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1) There is a statistically significant negative relationship between per pupil education expenditures and average SAT scores in a state. For every $1,000 increase in expenditures, SAT scores decrease by 2.2 points on average. 2) Around 22% of the variation in SAT scores is explained by differences in per pupil expenditures. 3) The standard error of the estimate from the regression model is around 60 points on the SAT scale.

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Regression in Stata Alicia Doyle Lynch Harvard-MIT Data Center (HMDC)
Documents for Today • Find class materials at: http://libraries.mit.edu/guides/subjects/data/ training/workshops.html – Several formats of data – Presentation slides – Handouts – Exercises • Let’s go over how to save these files together 2
Organization • Please feel free to ask questions at any point if they are relevant to the current topic (or if you are lost!) • There will be a Q&A after class for more specific, personalized questions • Collaboration with your neighbors is encouraged • If you are using a laptop, you will need to adjust paths accordingly
Organization • Make comments in your Do-file rather than on hand-outs – Save on flash drive or email to yourself • Stata commands will always appear in red • “Var” simply refers to “variable” (e.g., var1, var2, var3, varname) • Pathnames should be replaced with the path specific to your computer and folders
Assumptions (and Disclaimers) • This is Regression in Stata • Assumes basic knowledge of Stata • Assumes knowledge of regression • Not appropriate for people not familiar with Stata • Not appropriate for people already well- familiar with regression in Stata
Opening Stata • In your Athena terminal (the large purple screen with blinking cursor) type add stata xstata • Stata should come up on your screen • Always open Stata FIRST and THEN open Do- Files (we’ll talk about these in a minute), data files, etc. 6 HMDC Intro To Stata, Fall 2010
Today’s Dataset • We have data on a variety of variables for all 50 states – Population, density, energy use, voting tendencies, graduation rates, income, etc. • We’re going to be predicting SAT scores
Opening Files in Stata • When I open Stata, it tells me it’s using the directory: – afs/athena.mit.edu/a/d/adlynch • But, my files are located in: – afs/athena.mit.edu/a/d/adlynch/Regression • I’m going to tell Stata where it should look for my files: – cd “~/Regression” 8 HMDC Intro To Stata, Fall 2010
Univariate Regression: SAT scores and Education Expenditures • Does the amount of money spent on education affect the mean SAT score in a state? • Dependent variable: csat • Independent variable: expense
Steps for Running Regression • 1. Examine descriptive statistics • 2. Look at relationship graphically and test correlation(s) • 3. Run and interpret regression • 4. Test regression assumptions
Univariate Regression: SAT scores and Education Expenditures • First, let’s look at some descriptives codebook csat expense sum csat expense • Remember in OLS regression we need continuous, dichotomous or dummy-coded predictors – Outcome should be continuous
Univariate Regression: SAT scores and Education Expenditures csat Mean composite SAT score type: numeric (int) range: [832,1093] units: 1 unique values: 45 missing .: 0/51 mean: 944.098 std. dev: 66.935 percentiles: 10% 25% 50% 75% 90% 874 886 926 997 1024 expense Per pupil expenditures prim&sec type: numeric (int) range: [2960,9259] units: 1 unique values: 51 missing .: 0/51 mean: 5235.96 std. dev: 1401.16 percentiles: 10% 25% 50% 75% 90% 3782 4351 5000 5865 6738
Univariate Regression: SAT scores and Education Expenditures • View relationship graphically • Scatterplots work well for univariate relationships – twoway scatter expense scat – twoway (scatter scat expense) (lfit scat expense)
Univariate Regression: SAT scores and Education Expenditures • twoway (scatter scat expense) (lfit scat expense) 800 900 1000 1100 2000 4000 6000 8000 10000 Per pupil expenditures prim&sec Mean composite SAT score Fitted values Relationship Between Education Expenditures and SAT Scores
Univariate Regression: SAT scores and Education Expenditures • twoway lfitci expense csat
Univariate Regression: SAT scores and Education Expenditures • pwcorr csat expense, star(.05) | csat expense -------------+------------------ csat | 1.0000 expense | -0.4663* 1.0000
Univariate Regression: SAT scores and Education Expenditures • regress csat expense Source | SS df MS Number of obs = 51 -------------+------------------------------ F( 1, 49) = 13.61 Model | 48708.3001 1 48708.3001 Prob > F = 0.0006 Residual | 175306.21 49 3577.67775 R-squared = 0.2174 -------------+------------------------------ Adj R-squared = 0.2015 Total | 224014.51 50 4480.2902 Root MSE = 59.814 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | -.0222756 .0060371 -3.69 0.001 -.0344077 -.0101436 _cons | 1060.732 32.7009 32.44 0.000 995.0175 1126.447 ------------------------------------------------------------------------------
Univariate Regression: SAT scores and Education Expenditures • Intercept • What would we predict a state’s mean SAT score to be if its per pupil expenditure is $0.00? Source | SS df MS Number of obs = 51 -------------+------------------------------ F( 1, 49) = 13.61 Model | 48708.3001 1 48708.3001 Prob > F = 0.0006 Residual | 175306.21 49 3577.67775 R-squared = 0.2174 -------------+------------------------------ Adj R-squared = 0.2015 Total | 224014.51 50 4480.2902 Root MSE = 59.814 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | -.0222756 .0060371 -3.69 0.001 -.0344077 -.0101436 _cons | 1060.732 32.7009 32.44 0.000 995.0175 1126.447 ------------------------------------------------------------------------------
Univariate Regression: SAT scores and Education Expenditures • Slope • For every one unit increase in per pupil expenditure, what happens to mean SAT scores? Source | SS df MS Number of obs = 51 -------------+------------------------------ F( 1, 49) = 13.61 Model | 48708.3001 1 48708.3001 Prob > F = 0.0006 Residual | 175306.21 49 3577.67775 R-squared = 0.2174 -------------+------------------------------ Adj R-squared = 0.2015 Total | 224014.51 50 4480.2902 Root MSE = 59.814 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | -.0222756 .0060371 -3.69 0.001 -.0344077 -.0101436 _cons | 1060.732 32.7009 32.44 0.000 995.0175 1126.447 ------------------------------------------------------------------------------
Univariate Regression: SAT scores and Education Expenditures • Significance of individual predictors • Is there a statistically significant relationship between SAT scores and per pupil expenditures? Source | SS df MS Number of obs = 51 -------------+------------------------------ F( 1, 49) = 13.61 Model | 48708.3001 1 48708.3001 Prob > F = 0.0006 Residual | 175306.21 49 3577.67775 R-squared = 0.2174 -------------+------------------------------ Adj R-squared = 0.2015 Total | 224014.51 50 4480.2902 Root MSE = 59.814 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | -.0222756 .0060371 -3.69 0.001 -.0344077 -.0101436 _cons | 1060.732 32.7009 32.44 0.000 995.0175 1126.447 ------------------------------------------------------------------------------
Univariate Regression: SAT scores and Education Expenditures • Significance of overall equation Source | SS df MS Number of obs = 51 -------------+------------------------------ F( 1, 49) = 13.61 Model | 48708.3001 1 48708.3001 Prob > F = 0.0006 Residual | 175306.21 49 3577.67775 R-squared = 0.2174 -------------+------------------------------ Adj R-squared = 0.2015 Total | 224014.51 50 4480.2902 Root MSE = 59.814 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | -.0222756 .0060371 -3.69 0.001 -.0344077 -.0101436 _cons | 1060.732 32.7009 32.44 0.000 995.0175 1126.447 ------------------------------------------------------------------------------
Univariate Regression: SAT scores and Education Expenditures • Coefficient of determination • What percent of variation in SAT scores is explained by per pupil expense? Source | SS df MS Number of obs = 51 -------------+------------------------------ F( 1, 49) = 13.61 Model | 48708.3001 1 48708.3001 Prob > F = 0.0006 Residual | 175306.21 49 3577.67775 R-squared = 0.2174 -------------+------------------------------ Adj R-squared = 0.2015 Total | 224014.51 50 4480.2902 Root MSE = 59.814 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | -.0222756 .0060371 -3.69 0.001 -.0344077 -.0101436 _cons | 1060.732 32.7009 32.44 0.000 995.0175 1126.447 ------------------------------------------------------------------------------
Univariate Regression: SAT scores and Education Expenditures • Standard error of the estimate Source | SS df MS Number of obs = 51 -------------+------------------------------ F( 1, 49) = 13.61 Model | 48708.3001 1 48708.3001 Prob > F = 0.0006 Residual | 175306.21 49 3577.67775 R-squared = 0.2174 -------------+------------------------------ Adj R-squared = 0.2015 Total | 224014.51 50 4480.2902 Root MSE = 59.814 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | -.0222756 .0060371 -3.69 0.001 -.0344077 -.0101436 _cons | 1060.732 32.7009 32.44 0.000 995.0175 1126.447 ------------------------------------------------------------------------------
Linear Regression Assumptions • Assumption 1: Normal Distribution – The dependent variable is normally distributed – The errors of regression equation are normally distributed • Assumption 2: Homoscedasticity – The variance around the regression line is the same for all values of the predictor variable (X)
Homoscedasticity
Regression Assumptions • Assumption 3: Errors are independent – The size of one error is not a function of the size of any previous error • Assumption 4: Relationships are linear – AKA – the relationship can be summarized with a straight line – Keep in mind that you can use alternative forms of regression to test non-linear relationships
Testing Assumptions: Normality predict resid, residual label var resid "Residuals of pp expend and SAT" histogram resid, normal 0 .002 .004 .006 .008 Density -200 -100 0 100 200 Residuals of pp expend and SAT
Testing Assumptions: Normality swilk resid Note: Shapiro-Wilk test of normality tests null hypothesis that data is normally distributed Shapiro-Wilk W test for normal data Variable | Obs W V z Prob>z -------------+-------------------------------------------------- resid | 51 0.99144 0.409 -1.909 0.97190
Testing Assumptions: Homoscedasticity rvfplot -200 -100 0 100 200 Residuals 850 900 950 1000 Fitted values Note: “rvfplot” command needs to be entered after regression equation is run – Stata uses estimates from the regression to create this plot
Testing Assumptions: Homoscedasticity estat hettest Note: The null hypothesis is homoscedasticity Breusch-Pagan / Cook-Weisberg test for heteroskedasticity Ho: Constant variance Variables: fitted values of csat chi2(1) = 2.14 Prob > chi2 = 0.1436
Multiple Regression • Just keep adding predictors – regress dependent iv1 iv2 iv3…ivn • Let’s try adding some predictors to the model of SAT scores – Income (income), % students taking SATs (percent), % adults with HS diploma (high)
Multiple Regression . sum income percent high Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- income | 51 33.95657 6.423134 23.465 48.618 percent | 51 35.76471 26.19281 4 81 high | 51 76.26078 5.588741 64.3 86.6
Correlations with Multiple Regression . pwcorr csat expense income percent high, star(.05) | csat expense income percent high -------------+--------------------------------------------- csat | 1.0000 expense | -0.4663* 1.0000 income | -0.4713* 0.6784* 1.0000 percent | -0.8758* 0.6509* 0.6733* 1.0000 high | 0.0858 0.3133* 0.5099* 0.1413 1.0000
Multiple Regression . regress csat expense income percent high Source | SS df MS Number of obs = 51 -------------+------------------------------ F( 4, 46) = 51.86 Model | 183354.603 4 45838.6508 Prob > F = 0.0000 Residual | 40659.9067 46 883.911016 R-squared = 0.8185 -------------+------------------------------ Adj R-squared = 0.8027 Total | 224014.51 50 4480.2902 Root MSE = 29.731 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | .0045604 .004384 1.04 0.304 -.0042641 .013385 income | .4437858 1.138947 0.39 0.699 -1.848795 2.736367 percent | -2.533084 .2454477 -10.32 0.000 -3.027145 -2.039024 high | 2.086599 .9246023 2.26 0.029 .2254712 3.947727 _cons | 836.6197 58.33238 14.34 0.000 719.2027 954.0366 ------------------------------------------------------------------------------
Exercise 1: Multiple Regression
Multiple Regression: Interaction Terms • What if we wanted to test an interaction between percent & high? • Option 1: – generate a new variable – gen percenthigh = percent*high • Option 2: – Let Stata do your dirty work
Multiple Regression: Interaction Terms . regress csat expense income percent high c.percent#c.high Source | SS df MS Number of obs = 51 -------------+------------------------------ F( 5, 45) = 46.11 Model | 187430.399 5 37486.0799 Prob > F = 0.0000 Residual | 36584.1104 45 812.980232 R-squared = 0.8367 -------------+------------------------------ Adj R-squared = 0.8185 Total | 224014.51 50 4480.2902 Root MSE = 28.513 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | .0045575 .0042044 1.08 0.284 -.0039107 .0130256 income | .0887854 1.10374 0.08 0.936 -2.134261 2.311832 percent | -8.143001 2.516509 -3.24 0.002 -13.21151 -3.074492 high | .4240909 1.156545 0.37 0.716 -1.90531 2.753492 | c.percent#| c.high | .0740926 .0330909 2.24 0.030 .0074441 .1407411 | _cons | 972.525 82.5457 11.78 0.000 806.2694 1138.781 ------------------------------------------------------------------------------
Multiple Regression • Same rules apply for interpretation as with univariate regression – Slope, intercept, overall significance of the equation, R2, standard error of estimate • Can also generate residuals for assumption testing
Multiple Regression with Categorical Predictors • We can also test dichotomous and categorical predictors in our models • For categorical variables, we first need to dummy code • Use region as example
Dummy Coding ------------------------------------------------------------------------ region Geographical region ------------------------------------------------------------------------ type: numeric (byte) label: region range: [1,4] units: 1 unique values: 4 missing .: 1/51 tabulation: Freq. Numeric Label 13 1 West 9 2 N. East 16 3 South 12 4 Midwest 1 .
Dummy Coding • Option 1: Manually dummy code tab region, gen(region) gen region1=1 if region==1 gen region2=1 if region==2 gen region3=1 if region==3 gen region4=1 if region==4 NOTE: BE SURE TO CONSIDER MISSING DATA BEFORE GENERATING DUMMY VARIABLES • Option 2: Let Stata do your dirty work with “xi” command
Multiple Regression with Categorical Predictors . xi: regress csat expense income percent high i.region i.region _Iregion_1-4 (naturally coded; _Iregion_1 omitted) Source | SS df MS Number of obs = 50 -------------+------------------------------ F( 7, 42) = 51.07 Model | 190570.293 7 27224.3275 Prob > F = 0.0000 Residual | 22391.0874 42 533.121128 R-squared = 0.8949 -------------+------------------------------ Adj R-squared = 0.8773 Total | 212961.38 49 4346.15061 Root MSE = 23.089 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | -.004375 .0044603 -0.98 0.332 -.0133763 .0046263 income | 1.306164 .950279 1.37 0.177 -.6115765 3.223905 percent | -2.965514 .2496481 -11.88 0.000 -3.469325 -2.461704 high | 3.544804 1.075863 3.29 0.002 1.373625 5.715983 _Iregion_2 | 80.81334 15.4341 5.24 0.000 49.66607 111.9606 _Iregion_3 | 33.61225 13.94521 2.41 0.020 5.469676 61.75483 _Iregion_4 | 32.15421 10.20145 3.15 0.003 11.56686 52.74157 _cons | 724.8289 79.25065 9.15 0.000 564.8946 884.7631 ------------------------------------------------------------------------------
Regression, Categorical Predictors, & Interactions Source | SS df MS Number of obs = 50 -------------+------------------------------ F( 10, 39) = 44.49 Model | 195797.26 10 19579.726 Prob > F = 0.0000 Residual | 17164.1203 39 440.105648 R-squared = 0.9194 -------------+------------------------------ Adj R-squared = 0.8987 Total | 212961.38 49 4346.15061 Root MSE = 20.979 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | -.0053464 .0040912 -1.31 0.199 -.0136216 .0029287 income | .3045218 .9226456 0.33 0.743 -1.561705 2.170749 percent | -2.173732 .4101372 -5.30 0.000 -3.003313 -1.344151 high | 3.676953 1.063744 3.46 0.001 1.525327 5.828579 _Iregion_2 | -155.2988 100.0857 -1.55 0.129 -357.7412 47.14363 _Iregion_3 | (omitted) _Iregion_4 | 63.25404 16.12525 3.92 0.000 30.63764 95.87045 _Iregion_2 | (omitted) _Iregion_3 | 50.64898 21.39424 2.37 0.023 7.375034 93.92292 _Iregion_4 | (omitted) percent | (omitted) _IregXperc~2 | 2.90901 1.392714 2.09 0.043 .0919803 5.726039 _IregXperc~3 | -.6795988 .4419833 -1.54 0.132 -1.573594 .2143968 _IregXperc~4 | -1.421575 .5894918 -2.41 0.021 -2.613935 -.2292158 _cons | 729.9697 81.6624 8.94 0.000 564.7919 895.1475 ------------------------------------------------------------------------------ xi: regress csat expense income percent high i.region i.region*percent
How can I manage all this output? • Usually when we’re running regression, we’ll be testing multiple models at a time – Can be difficult to compare results • Stata offers several user-friendly options for storing and viewing regression output from multiple models
How can I manage all this output? • You can both store output in Stata or ask Stata to export the results • First, let’s see how we can store this info in Stata: regress csat expense income percent high estimates store Model1 regress csat expense income percent high region2 /// region3 region4 estimates store Model2
How can I manage all this output? • Now Stata will hold your output in memory until you ask to recall it esttab Model1 Model2 esttab Model1 Model2, label nostar
How can I manage all this output? ------------------------------------------------------------ (1) (2) (3) csat csat csat ------------------------------------------------------------ expense 0.00456 -0.00438 -0.00496 (1.04) (-0.98) (-1.16) income 0.444 1.306 0.978 (0.39) (1.37) (1.06) percent -2.533*** -2.966*** -7.643*** (-10.32) (-11.88) (-3.63) high 2.087* 3.545** 2.018 (2.26) (3.29) (1.63) region2 80.81*** 73.14*** (5.24) (4.83) region3 33.61* 32.24* (2.41) (2.42) region4 32.15** 37.87*** (3.15) (3.76) percenthigh 0.0635* (2.24) _cons 836.6*** 724.8*** 848.5*** (14.34) (9.15) (9.05) ------------------------------------------------------------ N 51 50 50 ------------------------------------------------------------
How can I manage all this output? ----------------------------------------------------------- (1) (2) (3) Mean compo~e Mean compo~e Mean compo~e ----------------------------------------------------------- Per pupil expendit~c 0.00456 -0.00438 -0.00496 (1.04) (-0.98) (-1.16) Median household~000 0.444 1.306 0.978 (0.39) (1.37) (1.06) % HS graduates tak~T -2.533 -2.966 -7.643 (-10.32) (-11.88) (-3.63) % adults HS diploma 2.087 3.545 2.018 (2.26) (3.29) (1.63) Northeast 80.81 73.14 (5.24) (4.83) South 33.61 32.24 (2.41) (2.42) Midwest 32.15 37.87 (3.15) (3.76) Percent*High 0.0635 (2.24) Constant 836.6 724.8 848.5 (14.34) (9.15) (9.05) ----------------------------------------------------------- Observations 51 50 50 ----------------------------------------------------------- t statistics in parentheses
Outputting into Excel • Avoid human error when transferring coefficients into tables regress csat expense income percent high outreg2 using csatprediction.xls • Now, let’s add some options regress csat expense income percent high outreg2 using csatprediction.xls, bdec(3) ctitle(Model 1) /// se title("Prediction of Average SAT scores") replace
How can I manage all this output? Prediction of Average SAT scores (1) (2) (3) VARIABLES Model 1 Model 2 Model 3 expense 0.005 -0.004 -0.005 (0.004) (0.004) (0.004) income 0.444 1.306 0.978 (1.139) (0.950) (0.920) percent -2.533*** -2.966*** -7.643*** (0.245) (0.250) (2.106) high 2.087** 3.545*** 2.018 (0.925) (1.076) (1.234) region2 80.813*** 73.141*** (15.434) (15.142) region3 33.612** 32.240** (13.945) (13.340) region4 32.154*** 37.865*** (10.201) (10.077) percenthigh 0.064** (0.028) Constant 836.620** * 724.829** * 848.521** * (58.332) (79.251) (93.787) Observations 51 50 50 R-squared 0.818 0.895 0.906 Standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1
What if my data are clustered? • Often, our data is grouped (by industry, schools, hospitals, etc.) • This grouping violates independence assumption of regression • Use “cluster” option as simple way to account for clustering and produce robust standard errors • DISCLAIMER: There are many ways to account for clustering in Stata and you should have a sound theoretical model and understanding before applying cluster options
What if my data are clustered? • We’ll review a simple way to produce robust standard errors in a multiple regression, but also see: • http://www.ats.ucla.edu/stat/stata/faq/cluste rreg.htm – Provides a complete description of various clustering options – Select option that best fits your needs
What if my data are clustered? . regress csat expense income percent high, cluster(region) Linear regression Number of obs = 50 F( 2, 3) = . Prob > F = . R-squared = 0.8141 Root MSE = 29.662 (Std. Err. adjusted for 4 clusters in region) ------------------------------------------------------------------------------ | Robust csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | .0072659 .0004267 17.03 0.000 .0059079 .0086238 income | .1136656 1.721432 0.07 0.952 -5.364701 5.592032 percent | -2.529829 .4536296 -5.58 0.011 -3.973481 -1.086177 high | 1.986721 1.0819 1.84 0.164 -1.456368 5.429809 _cons | 841.9268 79.55744 10.58 0.002 588.7395 1095.114 ------------------------------------------------------------------------------
Exercise 2: Regression, Categorical Predictors, & Interactions
Other Services Available • MIT’s membership in HMDC provided by schools and departments at MIT • Institute for Quantitative Social Science – www.iq.harvard.edu • Research Computing – www.iq.harvard.edu/research_computing • Computer labs – www.iq.harvard.edu/facilities • Training – www.iq.harvard.edu/training • Data repository – http://libraries.mit.edu/get/hmdc 55
Thank you! All of these courses will be offered during MIT’s IAP and again at Harvard during the Spring 2011 semester. • Introduction to Stata • Data Management in Stata • Regression in Stata • Graphics in Stata • Introduction to R • Introduction to SAS Sign up for MIT workshops at: http://libraries.mit.edu/guides/subjects/data/training/workshops.html Sign up for Harvard workshops by emailing: dataclass@help.hmdc.harvard.edu

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Alicia Doyle Lynch Regression in Stata pdf 55 slides.pdf

  • 1. Regression in Stata Alicia Doyle Lynch Harvard-MIT Data Center (HMDC)
  • 2. Documents for Today • Find class materials at: http://libraries.mit.edu/guides/subjects/data/ training/workshops.html – Several formats of data – Presentation slides – Handouts – Exercises • Let’s go over how to save these files together 2
  • 3. Organization • Please feel free to ask questions at any point if they are relevant to the current topic (or if you are lost!) • There will be a Q&A after class for more specific, personalized questions • Collaboration with your neighbors is encouraged • If you are using a laptop, you will need to adjust paths accordingly
  • 4. Organization • Make comments in your Do-file rather than on hand-outs – Save on flash drive or email to yourself • Stata commands will always appear in red • “Var” simply refers to “variable” (e.g., var1, var2, var3, varname) • Pathnames should be replaced with the path specific to your computer and folders
  • 5. Assumptions (and Disclaimers) • This is Regression in Stata • Assumes basic knowledge of Stata • Assumes knowledge of regression • Not appropriate for people not familiar with Stata • Not appropriate for people already well- familiar with regression in Stata
  • 6. Opening Stata • In your Athena terminal (the large purple screen with blinking cursor) type add stata xstata • Stata should come up on your screen • Always open Stata FIRST and THEN open Do- Files (we’ll talk about these in a minute), data files, etc. 6 HMDC Intro To Stata, Fall 2010
  • 7. Today’s Dataset • We have data on a variety of variables for all 50 states – Population, density, energy use, voting tendencies, graduation rates, income, etc. • We’re going to be predicting SAT scores
  • 8. Opening Files in Stata • When I open Stata, it tells me it’s using the directory: – afs/athena.mit.edu/a/d/adlynch • But, my files are located in: – afs/athena.mit.edu/a/d/adlynch/Regression • I’m going to tell Stata where it should look for my files: – cd “~/Regression” 8 HMDC Intro To Stata, Fall 2010
  • 9. Univariate Regression: SAT scores and Education Expenditures • Does the amount of money spent on education affect the mean SAT score in a state? • Dependent variable: csat • Independent variable: expense
  • 10. Steps for Running Regression • 1. Examine descriptive statistics • 2. Look at relationship graphically and test correlation(s) • 3. Run and interpret regression • 4. Test regression assumptions
  • 11. Univariate Regression: SAT scores and Education Expenditures • First, let’s look at some descriptives codebook csat expense sum csat expense • Remember in OLS regression we need continuous, dichotomous or dummy-coded predictors – Outcome should be continuous
  • 12. Univariate Regression: SAT scores and Education Expenditures csat Mean composite SAT score type: numeric (int) range: [832,1093] units: 1 unique values: 45 missing .: 0/51 mean: 944.098 std. dev: 66.935 percentiles: 10% 25% 50% 75% 90% 874 886 926 997 1024 expense Per pupil expenditures prim&sec type: numeric (int) range: [2960,9259] units: 1 unique values: 51 missing .: 0/51 mean: 5235.96 std. dev: 1401.16 percentiles: 10% 25% 50% 75% 90% 3782 4351 5000 5865 6738
  • 13. Univariate Regression: SAT scores and Education Expenditures • View relationship graphically • Scatterplots work well for univariate relationships – twoway scatter expense scat – twoway (scatter scat expense) (lfit scat expense)
  • 14. Univariate Regression: SAT scores and Education Expenditures • twoway (scatter scat expense) (lfit scat expense) 800 900 1000 1100 2000 4000 6000 8000 10000 Per pupil expenditures prim&sec Mean composite SAT score Fitted values Relationship Between Education Expenditures and SAT Scores
  • 15. Univariate Regression: SAT scores and Education Expenditures • twoway lfitci expense csat
  • 16. Univariate Regression: SAT scores and Education Expenditures • pwcorr csat expense, star(.05) | csat expense -------------+------------------ csat | 1.0000 expense | -0.4663* 1.0000
  • 17. Univariate Regression: SAT scores and Education Expenditures • regress csat expense Source | SS df MS Number of obs = 51 -------------+------------------------------ F( 1, 49) = 13.61 Model | 48708.3001 1 48708.3001 Prob > F = 0.0006 Residual | 175306.21 49 3577.67775 R-squared = 0.2174 -------------+------------------------------ Adj R-squared = 0.2015 Total | 224014.51 50 4480.2902 Root MSE = 59.814 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | -.0222756 .0060371 -3.69 0.001 -.0344077 -.0101436 _cons | 1060.732 32.7009 32.44 0.000 995.0175 1126.447 ------------------------------------------------------------------------------
  • 18. Univariate Regression: SAT scores and Education Expenditures • Intercept • What would we predict a state’s mean SAT score to be if its per pupil expenditure is $0.00? Source | SS df MS Number of obs = 51 -------------+------------------------------ F( 1, 49) = 13.61 Model | 48708.3001 1 48708.3001 Prob > F = 0.0006 Residual | 175306.21 49 3577.67775 R-squared = 0.2174 -------------+------------------------------ Adj R-squared = 0.2015 Total | 224014.51 50 4480.2902 Root MSE = 59.814 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | -.0222756 .0060371 -3.69 0.001 -.0344077 -.0101436 _cons | 1060.732 32.7009 32.44 0.000 995.0175 1126.447 ------------------------------------------------------------------------------
  • 19. Univariate Regression: SAT scores and Education Expenditures • Slope • For every one unit increase in per pupil expenditure, what happens to mean SAT scores? Source | SS df MS Number of obs = 51 -------------+------------------------------ F( 1, 49) = 13.61 Model | 48708.3001 1 48708.3001 Prob > F = 0.0006 Residual | 175306.21 49 3577.67775 R-squared = 0.2174 -------------+------------------------------ Adj R-squared = 0.2015 Total | 224014.51 50 4480.2902 Root MSE = 59.814 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | -.0222756 .0060371 -3.69 0.001 -.0344077 -.0101436 _cons | 1060.732 32.7009 32.44 0.000 995.0175 1126.447 ------------------------------------------------------------------------------
  • 20. Univariate Regression: SAT scores and Education Expenditures • Significance of individual predictors • Is there a statistically significant relationship between SAT scores and per pupil expenditures? Source | SS df MS Number of obs = 51 -------------+------------------------------ F( 1, 49) = 13.61 Model | 48708.3001 1 48708.3001 Prob > F = 0.0006 Residual | 175306.21 49 3577.67775 R-squared = 0.2174 -------------+------------------------------ Adj R-squared = 0.2015 Total | 224014.51 50 4480.2902 Root MSE = 59.814 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | -.0222756 .0060371 -3.69 0.001 -.0344077 -.0101436 _cons | 1060.732 32.7009 32.44 0.000 995.0175 1126.447 ------------------------------------------------------------------------------
  • 21. Univariate Regression: SAT scores and Education Expenditures • Significance of overall equation Source | SS df MS Number of obs = 51 -------------+------------------------------ F( 1, 49) = 13.61 Model | 48708.3001 1 48708.3001 Prob > F = 0.0006 Residual | 175306.21 49 3577.67775 R-squared = 0.2174 -------------+------------------------------ Adj R-squared = 0.2015 Total | 224014.51 50 4480.2902 Root MSE = 59.814 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | -.0222756 .0060371 -3.69 0.001 -.0344077 -.0101436 _cons | 1060.732 32.7009 32.44 0.000 995.0175 1126.447 ------------------------------------------------------------------------------
  • 22. Univariate Regression: SAT scores and Education Expenditures • Coefficient of determination • What percent of variation in SAT scores is explained by per pupil expense? Source | SS df MS Number of obs = 51 -------------+------------------------------ F( 1, 49) = 13.61 Model | 48708.3001 1 48708.3001 Prob > F = 0.0006 Residual | 175306.21 49 3577.67775 R-squared = 0.2174 -------------+------------------------------ Adj R-squared = 0.2015 Total | 224014.51 50 4480.2902 Root MSE = 59.814 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | -.0222756 .0060371 -3.69 0.001 -.0344077 -.0101436 _cons | 1060.732 32.7009 32.44 0.000 995.0175 1126.447 ------------------------------------------------------------------------------
  • 23. Univariate Regression: SAT scores and Education Expenditures • Standard error of the estimate Source | SS df MS Number of obs = 51 -------------+------------------------------ F( 1, 49) = 13.61 Model | 48708.3001 1 48708.3001 Prob > F = 0.0006 Residual | 175306.21 49 3577.67775 R-squared = 0.2174 -------------+------------------------------ Adj R-squared = 0.2015 Total | 224014.51 50 4480.2902 Root MSE = 59.814 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | -.0222756 .0060371 -3.69 0.001 -.0344077 -.0101436 _cons | 1060.732 32.7009 32.44 0.000 995.0175 1126.447 ------------------------------------------------------------------------------
  • 24. Linear Regression Assumptions • Assumption 1: Normal Distribution – The dependent variable is normally distributed – The errors of regression equation are normally distributed • Assumption 2: Homoscedasticity – The variance around the regression line is the same for all values of the predictor variable (X)
  • 26. Regression Assumptions • Assumption 3: Errors are independent – The size of one error is not a function of the size of any previous error • Assumption 4: Relationships are linear – AKA – the relationship can be summarized with a straight line – Keep in mind that you can use alternative forms of regression to test non-linear relationships
  • 27. Testing Assumptions: Normality predict resid, residual label var resid "Residuals of pp expend and SAT" histogram resid, normal 0 .002 .004 .006 .008 Density -200 -100 0 100 200 Residuals of pp expend and SAT
  • 28. Testing Assumptions: Normality swilk resid Note: Shapiro-Wilk test of normality tests null hypothesis that data is normally distributed Shapiro-Wilk W test for normal data Variable | Obs W V z Prob>z -------------+-------------------------------------------------- resid | 51 0.99144 0.409 -1.909 0.97190
  • 29. Testing Assumptions: Homoscedasticity rvfplot -200 -100 0 100 200 Residuals 850 900 950 1000 Fitted values Note: “rvfplot” command needs to be entered after regression equation is run – Stata uses estimates from the regression to create this plot
  • 30. Testing Assumptions: Homoscedasticity estat hettest Note: The null hypothesis is homoscedasticity Breusch-Pagan / Cook-Weisberg test for heteroskedasticity Ho: Constant variance Variables: fitted values of csat chi2(1) = 2.14 Prob > chi2 = 0.1436
  • 31. Multiple Regression • Just keep adding predictors – regress dependent iv1 iv2 iv3…ivn • Let’s try adding some predictors to the model of SAT scores – Income (income), % students taking SATs (percent), % adults with HS diploma (high)
  • 32. Multiple Regression . sum income percent high Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- income | 51 33.95657 6.423134 23.465 48.618 percent | 51 35.76471 26.19281 4 81 high | 51 76.26078 5.588741 64.3 86.6
  • 33. Correlations with Multiple Regression . pwcorr csat expense income percent high, star(.05) | csat expense income percent high -------------+--------------------------------------------- csat | 1.0000 expense | -0.4663* 1.0000 income | -0.4713* 0.6784* 1.0000 percent | -0.8758* 0.6509* 0.6733* 1.0000 high | 0.0858 0.3133* 0.5099* 0.1413 1.0000
  • 34. Multiple Regression . regress csat expense income percent high Source | SS df MS Number of obs = 51 -------------+------------------------------ F( 4, 46) = 51.86 Model | 183354.603 4 45838.6508 Prob > F = 0.0000 Residual | 40659.9067 46 883.911016 R-squared = 0.8185 -------------+------------------------------ Adj R-squared = 0.8027 Total | 224014.51 50 4480.2902 Root MSE = 29.731 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | .0045604 .004384 1.04 0.304 -.0042641 .013385 income | .4437858 1.138947 0.39 0.699 -1.848795 2.736367 percent | -2.533084 .2454477 -10.32 0.000 -3.027145 -2.039024 high | 2.086599 .9246023 2.26 0.029 .2254712 3.947727 _cons | 836.6197 58.33238 14.34 0.000 719.2027 954.0366 ------------------------------------------------------------------------------
  • 35. Exercise 1: Multiple Regression
  • 36. Multiple Regression: Interaction Terms • What if we wanted to test an interaction between percent & high? • Option 1: – generate a new variable – gen percenthigh = percent*high • Option 2: – Let Stata do your dirty work
  • 37. Multiple Regression: Interaction Terms . regress csat expense income percent high c.percent#c.high Source | SS df MS Number of obs = 51 -------------+------------------------------ F( 5, 45) = 46.11 Model | 187430.399 5 37486.0799 Prob > F = 0.0000 Residual | 36584.1104 45 812.980232 R-squared = 0.8367 -------------+------------------------------ Adj R-squared = 0.8185 Total | 224014.51 50 4480.2902 Root MSE = 28.513 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | .0045575 .0042044 1.08 0.284 -.0039107 .0130256 income | .0887854 1.10374 0.08 0.936 -2.134261 2.311832 percent | -8.143001 2.516509 -3.24 0.002 -13.21151 -3.074492 high | .4240909 1.156545 0.37 0.716 -1.90531 2.753492 | c.percent#| c.high | .0740926 .0330909 2.24 0.030 .0074441 .1407411 | _cons | 972.525 82.5457 11.78 0.000 806.2694 1138.781 ------------------------------------------------------------------------------
  • 38. Multiple Regression • Same rules apply for interpretation as with univariate regression – Slope, intercept, overall significance of the equation, R2, standard error of estimate • Can also generate residuals for assumption testing
  • 39. Multiple Regression with Categorical Predictors • We can also test dichotomous and categorical predictors in our models • For categorical variables, we first need to dummy code • Use region as example
  • 40. Dummy Coding ------------------------------------------------------------------------ region Geographical region ------------------------------------------------------------------------ type: numeric (byte) label: region range: [1,4] units: 1 unique values: 4 missing .: 1/51 tabulation: Freq. Numeric Label 13 1 West 9 2 N. East 16 3 South 12 4 Midwest 1 .
  • 41. Dummy Coding • Option 1: Manually dummy code tab region, gen(region) gen region1=1 if region==1 gen region2=1 if region==2 gen region3=1 if region==3 gen region4=1 if region==4 NOTE: BE SURE TO CONSIDER MISSING DATA BEFORE GENERATING DUMMY VARIABLES • Option 2: Let Stata do your dirty work with “xi” command
  • 42. Multiple Regression with Categorical Predictors . xi: regress csat expense income percent high i.region i.region _Iregion_1-4 (naturally coded; _Iregion_1 omitted) Source | SS df MS Number of obs = 50 -------------+------------------------------ F( 7, 42) = 51.07 Model | 190570.293 7 27224.3275 Prob > F = 0.0000 Residual | 22391.0874 42 533.121128 R-squared = 0.8949 -------------+------------------------------ Adj R-squared = 0.8773 Total | 212961.38 49 4346.15061 Root MSE = 23.089 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | -.004375 .0044603 -0.98 0.332 -.0133763 .0046263 income | 1.306164 .950279 1.37 0.177 -.6115765 3.223905 percent | -2.965514 .2496481 -11.88 0.000 -3.469325 -2.461704 high | 3.544804 1.075863 3.29 0.002 1.373625 5.715983 _Iregion_2 | 80.81334 15.4341 5.24 0.000 49.66607 111.9606 _Iregion_3 | 33.61225 13.94521 2.41 0.020 5.469676 61.75483 _Iregion_4 | 32.15421 10.20145 3.15 0.003 11.56686 52.74157 _cons | 724.8289 79.25065 9.15 0.000 564.8946 884.7631 ------------------------------------------------------------------------------
  • 43. Regression, Categorical Predictors, & Interactions Source | SS df MS Number of obs = 50 -------------+------------------------------ F( 10, 39) = 44.49 Model | 195797.26 10 19579.726 Prob > F = 0.0000 Residual | 17164.1203 39 440.105648 R-squared = 0.9194 -------------+------------------------------ Adj R-squared = 0.8987 Total | 212961.38 49 4346.15061 Root MSE = 20.979 ------------------------------------------------------------------------------ csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | -.0053464 .0040912 -1.31 0.199 -.0136216 .0029287 income | .3045218 .9226456 0.33 0.743 -1.561705 2.170749 percent | -2.173732 .4101372 -5.30 0.000 -3.003313 -1.344151 high | 3.676953 1.063744 3.46 0.001 1.525327 5.828579 _Iregion_2 | -155.2988 100.0857 -1.55 0.129 -357.7412 47.14363 _Iregion_3 | (omitted) _Iregion_4 | 63.25404 16.12525 3.92 0.000 30.63764 95.87045 _Iregion_2 | (omitted) _Iregion_3 | 50.64898 21.39424 2.37 0.023 7.375034 93.92292 _Iregion_4 | (omitted) percent | (omitted) _IregXperc~2 | 2.90901 1.392714 2.09 0.043 .0919803 5.726039 _IregXperc~3 | -.6795988 .4419833 -1.54 0.132 -1.573594 .2143968 _IregXperc~4 | -1.421575 .5894918 -2.41 0.021 -2.613935 -.2292158 _cons | 729.9697 81.6624 8.94 0.000 564.7919 895.1475 ------------------------------------------------------------------------------ xi: regress csat expense income percent high i.region i.region*percent
  • 44. How can I manage all this output? • Usually when we’re running regression, we’ll be testing multiple models at a time – Can be difficult to compare results • Stata offers several user-friendly options for storing and viewing regression output from multiple models
  • 45. How can I manage all this output? • You can both store output in Stata or ask Stata to export the results • First, let’s see how we can store this info in Stata: regress csat expense income percent high estimates store Model1 regress csat expense income percent high region2 /// region3 region4 estimates store Model2
  • 46. How can I manage all this output? • Now Stata will hold your output in memory until you ask to recall it esttab Model1 Model2 esttab Model1 Model2, label nostar
  • 47. How can I manage all this output? ------------------------------------------------------------ (1) (2) (3) csat csat csat ------------------------------------------------------------ expense 0.00456 -0.00438 -0.00496 (1.04) (-0.98) (-1.16) income 0.444 1.306 0.978 (0.39) (1.37) (1.06) percent -2.533*** -2.966*** -7.643*** (-10.32) (-11.88) (-3.63) high 2.087* 3.545** 2.018 (2.26) (3.29) (1.63) region2 80.81*** 73.14*** (5.24) (4.83) region3 33.61* 32.24* (2.41) (2.42) region4 32.15** 37.87*** (3.15) (3.76) percenthigh 0.0635* (2.24) _cons 836.6*** 724.8*** 848.5*** (14.34) (9.15) (9.05) ------------------------------------------------------------ N 51 50 50 ------------------------------------------------------------
  • 48. How can I manage all this output? ----------------------------------------------------------- (1) (2) (3) Mean compo~e Mean compo~e Mean compo~e ----------------------------------------------------------- Per pupil expendit~c 0.00456 -0.00438 -0.00496 (1.04) (-0.98) (-1.16) Median household~000 0.444 1.306 0.978 (0.39) (1.37) (1.06) % HS graduates tak~T -2.533 -2.966 -7.643 (-10.32) (-11.88) (-3.63) % adults HS diploma 2.087 3.545 2.018 (2.26) (3.29) (1.63) Northeast 80.81 73.14 (5.24) (4.83) South 33.61 32.24 (2.41) (2.42) Midwest 32.15 37.87 (3.15) (3.76) Percent*High 0.0635 (2.24) Constant 836.6 724.8 848.5 (14.34) (9.15) (9.05) ----------------------------------------------------------- Observations 51 50 50 ----------------------------------------------------------- t statistics in parentheses
  • 49. Outputting into Excel • Avoid human error when transferring coefficients into tables regress csat expense income percent high outreg2 using csatprediction.xls • Now, let’s add some options regress csat expense income percent high outreg2 using csatprediction.xls, bdec(3) ctitle(Model 1) /// se title("Prediction of Average SAT scores") replace
  • 50. How can I manage all this output? Prediction of Average SAT scores (1) (2) (3) VARIABLES Model 1 Model 2 Model 3 expense 0.005 -0.004 -0.005 (0.004) (0.004) (0.004) income 0.444 1.306 0.978 (1.139) (0.950) (0.920) percent -2.533*** -2.966*** -7.643*** (0.245) (0.250) (2.106) high 2.087** 3.545*** 2.018 (0.925) (1.076) (1.234) region2 80.813*** 73.141*** (15.434) (15.142) region3 33.612** 32.240** (13.945) (13.340) region4 32.154*** 37.865*** (10.201) (10.077) percenthigh 0.064** (0.028) Constant 836.620** * 724.829** * 848.521** * (58.332) (79.251) (93.787) Observations 51 50 50 R-squared 0.818 0.895 0.906 Standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1
  • 51. What if my data are clustered? • Often, our data is grouped (by industry, schools, hospitals, etc.) • This grouping violates independence assumption of regression • Use “cluster” option as simple way to account for clustering and produce robust standard errors • DISCLAIMER: There are many ways to account for clustering in Stata and you should have a sound theoretical model and understanding before applying cluster options
  • 52. What if my data are clustered? • We’ll review a simple way to produce robust standard errors in a multiple regression, but also see: • http://www.ats.ucla.edu/stat/stata/faq/cluste rreg.htm – Provides a complete description of various clustering options – Select option that best fits your needs
  • 53. What if my data are clustered? . regress csat expense income percent high, cluster(region) Linear regression Number of obs = 50 F( 2, 3) = . Prob > F = . R-squared = 0.8141 Root MSE = 29.662 (Std. Err. adjusted for 4 clusters in region) ------------------------------------------------------------------------------ | Robust csat | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- expense | .0072659 .0004267 17.03 0.000 .0059079 .0086238 income | .1136656 1.721432 0.07 0.952 -5.364701 5.592032 percent | -2.529829 .4536296 -5.58 0.011 -3.973481 -1.086177 high | 1.986721 1.0819 1.84 0.164 -1.456368 5.429809 _cons | 841.9268 79.55744 10.58 0.002 588.7395 1095.114 ------------------------------------------------------------------------------
  • 54. Exercise 2: Regression, Categorical Predictors, & Interactions
  • 55. Other Services Available • MIT’s membership in HMDC provided by schools and departments at MIT • Institute for Quantitative Social Science – www.iq.harvard.edu • Research Computing – www.iq.harvard.edu/research_computing • Computer labs – www.iq.harvard.edu/facilities • Training – www.iq.harvard.edu/training • Data repository – http://libraries.mit.edu/get/hmdc 55
  • 56. Thank you! All of these courses will be offered during MIT’s IAP and again at Harvard during the Spring 2011 semester. • Introduction to Stata • Data Management in Stata • Regression in Stata • Graphics in Stata • Introduction to R • Introduction to SAS Sign up for MIT workshops at: http://libraries.mit.edu/guides/subjects/data/training/workshops.html Sign up for Harvard workshops by emailing: [email protected]