ECON5115. Solution Proposal for Problem Set 4. Vibeke Øi and Stefan Flügel

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Edmund Parks
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1 ECON5115 Solution Proposal for Problem Set 4 Vibeke Øi (vo@nupi.no) and Stefan Flügel (stefanflu@gmx.de) Problem 1 (i) The standardized normal density of u has the nice property This helps to show that Proof: The expected value of the continuous random variable u given that u lies between a and b is Where is the density function of the truncated variable u. It equals the untruncated density function divided through the probability that u lies between a and b, which can be expressed by P(a<u<b) Thus From (2) follows: so that

2 (ii) pmax*u,a+ takes value a if u lies below a. Above the truncation point a, p takes values of u. This means that the distribution of p is truncated from below which implies that b is going to infinity. From (3) it follows that as and qmin[u,b], is the other way around. The distribution of q is single truncated from above. as and (iii) The binormally distributed variables ( are standardized by and So that ( where is the correlation coefficient The conditioned expectation of given is Respectively

3 First notice that as x is a linear transformation of u. Combining (9), (10), (11), (12) and it follows and Expressions for can be find straight forward combining (7), (8), (11), and (12) Expressions for (16). can be find by combining (9), (10), (13), (14), (15) and Accordingly (iv) Here the truncation point is the product of the observed covariates and the coefficients. As the product of a row vector and a column vector is a single number we suggest to proceed according to the formulas we used for the constants a and b (A and B). The expressions for are:

4 (v) The variable takes values of y if y>0 and values of 0 if y<0. We are facing a single truncation from below. The truncation point is 0, which in this case is also the mean of y (meaning that 50% of the density mass is left and 50% is right from 0). As takes values of y given y>0. We can state that We use the same calculation as above (combining (7) and (10)) and taking into account that 0, Also notice the fact that We get For the expression of we use the rule of iterated expectations Where is zero and equals 0,5 because is given ( is higher 0 if and only if y is higher 0, so occurs with probability 0,5). was just calculated in (23) Combining: 0,5 * The expected value of the variable which is single truncated from below at truncation point 0 is depending on the standard deviation of the underlying variable y. The higher the standard deviation of y the higher the expected value of will be. This is an intuitive result, taking into account that is never negative (the deviation from y in the negative numbers is cut of by the truncation). The probability of y being positive is also important for the expected value of The higher this probability is the higher will get. In our case this probability is 0,5. That yields straight forward

5 to the result that the unconditional expectation is half as high as the conditional expect Problem 2 (i) and (ii) The data can be load in with this command (adjust path accordingly) infile LFP WHRS KL6 K618 WA WE WW RPWG HHRS HA HE HW FAMINC MTR WMED WFED UN CIT AX using "M:\pc\Dokumenter\ECON5115_H08_DataSem05Prob2_only_rawdata.txt", clear where ECON5115_H08_DataSem05Prob2_only_rawdata.txt is a dataset which only contains the actual data.the variables description is cut out of this file, saved separately and is used as a template to give the variables their names. label var LFP "A dummy variable 1 if woman worked in 1975, else 0" label var WHRS "Wife's hours of work in 1975" label var KL6 "Number of children less than 6 years old in household" label var K618 "Number of children between ages 6 and 18 in household" label var WA "Wife's age" label var WE "Wife's educational attainment, in years" label var WW "Wife's average hourly earnings, in 1975 dollars" label var RPWG "Wife's wage reported at the time of the 1976 interview" label var HHRS "Husband's hours worked in 1975" label var HA "Husband's age" label var HE "Husband's educational attainment, in years" label var HW "Husband's wage, in 1975 dollars" label var FAMINC "Family income, in 1975 dollars" label var MTR "This is the marginal tax rate facing the wife" label var WFED "Wife's father's educational attainment, in years" label var UN " Unemployment rate in county of residence, in percentage points" label var CIT " Dummy variable 1 if live in large city (SMSA), else 0" label var AX " Actual years of wife's previous labor market experience"

6 we use the command sum to get an overview of basic statistics Notable for the interpretation might be that the age of the women is between years. The Mean of LFP (0,568) is also the probability the WHRS is higher than 0. Using the command browse to take a look at the micro data makes clear that 325 of the 753 women in the dataset are reported as not working (on the labor market) in the year The dummy variable LTF for these women is 0 and the WHRS (Wife s hours of work in 1975) is put to 0 as well. We say that the wives working hours have been truncated at 0 because we only have information about the working hours of the subset of wives working. The argument of truncation arises from the reasoning that we do not know how many hours these wives would have worked if they were employed. The above reasoning holds true when we consider the variable of wives working hours in isolation. Yet, the argument changes when we look at a model that includes more information about the observed wives. While the model we are putting up here leaves in information about age, education, and children, also for the 325 unemployed wives, we should think of the missing values as censored, not as truncated. As 0 is the lower boundary we face a left-censoring. (iii) We perform a Tobit Estimation explaining WHRS as a linear function of KL6 (Number of children less than 6 years old in household), WA (Wife's age), WE (Wife's educational attainment, in years) and CIT (Living in a large city, dummy). With the command help tobit we get some useful information about the implementation of the Tobit model in STATA. The left-censored limit at the point 0 can be implemented with ll(0) at the end of the command. tobit WHRS KL6 WA WE CIT, ll(0)

7 Looking at the regression coefficients we see that the KL6 ( children under 6 years in household ) and WA ( Wife s age ) have a negative and highly significant impact on WHRS (wives working hours), whereas the WE ( wife s education attainment in years ) have a positive and equally significant impact on WHRS. The constant and the dummy variable CIT (1live in a large city) are not significantly different from 0 (The Std. Errors are very high and consequently broad are the 95% confidence intervals). (iv) For a comparison we perform two OLS regressions. The first on the whole data set (753 women), neglecting the censoring. The second is regressed on the subset of 428 women, from whom we observe a labor supply (where the dummy variable LTF is 1). a) The command is regress WHRS KL6 WA WE CIT Qualitatively we get similar results as in the Tobit regression. Here the constant is significant as well. The coefficients are lower than in the Tobit-model, suggesting a less steep regression line. The reason is that the slope coefficient is calculated as if all observations were actually observed at zero, not censored to the value of zero.

8 b) Next we perform the same linear regress on the subset of employed wives. regress WHRS KL6 WA WE CIT if LFP1 Only the KL6 and the constant are now significant. The impact of the Wife s Education is now negatively correlated with the years of working (see graphs 1 and 2 as illustrations). Graph 1 (all observations)

9 Graph 2 (only if wife is working)

Estimating the return to education for married women mroz.csv: 753 observations and 22 variables

Return to education Estimating the return to education for married women mroz.csv: 753 observations and 22 variables 1. inlf =1 if in labor force, 1975 2. hours hours worked, 1975 3. kidslt6 # kids < 6