notes06 - Further Issues in Regression Analysis ECON 399...

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Unformatted text preview: Further Issues in Regression Analysis ECON 399 Neil Hepburn Contents 1 Introduction 1 2 Goodness of Fit in the k-Variable Model 1 3 Selecting Between Alternate Models 2 3.1 Log vs Linear Specifications . . . . . . . . . . . . . . . . . . . . . 3 4 Forecasts and Forecast Errors 5 1 Introduction Introduction • We now clean up our initial look at linear regression by touching on a couple of final topics • First we look at the R 2 in the multiple regression model and examine a method for preventing cheating by adding irrelevant variables • We conclude with the issue of forecasting, prediction, and confidence in- tervals for predictions. 2 Goodness of Fit in the k-Variable Model Goodness of Fit and Degrees of Freedom • In theory it is possible to make the unadjusted R 2 equal to one just by adding more regressors, even if they are meaningless • Doing so will certainly not make R 2 go down • We need to correct for this by taking into account the degrees of freedom. 1 Goodness of Fit and the Adjusted R 2 • We can compute an adjusted R 2 that prevents this little fudge from oc- curring • The adjusted ¯ R 2 is ¯ R 2 = 1- SSR / k SST /( n- 1) (1) • We can easily turn an unadjusted R 2 into an adjusted R 2 ¯ R 2 = 1- ( 1- R 2 ) ( n- 1) ( n- k- 1) (2) Goodness of Fit and the Adjusted R 2 • Imagine that we add several nonsense variables to our model. • There will be some change in R 2 , but it will be slight • If we look at Equation (2) we can see that ( 1- R 1 ) will become a bit smaller • However, ( n- k- 1) will become much smaller. The overall result in Equation (2) is that ¯ R 2 will become smaller. 3 Selecting Between Alternate Models Comparisons When y i is the Same in Both Models • When the dependent variable is the same in both models, we can compare the ¯ R 2 for both models • Since the dependent variable is the same in both models, the ¯ R 2 are telling us the same thing - the variation in y i that is explained by the model • If the dependent variables are different in the two models, then we cannot make a comparison this way Comparisons When y i is Not the Same • One case where this frequently comes up is when we have one model with y i and the other with ln( y i ) • Since the dependent variables are different we cannot make a comparison • There is a test that we will see later on that we can use to test for a linear vs logarithmic specification 2 3.1 Log vs Linear Specifications Comparing Log and Linear Models • There is a way that we can compare two specifications of the same under- lying model • Suppose that we are trying to choose between a model that uses y as the dependent variable and one that uses log ( y ) • Our two competing models are wage = β + β 1 belavg + β 2 abvavg + β 3 exper + β 4 expersq + β 5 service + β 6 educ + β 7 female + u log (wage) = β + β 1 belavg + β 2 abvavg + β 3 exper + β 4 expersq + β 5 service + β 6 educ + β 7 female + u Comparing Log and Linear Specifications...
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This note was uploaded on 10/04/2011 for the course ECONOMICS 399 taught by Professor Neil.h during the Spring '11 term at University of Alberta.

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notes06 - Further Issues in Regression Analysis ECON 399...

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