Problemsets for Applied Econometrics

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1 Department of Economics Problemsets for Applied Econometrics c Seminar of Statistics University of Fribourg Schwitzerland

2 Introduction Datasets All used datasets have been taken from the following book: Wooldridge J. M. (2006): Introductory Econometrics: A Modern Approach, 4th edition, Thomson South-Western. This book is the main reference manual for the exercises of the course Applied Econometrics. R Help The following manuals contain a wide range of R-functions related to the methods presented in the lectures and may be consulted for solving the problemsets: Kleiber C. and Zeileis A. (2008): Applied Econometrics with R (Use R!), Verlag Springer. Farnsworth G. V. (2008): Econometrics in R, Contributed documentation, Gross J. (2010): Grundlegende Statistik mit R, Vieweg und Teubner. Additional reading and help can be found in the library BP2 (FSES C50) as well as on the Moodle site of the course. Solving and Grading of the Problemsets The four exercise sessions give the students the opportunity to deepen the material covered in class and to become familiar with the statistics software R. Enrolled students will find the dates on Moodle. The organization is as follows: The first exercise session is devoted to an introduction to R (no homework). The following three exercise sessions consist of compulsory take-home problemsets (3 in total) containing R exercises related to the material covered in class. The take-home problemsets are compulsory: each student has to send her own solutions (including the R code) in due time to vincent.pochon@unifr.ch and lukas.seger@unifr.ch. The deadline for each problemset is communicated in every new exercise session. Important: If several students hand in exactly the same solution, their problemset will be graded as insufficient! A student is allowed to attend the exam if he or she has obtained at least 50 % of the points in 2 out of 3 problemsets. 1

3 Students having passed the 2 first problemsets do not need to hand in the 3rd one. Students obtaining 75% or more of the points in 2 problemsets are rewarded with 0.5 points on their final grade. The tutor communicates the grades in the exercise session following the hand-in deadline. Students not present at this session can request their grade by or ask a collegue to get it for them. The tutor solves parts of the problemset in class but does not distribute the corrected problemsets back to the students. Attendance to the tutorials is not compulsory. The solutions to the problemsets have to be written in English. Please respect the following points: (i) One single file (file format free); (ii) Single solution file contains 2 parts: A. Answers (including graphs, tables, etc.); B. R-code; (iii) Put your name in the file s name, e.g. "PS1_Vincent Pochon". Good luck! Exam The written exam will take place on the 15th of January Important: All material (theory, examples, etc.) covered in the tutorials and lectures except for the R-codes is relevant for the exam. 2

4 Problemset 1 Data Inspection, MLR and Hypothesis Testing Dataset: WAGE2.csv Hand-in Deadline: 10 November 2013, 24:00h The whole problemset is based on the following model 1 relating the exogenous variable weekly wage to different exogenous variables: WAGE =β 0 + β 1 IQ + β 2 EDUC + β 3 EXPER + β 4 EXPER 2 + β 5 TENURE + β 6 AGE + β 7 MARRIED + β 8 BLACK + β 9 SOUTH + β 10 BLACK SOUTH + β 11 URBAN + u. (1) 1. Data Inspection (a) How many variables and observations does the dataset WAGE2.csv contain? Which variables are dummy variables? (b) Represent the endogenous variable in a well-labelled histogram and describe its distribution. (c) Determine and interpret the average and the median of the variables IQ and MARRIED. (d) Plot all exogenous variables of model 1 against the endogenous variable (in one figure). Interpret the plots (EDUC, WAGE) and (MARRIED, WAGE). 2. Interpretation of the Regression Output (a) Perform the regression and interpret both the coefficient of determination R 2 and the F-statistic. (b) Interpret the estimates for β 0, β 1, β 3, β 4, β 5 and β 10. (c) How much more wage does a worker with 15 years education earn in comparison to a worker with 10 years education? (d) With how many years of experience is the weekly wage maximal? Is this value plausible? 3. Hypothesis Testing (at the α = 0.05 significance level) (a) Which exogenous variables have a statistically significant impact on the weekly wage? (Hint: use the p-values). (b) Test the null hypothesis: β 1 = 0. Interpret your result. (c) Test the null hypothesis: β 3 = 25. Interpret your result. (d) Test the null hypothesis: β 1 = β 4 = β 5 = β 11 = 0. Interpret your result. (e) Do urban workers earn significantly more than non-urban workers? Test this hypothesis with a Chow-test. (Hint: For this test you need to estimate model 1 without the variable URBAN). 3

5 Problemset 2 Model Specification, IV, Proxy- and Binary Dependent Variables Hand-in Deadline: 8 December 2013, 24:00h 1. Model Specification Data set: HPRICE1.csv We are interested in determining to what extent the price of a given house depends on its characteristics. The data set contains followings variables measured for 88 observations (houses): PRICE BDRMS LOTSIZE SQRFT Sales price of a house in 1000 USD Number of bedrooms Size of the property in squared feet Habitable surface in squared feet We want to estimate the relationship between a house s sales price (PRICE), the size of its property (LOTSIZE), its living space (SQRFT) and the number of bedrooms (BDRMS) with the following linear multiple regression models: PRICE = β 0 + β 1 LOTSIZE + β 2 SQRFT + β 3 BDRMS + u (2) log(price) = β 0 + β 1 log(lotsize) + β 2 log(sqrft) + β 3 BDRMS + u (3) (a) Estimate the models (2) and (3) with linear squares and interpret for both models the estimated coefficients β 0, β 1 and β 2. Which exogenous variables have a significant impact on the sales price at the α = 5% confidence level? (b) Which model do you prefer with respect to its goodness of fit, i.e. the share of the explained variance of PRICE? (c) Perform the Regression Equation Specification Error Test (RESET test) for both models (α = 0.05). Which model do you prefer based on this test? Compare your decision with your answer in b) and explain your results. (Hint: for the RESET test you need to install and load the package lmtest. The commands are resettest(regression, 2:3) and qf(1-α, df1, df2) ) 4

6 2. Proxy Variables, Instrumental Variables and Hausman Test Data set: WAGE2.1.csv We are interested in model 4 where ABIL describes a person s unobserved and intrinsic ability. WAGE =β 0 + β 1 EDUC + β 2 ABIL + u. (4) (a) Since ABIL cannot be observed, we decide to use IQ as proxy variable (model 5). Use the plug-in technique to estimate model 4 with the linear squares method. ABIL =δ 0 + δ 1 IQ + v. (5) i. Interpret the estimated coefficients, assuming that u is uncorrelated with IQ and that v is uncorrelated with all exogenous variables. ii. Would your interpretation change if one of the above-mentioned assumptions was not fulfilled? Explain. (b) Assume now that there is no appropriate proxy variable for ABIL. In this case, we have no other choice than to put ABIL in the error term. This leads us to model 6. However, since EDUC and ABIL are assumed to be correlated, a simple OLS estimation of β 1 would yield a biased and inconsistent estimator ˆβ 1. A solution to this problem is to use an instrumental variable (IV) for x = EDUC. We choose as instrument the number of siblings, z = SIBS. WAGE =β 0 + β 1 EDUC + w. (6) i. Estimate model 6 with the linear squares method. ii. Before using SIBS as an IV we must first check if SIBS satisfies the condition cov(x, z) 0. Is this condition fulfilled? (Hint: Do a simple regression and look at the significance of the coefficients) iii. Assuming cov(y, z) = 0, we can now estimate β IV. First calculate β IV by means of the covariances: β IV = cov(z, y)/cov(z, x). (Hint: The R command is cov() ) iv. Estimate β IV with the appropriate R function: ivreg(regression IV). Is the result different from (i)? (Hint: The function ivreg is in the package AER ) v. Interpret the estimator β OLS and β IV for β 1. (c) TSLS estimates are consistent but yield higher standard errors than OLS estimates. It is thus useful to check if there is really endogeneity in the model before to resort to TSLS. For this reason, we now want to check with a Hausman test if the variable EDUC is indeed endogenous. The Hausman test has the null hypothesis of the OLS estimates being consistent (i.e. β IV = β OLS ) against the alternative hypothesis of only β IV being consistent. Perform the Hausman test (α = 5%) and interpret your result. (Here s one way to do it in R: (i) install the package systemfit ; (ii) sm_ols <- systemfit(regression, data = Data, method = OLS ); (iii) sm_iv <- systemfit(regression, data = Data, method = 2SLS, inst = Instrument); (iv) hausman.systemfit(sm_iv, sm_ols)) 5

7 3. Binary Dependent Variables Data set: CHARITY.csv For their fundraising, charity organizations often contact their potential donators via direct mails. Direct mails are only efficient if they generate donations, so a charity organization is willing to address its direct mail to persons with a high donation probability. To analyze what this donation probability depends on, the present analysis focusses on a data set of observations (i.e. persons on the organization s address list) of a Dutch charity organization containing the following variables: RESPOND GIFT RESPLAST WEEKSLAST PROPRESP MAILSYEAR GIFTLAST AVGGIFT Dummy variable, equal to 1 if the person has responded with a donation to the last direct mail, 0 otherwise Amount of Dutch gulden donated in the last response Dummy, equal to 1 if the contacted person had responded (with a donation) to the previous (i.e. the second last) mail Number of weeks since the person s last reply Proportion of responds of the person to the direct mails Number of mails to the person per year Amount of the previous donation in gulden Average donation We want to analyze how a person s decision to donate (RESPOND=1) depends on the previous response (RESPLAST), the number of weeks since the last mail (WEEKSLAST), the person s response share (PROPRESP), the number of mails (MAILSYEAR) as well as the average amount of the donations (AVGGIFT). (a) Consider first the following linear probability model (LPM): RESPOND = β 0 + β 1 RESPLAST + β 2 WEEKSLAST + β 3 PROPRESP (7) + β 4 MAILSYEAR + β 5 AVGGIFT + u Estimate the LPM (7) and interpret the estimated coefficients of β 1, β 2 and β 3. (b) Calculate for all observations a prediction for the probability that a person responds positively to the direct mail (RESPOND=1) and check if all predictions make sense. (Hint: You can use the R commands P1 <- predict(regression) and summary(p1) ) (c) The endogenous variable RESPOND depends on the same variables as above, but their effect is now measured with the following simple probit model: P (RESPOND = 1 x) =Φ (β 0 + β 1 RESPLAST + β 2 WEEKSLAST (8) +β 3 PROPRESP + β 4 MAILSYEAR + β 5 AVGGIFT) Estimate the probit model (8) and interpret the estimated coefficients of β 1, β 2 and β 3. To what extent does the interpretation differ form the LPM in a)? (Hint: The R commands are glm(regression, family=binomial(link= probit ), data=data3) ) (d) Compare the effect of a 1-unit increase in MAILSYEAR on the response probability for models 7 and 8. Interpret your result. (Hint for model 8: The R command is M1 <- mean(dnorm(predict(regression))) ; M1*probit coeffcient (without dnorm() ).) 6

8 (e) Compare the LPM and the probit model with respect to their goodness of fit. To do this, generate the percentage of correct predicted RESPOND values of the LPM and probit model for the observations in the sample. Which model do you prefer for the prediction of the RESPOND values, according to their goodness of fit? Are there differences in the quality of the predicted RESPOND=1 values comparing to the predicted RESPOND=0 values? The following table summarizes the correct and incorrect predicted responses of both models. LPM Probit Forecast Forecast Response

9 Problemset 3 Heckman Selection, Heteroskedasticity and Time Series Analysis Hand-in Deadline: 22 December 2013, 24:00h 1. Heckman Sample Selection Method Data set: MROZ.csv The Heckman selection model (or heckit model) is a method for estimating regression models suffering from sample selection bias, i.e. when the dependent variable is only observable for a portion of the data. 1 The following exercise uses the dataset MROZ, containing the wage and the number of worked hours for 753 women in the U.S. for the year In this sample, we only observe the wage offer for 428 women (with a labor force participation of LFP= 1). Suppose that we want to investigate the impact of some factors on the female wage offer. We face a truncated sample, because the only data we can get are for the women who are actually working, while the other women in the population stay out of the job market, typically because the market wage is lower to their reservation wage. To account for the truncation, we make use of an additional model capturing the LFP of women. Consider the following wage model 9, where CITY is a dummy variable indicating that the woman lives in a large urban area: WAGE = β 0 + β 1 EXPER + β 2 EXPER 2 + β 3 EDUC + β 4 CITY + ε (9) as well as the following probit selection equation : LFP = γ 0 + γ 1 AGE + γ 2 AGE 2 + γ 3 FAMINC + γ 4 EDUC + γ 5 KIDS + u (10) where LFP is a binary variable taking a value of 1 if the woman is in the labor force, and 0 otherwise, AGE is her age, FAMINC is the level of household income not earned by the woman, and KIDS is a dummy variable for whether she has children or not. (a) Load the data file MROZ.csv either from Moodle or directly from the Internet via the R command read.csv(" header=t). In this data set, the WAGE series is wage, EXPER is exper, EDUC is educ, CITY is city, LFP is inlf, AGE is age, and FAMINC is faminc. There is no kids dummy variable, but 1 See: Heckman, J. J. (1976), The Common Structure of Statistical Models of Truncation, Sample Selection, and Limited Dependent Variables and a Simple Estimator for Such Models, Annals of Economic and Social Measurement, volume 5, pp Source: Mroz, T. A. (1987), The sensitivity of an empirical model of married women s hours of work to economic and statistical assumptions, Econometrica. volume 55, pp

10 there are two variables containing the number of children younger than 6 ( kidslt6 ), and the number of children between 6 and 18 ( kidsge6 ). After loading the data, you need to create a new (dummy) variable for KIDS. (Hint: in R, simply use MROZ$kids <-(MROZ$kidslt6+MROZ$kidsge6)>0 ) (b) Estimate the wage equation of model 9 with OLS and report your estimate on EDUC and its standard errors. What is the potential problem of using OLS? (c) Estimate now the return to education in 9 using Heckman s two-step method and the selection equation 10. Compare the estimated coefficient of education and its standard error to that obtained with OLS. (R-hint for heckit: install the package sampleselection and use the commands REGheckit <- heckit(probit selection equation, outcome equation, data = MROZ, method = 2step ) and summary(regheckit) ) (d) Is there evidence of a sample selection problem in estimating the wage offer equation? Interpret your result. Does it corroborate your findings in b) and c) with respect to the estimated standard errors for EDUC? (Hint: perform a simple t-test on the coefficient of the inverse Mills ratio) (e) Do you see a problem in the choice of variables for the selection equation 10? How could it be improved? 2. Heteroskedasticity Data set: HPRICE1.csv Consider models 2 and 3 from problemset 2 describing the relationship between the sales price of a house and its characteristics. We have concerns that the data contains heteroskedasticity and want to test and control for it. Recall that heteroskedasticity is present if the variance of the residuals is not constant over all observations (or time periods). As a consequence, the OLS estimator is inconsistent (but still unbiased). (a) Briefly explain White s test for heteroskedasticity. What is its null hypothesis? (b) Perform the White test for models 2 and 3 and interpret your result. Are there differences between the linear and the log-linear models? (R-hint: the function for the White test is bptest(extended regression), where you add the squared regressors and all interaction terms to the regression.) (c) Determine the White s heteroskedasticity-robust standard errors for model 2 and compare them to the standard errors obtained with standard OLS. Are there differences in the significance (with α = 10%) of the different estimated coefficients before and after controlling for heteroskedasticity? (Hint: install the package sandwich and simply use the command coeftest(regression, vcov = sandwich) ) 9

11 3. Time Series Analysis Data set: CPI2010.csv The National Consumer Price Index (English: CPI; German: LIK; French: IPC) measures the price evolution of key consumer goods and services for Swiss private households. It indicates to what extent Swiss consumer must increase or decrease their expenses to keep constant their consumption volume. The data set contains the monthly Swiss CPI with base period 2010 (index value = 100) from January 1983 to October 2011 and the corresponding inflation values (i.e. the percentage increase of the CPI). (a) Stationarity i. Represent the time series CPI and inflation in separate and well-labelled plots. (Hint: after loading the data, you first need to define the variables as time series with the following command: name <- ts(column, start=1983, freq=12) ) ii. Looking at the plots, would you say that the variables CPI and inflation follow stationary processes, and why? (b) Model selection We want to find out which AR(p) model describes best the time series inflation (INFL). i. Estimate the empirical autocorrelation and partial autocorrelation functions for inflation. Do they unequivocally give the order p of the AR(p) process? (R-hint: the functions are, respectively, acf(time series) and pacf() ) ii. The R function ar(time series) uses the AIC to estimate the optimal model. Of which order is this model? iii. Other criteria can be used for model selection. Explain (briefly and) only theoretically how to choose between different models according to the following criteria? A. AIC/BIC B. Significance of the estimated coefficients C. Unit roots (c) Prediction i. Predict the Swiss inflation for November 2011 and December 2011 using the AR model of 2.b.ii). (R-hint: use the function predict(ar(time series), n.ahead=2) ) ii. Estimate the level of the CPI in December 2011 using your just calculated predictions for inflation. 10

Problem Set # 1. Master in Business and Quantitative Methods

Problem Set # 1. Master in Business and Quantitative Methods Problem Set # 1 Master in Business and Quantitative Methods Contents 0.1 Problems on endogeneity of the regressors........... 2 0.2 Lab exercises on endogeneity of the regressors......... 4 1 0.1 Problems