Applied Econometrics Dummy Variables 1 Applied Econometrics Lecture 4: Use of Dummy Variables ‘Pure and complete sorrow is as impossible as pure and complete joy’ 1) Introduction The quantitative independent variables used in regression equations, which usually take values over some continuous range. Frequently, one may wish to include the quality independent variables, often called dummy variables, in the regression model in order to (i) capture the presence or absence of a ‘quality’, such as male or female, poor or rich, urban or rural areas, college degree or do not college degree, different stages of development, different period of time; (ii) to capture the interaction between them; and, (iii) or to take on one or more distinct values. 2) Intercept Dummy An intercept dummy is a variable, says D, has the value of either 0 or 1. It is normally used as a regressor in the model. For example, the consumption function (C) can be written as follows: C = b 0 + b 1 Y + b 2 D where Y is the gross national income D is equal to 1 for developing countries and 0 for developed countries Then, If D = 0, C = b 0 + b 1 Y If D = 1, C = b 0 + b 1 Y + b 2 D = (b 0 + b 2 )+ b 1 Y b 2 C = b 0 + b 1 Y C = (b 0 + b 2 )+ b 1 Y Y C Illustrative example 1 (Maddala, 308) We suppose that we regress the consumption (C) on income (Y) for household. We include the following quality variables in the form of dummy variables Written by Nguyen Hoang Bao May 22, 2004 Applied Econometrics Dummy Variables 2 ⎩ ⎨ ⎧ = femaleisgenderif0 maleisgenderif1 D 1 ⎩ ⎨ ⎧ < = otherwise0 25ageif1 D 2 ⎩ ⎨ ⎧ ≤≤ = otherwise0 50age25if1 D 3 ⎩ ⎨ ⎧ < = otherwise0 degreeschoolhigheducationif1 D 4 ⎩ ⎨ ⎧ <≤ = otherwise0 degreecollegeeducationdegreeschoolhighif1 D 5 Then we run the following regression equation C = α + βY + γ 1 D 1 + γ 2 D 2 + γ 3 D 3 + γ 4 D 4 + γ 5 D 5 The assumption made in the dummy variable method is that it is only the intercept that changes for each group but not the slope coefficient of Y. Illustrative example 2 (Maddala, 309) The dummy variable method is also used if one has to take care of seasonal factors. For example, if we have quarterly data on C and Y, we fit the regression equation C = α + βY + λ 1 D 1 + λ 2 D 2 + λ 3 D 3 where D 1 , D 2 , and D 3 are seasonal dummies defined by: ⎩ ⎨ ⎧ = othersfor0 quarterfirstthefor1 D 1 ⎩ ⎨ ⎧ = othersfor0 quartersecondthefor1 D 2 Written by Nguyen Hoang Bao May 22, 2004 Applied Econometrics Dummy Variables 3 ⎩ ⎨ ⎧ = othersfor0 quarterthirdthefor1 D 3 3) Slope Dummy The slope dummy is defined as an interactive variable. DY = D x Y D is equal to 1 for developing countries and 0 for developed countries Then, If D = 0, C = b 0 + b 1 Y If D = 1, C = b 0 + b 1 Y + b 2 D = b 0 +(b 1 + b 2 )Y C = b 0 + (b 1 + b 2 )Y C = b 0 + b 1 Y Y C 4) Combination of Slope and Intercept Dummies We may include both slope and intercept dummies in a regression model DY = D x Y D is equal to 1 for developing countries and 0 for developed countries The general model can be written as follows: Y = b 0 + b 1 Y + b 2 D + b 3 DY Then, If D = 0, C = b 0 + b 1 Y If D = 1, C = b 0 + b 1 Y + b 2 D = (b 0 +b 2 )+(b 1 + b 3 )Y b 2 C = (b 0 + b 2 ) +(b 1 + b 3 )Y C = b 0 + b 1 Y Y C 5) Piece – Linear Regression Model Most of the econometric models we have studied have been continuous, with small changes in one variable having a measurable effect on another variable. If we want to explain investment (I) as a function of interest rate (r), the two segments of the piecewise linear regression show in the below figure. Written by Nguyen Hoang Bao May 22, 2004 Applied Econometrics Dummy Variables 4 The general model can be written as follows: I = b 0 + b 1 r + b 2 (r – r * )D If r < r * , then D = 0: I = b 0 + b 1 r If r ≥ r * , then D = 1: I = b 0 – b 2 r * + (b 1 + b 2 )r where r * is obtained when we plot the dependent variable against the explanatory variables and observing if there seem to be a sharp change in the relation after a given value of r * . I r r * 6) Summary If a qualitative variable has m categories, we include (m – 1) dummy variables in the model. The coefficients attached to the dummy variables must always be interpreted in the relation to the base variable, that is, the group that gets the value zero. The use of dummy variables associated with two or more categorical variables allows us to study partial association and interaction effects in the context of multiple regression. Interactive dummies are obtained by multiplying dummies corresponding to the different categorical variables. This allows us to test formally whether interaction is present or not. References Bao, Nguyen Hoang (1995), ‘Applied Econometrics’, Lecture notes and Readings, Vietnam-Netherlands Project for MA Program in Economics of Development. Maddala, G.S. (1992), ‘Introduction to Econometrics’, Macmillan Publishing Company, New York. Mukherjee Chandan, Howard White and Marc Wuyts (1998), ‘Econometrics and Data Analysis for Developing Countries’ published by Routledge, London, UK. Wonnacott, Thomas H. and Ronald J. Wonnacott (1990). ‘Introductory Statistics’, Published by John Wiley and Sons, Inc., Printed in the United States of America. Written by Nguyen Hoang Bao May 22, 2004 Applied Econometrics Dummy Variables 5 Workshop 4: Use of Dummy Variables 1) To help firms determine which of their executive salaries might be out of line, a management consultant fitted the following multiple regression equation from data base of 270 executives under the age of 40: SAL = 43.3 + 1.23 EXP + 3.60 EDUC + 0.74 MALE (SE) (0.30) (1.20) (1.10) residual standard deviation s = 16.4 where SAL = the executive’s annual salary ($000) EDUC = number of years of post – secondary education EXP = number of years of experience MALE = dummy variable, coded 1 for male, 0 for female 1.1) From this regression, a firm can calculate the fitted salary of each of its executives. If the actual salary is much lower or higher, it can be reviewed to see whether it is appropriate. Fred Kopp, for example, is a 32 – year old vice president of a large restaurant chain. He has been with the firm since he obtained a 2 – year MBA at age 25, following a 4 – year degree in economics. He now earns $126,000 annually. 1.1.1) What is Fred’s fitted salary? 1.1.2) How many standard deviations is his actual salary away from his fitted salary? Would you therefore call his salary exceptional? 1.1.3) Closer inspection of Fred’s record showed that he had spent two years studying at Oxford as a Rhodes Scholar before obtaining his MBA. In light of this information, recalculate your answers to 5.1.1) and 5.1.2) 1.2) In addition to identifying unusual salaries in specific firms, the regression can be used to answer questions about the economy – wide structure of executive salaries in all firms. For example, 1.2.1) Is there evidence of sex discrimination? 1.2.2) Is it fair to say that each year’s education (beyond high school) increases the income of the average executive by $3,600 a year? Written by Nguyen Hoang Bao May 22, 2004 Applied Econometrics Dummy Variables 6 2) In an environment study of 1072 men, a multiple regression was calculated to show how lung function was related to several factors, including some hazardous occupations (Lefcoe and Wonnacott, 1974): AIRCAP = 4500 – 39 AGE – 9.0 SMOK – 350 CHEMW – 380 FARMW – 180 FIREW (SE) (1.8) (2.2) (46) (53) (54) where AIRCAP = air capacity (milliliters) that the worker can expire in one second AGE = age (years) SMOK = amount of current smoking (cigarettes per day) CHEMW = 1 if subject is a chemical worker, 0 if not FARMW = 1 if subject is a farm worker, 0 if not FIREW = 1 if subject is a firefighter, 0 if not A fourth occupation, physician, served as the reference group, and so did not need a dummy. Assuming these 1072 people were a random sample, 2.1) Calculate the 95% confidence interval for each coefficient Fill in the blanks, and choose the correct word in square brackets: 2.2) Other things being equal (things such as _____________), chemical workers on average have AIRCAP values that are _____________ milliliters [higher, lower] than physicians 2.3) Other things being equal, chemical workers on average have AIRCAP values that are _________ milliliters [higher, lower] than farm workers 2.4) Other things being equal, on average a man who is 1 year older has an AIRCAP value that is ___________ milliliters [higher, lower] 2.5) Other things being equal, on average a man who smokes one pack (20 cigarettes) a day has an AIRCAP value that is ____________ milliliters [higher, lower] 2.6) As far as AIRCAP is concerned, we estimate that smoking one package a day is roughly equivalent to aging ___________ years. But this estimate may be biased because of ________ Written by Nguyen Hoang Bao May 22, 2004 Applied Econometrics Dummy Variables 7 3) In an observation study to determine the effect of a drug on blood pressure it was noticed that the treated group (taking the drug) tended to weigh more than the control group. Thus, when treated group had higher blood pressure on average, was it because of the treatment or their weight? To untangle this knot, some regressions were computed, using the following variables: BP = blood pressure WEIGHT = weight D = 1 if taking the drug, 0 otherwise The data set is given by: D WEIGHT BP 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 180 150 210 140 160 160 150 200 160 190 240 200 180 190 220 81 75 83 74 72 80 78 80 74 85 102 95 86 100 90 3.1) How much higher on average would the blood pressure be: a) For someone of the same weight who is on the drug? b) For someone on the same treatment who is 10 lbs. heavier? 3.2) How would the simple regression coefficient compare to the multiple regression coefficient for weight? Why? Written by Nguyen Hoang Bao May 22, 2004 Applied Econometrics Dummy Variables 8 4) Use data file SRINA 4.1) Regress Ip on Ig 4.2) Repeat the regression using (i) an intercept dummy; (ii) a slope dummy; and, (iii) both slope and intercept dummies. Select the break point by looking at the scatter plot Ip against Ig 4.3) Draw scatter plot and fitted line on each regression 4.4) Comment on your results 5) Use data file LEACCESS 5.1) Regress LE on Y 5.2) Repeat the regression using (i) an intercept dummy; (ii) a slope dummy; and, (iii) both slope and intercept dummies. Use t test check whether they are significant or not. Select the break point by looking at the scatter plot LE against Y. 5.3) Draw scatter plot and fitted line on each regression 5.4) Comment on your results 6) Use data file AIDSAV 6.1) Regress S/Y on A/Y 6.2) Repeat the regression using dummy variable to take on the distinct value 6.3) Draw the scatter plot and fitted line on each regression 6.4) Comment on your results Written by Nguyen Hoang Bao May 22, 2004 Applied Econometrics Dummy Variables 9 7) Use data file TOT 7.1) Regress ln(TOT) on t 7.2) Repeat the regression using appropriate dummy 7.3) Draw the time graph of the TOT (not logged) and showing your two fitted line 7.4) Comment on your results 8) Use data file INDIA 8.1) Does your conclusion confirm that gender matter in terms of explaining earning differences? 8.2) Does your conclusion confirm that educational level in terms of explaining earning differences? 8.3) Regress ln(WI) on gender, education, and age using the appropriate dummy variables? Written by Nguyen Hoang Bao May 22, 2004 . Applied Econometrics Dummy Variables 1 Applied Econometrics Lecture 4: Use of Dummy Variables ‘Pure and complete sorrow is. United States of America. Written by Nguyen Hoang Bao May 22, 2004 Applied Econometrics Dummy Variables 5 Workshop 4: Use of Dummy Variables 1)