Empirical Economics Mock Exam 2023-24 PDF

EMPIRICAL ECONOMICS Mock Exam Time allowed: 1 hour and 45 minutes. Each mul ple choice is evaluated 1.5 points for correct answers and 0 points for incorrect answers. Each open ques on is evaluated up to a maximum of 6 points. PART 1 Sec on A: Mu...

EMPIRICAL ECONOMICS Mock Exam Time allowed: 1 hour and 45 minutes. Each mul ple choice is evaluated 1.5 points for correct answers and 0 points for incorrect answers. Each open ques on is evaluated up to a maximum of 6 points. PART 1 Sec on A: Mul ple choice ques ons A1.1 An econometrician regresses the sales of personal computers (PC_sales) on price (P) as well as the amount of expenditure on advertising (ADV) and its square. She runs OLS and obtains the following fitted regression equation: 𝑃𝐶_𝑠𝑎𝑙𝑒𝑠 = 109.7 − 7.6𝑃 + 12.2𝐴𝐷𝑉 − 2.8𝐴𝐷𝑉. Which of the following statements is correct? a) The marginal effect of adver sing is 12.2 + 2(−2.8)ADV [CORRECT] b) The marginal effect of adver sing is 12.2 c) The marginal effect of adver sing is constant d) A single t-test can be used to test the hypothesis that adver sing does not affect sales A1.2 What is the consequence of measurement error in one of the explanatory variables in the regression? a) The OLS estimator for the coefficient of the explanatory variable measured with error is biased, but that for the remaining coefficients is unbiased and consistent b) The OLS estimator for the coefficient vector is biased [CORRECT] c) The OLS estimator for the coefficient vector is inefficient d) The OLS estimator for the coefficient of the explanatory variable measured with error is not significant Sec on B: Open ques ons B1.1 Consider the following equa on: 𝑦 = 𝛽 + 𝛽 𝑥 + 𝛽 𝑥 + 𝜀 a) Assume that the researcher does not have data for 𝑥 and decides to es mate the following model: 𝑦 = 𝛽 + 𝛽 𝑥 + 𝜈, where 𝜈 is the error in the es mated equa on. Show that this implies that 𝑥 is endogenous and that the OLS es mator is inconsistent. This is the case of omi ed variables. DGP: 𝑦 = 𝛽 + 𝛽 𝑥 + 𝛽 𝑥 + 𝜀, where 𝑢 𝑖𝑖𝑑 (0, 𝜎 ) and 𝑐𝑜𝑣(𝜀, 𝑥 ) = 𝑐𝑜𝑣(𝜀, 𝑥 ) = 0 Es mated model: 𝑦 = 𝛽 + 𝛽 𝑥 + 𝜈, where 𝜈 = 𝛽 𝑥 + 𝜀, 𝑐𝑜𝑣(𝜈, 𝑥 ) = 𝑐𝑜𝑣(𝛽 𝑥 + 𝜀, 𝑥 ) = 𝛽 𝑐𝑜𝑣(𝑥 , 𝑥 ) ≠ 0 It follows that: 𝑐𝑜𝑣(𝜀, 𝑥 ) 𝑐𝑜𝑣(𝑥 , 𝑥 ) 𝛽 ⎯⎯ 𝛽 + = 𝛽 +𝛽 → 𝑣𝑎𝑟(𝑥 ) 𝑣𝑎𝑟(𝑥 )  Where the last term is different from zero and represents the asympto c bias. b) Assume also that 𝛽 < 0 and that 𝑥 and 𝑥 are posi vely correlated: in which direc on will the coefficient of 𝑥 be biased? In this case 𝛽 𝑐𝑜𝑣(𝑥 , 𝑥 ) < 0, hence the coefficient will be biased downwards c) Assume that the researcher has data for a variable 𝑧, which she decides to use as an instrument for 𝑥. What are the condi ons that 𝑧 needs to sa sfy in order to be a valid instrument? Which of these condi ons can she test and how? 𝑧 must sa sfy the following condi ons: 1. It is “external”, i.e. it does not affect 𝑦 directly 2. 𝑐𝑜𝑣(𝜀, 𝑧) = 0  it is exogenous 3. 𝑐𝑜𝑣(𝑥 , 𝑧) ≠ 0  it is relevant Condi on n. 3 can be tested with a significance test for the coefficient of 𝑧 in the reduced form equa on for the endogenous variable. Condi on n. 2 cannot be tested here because of exact iden fica on (one endogenous variable & one instrument). B1.2 You have a sample of 807 US citizens that includes the following information on each individual:  cigs = number of cigarettes smoked  cigpric = price, in cents, of a pack of cigarettes in the individual's state of residence  income = annual income, in dollars  educ = number of years of education  age = age You use these data to estimate the following demand function for cigarettes (note: lincome=log(income), lcigpric=log(cigpric), agesq=age2): a) How do you read the coefficients of lcigprice and lincome? Are they significant? What is your interpretation of the results? They represent, approximately, the expected change in the number of cigarettes smoked subsequent to a one percentage change in the price of cigarettes and in income, respectively. E.g. when price increases by one precent, the number of cigarettes smoked decreases by 2.8, on  average. None of the two coefficients is significant. Results suggest that cigarettes are necessity goods, because they are not sensitive to changes in price and income. This is consistent with smoke being addictive. b) What is instead the marginal effect of age on cigarettes' consumption? Is it constant? Does it change sign for some age value? Given the descriptive statistics below, is it relevant in the present sample? The marginal effect of age on cigarettes' consumption is: 0.78 − 2 ∙ 0.009𝑎𝑔𝑒 = 0.78 − 0.018𝑎𝑔𝑒,. which clearly changes with age and is positive for 𝑎𝑔𝑒 <. = 43,33. This is relevant because it just above average age in the sample, i.e. we do expect the marginal effect of age to have a different sign depending on age. c) After the OLS regression you perform the Ramsey test, the result of which is shown below. Specify the null hypothesis, the test statistic and its distribution. How do you interpret the result? Would you change your specification? 𝐻 : 𝑚𝑜𝑑𝑒𝑙 𝑖𝑠 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑙𝑦 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑒𝑑 𝐻 : 𝑚𝑜𝑑𝑒𝑙 𝑚𝑖𝑠𝑠𝑒𝑠 𝑓𝑢𝑟𝑡ℎ𝑒𝑟 𝑟𝑒𝑙𝑒𝑣𝑎𝑛𝑡 𝑛𝑜𝑛 𝑙𝑖𝑛𝑒𝑎𝑟𝑖𝑡𝑖𝑒𝑠 𝑜𝑟 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠 The test is based on the following auxiliary regression: 𝑦 = 𝑥 𝛽 + 𝛼 𝑦 + 𝛼 𝑦 + ⋯+ 𝛼 𝑦 + 𝑣 It’s simply an F-test on Q-1 restrictions (α’s are 0). The test just rejects at the 5% level, which is borderline. Specification should be changed to incorporate further variables or non-linearities in the already included variables. PART 2 Sec on A: Mul ple choice ques ons A2.1 In the dynamic panel data model, two-period lags of the dependent variable are a valid instrument as long as a) The transitory component of the error is iid over both i and t [CORRECT] b) T is large c) The data are subject to the “within” transformation d) Regressors are strictly exogenous A2.2 With a binary dependent variable, the OLS estimator is a) Biased and efficient b) Biased and inefficient c) Unbiased and inefficient [CORRECT] d) BLUE Sec on B: Open ques ons B2.1 Consider the following variables:  y: =1 if individual works, =0 if individual does not work  age: individuals’ age in years  educa on: number of years spent at school  south: =1 if the individual lives in Southern Italy, = 0 if in the North  married: =1 if the individual is married, =0 otherwise Looking at the STATA output below, can you tell: a) which model is es mated and what kind of informa on can be gained from the first table of coefficients appearing a er the probit command; Probit. The table reports Maximum Likelihood coefficients that we can interpret in terms of size and significance, while their quan ta ve interpreta on is less clearcut as they capture effects measured in the metric of the underlying latent propensity b) how much does age affect the propensity to work and is this effect significant; a unit increase in age reduces the probability of working by 0.3 percentage points, sta s cally significant c) whether the marginal effects at the mean would be the same as the marginal effects reported below and why. ( ) It would not. The reported effect is the average marginal effects, that is ∑ , while the effects at ( ̅ ) the mean would be ̅ which are different because F() (the cdf of the error) is non-linear. B2.2 Assume that you want to model employees job sa sfac on as a func on of their personal a ributes, and your sa sfac on data are on the following scale: 0 (= unhappy with the job); 1 (= neither unhappy or happy with the job); 2 (= happy with the job). a. Discuss what are the problems related to the use of the linear regression model in this context. Linear models would yield wrong predicted probabili es, heteroskedas c errors and highly non-normal errors b. Write down an econometric model that avoids the issues presented by the linear model. Assuming that errors are normally distributed, derive the likelihood func on of the model. What parameters are not iden fied with respect to the linear model, and why? The model is the ordered response one described in slides 5 and 6 of the slide set “Mul -response models”. You are expected to derive that framework and the corresponding likelihood The variance is not iden fied because we do not observe the con nuous propensity variable but only its discrete manifesta on. One of the thresholds is not iden fied because data refer to u lity orderings, not u lity levels.  c. How can you test the hypothesis that unhappiness and indifference represent the same feeling towards the job? H0: 2=0 (note: 2 is the threshold between indifference and happiness, while 0 is the threshold between unhappiness and indifference)