528 PBAF Modeling with Linear Regression
Assignment 3
The goal of this assignment is to think about appropriate ways of modeling relationships between our explanatory factors and the outcomes. We will go through the use and interpretation of dummy variables, interactions, and other nonlinear specifications. You can continue to work with the model you developed in the last assignment or pick a new one. Using this and other assignments to explore the data you will use in your report is a good idea. Assignment Requirements 1. Pick a model from Washington State Population Survey (or another data set you might havelike the one for your project!), which includes at least one dummy variable and one interaction term as explanatory variables. Keep in mind that you will have to discuss why this is an appropriate model of the relationship and outline your expectations for the coefficient values (positive or negative). 2. Create and run at least one dummy variable in a regression. 3. Create and run at least one interaction term in a regression. 4. Create and use a log or a quadratic term in a regression. 5. Perform and interpret the test of a hypothesis about a subgroup of coefficients. 6. Answer the discussion questions. In order to complete this exercise, you will have to create new variables, transform variables, or recode. If you do not, your regression results may be un- interpretable. Here are general directions for SPSS and Excel: In SPSS: To create a new variable that is a function of an old variable, click TRANSFORM>COMPUTE then type in the formula. Click OK. [Some on- line help at Evans:
http://www.evans.washington.edu/resources/Computing/SPSS/Basics.html#9 http://www.evans.washington.edu/resources/Computing/SPSS/Basics.html#10 ]
In Excel: In a column at the end of the dataset or in a column youve inserted, type the name of the new variable in the first row. Then type the formula for the new variable in the second row. Double click on the lower-right hand part of the box and the formula will fill all the way down the column. I. Dummy Variables Dummy variables are useful for modeling categorical differences or thresholds. In regression with an intercept term you must always have one less dummy variable than you do categories for each set of dummies, and thus there will be a reference category (also called the omitted category). In my model, amount paid in monthly rent (Q3P5) is a function of the persons income (PEARN01), the persons age (AGE), minutes required getting to school/work (Q8P4), whether the person lived in King County (KING), and married persons (MARRIED). Q3P5 = 0 + 1 (PEARN01) + 2 (AGE) + 3 (Q8P4) + 4(KING) + 5 (MARRIED) + Here I have created dummies for King County residents from the original categorical variable, REGION: one for having King County residence and zero for all other counties. The reference
group is people living outside King County. I also have a dummy indicating a married respondent (MARRIED)unmarried respondents are the reference group. The coefficients on the KING and MARRIED dummies will show the difference in monthly rent in each of these groups relative to those living outside of King County and unmarried respondents, respectively. The coefficient on KING shows how much higher or lower rent is for King County residents than for residents of other areas of the state. The coefficient on MARRIED will show how much higher or lower, on average, rent is for married householders than for unmarried people. Create at least one dummy variable and use it in a regression. II. Interaction Terms Interaction terms are used when you believe that the effect of one of your explanatory factors on the outcome depends on the level of another of your explanatory factors. The interactions can be between dummy variables and/or continuous variables. For my example, I believe that the effect of income on monthly rent may be different for King County residents and non-King County residents. I interacted the KING variable with PEARN01 variable (KINGINC) and included BOTH the new interaction and old variables in the regression. [What if I left out the old variables, what would I be assuming?] My new model is: Q3P5 = 0 + 1 (PEARN01) + 2 (AGE) + 3 (Q8P4)+ 4 (KING) + 5 (MARRIED) + 6(KINGINC)+ To create a new interaction variable in SPSS, use TRANSFORM>COMPUTE then type in the formula to multiply them or use the buttons create. to The formula for my interaction variable is KINGINC= KING* PEARN01. To create a new interaction variable in Excel, insert a column in your database and type the variable name in the first row. Press the = key and select the first variable included in the interaction, press * and then select the second variable in the interaction. Then press return (weve multiplied them). Double click on the lower right-hand corner to fill the formula down the column. I added this new variable to my regression. The coefficient on KING will give the effect of residency in King County on monthly rent paid; the coefficient on the interaction will give the difference in the effect of earnings for residents of King County and non-King County residents. The total effect of income on monthly rent for residents of different counties is 4 +6 . Create at least one interaction term and use it in a regression. III. Other nonlinear terms You may believe that one or more explanatory variables have a nonlinear relationship to the dependent variable [Note: these tricks dont work for dummy explanatory variables.]. Taking the natural log (LN) of an explanatory variable and/or the dependent variable is one way to
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model a nonlinear relationship. In my example, monthly rent (Q3P5) may not just have a linear relationship to the explanatory factors. One thing I tried is substituting the income variable (PEARN01) with logged income as an explanatory variable. This would be appropriate if monthly rent increased with income, but the effect leveled off at higher levels of income. [Note: logs will not work for 0 or negative values. If you have 0 values, add one to the variable prior to logging it.] In SPSS, use TRANSFORM> COMPUTE and choose LN as a function to create a logged variable. For example, my new variable is: LNINCOME=LN(PEARN01). In Excel, insert a column for your new variable, type the name in the first row, and type the cell formula [in this case =LN(CellAddress)]. Double click on the lower right-hand corner to fill the formula down the column. My new regression is: Q3P5 = 0 + 1 (LNINCOME) + 2 (AGE) + 3 (Q8P4) + 4 (KING) + 5 (MARRIED) + A quadratic model will also allow for an outcome that increases at a decreasing rate with increases in the explanatory variable. A quadratic specification includes both an X and X2 term in the regression. In my model, I also tried including income and a squared income variable in my model. Use the COMPUTE command to create a squared variable; my formula was INCOMESQ=(PEARN01)**2. Then add the squared term AND the original variable to the equation. My new equation is: Q3P5= 0 + ...