Mid-term exam Practice problems

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Alyson Elliott
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1 Mid-term exam Practice problems Most problems are short answer problems. You receive points for the answer and the explanation. Full points require both, unless otherwise specified. Explaining your answer is never a bad idea. A little practice on stuff from last semester in here; that stuff is always relevant, so I want you to make sure that knowledge hasn t left you. Plus, answering these questions should be good practice on how to answer test questions. Most questions rely on a mix of knowledge that you have obtained over the last two semesters. Question 16 (EKC question) is especially about issues we have not discussed this semester, but relies on knowledge you should have. If you want, you can skip this one and concentrate on other questions (but again, you should ultimately not find this to be a question that you cannot answer). 1 (5 points) True or false (no explanation necessary) In the multiple regression model, the goodness-of-fit measure R-squared always increases (or remains the same) when an additional explanatory variable is added. 2 (3 points) True or false (no explanation necessary) You estimate a regression model with three independent variables and get an R 2 = True or false: Such a high R 2 in a regression model with three independent variables indicates that each of the explanatory variables must be (individually) statistically significant. 3 (3 points) True or false (no explanation necessary) True or false: When using a probit model, the effect of a change in x on the probability that y = 1 depends on the value of x at which you evaluate the change. 1

2 4 (5 points) True or false (no explanation necessary) Estimated OLS regression coefficients (the β s) always have the same sign as the correlation between y and the respective x s. 5 (5 points) (this is a hard one; requires some thinking) Multiple choice circle all that apply (no explanation necessary) In an OLS model with a constant, endogeneity is characterized by: (a) E(u x) 0 (b) V ar(u x) 0. (c) V ar(u x) σ 2 (d) E(u x) = E(u) 6 (3 points) Multiple choice (no explanation necessary) In a linear probability model all of the following are true except: (a) the estimated coefficients can be interpreted directly as marginal effects (b) R 2 is a good measure of how well the model fits the data (c) the predicted probability can be negative (d) the errors are always heteroskedastic 7 (5 points) Multiple choice (no explanation necessary) In the probit model all of the following are lies except:... (a) β 0 cannot be negative since probabilities have to lie between 0 and 1 (b) β j tells you the effect of a unit increase in x j on the probability that y = 1 (c) β j does not have a simple interpretation (i.e. cannot be interpreted directly) (d) β 0 is the probability of observing y when all the x variables are 0 8 (5 points) provide an answer and an explanation You plan to run an instrumental variables regression (a two-stage least squares model). You run a first-stage regression in order to examine the relationship between your candidate instrument and the variable you suspect of being endogenous in the main regression. Which concern would you have if the first-stage regression results were as indicated: (a) the relationship between the candidate instrument and the endogenous regressor 2

3 seems weak the t-stat is low, the R 2 is low, and the F-stat is low. Consequently, you suspect that the instrument might not be valid. (b) the relationship between the candidate instrument and the endogenous regressor seems strong conditions opposite of those listed in (a). Consequently, you suspect that the instrument might not be valid. (c) the relationship between the candidate instrument and the endogenous regressor seems weak. Consequently, you suspect that the instrument might lead to imprecise estimates of the effect of interest in the second stage. 3

4 9 (6 points) Very short answer (don t get stuck with overly long explanations on this one) (Ballantine question) Suppose there are three variables, y, x 1, and x 2. The variation in y is represented by the circle made up of area A+B+C+D in the figure below; x 1 is represented by area B+C+E+F; x 2 is represented by area C+D+F+G. Three questions: (a) For the model y = β 0 + β 1 x 1 + β 2 x 2 + u, what area of the Ballantine figure represents u? (b) In the model x 1 = α 0 + α 1 x 2 + w, what area of the Ballantine figure represents w? (c) Finally, in the model y = γ 0 + γ 1 w + z, where w refers to the error term from the model in part (b), what area of the Ballantine figure represents z? Answer all three questions: (a)what area of the Ballantine figure represents u? (2 points) (b)what area of the Ballantine figure represents w? (2 points) (c)what area of the Ballantine figure represents z (2 points)? A B C D E F G 4

5 10 (5 points) Show your work or explain your answer Suppose you have data on y and x. You specify the following regression model y = β 0 + β 1 x + u and obtain the following results from OLS β 0 = 3 β1 = 4. You then add 10 to the y data. That is, you create y = y What will be the regression coefficients when you estimate the model That is, what will be α 0 and α 1? 11 (5 points) y = α 0 + α 1 x + w? There are a class of models sometimes referred to as polychotomous dependent variable models that can be used to analyze data when there are a number of possible discrete outcomes. Match the models listed below to the appropriate example. Model Ordered probit Multinomial logit Poisson model Use transportation choice number of bankruptcies Bond rating height 12 (10 points) Explain the difference between censored data and truncated data. Use the example of tickets to a sporting event (treat this as the y-variable) as an example of one of these two types of data. Use your own example of the other type of data. 13 (22 points) You are interested in estimating the impact of a household head s level of education on the food security status of his or her household. (Roughly stated, the issue of food security concerns whether a person or household can count on having access to a sufficient number of calories) You have data from a 2008 household survey run in Senegal. The survey is nationally representative and includes over 20,000 households and the following variables: 5

6 varname foodsecure educhd rural employhd members kids nmale nfem description dummy for households whose monthly income lies above the food poverty line (the line at which the household would achieve 2100 calories per person per day) years of education of household head dummy if household is situated in a rural area; zero otherwise dummy if household head is currently employed number of individuals living in the household number of children in the household number of men in the household number of women in the household (a) Write down a model that you would run using OLS. What is the marginal effect of educhd on the food security status of the household in this model? (3 points) (b) Specify an alternative model that you would run using maximum likelihood estimation (you need not write down a specific functional form based on a specific distribution; you may use the notion of a generic function G to represent the functional form you have named using words). How would you obtain the marginal effect of educhd on the food security status of the household in your new model? (3 points) (c) What are the pros and cons of model (a) vs. model (b)? Name all that you can. (3 points) (d) A colleague suggests that your estimate of the effect of household head s level of education suffers from omitted variable bias since you do not have a measure of household income. Describe the circumstances under which your colleague would be correct. Give an example and be specific about the nature of the bias that you would expect. (5 points) (e) Your colleague suggests using the education level of the household head s father as an instrumental variable for household head s level of education. Do you believe that this instrument is valid? Explain, being as specific as possible. (8 points) 14 (5 points) We estimate a regression of y on x: y = β 0 + β 1 x + u, and obtain estimates β 0 and β 1, and an R-squared coefficient of If you were to change the units of x by multiplying x by 10, what is the new R-squared? Explain why your answer makes intuitive sense. 6

7 15 (10 points) Suppose that you are attempting to build a model that explains aggregate savings behavior in the United States as a function of the level of interest rates. Would you rather construct your data sample during a period of fluctuating interest rates or a period in which interest rates were relatively constant? Why? 16 (15 points) The Environmental Kuznets Curve (EKC) is a theory relating income and productivity to environmental conditions. The theory (roughly) goes something like this. Environmental conditions degrade as income increases, because the act of producing lots of goods is taxing on the environment. At some point of development, however, the relationship changes. As people become richer, they start to develop preferences for a cleaner environment. (When you are poor, you d rather have food and shelter and smog than no food, no shelter, and no smog, but when you nail down the food and shelter thing, you prefer less smog to a fancier shelter). The (rough) shape of a theoretical EKC relationship is given in the figure below. Pollution GDP We gather data on sulfur-dioxide emissions and GDP in 100 countries and estimate an environmental Kuznet s curve model SO 2 = β 0 + β 1 GDP + β 2 GDP 2 + u, 7

8 where SO 2 represents sulfur-dioxide emissions, GDP is, well, GDP, and GDP 2 is GDPsquared. Write down the hypothesis test (or tests) that you would use to help you determine whether the data is consistent with the EKC theory. That is, choose the hypothesis test or tests that you feel would help you falsify the EKC theory, or, should you not reject the null, support it. 17 (15 points) We collect variables y (the dependent variable), x 1, and x 2. If we run a multiple regression of y on x 1, and x 2 y = β 0 + β 1 x 1 + β 2 x 2 + u we obtain estimated coefficients β 0, β 1, and β 2. The estimated slope coefficients β 1, and β 2 are interpreted as independent effects. That is, β1 is interpreted to be the effect of x 1 on y, independent of the effect of x 2 on y. And vice-versa. Another way to say the same thing is that β 1 is the effect of x 1 on y, controlling for the effect of x 2 on y. Use what is sometimes referred to as the partialling-out procedure to explain what, exactly, is meant by independent effect or control. That is, demonstrate what β 1 represents in the multiple-regression context by describing a series of bivariate regressions (regressions with one dependent variable and one independent variable). You should be able to arrive at the results of a bivariate regression in which one of the coefficient estimates is exactly equal to β 1 in the multiple regression above. Explain how this procedure highlights the independent effect interpretation of β1. [Hint: you don t need to remember the partialing-out procedure to complete this question. The β 1 coefficient in the multiple regression above represents the independent effect of x 1 on y, when the effect of x 2 on y has already been accounted for. Use this knowledge to reconstruct β 1 with a series of bivariate regressions involving y, x 1, and x 2.] 18 (15 points) Omitted variable bias is a term that refers to a situation when omitting an important independent variable (an x-variable ) from a regression equation causes us to mis-estimate the effect of another independent variable that is included in the regression. Explain why omitted variable bias is an example of endogeneity. (Endogeneity is when an independent variable is correlated with the unobservable, or error, term in a regression). Use the model below when constructing your explanation. y = β 0 + β 1 x 1 + β 2 x 2 + u Explain how omitting x 2 causes x 1 to be endogenous (i.e. correlated with the unobservable in a regression of y on x 1 ). (No algebra required! The answer can be given in 8

9 an intuitive discussion. If you find it easier to work with a real-variable example rather than generic x and y variables, that is fine) [Hint: If you don t know how to get started, work out how omitting x 2 would cause omitted variable bias in your estimation of β 1 (the relationship between y and x 1 ) under different scenarios describing the relationship between y, x 1, and x 2.] 9

Truncation and Censoring

Truncation and Censoring Laura Magazzini laura.magazzini@univr.it Laura Magazzini (@univr.it) Truncation and Censoring 1 / 35 Truncation and censoring Truncation: sample data are drawn from a subset of