficients :

Example with longitudinal data (3) • Adding our estimates together we find: • Later on we’ll test the constraint that  +  = 1

W The equation representing this relationship between unemployment and wage inflation is: Un Phillips Curve • The Phillips Curve is an example of ad-hoc variable inclusion (model based on economic reasoning)

Phillips Curve (2) • Estimation of Phillips Curve: transform the independent variable into an inverse (X into 1/X). Note, model is non-linear in variables, but linear in parameters. • With ad-hoc specification we don’t know what other variables are relevant • we need to make informed guesses determined by what we know of economic theory

The story so far • Functional form • Omitted variable bias • Examples of types of data & types of modeling approach (note, they are not exclusive): • Cross section: Return on education • Panel/longitudinal: Cobb-Douglas • Time-series: Phillips Curve

Variation in multivariate models • Let our model be • For test statistics we still want to calculate: • How to calculate these values.

Variation in multivariate models (2) • It still holds that the variance of the regression line is • It also still holds that:

Test statistics in multivariate models • We will start with the sum of squares identity, where: Total = Explained + Residual or • But, the composition of the ESS will be different - our sum of squares identity will look like this: • As you add more independent variables to the model, more terms get added to the ESS

Test statistics in multivariate models (2) • Our R2 is: • Now let’s look back to an example from an earlier lecture (L12-1.xls) • we looked at the returns to earnings of education (b1) and age (b2) • calculate the test statistics and consider model problems

Test statistics in multivariate models (3) • Remember the matrix of products and cross-products looks like this: • You will also need these these values:

Test statistics in multivariate models (4) • The regression line we calculated earlier is: • We can start our calculations with: • Taking the square root, we find the root mean square error:

Test statistics in multivariate models (5) • We can then calculate: • Taking the square root gives us

Test statistics in multivariate models (6) • We can then calculate: • Taking the square root gives

Hypothesis test on education • Form a null hypothesis • The t-ratio is calculated: • For a significance level of 5% we have a table t value of t/2,33 = 2.035 • Since |t| < t /2 , we cannot reject the null hypothesis • Recall that the purpose of the model was to examine whether or not education has an effect on earnings (allowing for age effects). Should we conclude that additional years of education have no effect on earnings?

Hypothesis test on age • We construct the hypothesis test: • The t-ratio is calculated: • For a significance level of 5% we have a table t value of t/2,33 = 2.035 • Since |t| > t /2 , we reject the null hypothesis

Looking at R2 • Coefficient of determination: R2: • This is a rather low R2 • This means that the regression equation doesn’t explain the variation well • The regression equation only explains about 1/5 of the variation in Y

Looking at R2 (2) • What should we do about the form of our estimated equation when years of education are shown to be statistically insignificant at our chosen significance level? • We chose a 5% significance level for our test, but we might have been able to reject the null at a different significance level • Remember: with hypothesis test we want to reduce the number of type I errors where we falsely reject a null

Testing explanatory power • What if we examined the regression equation as a whole? • We want to keep years of education for our model. Also want to retain the conditioning on age - it appears to be important! Are there other tests we can undertake? • Could examine the null hypothesis: H0 : b1 = b2 = 0 • This says that neither of the independent variables has any explanatory power • To test this, we will use an F test

Testing explanatory power (2) • The F statistic that we’re looking at can be found on the LINEST output • The F test comes from the ANOVA table for the multivariate case, which looks like this:

^ ^ ^ Testing explanatory power (3) • The F statistic will look like: • Using the F table, you choose a significance level and use the degrees of freedom in the numerator and denominator, or F0.05, 2, 33 • The 1st row in the table is df in the numerator • The 1st column is the df in the denominator • The 2nd column is the significance level

F table value H0: Cannot reject the null H1: Reject the null F Testing explanatory power (4) • If our calculated F statistic is greater than (to the right of) our F table value, we reject the null • If our calculated F statistic is less than (to the left of) our F table value, we cannot reject the null

Testing explanatory power (5) • Looking at the F table, we find that there is no value for exactly 33 df • We have to approximate using 30 df instead • Our approximated F value is F0.05, 2, 33  3.29 • We reject the null because F > F0.05, 2, 33 • Had we picked a 1% significance level, or F table value would be F0.01, 2, 33  5.27 • and we could not have rejected the null because F < F0.01, 2, 33

F* value 5% 1% F 3.29 3.81 5.27 Testing explanatory power (6) • In summary, we’re more likely to reject the null at a greater significance level • In this case, we rejected at a 5% significance level and could not reject the null at a 1% level • Graphically:

Testing explanatory power (7) • The t-test suggests that we should remove years of education from our regression • An F-test on the joint hypothesis rejects the null, but the test is weak. At a lower significance level (1 percent), we could not have rejected the null. • In this instance, we want to keep the years of education variable in the equation because of what we know of economic theory • What to do? Conclude that the economic theory is weak? Give up school? Or, obtain more data and try again! Look at results from other studies.

Adjustment to R2 • The more variables added to a regression, the higher R2 will be • R2 is important, but it isn’t the sole criteria for judging a model’s explanatory power • Adjusted R2 adjusts for the loss in degrees of freedom associated with adding independent variables to the regression

Adjustment to R2 (2) • Adjusted R2 is written as Adj R2 = 1 - (1 - R2)((n - 1)/(n - k)) n : sample size k : number of parameters in the regression

What’s next • Restricted least squares and the Cobb Douglas Production function • Including qualitative indicators into the regression equation (e.g. race, gender, marital status).

Multiple Regression Models