08 Jul ECO220Y5Y: Quantitative Methods in EconomicsFinal
ECO220Y5Y: Quantitative Methods in EconomicsFinal AssignmentReplacement for Exam Assessment on Regression1 Interactive Regression Exercise1.1 MotivationEconometrics is best understood by doing rather than by reading about what someone else hasdone. There are dicult choices and many pitfalls in arriving at the `correct’ model. Sometimesthe existing theory underlying the relationships in your model seem a bit o and you could build amuch `better’ model by including a di erent set of variables or transforming them (this gets at theinternal validity of the model). Sometimes choosing a model with the best t means you are makingdecisions that are ideal only for your sample and would not apply well to data outside your sampletime period or group of individuals (this gets at the external validity of the model). You need totrade o the internal validity with the external validity as a researcher. As a result, econometricscan sometimes feel more like an art than a science. However, you will be asked to follow a scienti capproach to making these model decisions and justifying these decisions in a scienti c way. Thisassignment requires that you make independent choices on speci cation, analyse the consequencesof these choices and adjust your choices to narrow in on a nal model. You will be asked to justifythe model you have selected and then provide some feedback on its economic implications.1.2 Overview & DataThe dependent variable for this interactive assignment is the Provincial Achievement Test (PAT)score earned by students in an Alberta high school. There are 70 observations for this data setmeasuring PAT scores and a number of possible causal factors have been randomly drawn out of apool of approximately 750 students over approximately one decade. The literature on PAT scoresindicates that scores are determined not only by ability and training but also various socio-economicfactors. Please see the attached article by James Fallows, `The Tests and the Brightest: How Fair Arethe College Boards.’ for a summary of views in the literature on how SAT performance in the USAmight be impacted by various socio-economic factors (PAT scores and SAT scores should be similarlydetermined). Measures of ability and training included here are the cumulative high school gradepoint average (GPA) and participation in advanced placement math and English courses (APMATHand APENG). Advanced placement courses may help students perform better on the PAT. Thisdata set also includes a number of dummy variables measuring qualitative socio-economic factorssuch as a student’s gender (MALE), ethnicity (WHITE), and native language (ENG). The data setalso includes a dummy variable indicating whether or not a student has attended a PAT preparationclass (PREP). The data set includes a variable indicating what year (YEAR) the students PAT scoreand other information was recorded. Finally there are several variables created as the product of twoother variables.Here is a detailed description of all variables in this assignment: PATi = the Provincial Achievement Test score of the ith student on a scale from 0 to 100 GPAi = the grade point average of the ith student on a scale from 0 to 5 APMATHi = a dummy variable equal to 1 if the ith student has taken AP Math, 0 otherwise APENGi = a dummy variable equal to 1 if the ith student has taken AP English, 0 otherwise APi = a dummy variable equal to 1 if the ith student has taken either AP Math and/or APEnglish, 0 otherwise MALEi = a dummy variable equal to 1 if the ith student is Male, 0 if Female WHITEi = a dummy variable equal to 1 if the ith student is Caucasian, 0 otherwise ENGi = a dummy variable equal to 1 if the ith student’s rst language is English, 0 otherwise PREPi = a dummy variable equal to 1 if the ith student has attended a PAT preparationcourse, 0 otherwise Y EARi = the year the Provincial Achievement Test was taken for the ith student recordedfrom 2007 to 2018 GPAMALEi = (GPAi)(MALEi) GPAWHITEi = (GPAi)(WHITEi) GPAENGi = (GPAi)(ENGi) WHITEMALEi = (WHITEi)(MALEi)1.3 Summary StatisticsIncluded below are the Means, Standard Deviations, and Correlation Coecients for the variablesin this assignmentMeans and Standard Deviations:Correlation Coecients:2 Section A: Building a Model of PAT Scores2.1 Choosing the best speci cationIn this section you will choose the speci cation you’d like to estimate from the list below, nd theregression number of that speci cation and then look at the regression results for your chosen spec-i cation in the appendix at the end. You can base your initial decision on the literature providedregarding potential discrimination in standardised testing design and also the summary statistics andcorrelation coecients for the variables. You should then decide if you are satis ed with your modelselection based on the results. If you are not satis ed you can use the information from the regressionyou ran to decide how to adjust the speci cation. You can now repeat the process until you decideon a nal selection of the `best’ speci cation. Once you decide on your preferred speci cation youwill answer the questions found below the regression model options.Regression Models:1. Model 1: PATi = 0 + 1GPAi + 2APMATHi + 3APENGi + i2. Model 2: PATi = 0 + 1GPAi + 2APMATHi + 3APENGi + 4ENGi + i3. Model 3: PATi = 0 + 1GPAi + 2APMATHi + 3APENGi + 4MALEi + i4. Model 4: PATi = 0 + 1GPAi + 2APMATHi + 3APENGi + 4PREPi + i5. Model 5: PATi = 0 + 1GPAi + 2APMATHi + 3APENGi + 4WHITEi + i6. Model 6: PATi = 0 + 1GPAi + 2APMATHi + 3APENGi + 4ENGi + 5MALEi + i7. Model 7: PATi = 0 + 1GPAi + 2APMATHi + 3APENGi + 4ENGi + 5PREPi + i8. Model 8: PATi = 0 + 1GPAi + 2APMATHi + 3APENGi + 4ENGi + 5WHITEi + i9. Model 9: PATi = 0 + 1GPAi + 2APMATHi + 3APENGi + 4MALEi + 5PREPi + i10. Model 10: PATi = 0 + 1GPAi + 2APMATHi + 3APENGi + 4MALEi + 5WHITEi + i11. Model 11: PATi = 0 + 1GPAi + 2APMATHi + 3APENGi + 4PREPi + 5WHITEi + i12. Model 12: PATi = 0 + 1GPAi + 2APMATHi + 3APENGi + 4ENGi + 5MALEi + 6PREPi + i13. Model 13: PATi = 0 + 1GPAi + 2APMATHi + 3APENGi + 4ENGi + 5MALEi + 6WHITEi + i14. Model 14: PATi = 0 + 1GPAi + 2APMATHi + 3APENGi + 4ENGi + 5PREPi + 6WHITEi + i15. Model 15: PATi = 0 + 1GPAi + 2APMATHi + 3APENGi + 4MALEi + 5PREPi + 6WHITEi + i16. Model 16: PATi = 0 + 1GPAi + 2APMATHi + 3APENGi + 4ENGi + 5MALEi + 6PREPi+ 7WHITEi + i17. Model 17: PATi = 0 + 1GPAi + 2APi + 3PREPi + i18. Model 18: PATi = 0 + 1GPAi + 2APi + 3PREPi + 4WHITEi + i19. Model 19: PATi = 0 + 1GPAi + 2APi + 3ENGi + 4PREPi + 5WHITEi + i20. Model 20: PATi = 0 + 1GPAi + 2APi + iSection A Questions:1. Write out the estimated model for your preferred speci cation including coecients and standarderrors.2. Evaluate your estimation results with respect to its economic meaning, overall model t, and the signsand signi cance of the individual coecients.3. What speci cation problems (omitted variables, irrelevant variables, multicollinearity) might yourregression have? Why?4. Do you have any possible suggestions to improve the model that you were not able to choose basedon the models provided?3 Section B: Correcting a Model of PAT Scores3.1 Understanding and correcting issuesIn this section you will assess the model you selected in the last section for heteroskedasticity andserial correlation and determine the desired approach to interpret and correct for these issues. Basedon your chosen model in Section A with its residual plot given in the last section appendix as wellas the scatter plots in the Section B appendix answer the following questions below. Provide a fewsentences to justify your answers.Section B Questions:1. Do you believe there might be a problem of heteroskedasticity in your chosen model? Do you believeit is pure or impure?2. Do you believe there might be a problem of serial correlation in your chosen model? Do you believe itis pure or impure?3. Based on the answers you gave to the two questions above, what would you suggest you do to improvethe estimated model and why?4 Section C: Interpreting a Model of PAT Scores4.1 Deciding what you can learn from the modelIn this section you will assume that a professional econometrician ran 2 models (model A & B) anddetermined the best speci cation is model B based on underlying theory. It is not your job in thiscase to question the model but rather to interpret the results. Based on the regression results formodel B answer all of the following questions below by providing your rough work in calculationsand at least a few sentences to support your argument. Note that LNPAT is the natural log of PATscores. Both models are given in the appendix under Section C.Section C Questions:1. Calculate the 98% two-sided con dence interval for the coecient on MALE. Interpret this coecientand what the con dence interval you calculated implies for your interpretation.2. Test whether the absolute value of the coecient on GPAWHITE is greater than the absolute valueof the coecient on GPAENG. Explain the meaning of this test result in terms of PAT scores.3. Draw and indicate the slope and intercept of the estimated models (lines of best t) relating GPA tothe natural log of PAT scores for white males vs. non-white females conditional on them having takenAdvanced Placement classes and speaking English as their rst language. Interpret the two estimatedlines in words.4. Solve for the impact on PAT scores of a student having a GPA of 2 rather than a GPA of 0, giventhey did not take AP courses, are non-white, male and do not speak English as a rst language. Showall your work in this calculation.5. Based on inference using the results from Model B but also taking into account both models, do youbelieve there is potential evidence of discrimination/bias in the way PAT’s are designed or adminis-tered?5 Section D: Working on a Model of PAT Scores in Stata5.1 Show you can generate your own results using codeIn this section you will indicate the code you would plan to use in Stata to achieve some basic tasks.This will draw on the sort of knowledge contained in labs, lectures, the data project and the helpsession you have received with Stata code that you can refer back to. For each question below youshould provide some basic Stata code that could be run and would achieve the results requested.There is often multiple correct ways to approach the coding, some more ecient than others, butthe only consideration will be if the actual desired outcome is achieved. Note, you do not need toactually run the code on a data set just indicate what you believe to be a correct approach but youcan assume you already have the variables indicated in this assignment loaded and ready in yourStata program.Section D Questions:1. Transform the GPA variable into a new variable measuring the natural log of GPA called LNGPA2. Run a regression of LNPAT on LNGPA3. Scatter LNPAT against LNGPA and display the line of best t (linear regression line) for the modelyou just estimated4. Calculate the residuals and create a new variable for them called RES5. Calculate the tted values and create a new variable for them called YHAT6. Scatter the residuals (RES) against the tted values (YHAT) to check for any issues7. Run a new regression of LNPAT on LNGPA , AP, MALE, ENG8. At the 1% level of sig, test whether the true coecient on MALE could be equal to ENG9. Test for speci cation error in the regression you ran10. Test for heteroskedasticity in the regression you ran6 Appendix:6.1 Section A Estimated ModelsRegression Model 1:-20-20120-1001000101012020R2e0sidualsResidualsR4e0siduals4045050560606707078080F8i0tted valuesFitted valuesFitted Regression Model 2:-20-20120-100100010101202023030R3e0sidualsResidualsR4e0siduals4045050560606707078080F8i0tted valuesFitted valuesFitted Regression Model 3:-20-20120-1001000101012020R2e0sidualsResidualsR4e0siduals4045050560606707078080F8i0tted valuesFitted valuesFittedRegression Model 4:-20-20120-1001000101012020R2e0sidualsResidualsR5e0siduals50560606707078080F8i0tted valuesFitted valuesFitted Regression Model 5:-20-20120-1001000101012020R2e0sidualsResidualsR4e0siduals4045050560606707078080F8i0tted valuesFitted valuesFitted Regression Model 6:-20-20120-10010001010120202303030 Residuals Residuals 40404505056060670707808089090F9i0tted valuesFitted valuesFittedRegression Model 7:-20-20120-100100010101202023030R3e0sidualsResidualsR4e0siduals4045050560606707078080F8i0tted valuesFitted valuesFitted Regression Model 8:-20-20120-1001000101012020R2e0sidualsResidualsR4e0siduals4045050560606707078080F8i0tted valuesFitted valuesFitted Regression Model 9:-20-20120-100100010101202023030R3e0sidualsResidualsR4e0siduals404505056060670707808089090F9i0tted valuesFitted valuesFittedRegression Model 10:-20-20120-1001000101012020R2e0sidualsResidualsR4e0siduals404505056060670707808089090F9i0tted valuesFitted valuesFitted Regression Model 11:-20-20120-1001000101012020R2e0sidualsResidualsR4e0siduals4045050560606707078080F8i0tted valuesFitted valuesFitted Regression Model 12:-20-20120-100100010101202023030R3e0sidualsResidualsR4e0siduals404505056060670707808089090F9i0tted valuesFitted valuesFittedRegression Model 13:-20-20120-1001000101012020R2e0sidualsResidualsR4e0siduals404505056060670707808089090F9i0tted valuesFitted valuesFitted Regression Model 14:-20-20120-1001000101012020R2e0sidualsResidualsR4e0siduals4045050560606707078080F8i0tted valuesFitted valuesFitted Regression Model 15:-20-20120-100100010101202023030R3e0sidualsResidualsR4e0siduals404505056060670707808089090F9i0tted valuesFitted valuesFittedRegression Model 16:-20-20120-1001000101012020R2e0sidualsResidualsR4e0siduals404505056060670707808089090F9i0tted valuesFitted valuesFitted Regression Model 17:-20-20120-1001000101012020R2e0sidualsResidualsR4e0siduals4045050560606707078080F8i0tted valuesFitted valuesFitted Regression Model 18:-20-20120-1001000101012020R2e0sidualsResidualsR4e0siduals4045050560606707078080F8i0tted valuesFitted valuesFittedRegression Model 19:-20-20120-1001000101012020R2e0sidualsResidualsR4e0siduals4045050560606707078080F8i0tted valuesFitted valuesFitted Regression Model 20:-20-20120-1001000101012020R2e0sidualsResidualsR4e0siduals4045050560606707078080F8i0tted valuesFitted valuesFitted6.2 Section B Scatter Plots40404505056060670707808089090P9A0TPATP1AT11222333444555GPAGPA4G0PA405060708090PATP0AT00.2.2.24.4.46.6.68.8.1811APMATHAPMATH4A0PMATH405060708090PAT0.2.4.6.811APENGAPENG4A0PENG405060708090PAT0.2.4.6.811APAP4A0P405060708090PAT0.2.4.6.811MALEMALE4M0ALE405060708090PAT0.2.4.6.811WHITEWHITE4W0HITE405060708090PAT0.2.4.6.811ENGENGE4N0G405060708090PAT0.2.4.6.811PREPPREPP4R0EP405060708090PATP2A0T05200520010520102010520152012052020Y2E0A2R0YEARYEAR6.3 Section C Additional Model EstimationsRegression Model A:Regression Model B:
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