Final Exam, section 1

Transcription

1 San Francisco State University Michael Bar ECON 312 Fall 2015 Final Exam, section 1 Monday, December 14, 2015 Time: 1 hour, 30 minutes Name: Instructions: 1. This is closed book, closed notes exam. 2. No calculators of any kind are allowed. 3. Show all the steps and calculations. 4. If you need more space, use the back of the page. 5. Fully label all graphs. Good Luck 1

2 1. (50 points). Jose wants to study the factors a ecting economic growth across countries, with particular interest in the role of government sector. He collected data on 171 countries, with the following variables: Variable Description gdpg5 5 Year average GDP Growth Rate (in %) govshare Gov t Expenditure as % of GDP govshare2 govshare 2 (i.e., govshare squared) debt Public Debt (% of GDP) "NA" =1 if North America, 0 otherwise "AP" =1 if Asia and the Paci c, 0 otherwise region dummies "EU" =1 if Europe, 0 otherwise "ME" =1 if Middle East and North Africa, 0 otherwise "SCA" =1 if South and Central America and Caribbean, 0 otherwise "SSA" =1 if Sub-Saharan Africa, 0 otherwise pop Population (Millions) Jose s R command and regression output are presented below: * *Jose is using the country s population as weight on each observation, so that country like China has more weight in the sample than Singapore. (a) (5 points). What is the dependent variable in the above regression? gdpg5 2

3 (b) (5 points). What is the omitted (reference) category for region? North America. (c) (10 points). Interpret the estimated coe cients on govshare and govshare2. The impact of a 1% increase in government share on growth, holding all other regressors xed, = b 2 + 2b 3 govshare = 0:268 0: govshare This means that the predicted growth rate of a country is increasing with government share, up to certain level of govshare, and once government share is beyond certain threshold level, the growth rate decreases in govshare. We can nd the "optimal" size of government for economic growth, = b 2 + 2b 3 govshare = 0 govshare = b 2 2b 3 = 36:41581% The above analysis is performed while holding all other regressors, besides govshare, xed. (d) (5 points). Interpret the estimated coe cient on debt. b 4 = 0:0335 means that an increase in public debt of a country by 1% of its GDP, is predicted to slow the growth rate of country s GDP by 0.03%, holding all other regressors xed. Remark: this means that when compared to countries with no public debt, a country with debt of 100% of GDP (such as U.S.) is predicted to grow slower by 3.35% per year. 3

4 (e) (5 points). Interpret the estimated coe cient on EU. b 6 = 1:466 means that the predicted growth rate of a European country, is 1.466% lower than that of a North American country, holding all other regressors the same. (f) (5 points). Suppose that Jose wants to test whether Asian and Paci c countries grow faster than North American, holding all other characteristics xed. Write the null and alternative hypotheses for this test. H 0 : 5 = 0 H 1 : 5 > 0 (g) (5 points). Based on the regression output, what is your conclusion about the test in the last section, assuming signi cance level of = 0:05? Explain your answer. The reported t-statistic is positive, which is consistent with upper tail alternative. The reported p-value is 1:16e 12 is for the two-tailed test, which means that it is divided by 2 for upper tail test. Therefore, we reject the null hypothesis at signi cance level (1:16e 12) =2 or higher, and conclude that Asian and Paci c countries grow faster than North American. 4

5 (h) (5 points). Suppose that Jose wants to test if growth of GDP in Middle Eastern countries is di erent from that of North American. Write the null and alternative hypotheses for this test. H 0 : 7 = 0 H 1 : 7 6= 0 (i) (5 points). In order to conduct the above test, Jose calculated the 95% con dence intervals for all coe cients. Based on the above con dence intervals, what is your conclusion about the test in the last section? Explain your answer. The reported 95% con dence interval for 7 is [ 1:1484; 1:8893]. This interval contains all the null values of 7, which cannot be rejected at signi cance level of = 5%, based on the sample at hand. Since 0 is inside the CI, we fail to reject the null hypothesis, and conclude that the growth rate of GDP in Middle Eastern countries does not di er from that of North American. 5

6 2. (20 points). Oscar is a banker, who wants to study the factors a ecting the probability of mortgage delinquency 1. He collected data on 1000 mortgages from Las Vegas, Nevada, single family dwellings in year 2008, with the following variables: Variable Description delinquent dummy (= 1 if payment late by 90+ days, 0 otherwise) arm dummy (= 1 if adjustable rate mortgage, 0 if xed) ref dummy (= 1 if for a re nance, 0 if for purchase) insur dummy (= 1 if borrower has mortgage insurance, 0 otherwise) lvr leverage (ratio of loan amount to value of property, in %) rate initial interest rate, in % amount loan amount in $100,000 units credit credit score of the borrower term loan term in years (a) Oscar s rst attempt was to estimate the linear probability model. The summary of the predicted values is as follows: Based on the above summary statistics, explain brie y what is wrong with the above predictions. The linear probability model predicts negative probabilities of delinquency. In particular, the above summary statistics shows a minimum value of , which is not acceptable value for probability. (b) Oscar s next attempt was the probit model. The summary of the tted values is as follows: Based on the above summary statistics, explain brie y why Oscar prefers the probit model over the linear probability model. The predicted probabilities from the probit model are always in [0; 1]. In the above summary statistics, we see that the predicted probability of delinquency is between and A mortgage is considered delinquent if the borrower missed at least 3 monthly payments (90+ days late). 6

7 (c) Oscar s R commands and output are as follows: Interpret the estimated marginal e ect of arm. The predicted probability of delinquency on adjustable rate mortgage is 13.5% higher than on a xed rate mortgage, holding all other regressors at their sample mean values. (d) Interpret the estimated marginal e ect of amount. A $100,000 increase in the mortgage amount is predicted to increase the probability of delinquency by 2.5%, holding all other regressors at their sample mean values. 7

8 3. (10 points). The following gure shows the predicted probability that a woman works for wages, as a function of the number of small children (younger than 10 years old) in her house. (a) Based on the above gure, approximately, what is the predicted probability that a woman with 2 small children works for wages? Approximately 40%. (b) Based on the above gure, what is the predicted marginal e ect of additional small child, on the probability to work of a woman that has 3 children? Approximately -0.1 (or -10%), i.e. a decline in probability to work from 20% (with 3 children) to 10% (with 4 children). 8

9 4. (10 points). The following gure shows the time series of personal consumption and federal tax receipts for the U.S. (a) Based on the above gure, regression of consumption on taxes is likely to result in (circle the correct answer): i. perfect multicollinearity ii. heteroschedasticity iii. imperfect multicollinearity iv. spurious regression (b) Brie y describe one solution to avoid the problem in the previous section. i. Including time as a regressor. ii. Detrending (removing the trends from the variables) before using them in regression. iii. Normalizing expressing the variables in terms of ratios of the original variable to some other key variable. For example, both consumption and tax receipts can be expressed as a fraction of GDP. 9

10 5. (5 points). What are the consequences of omitting a relevant variable from a regression model? (a) The OLS estimators are unbiased, consistent and e cient (BLUE). (b) The OLS estimators are biased but consistent. (c) The OLS estimators are biased and inconsistent. (d) The OLS estimators are unbiased and consistent, but ine cient. (e) The OLS estimators are biased and inconsistent, but e cient. 6. (5 points). In general, what is the most important source of guidance for data collection and model speci cation in applied research project? Economic (or other) theory. 10