CHAPTER 4 & 5 Linear Regression with One Regressor. Kazu Matsuda IBEC PHBU 430 Econometrics

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1 CHAPTER 4 & 5 Linear Regression with One Regressor Kazu Matsuda IBEC PHBU 430 Econometrics

2 Introduction Simple linear regression model = Linear model with one independent variable. y = dependent variable x = independent variable β 0 = y-intercept β 1 = slope of the line ε = Error variable β 0 and β 1 are population parameters to be estimated.

4 Estimating the coefficients y x Y X Simple linear regression = Draw a (best-fitted) straight line through the sample data.

5 Estimation with Excel Y X Y Predicted Y

6 Method = Y Y Predicted Y X Xi Yi Predicted Y Residuals (Residuals)^

7 Where are these numbers from? The coefficients b 0 and b 1 are calculated so that the sum of squared deviations is minimized. b = cov xy, 1 2 sx 0 1 ( ) b = y bx ( xy) cov, s = n 2 i= 1 x = i= 1 ( x x) i n 1 n ( x x)( y y) 2 i n 1 i Check this out!

8 Required Conditions for Error Variable 6 4 Residuals X Variable 1. The probability distribution of ε is?. 2. The mean of the error distribution is?. 3. The standard deviation of ε is σ ε, which is? regardless of the value of x. 4. The value of ε associated with any particular value of y is independent of ε associated with any other value of y.

9 Assessing the Model Y X Y Predicted Y Xi Yi Predicted Y Residuals (Residuals)^

<1> Standard Error of Estimate We learned that the least squares method assumes that the error variable ε is? distributed with mean? and standard deviation σ ε.

10 <1> Standard Error of Estimate We learned that the least squares method assumes that the error variable ε is? distributed with mean? and standard deviation σ ε. If σ ε is large, some of the errors will be large, which implies that the model s fit is?. If σ ε is small, the errors tend to be close to the mean (which is zero), and, as a result, the model fits?. Since σ ε is a population parameter (unknown). Thus, we estimate σ ε from the data. s ε =

11 <2> Testing the Slope

12 <2> Testing the Slope: Two Sided The null hypothesis specifies that there is no linear relationship. H 0 : H 1 :

13 <2> Testing the Slope: Two Sided The test statistic is: Where s b1 is the standard deviation of b 1 (also called the standard error of b 1 ).

14 <2> Testing the Slope: One Sided The null hypothesis specifies that there is no linear relationship. H 0 : H 1 :

15 <3> Coefficient of Determination From the t test of β 1, we know already that there is evidence of a linear relationship. R 2 merely supplies us with a measure of the? of that relationship. A Measure of explanatory power of the model.

16 <3> Coefficient of Determination Interpret? % of the variation y is explained by the variation in x. The remaining? is unexplained by the variation in x. Unlike the value of a test statistic, R 2 does not have a critical value that enables us to draw conclusions. In general, the? the value of R 2, the better the model fits the data.

Textbook Example The superintendent of an elementary school district must decide whether to hire additional teachers and she wants your advice.

17 Textbook Example The superintendent of an elementary school district must decide whether to hire additional teachers and she wants your advice. If she hires the teachers, she will reduce the number of students per teacher (the student-teacher ratio) by two. She faces a tradeoff. Hiring more teachers means spending more money, which is not to the liking of those paying the bill! So she asks you: If she cuts class sizes, what will the effect be on student performance? If she reduces the average class size by two students, what will the effect be on standardized test scores in her district?

18 What is the OLS regression line for these 420 observations? If she reduces the average class size by two students, what will the effect be on standardized test scores in her district? o Predict the districtwide test score for a district with 20 students per teacher.

19 Is the estimated slope significantly different from zero at 5% level? (Two sided test and one sided test) What can you say about the explanatory power of the model?? % of the variation y=testscr is explained by the variation in x=str.

20 Regression Diagnostics Residual Analysis Required Conditions for Error Variable (Again) 6 4 Residuals X Variable 1. The probability distribution of ε is?. 2. The mean of the error distribution is?. 3. The standard deviation of ε is σ ε, which is? regardless of the value of x. 4. The value of ε associated with any particular value of y is independent of ε associated with any other value of y.

21 <1> Check for constant standard deviation of error term σ ε

22 SAS plot of predicted values vs. residuals for California school case

25 <2> Check for Normality of ε

26 Examples: Nominal Independent variables o Alternative names are indicator variables, binary variables, dummy variables, categorical variables, qualitative variables, dichotomous variables. o A regression model may contain explanatory variables that are exclusively dummy in nature. Such models are called? models.

27 Data on starting salaries of ONU business graduates by sex (hypothetical, 1985) Salary Sex Population regression model Di 1 if male graduate = 0 if female graduate

28 How to interpret coefficients Y = β + β D + u i 0 1 i i Di Population mean salary of male graduates: 1 if male graduate = 0 if female graduate Population mean salary of female graduates: β 1 =

29 Estimation results What is the point estimator of population mean salary of female graduates? What is the point estimator of population mean salary of male graduates? What is the interval estimator of population gender gap with 95% confidence level?

Linear Regression with one Regressor

1 Linear Regression with one Regressor Covering Chapters 4.1 and 4.2. We ve seen the California test score data before. Now we will try to estimate the marginal effect of STR on SCORE. To motivate these