Midterm Exam 1, section 1. Thursday, September hour, 15 minutes

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1 San Francisco State University Michael Bar ECON 312 Fall 2018 Midterm Exam 1, section 1 Thursday, September 27 1 hour, 15 minutes Name: Instructions 1. This is closed book, closed notes exam. 2. You can use one double-sided sheet of paper, letter size (8½ 11 in or mm), with any content you want. 3. No calculators of any kind are allowed. 4. Show all the calculations, and explain your steps. 5. If you need more space, use the back of the page. 6. Fully label all graphs. Good Luck

2 1. (10 points). Let XX be a random variable with mean μμ and variance 2, and let XX μμ YY = be the standardized transformation of XX. a. Using the rules of expected values show that the mean of YY is 0. XX μμ EE(YY) = EE given form of YY = 1 EE(XX μμ) constants factor out of EE = 1 (EE(XX) μμ) EE of sum = sum of EE = 1 (μμ μμ) it is given that EE(XX) = μμ = 0 b. Using the rules of variances, show that the variance of YY is 1. XX μμ vvvvvv(yy) = vvvvvv given the form of YY = 1 vvvvvv(xx μμ) constants factor out of vvvvvv squared 2 = 1 vvvvvv(xx) adding constant does not affect vvvvvv 2 = = 1 given vvvvvv(xx) 1

3 2. (10 points). Let XX and YY be two random variables. Prove that vvvvvv(xx + YY) = vvaaaa(xx) + vvvvvv(yy) + 2cccccc(XX, YY) Using the property that variance is equal to the covariance between a random variable and itself: vvvvvv(xx + YY) = cccccc(xx + YY, XX + YY) = cccccc(xx, XX) + cccccc(xx, YY) + cccccc(yy, XX) + cccccc(yy, YY) distr. property of cov vvvvvv(xx) vvvvvv(yy) = vvvvrr(xx) + vvvvvv(yy) + 2cccccc(XX, YY) from def. of cov, cccccc(xx, YY) = cccccc(yy, XX) 2

4 3. (20 points). Let XX 1, XX 2,, XX be a random sample from population XX, with population mean μμ and variance 2. a. Prove that (XX ii XX ) = 0 where XX is the sample average. Your answer must start with a definition of sample average. Sample average is defined as follows: Thus, XX = 1 XX ii (XX ii XX ) = XX ii XX = XX XX = 0 b. Supposed that EE(XX 1 ) = 7. Find EE(XX 1 XX 2 ). Since XX 1, XX 2,, XX is a random sample, all observations must be independent random variables, which means they are uncorrelated: cccccc(xx 1, XX 2 ) = EE(XX 1 XX 2 ) EE(XX 1 )EE(XX 2 ) = 0 Therefore, EE(XX 1 XX 2 ) = 7 7 = 49 3

5 4. (20 points). In order to estimate the population mean, a random sample of observations was collected XX 1, XX 2,, XX, and the sample average XX = 1 XX ii is proposed as an estimator. a. Prove that XX is an unbiased estimator of the population mean μμ. EE(XX ) = EE 1 XX ii = 1 EE(XX ii) = 1 μμ = 1 = μμ b. Let 2 denote the population variance. Prove that XX is a consistent estimator of the population mean μμ. Since we proved that XX is unbiased, we only need to prove that lim vvvvvv(xx ) = lim 2 = 0. vvvvvv(xx ) = vvvvvv 1 XX ii = 1 2 vvvvvv(xx ii) = = 2 lim vvvvvv(xx ) = lim 2 = 0 4

6 5. (20 points). Consider the simple regression model YY ii = ββ 1 + ββ 2 XX ii + uu ii. a. Suppose that YY ii is crime rate in state ii (number of crimes per 100,000 population), and XX ii is poverty rate in state ii (% of population below poverty rate). What is the interpretation of the error term uu ii? Your interpretation must contain one relevant example. The error term uu ii represents all the factors, other than poverty rate, which affect crime rate. For example, uu ii may characteristics of law enforcement and judicial system in state ii. b. Define the OLS estimators of the unknown parameters ββ 1, ββ 2 and denote them by bb 1 OOOOOO, bb 2 OOOOOO. Let the fitted model be YY ii = bb 1 + bb 2 XX ii, where bb 1 and bb 2 are some estimates of ββ 1 and ββ 2. The residual of observation ii (or prediction error) is ee ii = YY ii YY ii. The OLS estimators bb OOOOOO OOOOOO 1, bb 2 are values of bb 1, bb 2 which minimize the Residual Sum of Squares, i.e. solve the following problem: 2 min RRRRRR = ee ii = (YY bb 1,bb 2 ii bb 1 bb 2 XX ii ) 2 5

7 c. Suppose that Ariana estimated bb 1 = 3500 and bb 2 = 100. What is the predicted crime rate of a state with poverty rate of 15%? (In the data 15% appears as 15). Substituting the given values into the fitted equation: YY ii = bb 1 + bb 2 XX ii = = 5000 cccccccccccc pppppp 100kk pppppppppppppppppppp d. If the average crime rate in the sample is 6000, what is the average poverty rate in the sample? Using the fact that the fitted equation must pass through the point of sample averages, YY = bb 1 + bb 2 XX 6000 = XX 100 XX = 2500 XX = 25 percent 6

8 6. (20 points). Maria is studying the relationship between people s age and medical expenditures. She collected a random sample on 5574 individuals ages 0-63, with the following key variables: med aual out of pocket medical expenditures (in $) age age of individuals in years Maria s R output is presented below: lm(med ~ age, data = MedExp) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) age <2e-16 *** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: 798 on 5572 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 1 and 5572 DF, p-value: < 2.2e-16 a. What is the dependent variable in the above regression model? med b. What is the independent variable (regressor) in the above regression model? age 7

9 c. Interpret the estimated regression coefficients. bb 2 = 5.29 means that with each year increase in individual s age, the aual out of pocket medical expenditures are predicted to increase by $5.29. bb 1 = is the predicted aual out of pocket medical expenditures on children age 0 (infants). This is one of a few examples where the estimated intercept has real life meaning. d. Explain the meaning of the reported RR 2, and comment on its magnitude. Your comment must contain at least one relevant example. RR 2 = means that only 1.2% of the variation in aual out of pocket medical expenditures in the sample can be explained by this model, with age as the only regressor. This magnitude is very small, meaning that nearly 99% of the variation in medical expenditures are explained by factors other than age. Such factors could include for example the type of medical insurance that people have (with good insurance there could be very few out-of-pocket expenditures). Other factors could be the health condition of individuals, genetics, lifestyle (eating and smoking habits, exercising, etc.). The fact that age explains so little of the variation in healthcare expenditures in this sample is not surprising. Recall that the sample consists of people ages The steep increase in medical costs occurs after age of 65. And even after that age, the type of insurance people have, explains most of the difference in out-of-pocket expenditures. 8