Department of Economics Columbia University W3412 Fall 2015 Problem Set 3 Introduction to Econometrics Profs. Seyhan Erden and Miikka Rokkanen for all sections. Part I. True, False, Uncertain with Explanation: (a) \"Dummy" variables are variables added to the regression that have no explanatory power but serve only to increase the number of degrees of freedom. (b) tests and t tests on coefficients in a regression are equivalent in the sense that dropping all variables with small (insignificant) t statistics always results in the same final equation as performing the appropriate F tests. Part II. 1. Let R be the expected return on a risky investment and Rf be the return on a risk-free investment. The fundamental idea of modern finance is that an investor needs a financial incentive to take a risk. Hence, R must exceed Rf. According to the capital asset pricing model (CAPM) the expected excess return on an asset is proportional to the expected excess return on a portfolio of all available assets (the \"market portfolio\") That is, the CAPM says that R - Rf = (Rm - Rf) + u where Rm is the expected return on the market portfolio and is the coefficient in the population regression of R - Rf on Rm - Rf. In the following STATA output, variable freturn is the excess returns for two firms in computer chip industry and mreturn is the excess returns for the market. Linear regression Number of obs = 384 F( 1, 382) = 104.52 Prob > F = 0.0000 R-squared = 0.2175 Root MSE = .13447 -----------------------------------------------------------------------------| Robust freturn | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------mreturn | 1.608313 .1573139 10.22 0.000 1.299004 1.917623 _cons | .0031122 .0071605 0.43 0.664 -.0109666 .0171911 ------------------------------------------------------------------------------ (a) According to CAPM, the true intercept must be zero and the true slope must be one. Using hypothesis testing at 10% significance level test if CAPM is correct according to above results. (b) What is the meaning of F test in this regression? What is it testing? and how is that statistic related to t test statistic on the same output (c) In a given year, the rate of return on 3-month Treasury bills is 2.1% and the rate of return on a large diversified portfolio of stocks (the S&P 500) is 6.2%. For each company listed below, use the estimated value of to estimate the stock's expected rate of return. 1 Company Estimated Expected rate of return Kellogg (breakfast cereal) -0.03 Amazon (online retailer) 2.65 Barnes and Noble (book retailer) 1.02 2. Ordinary Least Squares(OLS) estimator is the most common estimator used in introductory econometrics courses. For a population regression given as = 0 + 1 + Let the OLS estimator for the sample intercept and the sample slope be denoted as 0and 1; (a) Explain in words (no equations here) what is the idea behind OLS estimators (b) Derive the formula for 0 by using the method you have described in part (a) 1 by using the method you have described in part (a) (c) Derive the formula for Part III. 1. Use Table 2 and GPA4.dta data file to answer the following questions. Table 2 presents the results of three regressions, one in each column. Estimate the indicated regressions and fill in the values (you may either handwrite or type the entries in, at you convenience; if you choose to type up the table, an electronic copy of Table 2 in .doc format is available on the course Web site). For example, to fill in column (1), estimate the regression with colGPA as the dependent variable and hsGPA and skipped as the independent variables, using the \"robust\" option, and fill in the estimated coefficients (a) Write the regression in column (1) in \"equation form,\" with the standard error below the respective regression coefficient. (b) Explain in words what the coefficient on hsGPA means in regression (1), holding skipped unchanged. (c) Using regression (1), test the hypothesis that the coefficient on skipped is zero, against the alternative that it is nonzero, at the 5% significance level. In everyday words (not statistical terms), what precisely is the hypothesis that you are testing? (d) Test the hypothesis that the coefficient on skipped is zero in regressions (1), (2), and (3) at 1% using the p value, does your answer change depending on what other variables are included in the regression? (e) Using regression (3), consider the coefficient on campus. Does the sign and magnitude make sense? Explain. (f) Using regression (3), consider the coefficient on bgfriend. Does the sign and magnitude make sense? Explain In regression (3), is the coefficient on campus statistically significant at the 1% significance level? Is the coefficient on bgfriend statistically significant at the 1% significance level? 2 Table 2 College GPA Results Dependent variable: colGPA Regressor (1) (2) (3) hsgpa ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) skipped PC __ bgfriend __ campus __ __ __ Intercept ( ) ( ) Regression summary statistics R 2 Regression RMSE n The remaining problems consider one of the big questions in economics - what are the determinants of economic growth? One theory holds that trade spurs growth: an economy that is open to foreign trade and to foreign investment can introduce new industries, so workers can get better-paying industrial jobs instead of low-paying agricultural jobs. In this problem set, you will quantify some basic relationships - correlations and bivariate linear regressions - between 3 growth and trade; we will return to richer specifications and relationships later in the course. The data set for this problem, growth.dta, is described at the end of this problem set. 2. Using STATA, compute the sample mean and standard deviation of growth and tradeshare.. 3. Estimate a regression of growth on tradeshare, using the \"robust\" option. (a) What is the coefficient on tradeshare? Explain in words what it means. Is the numerical value of your estimate large or small in an economic (real-world) sense? (b) Graph the data points and the estimated regression line. (c) Is the slope coefficient statistically significantly different from zero at the 5% significance level? Show how you reach this conclusion. (d) Report the 95% confidence interval for 1, the slope of the population regression line. (e) What is the R2 of this regression? What does this mean? (f) Compute the correlation coefficient between growth and tradeshare, and compare its square to the R2. How are the correlation coefficient and the R2 related? (g) What is the value of the root mean squared error of the regression? What does this mean? (h) Graph the data points and the estimated regression line in part (a), does the regression error appear to be homoskedastic or heteroskedastic? (i) Run the regression again without the \"robust\" option. Compare the results to what you obtained with the \"robust\" option. What is different? (j) You should see an outlier in the data set. Rerun the regression (with the \"robust\" option), dropping the outlier. Does dropping the outlier make a qualitative difference to your results? Explain. (k) What is the outlying observation? Considering the economics of the relation you are investigating, in your judgment should that outlier be omitted from the regression? (You might need to do a bit of research about that outlier to answer this question properly.) Comment on the answer to question 4: the response below includes the outlier, but it is acceptable to report results excluding it. 4. Construct a new variable, lorgdp60, which equals one if the country's GDP is in the bottom quartile of GDP for 1960 and equals zero otherwise. Estimate a regression of growth on lorgdp60, using the \"robust\" option. (a) What is the coefficient on lorgdp60? Explain in words what this means. Is the numerical value of your estimate large or small in an economic (real-world) sense? (b) Test the hypothesis that the mean growth rate from 1960-1995 is the same for countries with lorgdp60 = 1 as it is for countries with lorgdp60 = 0, against the alternative that they differ, at the 5% significance level. (c) Using the \"summarize\" command, compute the sample average of growth for countries with lorgdp60 = 1 and then again for countries with lorgdp60 = 0; from this compute the difference in mean GDP growth rates for the two groups and construct the differences-of-means t-statistic testing the hypothesis that the mean growth rates are the same. (d) Reestimate the regression of growth on lorgdp60, without the \"robust\" option. How does the t-statistic computed in (c) compare to the t-statistic on the slope coefficient in the regression of growth on lorgdp60 obtained with and without the \"robust\" option? Explain. 4 DATA DESCRIPTION, FILE: growth.dta The data are a cross-sectional sample of n = 65 non-Communist countries, excluding economies dominated by oil exports. In this problem set, you will focus on three of the variables in the data set: economic growth, the share of trade, and initial GDP per capita. Definitions of Selected Variables in growth.dta Variable growth tradeshare rgdp60 Definition Average annual percentage growth of real per capita (i.e. divided by the total population) Gross Domestic Product (GDP)* from 1960 to 1995. The average share of trade in the economy from 1960 to 1995, measured as the sum of exports plus imports, divided by GDP; that is, the average value of (X + M)/GDP from 1960 to 1995, where X = exports and M = imports (both X and M are positive). The value of real GDP* per capita in 1960, converted to 1960 US dollars *Annual GDP is the market value of all final goods and services produced in a country that year. Real GDP is GDP, expressed in constant (inflation-adjusted) dollars. STATA HINTS STATA command summarize rgdp60, detail summarize growth if lorgdp60==1 regress growth tradeshare, robust predict pgrowth twoway scatter growth tradeshare || lfit growth tradeshare correlate growth tradeshare What it does: computes detailed summary statistics of rgdp60 including percentiles Compute the sample mean of growth for countries for which lorgdp60 = 1. Estimate OLS regression of growth on tradeshare, with heteroskedasticity-robust standard errors Create the variable pgrowth containing the predicted values for the previous regression Graph growth and pgrowth (on the vertical axis) against tradeshare (on the horizontal axis), not connecting the points for growth but connecting the points for pgrowth. (This is STATA 7 syntax, which is grandfathered.) Compute the correlation between growth and tradeshare 5 Following questions will not be graded, they are for you to practice and will be discussed at the recitation: 5. SW Empirical Exercise 5.2 6. SW Empirical Exercise 5.3 7. SW Empirical Exercise 6.1 (a,b and d only) 8. SW Empirical Exercise 6.2 6