Statistics and Data Analysis

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Statistics and Data Analysis Professor William Greene Phone: 212.998.0876 Office: KMC 7-90 Home page: http://people.stern.nyu.edu/wgreene Email: wgreene@stern.nyu.edu Course web page: http://people.

Cupcake Wars: This Time It s Personal When last we met the cupcake warriors, Stern consultant Peter had determined that Allison s new cupcake store had significantly damaged Betsy s business, so much

1 Statistics and Data Analysis Professor William Greene Phone: Office: KMC 7-90 Home page: Course web page: Assignment 5 Multiple Regression Model Part I. Cupcake Wars: This Time It s Personal When last we met the cupcake warriors, Stern consultant Peter had determined that Allison s new cupcake store had significantly damaged Betsy s business, so much so that she fired one of her bakers (Julie). The economy being what it is, jobs for bakers are hard to find, and Julie took it personally. She wants her job back. A year has gone by and Julie s consultants, Stanley and Larry (from Stern, Inc.) have gathered a year s data on some variables they think will help resolve the question (for good?). Their data file is waiting for you to analyze at There are 365 days of data in the file, consting of IOMS Department BETSY = Betsy s sales of cupcakes ALLISON = Allison s sales ECONOMY = An index of how good the economy is in the Gorgetown area WARM = Dummy variable, 1 if day is in a warm month, 0 if not DAY = Day of week, coded 1,2,3,4,5,6,7 for the 7 days WEEKEND = Dummy variable, 1 if day is a Saturday or a Sunday, 0 if not. Use multiple regression and your knowledge of statistics to answer Julie s question did Allison pull a significant amount of business away from Betsy? 1

2 Part II. Does Gross Domestic Product Buy Happiness? Several years ago, Forbes Magazine compiled a study of the world s happiest (and least happy) countries. I have collected for you the variables for the 155 countries listed, Percent Thriving which is their happiness measure (higher is happier) and the Rank by % thriving (lower is better, 1=Denmark is best). In fact, we have some other interesting data on 146 of these countries in our own World Health Organization (WHO) data that we have used at several points in class. We can (perhaps) use these data to learn something about Gross National Happiness. Are there country characteristics that help to explain either of these variables? Our data base contains COUNTRY = country name COMP = health success index HLTHEXP = per capita health expenditure EDUC = average years of education DALE = life expectancy GINI = index of income distribution higher is worse (less equal) TROPICS = 1 if country is near the equator, 0 if not POPDEN = population density in people per square kilometer PUBTHE = percent of public health bill paid by government GDPC = per capita GDP GEFF = world bank goernment effectiveness index (higher is better) VOICE = democratization index (higher is more) OECD = 1 for OECD countries, 0 for others HAPPY = Forbes happiness index HAPPYRNK = Forbes happiness rank Using multiple regression modeling, explore these data to learn whether it appears that there are variables that help to explain how happy the people in a country are, or what their happiness rank is. 2

3 Part III. Multiple Regression Model: Theory In a carefully worded paragraph, explain how the multiple linear regression model explains the relationships between a dependent variable and several independent variables. As part of your explanation, describe how the coefficients in the model are interpreted. Also as part of your explanation, describe how to interpret the R 2 in a multiple regression equation. NOTE: This question will appear, exactly as above, as a 10 point question on the final exam for this course. Part IV. The Multiple Regression Model of Income and Education (The data for this exercise are in the project file GermanHealth.mpj.) This exercise is based on a subset of the German Health care data that we have discussed at a few points in class. A linear regression of Income on Age and Education produces the following results Regression Analysis: INCOME versus AGE, EDUC The regression equation is INCOME = AGE EDUC Predictor Coef SE Coef T P Constant AGE EDUC S = R-Sq = 12.0% R-Sq(adj) = 11.8% Analysis of Variance Source DF SS MS F P Regression Residual Error Total a. Form a confidence interval for the coefficient on education in the model. b. Test the hypothesis that neither AGE nor EDUCATION have an influence on income. That is, test the hypothesis that both coefficients are zero. c. Test the hypothesis that AGE and INCOME are unrelated. d. How many observations are there in the sample? e. What is the estimate of the standard error of the regression? f. What is the sample standard deviation of the variable INCOME? g. What is the R 2 in the model? h. Does this set of results give a convincing description of a relationship between INCOME and (AGE,EDUC) i. How do you interpret the coefficient on EDUC? j. Since one must grow a year older in order to obtain an additional year of education, how are we able to compute coefficients for both education and for age? (Hint: This is what multiple regression is all about. If you understand the answer to this question, you understand multiple regression.) 3

4 Part V. Interpreting A Model for Self Assessed Health Using the same data as in Part IV, we now examine a different equation. Using Minitab, obtain the regression results for the model HEALTH = α + β 1 INCOME + β 2 MARRIED + β 3 FEMALE + β 4 KIDS + β 5 WORKING + ε The variables are HEALTH = survey question: On a scale of 0 to 10, how do you feel about your health? INCOME = household income MARRIED = 0 if the person is not married, 1 if they are FEMALE = 0 if the person is not female, 1 if they are KIDS = 0 if there are no children in the household, 1 if there are WORKING = 0 if the person is not working, 1 if they are. a. Report the regression results. b. How good is the fit of the regression? (I.e., comment on R 2 ) c. Are the coefficient estimates significant? Is there a contradition between b and c? d. Notice that 4 of the variables in the model take only the values 0 and 1. These variables shift the regression line up or down. For example, the relationship between HEALTH and INCOME for people who are (MARRIED,MALE,noKIDS, not WORKING) is HEALTH = α + β 1 INCOME + β 2 (1) + β 3 (0) + β 4 (0) + β 5 (0) + ε = (α+ β 2 ) + β 1 INCOME + ε. Draw a sketch (or use some other drawing tools if you like) of the relationship between HEALTH and INCOME for the following groups of people: Draw all three relationships on the same graph. Also, write out the corresponding equation. (1) MARRIED, WOMEN, nokids, WORKING (2) notmarried, MALE, KIDS, notworking (3) MARRIED, MALE, nokids, WORKING e. In part b., you should have found an extremely low value of R 2 for the regression. Now, locate the F statistic for the model you have fit. What is the value? What is the P value associated with F? Does this seem like a contradiction? How would you explain it? 4

5 Part VI. Multiple Regression Model for Electronic Store Sales (The data are given in UKElectronics.mpj) a. In class, we examined the relationship between camera sales and the two inputs FLOOR and STAFF. To continue that analysis, use multiple regression to fit the model Videos = α + β 1 Floor + β 2 Staff + ε Report all the relevant regression results, then determine whether Floor or Staff appears to be more important in explaining variation in sales of Videos. b. Since there is a units of measurement problem in comparing the coefficients on Floor and Staff in the model, we can use logs instead, and estimate coefficients which are elasticities rather than rates of change. Reestimate the model in part a, but use logs for all three variables. Once again, report all the regression results. c. Something peculiar has emerged in part b. How do you interpret the negative coefficient on logfloor? To investigate this a bit further, compute the coefficients of a model that contains only logfloor that is, refit the model, but now leave logstaff out of the eqauation. What happened? Can you explain this? Does it make sense? d. Suppose we try to build a more complete model. The store sells three things. Let s aggregate them: Suppose the average prices are 200 for cameras, 300 for videos and 20 for warranties. Then Sales = 200 Cameras Videos + 20 Warranties. Compute total sales. Now, estimate a new model, logsales = α + β 1 logfloor + β 2 logstaff + ε Report your regression results. What is the R 2 for the model? Based on the statistics reported and on your intuition, do you suspect that this is a reasonable model for total sales of a UK electronics store? 5

Part VII. Dealing with Categories. (The data are given in Macs&Movies.mpj (More Movie Madness data)) The movie data used in the Douglas/Craig/Greene study were discussed in class.

6 Part VII. Dealing with Categories. (The data are given in Macs&Movies.mpj (More Movie Madness data)) The movie data used in the Douglas/Craig/Greene study were discussed in class. A regression model that is based on these data appears in the figure below (which appears in your class notes). a. Using these movie data, fit a simpler model: logcountrybox = α + β 1 logusbox + β 2 logpercapitaincome + β 3 logpcmacs + β 4 Argentina +β 5 Chile + β 6 Spain + β 7 Mexico + β 8 Germany + ε Report the regression results. b. Does the regression appear to be significant? What statistics help you reach your conclusion? c. How do you interpret the country variables in your model? d. Now, interpret the results given in the display of the full model below. 6

Econometric Analysis of Panel Data. Final Examination: Spring 2018

Department of Economics Econometric Analysis of Panel Data Professor William Greene Phone: 212.998.0876 Office: KMC 7-90 Home page: people.stern.nyu.edu/wgreene Email: wgreene@stern.nyu.edu URL for course