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CORNELL ECON 3120 - I. Goodness of Fit II. Unbiasedness

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Econ 3120 1st Edition Lecture 11Outline of Current Lecture I. RegressionCurrent LectureI. Goodness of FitII. UnbiasednessGoodness-of-fit First, recall from above that y can be decomposed as follows: yi = yˆi +uˆi 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) = ∑(yˆi −y¯) 2 residual sum of squares (SSR) = ∑uˆ 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 = β0 +β1 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: ˆβ0 = 0.9640 ˆβ1 = −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 = β0 +α1(pricei/100) +ui This yields the estimates ˆβ0 = 0.9640 αˆ 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 α1 makes sense because the above equations are equivalent when β1 = α1/100. 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 R-squared 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 = β0 +β1x+u In terms of β0 and β1 , find α0 and α1 in the regression z = α0 +α1x+ε where z = y/10. 5.1 Basic log-transformations By now you should be familiar with the regression of earnings on schooling wage = β0 +β1educ+u In this case, β1 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) = β0 +β1educ+u In this case, β1 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 (β0 +β1educ+u) 1 wage d wage d educ = β1 % change in wage 1 unit change ineduc = β1 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) = β0 +β1log(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 thatβ1 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 β0 and β1 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 = β0 +β1x+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 ˆβ1 • 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 ˆβ1, 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( ˆβ1|x). All of the derivations in this lecture are done conditional on x. 3 ˆβ1 = ∑(xi −x¯)yi ∑(xi −x¯) 2 = ∑(xi −x¯)yi SSTx E( ˆβ1|x) = E ∑(xi −x¯)yi SSTx = 1 SSTx E ∑(xi −x¯)yi = 1 SSTx E∑(xi −x¯)(β0 +β1xi +ui) = ∑(xi −x¯)(β0 +β1xi +E(u)) SSTx = 0+β1 ∑(xi −x¯)xi +0 SSTx = β1 3This shows that under our assumptions, E( ˆβ1|x) = β1. It is

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