Fluent Essays

for the stata assignment just do the first question Problem Set 3, Econ 120B Prof. Yixiao Sun 1. Consider the simple linear regression model with one explanatory variable, Y = β0 + β1X + u. Suppose we estimate the coeffi cients using OLS on a sample of observations (Xi,Yi), i = 1, · · · ,n. Recall the following definitions of the sample covariance, sample variances, and sample correlation: ĉov (X,Y ) = 1 n n∑ i=1 (Xi − X̄)(Yi − Ȳ ), v̂ar (X) = 1 n n∑ i=1 (Xi − X̄)2, v̂ar (Y ) = 1 n n∑ i=1 (Yi − Ȳ )2, ρ̂XY = ĉov (Y,Y )√ v̂ar (X) √ v̂ar (Y ) . (a) Using the facts that β̂0 = Ȳ − β̂1X̄ and β̂1 = ĉov (X,Y ) /v̂ar (X), show that 1 n SSE = [ĉov (X,Y )] 2 v̂ar (X) . (b) Show that R2 = ρ̂2XY . 2. Consider the following causal model y ← bx + v, x ← cy + u, where b 6= 0,c 6= 0 and bc 6= 1. Suppose that the values of (u,v) are generated from (U,V ) ∼ N [ 0, ( σuu σuv σuv σvv )] . We do not observe (U,V ) but we could observe the equilibrium solution (X,Y ). 1 (a) Consider the special case with σuu = 0 but σvv 6= 0 so that the model becomes y ← bx + v, x ← cy. What is best linear prediction of Y given X (i.e., suppose I give you the value of X but withhold the value of Y, what would be your best guess of Y according to the MSE criterion?). What is best linear prediction of X given Y ? (b) Consider the special case with σvv = 0 but σuu 6= 0 so that the model becomes y ← bx, x ← cy + u What is best linear prediction of Y given X? What is best linear prediction of X given Y ? (c) Now suppose σuu 6= 0 and σvv 6= 0. Under what condition(s) is X useless as a predictor of Y ? Under what condition(s) is Y useless as a predictor of X? Explain. 3. Consider the following model Yi = Xiβ1 + ui, i = 1, ...,n withoutan intercept. Thismodel isusedonlyunder theassumption that E(Y ) = E(X) = 0. We maintain this assumption here. (a) Derive the OLS estimator β̂1 for β1. Show your steps. (b) Given a dataset, suppose we know that the sample correlation coeffi cient between X and Y is 0.5, the sample variance of X is 4 and that of Y is 60. Compute β̂1. (c) Now suppose we regress X on Y without an intercept. Given the information in (b), can you find the OLS estimate for the coeffi cient on Y ? 4. Suppose a researcher, using wage data on 250 randomly selected male workers and 280 female workers, obtained the OLS regression estimates (standard errors are given in paren- theses). ŵage = 12.52 (0.23) + 2.12 (0.36) male, R2 = 0.06. (1) Here wage is measured in dollars/hour and male is a binary variable which equals 1 if the person is male. Define the gender wage gap as the difference in earnings between males and females. (a) What is the estimated gender wage gap? (b) In the sample, what is the mean wage for men and women? (c) Another researcher uses the same data but regresses wage on female, a variable that equals 1 if the person is female. what are the regression estimates in this new regres- sion? Also compute the R2 for this new regression. 2 5. In your sample, all individuals have received a stimulus check of $1,200/month. Let Y be the monthly income (measured in dollars) and X be the total hours worked in a month. You decide to run a regression of Y net, the monthly income on hours worked, net of the $1,200 stimulus check (Y net = Y − 1200). The estimated coeffi cients are reported below: Ŷ net = 1256 + 30 X. (a) What would happen to the estimates if instead you ran the regression using gross income (hence including the $1,200 for all individuals) on hours worked? Explain. (b) Suppose you construct a new variable, the total hours worked in a week Xweek = X/4. What would be the corresponding slope estimate from a regression of Y net on Xweek? 6. Work on the the Stata Assignment. 3 Stata Questions Econ 120B, UCSD Prof. Yixiao Sun • clear all // clear the environment/memory set more off sysuse nlsw88 // load the built-in dataset nlsw88 Please make sure your do-file is clearly documented to help us understand your code. • nlsw88 is a built-in dataset that comes with Stata. It is an extract from the 1988 round of the National Longitudinal Survey of Mature and Young Women. Following is a summary of the variables in this dataset. idcode survey id age age race race, can take three values, white, black or other married = 1 if is currently marries, = 0 otherwise never married = 1 if never married, = 0 otherwise grade current grade completed collgrad = 1 if graduated from college, = 0 otherwise south = 1 if lives in southern states, = 0 otherwise smsa = 1 if lives in standard metropolitan statistical area, = 0 otherwise c city = 1 if lives in central city, = 0 otherwise industry industry, use tab industry to see the categories occupation occupation, use tab occupation to see the categories union = 1 if is in a union, , = 0 otherwise wage hourly wage, measured in $ hours hours worked per week ttl exp total work experience, measured in years tenure current job tenure, measured in years More information on the original data can be found here: https://www.bls.gov/nls/orginal-cohorts/mature-and-young-women.htm 1 https://www.bls.gov/nls/orginal-cohorts/mature-and-young-women.htm 1. In this exercise you will re-label variables and create some new variables which will be used later. (a) Re-label the variable smsa to “lives in urban area”so that it is more in- formative. Note that SMSA stands for “standard metropolitan statistical area.” (b) Re-name the variable smsa to urban. (c) Generate a new variable called wageofc taking the same values as the variable wage, so that we can modify the wage data without loosing the original variable. (d) The minimum wage in 1988 was $3.35 an hour. Let’s say our fictional bosses at the Bureau of Labor Statistics will be mad if they see evidence of minimum wage law violations in the dataset. Re-classify those earning below minimum wage as “volunteers.” To be more specific, In wageofc, replace wageofc with 0 for workers that earned strictly less than $3.35 an hour. Note that we often find evidence of statutes not being followed in datasets. (e) How many observations are in this dataset? (f) How many non-missing observations are in wageofc? (g) Generate a variable called lnwageofc which is the natural logarithm of wageofc. (h) How many non-missing observations are in lnwageofc? Why does this make sense? 2. In this exercise, you are asked to compute some simple summary statistics using the binary variable collgrad, contained in the dataset. (a) Use thecommandtabulate to showthecategoriesof thevariablecollgrad and their frequencies. What is the relative frequency of the category col- lege grad? Please report a number between 0 and 1. (b) Use the same command, this time specifying the option nolabel, to vi- sualize the numeric values corresponding to the different categories of collgrad. Which numeric value corresponds to the label college grad? (c) Use the command summarize to compute the sample mean of collgrad. After executing summarize, Stata stores temporarily the sample mean in the object r(mean). To see this, generate a scalar variable collgrad mean equal to r(mean), by typing scalar collgrad mean = r(mean) in the line just after the command summarize. Finally, display the variable value by typing display collgrad mean, and verify that the value displayed is the same as the one returned by the command summarize. What is the sample mean of collgrad? What is its relation to your answer in 2(a)? 2 (d) Repeat the stepsof2(c), this timetocreatea scalarvariable, collgrad var, containing the sample variance of collgrad. What is the sample variance of collgrad? (e) Compute the sample variance of collgrad without the summarize com- mand, using only the variable collgrad mean. (Hint: you can think of collgrad as drawn from a Bernoulli distribution with parameter p, where p is the probability of having graduated from college. The (population) variance of a Bernoulli is p(1−p). What is the relation between p and the sample mean collgrad_mean? Finally, remember that the sample vari- ance can be obtained starting from the formula of the population variance by replacing the population mean with the sample mean.) 3. The following problems provide more practice using conditional statements to tabulate and summarize variables. (a) How many unmarried people in the dataset were married before? (Hint: use the variables, married and never married.) (b) What is the difference in average hours worked for married and unmar- ried workers? Please report a positive number. (Hint: use the variables married and hours.) (c) What is the average hours worked for married college graduates with strictlymore than10yearsof experience? (Hint: use thevariablesmarried, collgrad, ttl_exp, and hours.) (d) What fraction of laborers or craftsman that live in urban areas are black? Please reportanumberbetween0and1. (Hint: use thevariablesoccupation = 8 and 5, urban, and race.) (e) Using the variable wageofc, what fraction of workers that earn strictly more than $7 an hour are in a union? Please report a number between 0 and 1. (Be careful about missing values.) (f) Using the variable lnwageofc, what fraction of workers that earn strictly more than $7 an hour are in a union? Please report a number between 0 and 1. (That is, you should compare the variable, lnwageofc, to ln7. Be even more careful about missing values.) 4. This exercise refers to the following model: wagei = β0 + β1gradei + ui, 3 where the wage of individual i is regressed on his/her highest grade completed and a constant term. You are asked to compute the intercept and slope esti- mates in a variety of ways, and compare your results in each case. First, use the command keep if !missing(wage, grade) to drop people with missing wage or grade from the dataset. How many ob- servations were dropped? (a) Use the regress command to estimate the OLS coeffi cients β̂0 and β̂1. What is the value of β̂0? What is the value of β̂1? (Hint: type regress wage grade, the constant term will be added automatically to the regres- sion.) (b) You are now asked to compute the same estimates using the formulas we derived in the lecture. Adopt the following procedure: • Compute the sample covariance between wage and grade, and the sample variance of grade, and save them in two scalars, cov wg and var g. (Hint: you can compute the variance-covariance matrix us- ing the corr command, with the option covariance. For instance, if you type corr wage grade, covariance, the output will be a ma- trix containing the variance of wage, the variance of grade and the covariance between wage and grade; the three values will be stored in r(Var 1), r(Var 2) and r(cov 12), respectively. You can check the list of stored objects by typing return list just after running the corr command.) • Generate the scalar beta 1 equal to cov wg/var g and display it by typing display beta 1. What is the relation between this estimate for β1 and the one in 4(a)? • Create two scalars, grade mean and wage mean, equal to the sample means of grade and wage. • compute your estimate for β0 by typing scalar beta 0 = wage mean - beta 1 * grade mean, and then display beta 0. What is the re- lation between this estimate for β0 and the one in 4(a)? (c) Finally, you can compute β̂1 using a “centered”regression. For this part, Adopt the following procedure: • Define a new variable, wage 0 as wage - wage mean, so that this new variable has a sample mean of 0. Similarly, define grade 0 as grade - grade mean. This is called “demeaning”or “centering”a variable. • Regress the centered variable, wage 0, on the other centered variable, grade 0. What are the intercept and slope estimates in this new regression? 4