Simple Linear Regression
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1 Chapter 2 Simple Liear Regressio 2.1 Simple liear model The simple liear regressio model shows how oe kow depedet variable is determied by a sigle explaatory variable (regressor). Is is writte as: Y i = β 1 + β 2 X i + u i. (2.1) The subscript i refers to the observatio,2,..., ad Y i is the depedet variable. We break dow Y i ito two compoets, the determiistic (oradom) compoet β 1 + β 2 X i ad the stochastic (radom) compoet u i. The explaatory variable is X i ad the populatio parameters we wat to estimate are give by itercept β 1 ad the slope β 2. The term u i is the disturbace term. Figure 2.1 shows a graphical represetatio of the problem. The regressio lie Y i = β 1 +β 2 X i +u i is show as the upward slopig blue lie. Oly a sigle observatio poit at (X i, Y i ) is illustrated. We ca see how for this observatio i, we break dow Y i ito the disturbace term u i give by the vertical distace betwee Y i ad Ŷ i ad the height of the regressio lie at poit X i, give by β 1 + β 2 X i. 2.2 Least squares regressio The mai idea i ecoometric aalysis is to estimate the parameters β 1 ad β 2. The most popular estimator for these populatio parameters is the Ordiary Least Squares (OLS) estimator. Let the OLS estimators of β 1 ad β 2 be b 1 ad b 2, respectively. The, the fitter regressio equatio is: Y i = b 1 + b 2 X i + e i. (2.2) The differece betwee Equatios 2.1 ad 2.2 is that the first correspod to the populatio, while the secod is the sample couterpart. The idea i the OLS es- 17
2 18 2 Simple Liear Regressio Fig. 2.1 Regressio lie Y i = β 1 + β 2 X i + u i. timator is simple, we wat to pick values for the itercept b 1 ad slope b 2 coefficiets that are as close as possible to the actual data poits. That is, we wat to e i (e i = Y 1 b 1 b 2 X i ) to be small. Because some of the e i are positive ad some are egative, we will first square them to have all positive umbers. The, to take ito accout all data poits we will sum across all observatios. That is how our objective is to pick b 1 ad b 2 to miimize the followig residual sum of squares: RSS=e 2 1+ e e 2 = This miimizatio exercise yields the OLS estimators: for the slope coefficiet, ad e 2 i. (2.3) b 2 = (X i X)(Y i Ȳ) (X i X) 2 (2.4) b 1 = Ȳ b 2 X (2.5) for the itercept. The derivatio of the least squares coefficiet estimators (Equatios 2.4 ad 2.5) has the followig steps. We start with the regressio equatio: Y i = b 1 + b 2 X i + e i Ŷ i = b 1 + b 2 X i
3 2.3 Iterpretatio of the regressio coefficiets 19 For observatio i we obtai the residual, the square it ad fially sum across all observatios to obtai the residual sum of squares: e i = Y i Ŷ i (2.6) e 2 i = (Y i Ŷ i ) 2 e 2 i = (Y i Ŷ i ) 2 The coefficiets b 1 ad b 2 are chose to miimize the residuals sum of squares: mi b 1,b 2 (Y i Ŷ i ) 2 (2.7) mi (Y i b 1 b 2 X i ) 2 b 1,b 2 The first order ecessary coditio are: 2 2 (Y i b 1 b 2 X i ) = 0 w.r.t. b 1 (2.8) X i (Y i b 1 b 2 X i ) = 0 w.r.t. b 2 (2.9) Dividig Equatio 2.9 by ad workig through some math we obtai the OLS estimators for the costat: b 1 = Ȳ b 2 X. Pluggig this result ito Equatio 2.9 we obtai: b 2 = i=0 (X i X)(Y i Ȳ) i=0 (X i X) Iterpretatio of the regressio coefficiets If the estimated regressio equatio is give by: ŵage i = exper i, (2.10) where wage is the hourly wage measured i dollars, ad exper is the umber of years of experiece, the the iterpretatio of the slope coefficiet is the followig: wage exper = 0.09.
4 20 2 Simple Liear Regressio Therefore, if the chage i the umber of years of experiece is oe, exper, the the chage i the hourly wage i dollars is give by wage = I words, a additioal year of experiece will icrease your hourly wage by 0.09 dollars (or 9 cets). For the iterpretatio of the itercept, just cosider the case where someoe has ot experiece, exper = 0. The, this perso s predicted wage will be 4.64 dollars. If the estimated regressio equatio takes the form: logwage i = exper i, (2.11) where the log wage is the atural logarithm of wage, the the iterpretatio is differet. Here, if the umber of years of experiece icreases by oe, the wage icreases by 2% ( percet). Fially, for the folowig estimated equatio: logwage i = logexper i. (2.12) A oe percet icrease i exper will icrease wage by 0.25 percet. The 0.26 is iterpreted as a elasticity. 2.4 Goodess of fit How good is the regressio equatio i explaiig the variatio i variable Y? First we eed a way to measure the total variatio i Y. Let s try the sum of squared deviatios about the sample mea of Y. That is, Now, let s start with a simple equality: (Y i Ȳ) 2 (2.13) Y i Ȳ = Y i Ȳ. If we add ad subtract Ŷ i o the right had side of the above equality, we have Y i Ȳ = Y i Ȳ +Ŷ i Ŷ i Y i Ȳ = (Ŷ i Ȳ)+(Y i Ŷ i ) Squarig both sides of the equatio ad the summig across all observatios i we obtai: (Y i Ȳ) 2 = (Ŷ i Ȳ) 2 + (Y i Ŷ i ) 2 (2.14) T SS = ESS+RSS. (2.15)
5 2.4 Goodess of fit 21 Fig. 2.2 Decompositio of Ŷ i Ȳ. Notice that the sum of deviatios from the mea is zero, that is why there are oly two compoets o the right had side. The T SS is the Total Sum of Squares, as preseted i Equatio The first term o the right had side is ESS, the Explaied Sum of Squares, ad the secod term o the right had side is the RRS, Residual Sum of Squares. This decompositio of the variable Y ito two compoets ca be appreciated i Figure 2.2. For every observatio Y i i the sample, the distace betwee Y i ad Ȳ ca be decomposed i two, the part that the regressio equatio ca explai, Ŷ i Ȳ, ad the part that the regressio equatio caot explai, Y i Ŷ i. What is the proportio of the variatio i Y that is explai by the regressio equatio? We just eed to divide Equatio 2.15 by T SS ad defie the ratio of ESS to T SS as the proportio of the explaied variatio i Y, the R 2 : 1 = ESS T SS + RSS T SS (2.16) R 2 = ESS RSS = 1 T SS T SS (2.17) R 2 = (Ŷ i Ȳ) 2 (Y i Ȳ) 2 = 1 e2 i (Y i Ȳ) 2 (2.18) The R 2 is a umber betwee zero ad oe, beig higher whe the model explais more of the variatio i Y. Figures 2.3 ad 2.4 illustrate how the regressio lie explai the variatio i Y whe the R 2 is low ad high, respectively.
6 22 2 Simple Liear Regressio Fig. 2.3 Low R 2. Fig. 2.4 High R 2.
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