Econ 3120 1st Edition Lecture 11 Outline of Current Lecture I Regression Current Lecture I II Goodness of Fit Unbiasedness Goodness of fit First recall from above that y can be decomposed as follows yi y i u i 8In order to analyze goodness of fit how well the regression fits the data it is useful to define the following total sum of squares SST yi y 2 explained sum of squares SSE y i y 2 residual sum of squares SSR u 2 i Note that the the explained sum of squares is sometimes called the regression sum of squares or model sum of squares The total sum of squares can be decomposed into the explained plus the residual sum of squares SST SSE SSR 4 3 R squared The R squared of a regression gives us a measure of goodness of fit It is defined as R 2 SSE SST 1 SSR SST In words this is the fraction of the variation in y that is explained by x Note that the definition implies that 0 R 2 1 Note that in economics it is not uncommon to have an R squared close to 0 In our regression of wages on schooling the R squared equals 0 140 While this implies that variation in schooling does not explain much of the variation in wages it does not necessarily mean that we have not done a good job estimating the relationship between schooling and earnings 95 Units of Measurement When running regressions sometimes it s convenient to change the units of measurement so that the regression estimates are easy to read Consider the following example Ashraf Berry and Shapiro 2010 analyze the results of a field experiment in Zambia which estimated the demand for bottles of water purification solution among a sample of 1004 urban households Bottles were offered for sale to individual households at prices between 300 and 800 Zambian Kwacha 3600 Kwacha 1 The authors estimate the following demand equation purchasei 0 1 pricei ui where purchasei is a variable which equals 1 if the household purchased the bottle and 0 otherwise and pricei is the price in Kwacha 1 Estimation of this equation yields the following estimates 0 0 9640 1 0 000664 This implies that an increase in the price of the bottle by 1 Kwacha lowered the purchase probability by 0 066 percentage points However 1 Kw is a very small amount relative to the price at which the bottles were offered Thus it might be more useful to estimate purchasei 0 1 pricei 100 ui This yields the estimates 0 0 9640 1 0 0664 The estimates imply that a price increase of 100 kwacha lowered the purchase probability by 6 6 percentage points This is the same as the estimate above but it is more clear from a presentation standpoint The estimate of 1 makes sense because the above equations are equivalent when 1 1 100 1This type of model is called the linear probability model Instead of predicting purchase or no purchase as a 0 1 variable the model predicts purchase d i as the estimated probability These notes represent a detailed interpretation of the professor s lecture GradeBuddy is best used as a supplement to your own notes not as a substitute of purchase We ll study this model in more detail at the end of the course 10The key here is that multiplying a variable or both by a constant simply changes the magnitudes of the coefficients but it does not change the underlying relationship between the variables It also does not change the Rsquared of the regression It is tedious to prove this but the intuition is that by multiplying x by a scalar we are not changing the underlying variability which can explain the variance in y Example Start with the basic model y 0 1x u In terms of 0 and 1 find 0 and 1 in the regression z 0 1x where z y 10 5 1 Basic log transformations By now you should be familiar with the regression of earnings on schooling wage 0 1educ u In this case 1 can be interpreted as the expected change in wages for a 1 year change in schooling It is possible however that the relationship between schooling and wages the CEF is not linear If the CEF is not linear it is often possible to improve the model with a simple transformation It turns out that the relationship between log earnings and education is thought to be linear log wage 0 1educ u In this case 1 is interpreted as the percentage change in wages for a 1 year change in schooling 11This can be seen by differentiating d d educ log wage d d educ 0 1educ u 1 wage d wage d educ 1 change in wage 1 unit change in educ 1 where we assume that d u d educ 0 2Using our NLSY79 data we obtain the following estimates log wage 1 68 0 089 educ R2 0 130 Example Suppose you have the following model of an individual s cigarette consumption log cigs 0 1log price u where cigs is the quantity of cigarettes smoked per day and price is the average price of cigarettes in the state of residence Show that 1 can represent the elasticity of demand for cigarettes 6 Sampling distributions of regression estimates 6 1 Unbiasedness of OLS In order to show that OLS estimators of 0 and 1 are unbiased we need to make use of the assumptions that we have been using so far We can formalize them as follows SLR 1 Linear model y 0 1x u 2We assume this because we would like to know the percentage change in wage for a one unit change in educ holding u fixed 12 SLR 2 Random sampling our sample x y i 1 n is a random sample following the population model in SLR 1 SLR 3 Sample variation in the explanatory variable x SSTx xi x 2 6 0 As we have seen this is necessary in order to obtain an estimate of 1 SLR 4 Zero conditional mean E u x 0 With these assumptions we can show that OLS estimates are unbiased To show unbiasedness of For 1 we need to take the expectation conditional on x but to keep the notation neat we will suppress the conditioning Just keep in mind that the below algebra is finding E 1 x All of the derivations in this lecture are done conditional on x 3 1 xi x yi xi x 2 xi x yi SSTx E 1 x E xi x yi SSTx 1 SSTx E xi x yi 1 SSTx E xi x 0 1xi ui xi x 0 1xi E u SSTx 0 1 xi x xi 0 SSTx 1 3This shows that under our assumptions E 1 x 1 It is straightforward to show that the unconditional expectation of 1 is 1 If we take the expectation of both sides and use the law of iterated expectations E 1 E E 1 x E 1 E 1 1 13