Regression Analysis: Studying the Relationship between Hourly Wage and Years of Schooling, Study notes of Sociology

Download Regression Analysis: Studying the Relationship between Hourly Wage and Years of Schooling and more Study notes Sociology in PDF only on Docsity! sociology regression Regression is a means of studying how the conditional distribution of a response variable (say, Y) varies for different values of one or more independent explanatory variables (say, X). The feature of the response variable distribution that most work on regression looks at is the mean. The response variable is frequently quantitative and measured on a true metric, but it doesn’t have to be: we’ll do regression with qualitative, categorical response variables. The independent variables (aka regressors) are frequently quantitative, but they don’t have to be: we’ll do regressions with qualitative, categorical independent variables. But for the time being we’ll work exclusively with regression models in which the dependent variable and the independent variable are both quantitative. Below we use data from 515 respondents to the 1985 current population survey (cps) to look at how the mean of the sample conditional distribution of hourly wage varies across distinct values of years of schooling. Let’s begin by graphing Y against X, wages (vertical axis) against schooling (horizontal axis). h o u rl y w a g e i n d o lla rs figure 1. conditional distributions of wage by schooling years of schooling 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 5 10 15 20 25 30 Let’s begin by looking at a model for wages that totally ignores schooling: So the equation for the ith observation at the jth year of schooling is ayij =ˆ ijij eay += docsity.com 1. regress hrwage Source | SS df MS Number of obs = 515 ---------+------------------------------ F( 0, 514) = . Model | 0.00 0 . Prob > F = . Residual | 12374.963 514 24.0758035 R-squared = 0.0000 ---------+------------------------------ Adj R-squared = 0.0000 Total | 12374.963 514 24.0758035 Root MSE = 4.9067 ------------------------------------------------------------------------------ hrwage | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- _cons | 9.088874 .2162155 42.036 0.000 8.664099 9.513649 ------------------------------------------------------------------------------ So a = $9.0889. This will be our predicted or fitted value of wage for everyone in the sample, no matter how many years of schooling they have. 2. pred grand Here’s the graph of the fitted line against years of schooling. ho ur ly w ag e in d ol la rs figure 2. fitting constant function years of schooling hrwage grand 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 5 10 15 20 25 30 Now lets fit a model in which the fitted/predicted values of y are equal to the mean wage at each value of years of schooling. Hence, there will be as many different predictions as there are different values of schooling, in this case, eleven. So the model for the mean of y is And the equation for the ith observation at the jth years of schooling is jij ay =ˆ ijjij eay += docsity.com Here’s the graph of all the fitted models. The linear regression does a good job of tracking the exact fitted conditional mean function. To see how good, compare the mean square residuals from the different models. ho ur ly w ag e in d ol la rs figure 5. constant, mean, and blp functions years of schooling grand mean_y blp 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 5 10 15 20 25 30 model comparisons constant model conditional mean linear regression SST total sum of squares 12734.96 12734.96 12734.96 SSresidual residual sum of squares 12734.96 10172.98 10394.90 SSregression Regression sum of squares 0 2201.98 1980.06 df residual degrees of freedom (n-1) = 514 (n-11) = 504 (n-2) = 513 MSres mean square residual (12734.9/514)=24.07 (10172.98/504)=20.18 (10394.9/513)=20.26 docsity.com Root MSres sqrt(24.07)= 4.90 sqrt(20.18)= 4.49 sqrt(20.26)= 4.50 Other statistics for wages and schooling total variation in y: standard deviation of y: mean of y: _____________________________________________________________________________ total variation in x: standard deviation of x: mean of x: _____________________________________________________________________________ covariation of y and x: covariance of y and x: _____________________________________________________________________________ correlation of x and y: 963.12374)( 2 =−∑ yyij 088.9=y 967.2919)( 2 =−∑ xxij 192.13=x 5228.2404))(( =−−∑ yyxx ijij 91.4514/963.12374 ==ys 38.2514/967.2919 ==xs 68.4514/5228.2404 ==xys 40.)91.4)(38.2/(68.4/ === yxxyxy sssr docsity.com