Exercise 2 MLR Intro v13

Exercise 2: MLR Analysis I Note : Feel free to work in Stata and/or Excel. Be sure to include snapshots of your final Excel/Stata analyses with your Answers… but no need to include copies of your datasets. MLR I: KBO and Moneyball – Take II In Exercise #1, you estimated a SLR model that explored the relationship between RS/RA ( runs scored and runs allowed ) and %wins in Korean baseball. That model, for which 2 .90 R = , assumed a linear relationship, and yielded the following SRF:  ( ) % .03 .46 / wins RS RA = + . On closer inspection (see chart below), however, it sure looks like there’s a bit of curvature to the relationship, with %wins slightly concave in RS/RA. You’ll now be adding a second explanatory variable, ( ) 2 / RS RA , to the model to capture some of these non-linear effects. More specifically, you’ll be estimating a MLR model is which % wins is now regressed on two explanatory variables, / x RS RA = and ( ) 2 / z RS RA = . To fix notation, 0 ˆ ˆ ˆ ˆ x z y x z β β β = + + ( ) ( ) 2 0 ˆ ˆ ˆ / / x z RS RA RS RA β β β = + will be the estimated SRF for your MLR model. 1. Grab your KBO dataset from Exercise #1 and estimate the SLR (from Exercise #1) and MLR (above) models. Use eststo/esttab to capture and present the results (include the 2 ' R s , adjusted 2 ' R s and RMSE’s in the esttab output), 1 and record the 2 ' R s , adjusted 2 ' R s and RMSE’s on the Answer Sheet. 2. Plot the SLR and MLR SRF’s using one scatterplot (so you can compare the results), with ( ) / x RS RA = on the horizontal axis. 2 Looking at your scatterplot, would you say that the SRF changed in a meaningful way when ( ) 2 / z RS RA = was added to the model? (We call this the eyeball test .) 3. Under one interpretation of the MLR coefficients, the estimated coefficients capture the average relationship between changes in the specific explanatory variable and the 1 Your Stata syntax might be something like: esttab, r2 ar2 scalar(rmse) . 2 Your Stata syntax might be something like: scatter yhat1 yhat2 x1 . .3 .4 .5 .6 .7 %wins .6 .8 1 1.2 1.4 RS/RA

Exercise #2: MLR Analysis 2 dependent variable, ceteris paribus . Explain why that particular interpretation makes zero sense for your MLR model. 4. For your MLR model, estimate the two associated SLR collinearity regressions (regress ( ) / x RS RA = on ( ) 2 / z RS RA = , and vice-versa ) and record the estimated slopes and the 2 ' R s of the regressions on your Answer Sheet. a. You should have found that the 2 ' R s for the two collinearity regressions are identical. But you are not surprised. Explain why. b. Use the results from the estimated MLR model and the collinearity regressions to estimate the change in the z coefficient estimate if x is dropped (omitted) from the OLS/MLR model. Enter the full expression on the Answer Sheet… do not simplify the expression (we want to make sure you know how to compute omitted variable bias ). 5. Continuing to focus on the MLR results: Elasticity:  % (% ) % ( / ) wins RS RA ε ∆ = ∆ a. Compute the means of % wins and / RS RA , and record those on the Answer Sheet. b. Compute the value of the MLR SRF at the mean of / RS RA … so when ( ) ( ) 2 / / x RS RA and z RS RA   = =   (the square of the mean of RS/RA). 3 c. Use the fact that 2 0 ˆ ˆ ˆ ˆ x z y x x β β β = + + , where / x RS RA = , and ˆ ˆ ˆ 2 x z dy x dx β β = + to compute ˆ dy dx at ( ) / x RS RA = , which is ˆ ˆ 2 x z x β β + . d. Once you have ˆ dy dx at x , estimate the elasticity of the SRF at x , and record your answer on the Answer Sheet. Hint: Avoid the margins command here… it’ll only get you into trouble. 3 Note: This is not the same as the mean of [(RS/RA) 2 ].

Exercise #2: MLR Analysis 4 Do you think that maybe Apple Music streaming might have impacted Apple iTunes track sales? Just maybe! We did find an apparent Spotify displacement effect in Exercise #1. But maybe we need to consider an Apple Music impact as well, and worry about Omitted Variable Bias ( Endogeneity ) and the possibility that our estimated displacement effect in Exercise #1 was biased by the omission of an Apple Music effect in the SLR analysis. 9 Just thinking. And indeed, Apple Music proved ultimately to be the death of iTunes. In early June, 2019, Apple announced that beginning with the rollout of the new Catalina Mac operating system, which debuted in October, 2019, iTunes was being parsed into three apps, a music app, AppleMusic, an Apple TV app and an Apple Podcast app (oh, and there's a fourth book app… don’t get too excited!). 10 Under the new system, users can still access iTunes through AppleMusic, though that's a bit of work, and some claim that the iTunes Store is hidden by default with the new operating system. Oh iTunes, lost to history… How quickly they forget you! So let’s look into this Apple Music effect. But there’s a data problem. While we have weekly data for US Spotify streams and iTunes track sales, we have zero zilch nada Apple Music streams data. You’ll be dealing with this deficiency in two ways: • Way #1 - Dummy Variable (aplm): Introduce an Apple-Music-in-the-house dummy variable, which is equal to 0 through June 2015, and takes on the value 1 thereafter (so it takes on the value one starting the week of July 2, 2015). • Way #2 – Proxy Trend Variable (aplmtrend): Assume that US Apple Music streaming during the relevant time period is proportional to the number of Apple Music’s worldwide subscribers. And since the chart above suggests a fairly linear increase in subscriptions over time, create a trend variable that is 0 through June 2015, and increasing in a linear fashion through October, 2019. So assuming a linear relationship, the trend variable takes on the value 0 prior to the week of 7/2/2015, 1 in that week, 2 in the following week… and continues to increment by one unit each week moving forward. ( Be careful : Make sure your trend variable increments by 2 units if there's a skipped week in the data.) 9 I got the idea to add Apple Music to this Exercise from Kasey Hague, Samantha Quinn and Will Torsiglieri, who did just that in their EC228 research project in the Fall of 2016. 10 https://support.apple.com/en-us/HT210200

Exercise #2: MLR Analysis 5 Let’s get busy! You’ll be re-estimating your Exercise #1 SLR model (using the full time frame), which had but one explanatory variable, Spotify streams. And then considering several MLR models, which include a Way #1 – Apple Music Dummy ( aplm ) effect and a Way #2 – Apple Music Trend ( aplmtrend ) effect. Grab your Spotify v. iTunes dataset from Exercise #1, and add in your two new variables, aplm and aplmtrend . Grab the data through May, 2019... stopping just prior to Apple's announcement that iTunes would become an Apple Music sideshow with the release of Catalina in October. 6. Estimate the Exercise #1 SLR model and the three new MLR models: a. MLR A : Add aplm - reg itunes spotify aplm b. MLR B : Add aplmtrend - reg itunes spotify aplmtrend c. MLR C : Add both - reg itunes spotify aplm aplmtrend As before, use eststo/esttab to capture and present the results, and record the 2 ' R s , adjusted 2 ' R s and RMSE’s on the Answer Sheet. 7. You see that 2 R always increases when you add new explanatory variables to your models. But you knew that would happen. Explain why. 8. Which of the four estimated models ( SLR and MLR A-C ), do you think is best for predicting iTunes sales, and why? 9. Turning to MLB C , estimate the collinearity regression in which the explanatory variable aplmtrend is regressed on spotify and aplm, the other explanatory variables in the model. a. Use the results from the estimated MLR C model and the just estimated collinearity regression to predict the change in the estimated spotify coefficient estimate when aplmtrend is dropped (omitted) from MLR C . Enter the full expression on the Answer Sheet… do not simplify the expression (we want to make sure you know how to compute omitted variable bias ). b. Of course you have the spotify coefficients in the MLR A and MLR C models. What was the actual change in that estimated coefficient in going from MLR C to MLR A, when aplmtrend was dropped from MLR C? Just as you predicted, yes? 10. Turning to MLR B, use the margins command in Stata to estimates elasticities at the means for the two RHS variables. Record those on your Answer Sheet. Which of the two effects has the larger (in magnitude) elasticity… and accordingly, the more meaningful estimated impact on the predicted values of the dependent variable? 11. Staying with MLR B, run a beta regression to get a sense of which of the two explanatory variables has more explanatory power. Record the estimated coefficients on your Answer Sheet. Which of the two RHS variables has the larger estimated coefficient…

Exercise #2: MLR Analysis 7 largely driven by the controversial and subjective Corruption Perceptions Index . 12 The title of his piece hints at Nate Silver’s findings. The data for this analysis are in the posted sovdebt Excel workbook. 13 In your regressions below: • the dependent variable will be NSrate, • your first explanatory variable will be corrupt ( www.transparency.org ’s Corruption Perception Index ; note that higher values of the index correspond to less corruption ( go figure! )), • and the other RHS variables are lngdp , inflation deficit_gdp, debt_gdp . I have separately posted details on the construction of this dataset. This is a favorite coefficient model, with corrupt being the explanatory variable in the spotlight . Working with the posted workbook, build your final Stata dataset by merging together the data from the various worksheets. Start with the crosswalk worksheet data, which matches up the different country names from the different data sources (S&P, transparency.org and the IMF), and merge on the other data using the appropriate country name. You can merge the datasets in Excel using the vlookup (.) command, or in Stata using the merge command. Note that you’ll need to: • LHS variable: NSRate . Convert the alpha S&P debt ratings ( Foreign Currency Ratings ) to a numeric NSrate score using the provided worksheet… courtesy of Nate Silver. • RHS variables: corrupt , lngdp , inflation , deficit_gdp , and debt_gdp . Create lngdp , inflation , deficit_gdp , and debt_gdp working with the IMF data: - gdp = NGDPD (GDP (current prices; $US))… ln( ) lngdp gdp = , - inflation = PCPIPCH (average consumer prices; annual percent change), - deficit_gdp = -GGXCNL_NGDP (Net Lending per GDP; the negative of Deficit per GDP), and - debt_gdp = GGXWDG_NGDP (Gross Debt per GDP). 14. Estimate the SLR (simple favorite coefficient) model with only corrupt , the variable in the spotlight , on the RHS. Record your results on the Answer Sheet. 15. Estimate the complete MLR model (include all of the other RHS variables) and record the new estimated (favorite) coefficient for the variable in the spotlight ( corrupt ). 12 http://www.transparency.org/ 13 Standard and Poors now provides a sense of the type of data it considers when rating sovereign debt: https://www.spratings.com/sri/ … and has posted a general description of their methodology: https://www.spratings.com/documents/20184/4432051/Sovereign+Rating+Methodology/5f8c852c-108d-46d2-add1- 4c20c3304725 . The company was not always so forthcoming.

Exercise #2: MLR Analysis 8 a. Would you say that your estimated favorite coefficient was impacted much by the inclusion of additional explanatory variables in the analysis? b. Did adjusted R-squared increase or decrease in going from the SLR model to the full MLR model? And the same question for MSE/RMSE? c. You should find that adjusted R-squared and RMSE moved in opposite directions. Did you? 16. Run vif after the full MLR regression to generate the 2 ' j R s for the RHS variables. a. Which explanatory variable is most collinear with the other RHS variables? What is its 2 j R ? b. And which is the least collinear with the other RHS variables? … and what is its 2 j R ? 17. Run the collinearity regression for the lngdp and use the regression results to predict the change in the OLS coefficient for the variable in the spotlight ( corrupt ) if lngdp is dropped from the full MLR model. Enter you answer on the Answer Sheet (do not simplify the expression… we want to make sure you know how to compute omitted variable bias ).