{"text_format":"1","workbook":"xlsx","n_sheets":15,"sheets":["Intro","Geometry","OLS","Ortho","Doc","SolverOLS","SolverOrtho","MCSim6","CEM","EIV","TheEnd","OneVarMCSim","TwoVarMCSim","MCSim7","MCSim8"],"not_rendered":{"charts":10,"images":6,"macros":1,"external_links":1,"defined_names":170,"comments":0,"pivot_tables":0},"unopened":[],"bounds":{"document_bytes":16777216,"sheet_cells":100000,"rows_first":2000,"rows_last":500,"rows_between":2000,"halvings":0,"cell_chars":16384,"cell_head_chars":4096},"cells_cut":0,"cells_raw":0,"read":"whole","date_system":"1900"}

{"sheet":"Intro","index":0,"n_rows":47,"n_cols":7,"rows_kept":47,"cut":false,"hidden":false,"merged":[]}
[null,null,null,null,null,null,"One-shot Introduction to Orthogonal Regression"]
[null,null,null,null,null,null,"by"]
[null,null,null,null,null,null,"Humberto Barreto"]
[null,null,null,null,null,null,"hbarreto@depauw,edu"]
[]
["Orthogonal regression is related to ordinary least squares (OLS) regression in that both minimize the sum of squared residuals (SSR)."]
["It is most commonly known as total least squares (TLS) (there are other names), but orthogonal regression clearly tells you what is going on."]
["From Greek, ortho means right (like orthodox) and gonal means angle (like diagonal) and that is exactly what orthogonal regression does."]
[]
["Orthogonal regression is an alternative to OLS and this workbook will explain how it fits a line and explore how it performs under varying data generating processes (DGP)."]
[]
["This Excel workbook is completely self-contained and all you have to do is read (scrolling down as needed), follow instructions (especially obey the CLICK commands), and pay close attention."]
[]
["Student Background:"]
[null,"This material is accessible to you only if you know what regression analysis is and can interpret coefficients and regression statistics (such as SE, RMSE, and R2)."]
[null,"Remember that the OLS fitted line is found by minimizing the SSR, but there are many other ways (recipes) to fit lines to data."]
[null,"The Gauss-Markov Theorem says that OLS is BLUE if certain conditions apply, such as a DGP with fixed Xs and independently, identically distributed errors (iid). "]
[]
[null,"If the statements above mean nothing to you, this workbook is not appropriate for your knowledge level."]
[]
["Software Requirements:"]
[null,"You need Microsoft EXCEL, WINDOWS DESKTOP version 2007 or greater to get full functionality from this file."]
[null,null,"The file can be opened with Mac Excel, but it uses Visual Basic code that may not be fully supported on a Mac. "]
[null,null,"This file will NOT work with a browser-version of Excel because it contains macros. You must open it with a DESKTOP-version of Excel."]
[]
[null,"You must be sure to Enable Content or the buttons and code in this macro-enabled workbook will not work."]
[]
["Let's see if you are ready to go -- CLICK the button below."]
[]
[]
["If the button above does not display a message on your screen when you click it, please close the workbook and open it again, making sure that you Enable Content."]
[null,"Search \"enable macros\" in Excel's Help for more information on how to do this."]
[null,"You can also click the link below for help from the internet:"]
[null,"www.google.com/search?q=excel+enable+macros"]
[]
["You also need Excel's Solver. Click the Data tab in the menu and make sure it is in the top-right corner."]
[null,"If Solver is not installed, press alt (release), then t (release), then I to bring up the Add-ins Manager and check Solver and click OK. "]
[]
["Once you get the Click to Test button (above) to work and you have Solver installed, you are ready to begin."]
[]
["You will reveal new sheets in the workbook as you proceed through the material. Your job is to read carefully and pay attention to the charts and think about what you are doing."]
[]
["Keep an eye out for bold, red text -- it always means you are supposed to do something!"]
[]
["Also, answer the questions as best you can -- they are not meant to test you, but to keep you focused and engaged. The answer is usually pretty easy or revealed right after the question."]
[]
["CLICK on the button below to proceed to the Geometry sheet when you are ready."]

{"sheet":"Geometry","index":1,"n_rows":11,"n_cols":9,"rows_kept":11,"cut":false,"hidden":true,"merged":[]}
["As they say, a picture is worth a thousand words."]
[]
[]
[null,null,null,null,null,null,null,null,"OLS: Orange"]
[null,null,null,null,null,null,null,null,"Residuals measured as VERTICAL distances."]
[null,null,null,null,null,null,null,null,"So, the residual of the third point is a little over 2 units of Y (5 - a little less than 3)."]
[]
[null,null,null,null,null,null,null,null,"Orthogonal regression: Green"]
[null,null,null,null,null,null,null,null,"Residuals measured as PERPENDICULAR distances."]
[null,null,null,null,null,null,null,null,"The residual is the CLOSEST distance to the fitted line"]
[null,null,null,null,null,null,null,null,"That green line is at a RIGHT ANGLE (90 degrees) from the blue point to the fitted line."]

{"sheet":"OLS","index":2,"n_rows":28,"n_cols":15,"rows_kept":28,"cut":false,"hidden":true,"merged":[]}
["OLS","X","Y",null,"OLS b1","OLS b0",null,0,"OLS b0","OLS Pred","OLS res","OLS res^2",null,"The OLS residual is simply actual - predicted."]
[null,1,2,null,{"$f":"LINEST(C2:C6,B2:B6,1,1)","v":0.19999999999999996},2.2400000000000002,null,0.5,"OLS b1",{"$f":"$H$1+$H$2*B2","v":0.5},{"$f":"C2-J2","v":1.5},{"$f":"K2^2","v":2.25},null,"Click on cell K2 to see its formula (K3:K7 are the same)"]
[null,2,2.1,null,0.45062918976323163,1.4945679420265019,null,null,null,{"$f":"$H$1+$H$2*B3","v":1},{"$f":"C3-J3","v":1.1000000000000001},{"$f":"K3^2","v":1.2100000000000002},null,"This sheet offers FOUR ways to run an OLS regression:"]
[null,3,5,null,6.1614294516327842E-2,1.4250146198080447,null,null,null,{"$f":"$H$1+$H$2*B4","v":1.5},{"$f":"C4-J4","v":3.5},12.25,null,"1) LINEST -- OLS b1 in cell E2"]
[null,4,2.1,null,0.19697964543663843,3,null,null,null,{"$f":"$H$1+$H$2*B5","v":2},{"$f":"C5-J5","v":0.10000000000000009},1.0000000000000018E-2,null,"2) Matrix algebra -- OLS b1 in cell A28"]
[null,5,3,null,0.40000000000000036,6.0919999999999996,"OLS SSR",null,null,{"$f":"$H$1+$H$2*B6","v":2.5},{"$f":"C6-J6","v":0.5},0.25,null,"Run Solver (in the Data tab) to see how Solver's optimization algorithm can be used to minmize SSR"]
[null,null,null,null,null,null,null,null,null,null,null,{"$f":"SUM(L2:L6)","v":15.97},"SSR","3) Solver -- OLS b1 in cell H2"]
["Include an intercept term for matrix algebra computations:",null,null,null,null,null,null,null,null,null,null,null,null,"The orange fitted line in the chart is now minimizing SSR"]
[null,"int","X","Y",null,null,null,null,null,null,null,null,null,"Right-click a blue dot in the chart and ADD a Trendline, report the equation"]
[null,1,{"$f":"B2","v":1},{"$f":"C2","v":2},null,null,null,null,null,null,null,null,null,"4) Trendline"]
[null,1,{"$f":"B3","v":2},2.1]
[null,1,3,5,null,null,null,null,null,null,null,null,null,null,"Q1: Is the Trendline you added the same as the orange line? What does that mean?"]
[null,1,4,2.1]
[null,1,5,3]
[]
[null,"(X'X)"]
[null,{"$f":"MMULT(TRANSPOSE(B10:C14),B10:C14)","v":5},15]
[null,15,55]
[null,"(X'X)-1"]
[null,{"$f":"MINVERSE(B17:C18)","v":1.1000000000000005},-0.3000000000000001]
[null,-0.3000000000000001,0.10000000000000003]
[]
[null,"(X'X)-1X'"]
[null,{"$f":"MMULT(B20:C21,TRANSPOSE(B10:C14))","v":0.80000000000000049},0.50000000000000033,0.20000000000000018,-9.9999999999999867E-2,-0.39999999999999991]
[null,-0.20000000000000007,-0.10000000000000003,0,0.10000000000000003,0.20000000000000012]
[null,"(X'X)-1X'y"]
[null,{"$f":"MMULT(B24:F25,D10:D14)","v":2.2400000000000033},"OLS b0"]
[null,0.20000000000000018,"OLS b1","agrees with LINEST"]

{"sheet":"Ortho","index":3,"n_rows":28,"n_cols":19,"rows_kept":28,"cut":false,"hidden":true,"merged":[]}
["Ortho reg","X","Y",null,"OLS b1","OLS b0",null,2.2399923496177387,"OLS b0","OLS Pred","OLS res","OLS res^2",null,0.5,"Ortho b0","Ortho Pred","Ortho res","Ortho res^2"]
[null,1,2,null,{"$f":"LINEST(C2:C6,B2:B6,1,1)","v":0.19999999999999996},2.2400000000000002,null,0.2000020429087801,"OLS b1",{"$f":"$H$1+$H$2*B2","v":2.4399943925265188},{"$f":"C2-J2","v":-0.43999439252651884},{"$f":"K2^2","v":0.19359506545478034},null,0.5,"Ortho b1",{"$f":"$N$1+$N$2*B2","v":1},{"$f":"(C2-P2)/(SQRT(1+$N$2^2))","v":0.89442719099991586},{"$f":"Q2^2","v":0.79999999999999993}]
[null,2,2.1,null,0.45062918976323163,1.4945679420265019,null,null,null,{"$f":"$H$1+$H$2*B3","v":2.6399964354352989},{"$f":"C3-J3","v":-0.53999643543529885},{"$f":"K3^2","v":0.29159615028282887},null,null,null,{"$f":"$N$1+$N$2*B3","v":1.5},{"$f":"(C3-P3)/(SQRT(1+$N$2^2))","v":0.53665631459994956},{"$f":"Q3^2","v":0.28800000000000003}]
[null,3,5,null,6.1614294516327842E-2,1.4250146198080447,null,null,null,{"$f":"$H$1+$H$2*B4","v":2.839998478344079},{"$f":"C4-J4","v":2.160001521655921},4.6656065735558938,null,null,null,2,2.6832815729997477,7.2]
[null,4,2.1,null,0.19697964543663843,3,null,null,null,{"$f":"$H$1+$H$2*B5","v":3.0400005212528591},{"$f":"C5-J5","v":-0.94000052125285904},0.88360097995564668,null,null,null,2.5,-0.35777087639996624,0.12799999999999992]
[null,5,3,null,0.40000000000000036,6.0919999999999996,"OLS SSR",null,null,{"$f":"$H$1+$H$2*B6","v":3.2400025641616392},{"$f":"C6-J6","v":-0.24000256416163923},5.7601230804161754E-2,null,null,null,3,0,0]
[null,null,null,null,null,null,null,null,null,null,null,{"$f":"SUM(L2:L6)","v":6.0920000000533117},"SSR",null,null,null,null,{"$f":"SUM(R2:R6)","v":8.4160000000000004},"SSR"]
["Include an intercept term for matrix algebra computations:",null,null,null,null,null,null,null,null,null,null,null,null,"To fit the Orthogonal regression line, we need to modify the residual"]
[null,"int","X","Y",null,null,null,null,null,null,null,null,null,"Click on cell Q2 to see how the residual is being divided by an adjustment term (Q3:Q7 are the same)"]
[null,1,{"$f":"B2","v":1},{"$f":"C2","v":2},null,null,null,null,null,null,null,null,null,"It's complicated, but the intuition is that we need to use the Pythagorean Theorem, hence the SQRT function"]
[null,1,{"$f":"B3","v":2},2.1,null,null,null,null,null,null,null,null,null,"We don't really care right now about the mechanics of actually computing the orthogonal fitted line."]
[null,1,3,5,null,null,null,null,null,null,null,null,null,"Software will do that for us. We do want to understand that it is a different recipe than OLS."]
[null,1,4,2.1,null,null,null,null,null,null,null,null,null,"Run Solver to fit the green orthogonal fitted line."]
[null,1,5,3]
[]
[null,"(X'X)",null,null,null,null,null,null,null,null,null,null,null,null,"Q2: Are OLS and orthogonal regression different? Why?"]
[null,{"$f":"MMULT(TRANSPOSE(B10:C14),B10:C14)","v":5},15]
[null,15,55]
[null,"(X'X)-1"]
[null,{"$f":"MINVERSE(B17:C18)","v":1.1000000000000005},-0.3000000000000001]
[null,-0.3000000000000001,0.10000000000000003]
[]
[null,"(X'X)-1X'"]
[null,{"$f":"MMULT(B20:C21,TRANSPOSE(B10:C14))","v":0.80000000000000049},0.50000000000000033,0.20000000000000018,-9.9999999999999867E-2,-0.39999999999999991]
[null,-0.20000000000000007,-0.10000000000000003,0,0.10000000000000003,0.20000000000000012]
[null,"(X'X)-1X'y"]
[null,{"$f":"MMULT(B24:F25,D10:D14)","v":2.2400000000000033},"OLS b0"]
[null,0.20000000000000018,"OLS b1","agrees with LINEST"]

{"sheet":"Doc","index":4,"n_rows":29,"n_cols":2,"rows_kept":29,"cut":false,"hidden":true,"merged":[]}
["2025-09-02"]
["Ideas"]
[]
["Start with SolverOLS"]
[null,"Shows how to compute OLS fitted line with Solver, LINEST, matrix"]
[]
["SolverOrtho shows Solver for orthogonal reg"]
[null,"Ortho residual = (actual - pred)/sqrt(1+b1^2)"]
[null,"It can be found by the eigenvector of the X-Y var-covar matrix"]
[null,"Should be possible to make a function out of this (SEE BELOW)"]
[null,"Could extend to multivar case but not needed if just a demo of OLS v Ortho"]
[]
["LiveOLSOrtho is fixed Xs and LiveOLSOrtho (2) is random Xs (Name Changed to LiveOLSOrthoEIV)"]
[null,"Puzzling sim results: Ortho OLS biased with fixed Xs? Does even worse with random Xs?"]
[null,"THIS HAPPENED BECAUSE DGP WAS WRONG, I HAD Y DEPENDING ON OBS X, NOT TRUE X"]
["Ordered a book to get some help on this."]
[]
["2025-09-11"]
["ChatGPT a big help, book is OK"]
["The total least squares problem : computational aspects and analysis by Sabine van Huffel; J Vandewalle"]
[]
["Key is the DGP, for ortho, y is generated by true X, not observed X"]
[]
["LiveOLSOrtho shows OLS better when no error in fixed Xs (classical model)"]
[]
["LiveOrthoOLSEIV has measurement error in X (errors in variables) so observed X is random"]
["Results show clearly that OLS b1 and est SE are biased and that Ortho is unbiased"]
[]
["The function orthoreg(known Y, known X) produces intercept and slope for orthogonal bivariate regression"]

{"sheet":"SolverOLS","index":5,"n_rows":36,"n_cols":12,"rows_kept":36,"cut":false,"hidden":true,"merged":[]}
[null,"X","Y",null,{"$f":"LINEST(C2:C6,B2:B6,1,1)","v":9.3695665584804342},19.474780694340012,null,19.474817374102013,"OLS b0","OLS Pred","OLS res","OLS res^2"]
[null,1,29.256106234167866,null,2.7332815618059483,9.0652693869069321,null,9.3695554488582733,"OLS b1",{"$f":"$H$1+$H$2*B2","v":28.844372822960288},{"$f":"C2-J2","v":0.41173341120757811},{"$f":"K2^2","v":0.16952440190462861}]
[null,2,32.478159422990274,null,0.79662225344899384,8.6433952218490866,null,null,null,{"$f":"$H$1+$H$2*B3","v":38.21392827181856},{"$f":"C3-J3","v":-5.7357688488282861},{"$f":"K3^2","v":32.899044287188964}]
[null,3,59.116984945150399,null,11.750876390734394,3,null,null,null,47.583483720676838,11.533501224473561,133.02165049493314]
[null,4,49.446265017065684,null,877.88777493794942,224.12484288325089,null,null,null,56.953039169535103,-7.5067741524694185,56.351658176182958]
[null,5,67.619886229532341,null,null,null,null,null,null,66.322594618393381,1.2972916111389594,1.6829655243315171]
[null,null,"=10*X+RAND*20",null,null,null,null,null,null,null,null,{"$f":"SUM(L2:L6)","v":224.12484288454121}]
[null,{"$f":"AVERAGE(B2:B6)","v":3},{"$f":"AVERAGE(C2:C6)","v":47.583480369781313}]
[null,"centered (subtract means)"]
[null,"x","y",null,{"$f":"LINEST(C11:C15,B11:B15,1,1)","v":9.3695665584804342},0]
[null,{"$f":"B2-B$8","v":-2},{"$f":"C2-C$8","v":-18.327374135613447},null,2.7332815618059483,3.8654438544902865]
[null,{"$f":"B3-B$8","v":-1},-15.105320946791039,null,0.79662225344899384,8.6433952218490866]
[null,0,11.533504575369086,null,11.750876390734394,3]
[null,1,1.8627846472843714,null,877.88777493794942,224.12484288325089]
[null,2,20.036405859751028,null,"to recover the intercept: avg y - b1*avg x"]
[null,null,null,null,{"$f":"C8-E10*B8","v":19.474780694340012},"check"]
["int","X","Y"]
[1,{"$f":"B2","v":1},{"$f":"C2","v":29.256106234167866}]
[1,{"$f":"B3","v":2},32.478159422990274]
[1,3,59.116984945150399]
[1,4,49.446265017065684]
[1,5,67.619886229532341]
[]
["(X'X)"]
[{"$f":"MMULT(TRANSPOSE(A18:B22),A18:B22)","v":5},15]
[15,55]
["(X'X)-1"]
[{"$f":"MINVERSE(A25:B26)","v":1.1000000000000005},-0.3000000000000001]
[-0.3000000000000001,0.10000000000000003]
[]
["(X'X)-1X'"]
[{"$f":"MMULT(A28:B29,TRANSPOSE(A18:B22))","v":0.80000000000000049},0.50000000000000033,0.20000000000000018,-9.9999999999999867E-2,-0.39999999999999991]
[-0.20000000000000007,-0.10000000000000003,0,0.10000000000000003,0.20000000000000012]
["(X'X)-1X'y"]
[{"$f":"MMULT(A32:E33,C18:C22)","v":19.474780694340041}]
[9.3695665584804431,"check"]

{"sheet":"SolverOrtho","index":6,"n_rows":52,"n_cols":31,"rows_kept":52,"cut":false,"hidden":true,"merged":[]}
[null,"X","Y",null,{"$f":"LINEST(C2:C6,B2:B6,1,1)","v":0.5706943782303332},2.8112974337403154,null,19.474817374102013,"OLS b0","OLS Pred","OLS res","OLS res^2",null,1.7449855592468633,"Ortho b0","Ortho Pred","Ortho res","Ortho res^2",null,"Ortho Pred","Ortho res","Ortho res^2",null,"xx","xy","yx","yy",null,"x-avgx","y-avgy"]
[null,1,3.8003255384492616,null,0.44220557534323501,1.4666299736167456,null,9.3695554488582733,"OLS b1",{"$f":"$H$1+$H$2*B2","v":28.844372822960288},{"$f":"C2-J2","v":-25.044047284511027},{"$f":"K2^2","v":627.2043043888242},null,0.92613347987541939,"Ortho b1",{"$f":"$N$1+$N$2*B2","v":2.6711190391222828},{"$f":"(C2-P2)/(SQRT(1+$N$2^2))","v":0.82848159356111872},{"$f":"Q2^2","v":0.68638175086957076},null,{"$f":"$N$1+$N$2*B2","v":2.6711190391222828},{"$f":"(C2-T2)/(SQRT(1+$N$2^2))","v":0.82848159356111872},{"$f":"U2^2","v":0.68638175086957076},null,{"$f":"(B2-$B$8)*(B2-$B$8)","v":4},{"$f":"(B2-$B$8)*(C2-$C$8)","v":1.4461100599641066},{"$f":"(C2-$C$8)*(B2-$B$8)","v":1.4461100599641066},{"$f":"(C2-$C$8)*(C2-$C$8)","v":0.52280857638234801},null,{"$f":"B2-B$8","v":-2},{"$f":"C2-C$8","v":-0.72305502998205329}]
[null,2,2.2811293826353611,null,0.35698956798260123,1.398376812109817,null,null,null,{"$f":"$H$1+$H$2*B3","v":38.21392827181856},{"$f":"C3-J3","v":-35.932798889183196},{"$f":"K3^2","v":1291.1660360104852},null,null,null,{"$f":"$N$1+$N$2*B3","v":3.5972525189977018},{"$f":"(C3-P3)/(SQRT(1+$N$2^2))","v":-0.96561948056977343},{"$f":"Q3^2","v":0.93242098125583905},null,{"$f":"$N$1+$N$2*B3","v":3.5972525189977018},{"$f":"(C3-T3)/(SQRT(1+$N$2^2))","v":-0.96561948056977343},{"$f":"U3^2","v":0.93242098125583905},null,{"$f":"(B3-$B$8)*(B3-$B$8)","v":1},{"$f":"(B3-$B$8)*(C3-$C$8)","v":2.2422511857959537},{"$f":"(C3-$C$8)*(B3-$B$8)","v":2.2422511857959537},{"$f":"(C3-$C$8)*(C3-$C$8)","v":5.027690380203361},null,{"$f":"B3-B$8","v":-1},{"$f":"C3-C$8","v":-2.2422511857959537}]
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[null,4,5.1525218608636898,null,3.2569207334370684,5.8663731259392442,null,null,null,56.953039169535103,-51.800517308671417,2683.293593445967,null,null,null,5.4495194787485408,-0.21790262444978015,4.7481553742101924E-2,null,5.4495194787485408,-0.21790262444978015,4.7481553742101924E-2,null,1,0.62914129243237493,0.62914129243237493,0.39581876584347914,null,1,0.62914129243237493]
[null,5,5.2181011904867631,null,null,null,null,null,null,66.322594618393381,-61.104493427906618,3733.7591170810833,null,null,null,6.3756529586239594,-0.84927808515747749,0.72127326592875163,null,6.3756529586239594,-0.84927808515747749,0.72127326592875163,null,4,1.3894412441108965,1.3894412441108965,0.482636742709109,null,2,0.69472062205544827]
[null,null,"=10*X+RAND*20",null,null,null,null,null,null,null,null,{"$f":"SUM(L2:L6)","v":10050.928351938181},null,null,null,null,null,{"$f":"SUM(R2:R6)","v":3.8378928514967283},null,null,null,{"$f":"SUM(V2:V6)","v":3.8378928514967283},null,{"$f":"SUM(X2:X6)","v":10},{"$f":"SUM(Y2:Y6)","v":5.7069437823033322},5.7069437823033322,9.1232938593763127,null,"Z'Z"]
[null,{"$f":"AVERAGE(B2:B6)","v":3},{"$f":"AVERAGE(C2:C6)","v":4.5233805684313149},"averages",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"X7/4","v":2.5},{"$f":"Y7/4","v":1.4267359455758331},1.4267359455758331,2.2808234648440782,null,{"$f":"MMULT(TRANSPOSE(AC2:AD6),AC2:AD6)","v":10},5.7069437823033322]
[null,"centered (subtract means)",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,5.7069437823033322,9.1232938593763127]
[null,"x","y",null,{"$f":"LINEST(C11:C15,B11:B15,1,1)","v":0.5706943782303332},-3.5527136788005011E-16,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Var-cov matrix of X and Y",null,null,null,"1/4Z'Z"]
[null,{"$f":"B2-B$8","v":-2},{"$f":"C2-C$8","v":-0.72305502998205329},null,0.44220557534323501,0.62537312200740036,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"X8","v":2.5},{"$f":"Y8","v":1.4267359455758331},null,null,{"$f":"AC8/4","v":2.5},{"$f":"AD8/4","v":1.4267359455758331}]
[null,{"$f":"B3-B$8","v":-1},-2.2422511857959537,null,0.35698956798260112,1.398376812109817,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"Z8","v":1.4267359455758331},{"$f":"AA8","v":2.2808234648440782},null,null,{"$f":"AC9/4","v":1.4267359455758331},{"$f":"AD9/4","v":2.2808234648440782},"check"]
[null,0,1.641444301290182,null,1.6655541661862563,3,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"eigenvector gives Ortho OLS b1"]
[null,1,0.62914129243237493,null,3.2569207334370667,5.8663731259392442,null,null,null,null,null,null,null,null,null,null,{"$f":"SQRT(1+X16^2)","v":1.4142135623730951},"perpendicular",null,null,null,null,null,{"$f":"MatEigenvector_max(Y11:Z12)","v":1},0.92613511006666682,null,null,null,{"$f":"SQRT(AC11)","v":1.5811388300841898},"sqrt vax(x)"]
[null,2,0.69472062205544827,null,"to recover the intercept: avg y - b1*avg x",null,null,null,null,null,null,null,null,null,null,null,{"$f":"K4/Q14","v":-29.287414541162736},"perpendicular/vertical",null,null,null,null,null,null,null,null,null,null,{"$f":"_xlfn.STDEV.S(B2:B6)","v":1.5811388300841898},"sample SD Excel formula"]
[null,null,null,null,{"$f":"C8-E10*B8","v":2.8112974337403154},"check",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"1/X14","v":1},"CHECK!",null,"by orthoreg function!"]
["int","X","Y",null,null,null,null,null,null,null,null,null,null,null,null,null,"Key is that fitted has a perpendicular line that is -1/b1",null,null,null,null,null,null,null,null,null,null,"CHECK!"]
[1,{"$f":"B2","v":1},{"$f":"C2","v":3.8003255384492616},null,null,null,null,null,null,null,null,null,null,null,null,null,"You compute that distance as hypotenuse of a right triangle",null,null,null,null,null,null,{"$f":"C8-X16*B8","v":1.5233805684313149},"CHECK!",null,null,"CHECK!"]
[1,{"$f":"B3","v":2},2.2811293826353611,null,null,null,null,null,null,null,null,null,null,null,null,null,"This makes Ortho residual = (yi - y-hat)/(sqrt(1+b1^2))"]
[1,3,6.1648248697214969]
[1,4,5.1525218608636898]
[1,5,5.2181011904867631,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Produced by ChatGPT"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"different numerical example (look at the axes)"]
["(X'X)"]
[{"$f":"MMULT(TRANSPOSE(A18:B22),A18:B22)","v":5},15]
[15,55,null,null,"var-covar of coefficients"]
["(X'X)-1",null,null,null,"s^2(X'X)-1"]
[{"$f":"MINVERSE(A25:B26)","v":1.1000000000000005},-0.3000000000000001,null,null,{"$f":"$F$12^2*A28","v":2.1510034795110569},{"$f":"$F$12^2*B28","v":-0.58663731259392449}]
[-0.3000000000000001,0.10000000000000003,null,null,{"$f":"$F$12^2*A29","v":-0.58663731259392449},{"$f":"$F$12^2*B29","v":0.1955457708646415},{"$f":"SQRT(F29)","v":0.44220557534323501},"check"]
[]
["(X'X)-1X'"]
[{"$f":"MMULT(A28:B29,TRANSPOSE(A18:B22))","v":0.80000000000000049},0.50000000000000033,0.20000000000000018,-9.9999999999999867E-2,-0.39999999999999991]
[-0.20000000000000007,-0.10000000000000003,0,0.10000000000000003,0.20000000000000012]
["(X'X)-1X'y"]
[{"$f":"MMULT(A32:E33,C18:C22)","v":2.8112974337403194}]
[0.57069437823033353,"check"]
[]
[]
[]
[]
[]
[]
[]
[]
[]
[]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"damn, so the chart with x axis 1,2,3,4 is"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"supper misleading."]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"The ortho residual REALLY is tiny b/c"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"fitted line is so steep. "]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"You can draw a perpendicular to the line"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"that is almost horizontal."]

{"sheet":"MCSim6","index":7,"n_rows":103,"n_cols":39,"rows_kept":103,"cut":false,"hidden":true,"merged":[[1,5,1,6],[3,8,3,9],[3,10,3,11],[3,12,3,13]]}
["Results of Monte Carlo Simulation",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"LiveOLSOrtho!$E$1","LiveOLSOrtho!$H$16"]
["Sample Number","LiveOLSOrtho!$E$1","LiveOLSOrtho!$H$16",null,null,"Simulation Stats",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-3500,null,null,-3500,0,0]
[1,10.512698537630493,19.247484254602174,null,null,1000,"repetitions",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-3400,null,null,-3500,0,1]
[2,7.478853624422201,9.1924835844561414,null,null,0.99999960511922836,"secs",null,"LiveOLSOrtho!$E$1",null,"LiveOLSOrtho!$H$16",null,"Notes",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-3300,null,null,-3400,0,1]
[3,11.850260058594886,22.267933084409691,null,null,null,null,null,"Average",9.9751535679350276,"Average",14.357400103379327,"OLS wins, Ortho  biased",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-3200,null,null,-3400,0,0]
[4,17.311836472144087,23.552977304754759,null,null,null,null,null,"SD",4.7197902434018442,"SD",113.76112356959388,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-3100,null,null,-3300,0,0]
[5,8.361196836917216,22.804372897584724,null,null,null,null,null,"Max",25.743244245060882,"Max",267.54376296888745,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-3000,null,null,-3300,0,0]
[6,6.296273689582474,19.473440206760458,null,null,null,null,null,"Min",-4.5453699027034435,"Min",-3499.4400623821134,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-2900,null,null,-3200,0,0]
[7,6.3040524698162423,8.5436909709405757,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-2800,null,null,-3200,0,0]
[8,12.366332045471655,13.545003833735979,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-2700,null,null,-3100,0,0]
[9,13.179435881773859,17.112803116372564,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-2600,null,null,-3100,0,0]
[10,12.86237916223468,14.972545180961939,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-2500,null,null,-3000,0,0]
[11,14.491209454821421,17.643929781159624,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-2400,null,null,-3000,0,0]
[12,10.696291890585316,14.537315894055045,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-2300,null,null,-2900,0,0]
[13,8.4130915785706328,8.9523545835576464,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-2200,null,null,-2900,0,0]
[14,14.501679465636039,16.73498189059692,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-2100,null,null,-2800,0,0]
[15,5.652574307303194,21.46310259626944,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-2000,null,null,-2800,0,0]
[16,9.805598584303695,17.224917153953843,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1900,null,null,-2700,0,0]
[17,2.5160084477049907,29.742100507888861,null,null,1,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1800,null,null,-2700,0,0]
[18,12.088915699587037,12.311046570481608,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1700,null,null,-2600,0,0]
[19,13.665141654991547,15.694398167757624,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1600,null,null,-2600,0,0]
[20,15.340922212861775,24.059252735160587,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1500,null,null,-2500,0,0]
[21,13.141998048050132,21.257372308354544,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1400,null,null,-2500,0,0]
[22,9.8719791096165839,11.873731429153656,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1300,null,null,-2400,0,0]
[23,5.934390143363709,14.151611477395724,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1200,null,null,-2400,0,0]
[24,5.8875452801285473,18.909445035654972,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1100,null,null,-2300,0,0]
[25,10.36183847297985,19.07472318585085,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1000,null,null,-2300,0,0]
[26,12.798179321850409,14.603822633823098,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-900,null,null,-2200,0,0]
[27,9.8835296220020794,10.600732060706298,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-800,null,null,-2200,0,0]
[28,16.781824277828857,20.361770069151067,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-700,null,null,-2100,0,0]
[29,-1.2561298597338675,-36.394522432818626,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-600,null,null,-2100,0,0]
[30,8.2569531672347143,11.266007385501679,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-500,null,null,-2000,0,0]
[31,12.291314475654946,17.014712293211939,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-400,null,null,-2000,0,0]
[32,6.6293091943685107,30.075518219451272,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-300,null,null,-1900,0,0]
[33,19.015013177381554,21.88406263762414,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-200,null,null,-1900,0,0]
[34,14.197980674543373,17.870107254098684,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-100,null,null,-1800,0,0]
[35,5.1904091037417359,22.677375601280158,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0,null,null,-1800,0,0]
[36,6.3025338207178772,13.306700190627922,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,100,null,null,-1700,0,0]
[37,-1.8139402373060678,-3.3654008798062867,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1700,0,0]
[38,10.601846141208689,21.4329729441274,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1600,0,0]
[39,15.752597225631687,20.747230360705174,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1600,0,0]
[40,14.005046261827333,18.584883084380085,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1500,0,0]
[41,14.329466430657609,16.9644801349021,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1500,0,0]
[42,12.981340871142947,16.837508898393629,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1400,0,0]
[43,16.042900175363897,20.048608688367118,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1400,0,0]
[44,12.646569845375145,26.839614330794493,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1300,0,0]
[45,5.5382955210301068,6.399863339916612,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1300,0,0]
[46,17.200094900903217,30.908061104436243,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1200,0,0]
[47,2.1503220405621986,33.744067682800676,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1200,0,0]
[48,12.924249586591468,22.73165411758773,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1100,0,0]
[49,15.030551049792402,15.423098573598731,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1100,0,0]
[50,6.8170000425540698,7.3308530582356397,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1000,0,0]
[51,9.4704648866736161,9.6504537827756884,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1000,0,0]
[52,3.9366775314821525,25.768230431096764,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-900,0,0]
[53,16.793948083036145,17.575416140838922,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-900,0,0]
[54,5.088044783279237,7.5785746528206879,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-800,0,0]
[55,16.685030414281623,18.278179401327929,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-800,0,0]
[56,10.114371769890809,13.058671219858621,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-700,0,0]
[57,11.055576448870013,12.634466316279228,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-700,0,0]
[58,7.5647028101486908,19.155662504721871,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-600,0,0]
[59,9.6699357834751662,27.948170111451702,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-600,0,0]
[60,4.6409881132354398,14.949485537186753,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-500,0,0]
[61,8.868497960547419,19.828696903628909,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-500,0,0]
[62,10.828857312396176,24.933121243862747,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-400,0,0]
[63,21.729997003211288,26.392506072136396,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-400,0,1]
[64,11.336719327217452,15.330559233321996,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-300,0,1]
[65,6.7251439645258477,16.323960419251556,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-300,0,1]
[66,8.4822306402472076,12.961056072349068,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-200,0,1]
[67,11.534055727090855,23.991305533617329,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-200,0,3]
[68,8.3389380351379163,10.845213403145809,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-100,0,3]
[69,13.027090793974866,17.895757872338272,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-100,22,16]
[70,13.267337432142739,19.181356762030934,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0,22,16]
[71,0.27053935731785322,111.57123841736014,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0,978,971]
[72,11.806089049917496,13.280884335776527,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,100,978,971]
[73,12.586122884603439,29.882362390018663,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,100,0,5]
[74,8.174532651796115,15.226127647273241,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,200,null,5]
[75,5.3642714818526311,24.142051010899461,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,200,null,2]
[76,2.3260989826174057,17.080154319612284,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,300,null,2]
[77,12.976926329545341,14.246798333526208,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,300,null,0]
[78,13.767885547539597,15.927222127157521]
[79,4.5948687665443098,13.856598481049366]
[80,16.805740007982195,17.697340103479032]
[81,10.67655120197748,31.475695542606196]
[82,14.158207239229473,19.160852704901231]
[83,2.1546149415894691,20.430358257495161]
[84,18.082592130802951,19.119550785967121]
[85,5.8411628670803202,15.640509887135705]
[86,-0.85847213135231459,-349.5948750521033]
[87,12.708798857742361,21.475264058872316]
[88,10.71213725278429,15.005978221925607]
[89,8.2562210504974107,29.130442747202387]
[90,9.814921167051418,12.178391377322436]
[91,13.571288563402705,18.907894279910316]
[92,11.391211295758451,18.592116850211951]
[93,8.7082172356335068,11.308511715442856]
[94,8.5848474564381316,13.177691665831334]
[95,8.4787666444599505,9.1418148272505171]
[96,13.195759919460766,18.747625490078967]
[97,9.0069958327784043,11.675690391504684]
[98,6.1060673622297443,28.744680560069202]
[99,18.672731849296568,20.002797384020806]
[100,11.374605443309861,21.375794862361815]
["Only the first 100 repetitions are displayed on this worksheet."]

{"sheet":"CEM","index":8,"n_rows":30,"n_cols":19,"rows_kept":30,"cut":false,"hidden":true,"merged":[]}
["Classical Econometric Model DGP: Y = b0 + b1X + e~N(0,sigma), fixed Xs",null,null,null,null,null,null,null,null,null,null,null,null,"We can't use Solver for a sim because we need instant update of the new solution for new Ys."]
[0,"beta0",null,null,null,null,null,null,null,null,null,null,null,"This workbook has a way to get the orthogonal regression coefficients without Solver."]
[1,"beta1",null,null,null,null,null,null,null,null,null,null,null,"The computation is quite advanced."]
[1,"sigma",null,null,null,null,null,null,null,null,null,null,null,"Below is a brief explanation of how the ortho coefficients are computed"]
[null,null,"X","Y","OLS b1",{"$f":"LINEST(D6:D10,C6:C10,1,1)","v":1.0761272456892674},-1.8773318207254697E-2,null,"xx","xy","yx","yy",null,"and the orthoreg(known Ys, known Xs) user-defined array function is in cells I29:I30."]
[null,null,1,{"$f":"beta0+beta1*C6+_xlfn.NORM.INV(RAND(),0,sigma)","v":1.7429093105972862},null,0.20217554614335137,0.67054042834268113,null,{"$f":"(C6-$C$12)*(C6-$C$12)","v":4},{"$f":"(C6-$C$12)*(D6-$D$12)","v":2.9333982165265224},{"$f":"(D6-$D$12)*(C6-$C$12)","v":2.9333982165265224},{"$f":"(D6-$D$12)*(D6-$D$12)","v":2.1512062741802458}]
[null,null,2,{"$f":"beta0+beta1*C7+_xlfn.NORM.INV(RAND(),0,sigma)","v":1.5620285289864548},null,0.90424980359174523,0.63933521300146134,null,{"$f":"(C7-$C$12)*(C7-$C$12)","v":1},{"$f":"(C7-$C$12)*(D7-$D$12)","v":1.6475798898740925},{"$f":"(D7-$D$12)*(C7-$C$12)","v":1.6475798898740925},{"$f":"(D7-$D$12)*(D7-$D$12)","v":2.714519493517527},null,"We use the variance-covariance matrix and then something called an eigenvector to compute the coefficients."]
[null,null,3,{"$f":"beta0+beta1*C8+_xlfn.NORM.INV(RAND(),0,sigma)","v":2.6784423772980857},null,28.331528420148125,3,null,{"$f":"(C8-$C$12)*(C8-$C$12)","v":0},{"$f":"(C8-$C$12)*(D8-$D$12)","v":0},{"$f":"(D8-$D$12)*(C8-$C$12)","v":0},{"$f":"(D8-$D$12)*(D8-$D$12)","v":0.28213736370913484}]
[null,null,4,{"$f":"beta0+beta1*C9+_xlfn.NORM.INV(RAND(),0,sigma)","v":4.3202041477681163},null,11.580498489147693,1.2262485437508719,null,{"$f":"(C9-$C$12)*(C9-$C$12)","v":1},{"$f":"(C9-$C$12)*(D9-$D$12)","v":1.110595728907569},{"$f":"(D9-$D$12)*(C9-$C$12)","v":1.110595728907569},{"$f":"(D9-$D$12)*(D9-$D$12)","v":1.2334228730677343},null,"The estimated beta1 via orthogonal regression is in the green shaded cell."]
[null,null,5,{"$f":"beta0+beta1*C10+_xlfn.NORM.INV(RAND(),0,sigma)","v":5.744457729652793},null,null,null,null,{"$f":"(C10-$C$12)*(C10-$C$12)","v":4},{"$f":"(C10-$C$12)*(D10-$D$12)","v":5.0696986215844912},{"$f":"(D10-$D$12)*(C10-$C$12)","v":5.0696986215844912},{"$f":"(D10-$D$12)*(D10-$D$12)","v":6.4254610284239231},null,"Try it! Press F9 and watch it bounce!"]
[null,null,null,null,null,null,null,null,{"$f":"SUM(I6:I10)","v":10},{"$f":"SUM(J6:J10)","v":10.761272456892675},10.761272456892675,12.806747032898565]
[null,null,{"$f":"AVERAGE(C6:C10)","v":3},{"$f":"AVERAGE(D6:D10)","v":3.2096084188605474},"averages",null,null,null,{"$f":"I11/4","v":2.5},{"$f":"J11/4","v":2.6903181142231687},2.6903181142231687,3.2016867582246413,null,"Before we sim, think a bit -- what is the true value of beta1? Why can't we directly observe it?"]
[null,null,null,null,null,null,null,null,"Variance-covariance matrix:"]
[null,null,null,null,null,null,null,null,null,{"$f":"I12","v":2.5},{"$f":"J12","v":2.6903181142231687},null,null,"Now that you know what we are looking for, we can sim!"]
[null,null,null,null,null,null,null,null,null,{"$f":"K12","v":2.6903181142231687},{"$f":"L12","v":3.2016867582246413},null,null,"Q3: Race OLS against ortho. What do you find?",null,null,null,null,"(Click the Help button if you don't know what to sim.)"]
[]
[null,null,null,null,null,null,null,null,"eigenvector"]
[null,null,null,null,null,null,null,null,{"$f":"MatEigenvector_max(J14:K15)","v":0.8780578601111656},1]
[]
[null,null,null,null,null,null,null,null,"Orthogonal regression fitted line"]
[null,null,null,null,null,null,null,null,{"$f":"IF(J14>K15,1/J18,1/I18)","v":1.1388771121226522},"Ortho b1","Pred Ortho"]
[null,null,null,null,null,null,null,null,{"$f":"D12-I21*C12","v":-0.20702291750740898},"Orthob0",{"$f":"$I$22+$I$21*C6","v":0.93185419461524321}]
[null,null,null,null,null,null,null,null,null,null,{"$f":"$I$22+$I$21*C7","v":2.0707313067378954}]
[null,null,null,null,null,null,null,null,null,null,{"$f":"$I$22+$I$21*C8","v":3.2096084188605474}]
[null,null,null,null,null,null,null,null,null,null,{"$f":"$I$22+$I$21*C9","v":4.3484855309831998}]
[null,null,null,null,null,null,null,null,null,null,{"$f":"$I$22+$I$21*C10","v":5.4873626431058522}]
[]
[null,null,null,null,null,null,null,null,"Here's the orthoreg UDF, it outputs the coefficients in usual b0, b1 order:"]
[null,null,null,null,null,null,null,null,{"$f":"orthoreg(D6:D10,C6:C10)","v":-0.20702291750740986},"Ortho b0"]
[null,null,null,null,null,null,null,null,1.1388771121226524,"Ortho b1"]

{"sheet":"EIV","index":9,"n_rows":112,"n_cols":15,"rows_kept":112,"cut":false,"hidden":true,"merged":[]}
["Errors in Variables (EIV) DGP: Y = b0 + b1True X + e~N(0,sigmaY)",null,null,null,null,null,null,"The DGP should really be called Errors in X Variables, but since Y already has error, it is obviously a situation where we are adding error to X."]
["Obs X = True X + n~N(0,sigmaX)",null,null,"(n is Greek letter nu)",null,null,null,"This situation, where both X and Y are measured with error is common in regressions that involve classification and computer vision."]
["sigmaX=sigmaY is known as total least squares (TLS)"]
["If different, then Deming regression, residuals adjusted with lambda=s2e /s2n",null,null,null,null,null,null,"CEM has points that move vertically, up and down"]
["We will stay in the simple case of sigmaX = sigmaY",null,null,null,null,null,null,"EIV has points that move vertically and horizontally, up and down and right and left"]
[null,null,null,null,null,null,null,"Press F9 a few times and watch the chart. There is up and down movement (Y), but also side-to-side movement (X)."]
[10,"beta0",null,null,null,null,null,null,"The side-to-side movement seems bigger than up and down because of the X axis scale, but they have the same sigma"]
[10,"beta1",null,null,null,null,null,"Evaluate the performance of OLS with a sim (track cells F13 and F14)."]
[1,"sigmaY","Data below are 20 sets of X=1 to 5"]
[{"$f":"sigmaY","v":1},"sigmaX",null,null,null,null,null,"Race OLS against Ortho with a sim (track cells F13 and I14)."]
[null,null,null,"Y based on True X, but we see Obs X"]
[null,"True X","Obs X","Y",null,"OLS b1","OLS b0",null,"Using orthoreg UDF",null,"Pred Ortho","residual","res^2","SSR"]
[null,1,{"$f":"B13+_xlfn.NORM.INV(RAND(),0,sigmaX)","v":1.4855670104096477},{"$f":"beta0+beta1*B13+_xlfn.NORM.INV(RAND(),0,sigmaY)","v":20.984355230900462},null,{"$f":"LINEST(D13:D112,C13:C112,1,1)","v":6.8776242077744127},18.934265804121797,null,{"$f":"orthoreg(D13:D112,C13:C112)","v":7.024097718108524},"ortho b0",{"$f":"$I$13+$I$14*C13","v":23.027207576057684},{"$f":"(D13-K13)/(SQRT(1+$I$14^2))","v":-0.1888259210384195},{"$f":"L13^2","v":3.5655228456007437E-2},{"$f":"SUM(M13:M112)","v":95.7500372872708}]
[null,2,1.9339598576535042,31.354126251842839,null,0.52633017721394471,1.8232330692788539,null,10.77239178429001,"ortho b1",{"$f":"$I$13+$I$14*C14","v":27.857470999841809},{"$f":"(D14-K14)/(SQRT(1+$I$14^2))","v":0.32320453804609489},{"$f":"L14^2","v":0.1044611734135896}]
[null,3,4.4278903285474991,41.152555006373575,null,0.63534863994970925,8.5653091758875632,null,null,null,54.723067115090792,-1.2543561721267431,1.5734094065524555]
[null,4,4.7490138277999732,49.22704380488814,null,170.74985461862627,98,null,null,null,58.182335260180601,-0.82775985313953038,0.68518637446957686]
[null,5,4.4982152870086489,57.982153355098212,null,12526.981342476447,7189.7230852972798,"OLS SSR",null,null,55.480635119848216,0.23122154955801724,5.346340498001062E-2]
[null,{"$f":"B13","v":1},1.112906346647035,19.038715947830234,null,null,null,null,null,null,19.012760903413255,2.399089282809532E-3,5.7556293868915548E-6]
[null,{"$f":"B14","v":2},1.5548376050432338,30.659528879621604,null,null,null,null,null,null,23.773417560581411,0.63650038891615468,0.40513274509041619]
[null,3,3.1567304094108133,39.873785731577804,null,null,null,null,null,null,41.029634445664009,-0.1068379702212903,1.1414351881005312E-2]
[null,4,4.0617200045941733,49.870010433050176,null,null,null,null,null,null,50.778536925685174,-8.3977362419898852E-2,7.0521973990030403E-3]
[null,5,3.7878149915919996,59.211475526574119,null,null,null,null,null,null,47.827924813944712,1.0522098932384534,1.1071456594288775]
[null,1,0.23111234033898453,20.024445340058339,null,null,null,null,null,null,9.513730394424238,0.97153151332084875,0.94387348137549854]
[null,2,2.7678521901535915,31.215452420881633,null,null,null,null,null,null,36.840485911448184,-0.51993583003985522,0.27033326735923319,null,"Q4: If EIV is the DGP, would you use OLS or Orthogonal regression? Why?"]
[null,3,4.6665874851766258,40.123750561387041,null,null,null,null,null,null,57.294406404095781,-1.587126407847977,2.5189702344884228]
[null,4,4.2982254142546301,50.219671903629923,null,null,null,null,null,null,53.32626585765162,-0.28715020253477314,8.2455238815761236E-2]
[null,5,3.8460963704090538,59.515295956444589,null,null,null,null,null,null,48.455754660290637,1.0222608973474214,1.0450173422455553]
[null,1,0.91009509943400657,20.639891173142406,null,null,null,null,null,null,16.827998690174017,0.35234269902190996,0.12414537755404423]
[null,2,1.1082511406079103,29.826314904075566,null,null,null,null,null,null,18.962613200123208,1.0041589569595524,1.0083352108420962]
[null,3,1.4810154721918916,40.83422330457762,null,null,null,null,null,null,22.978176623154845,1.6504787870340079,2.7240802264492499]
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{"sheet":"TheEnd","index":10,"n_rows":56,"n_cols":1,"rows_kept":56,"cut":false,"hidden":true,"merged":[]}
["Conclusion"]
[]
["Errors in X variables, called just errors in variables, means you have a different DGP for which OLS is no longer BLUE -- in fact, it isn't even U!"]
["Orthogonal regression is a different recipe than OLS -- the residual is measured as a perpendicular distance to the fitted line."]
["Orthogonal regression is unbiased for an EIV DGP, but is it BLUE? That's too hard to answer here."]
[]
["What are things have you learned? What other questions do you have? Let's BRAINSTORM! Jot down ideas below."]
["(Scroll down to see some of mine.)"]
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["Does OLS get worse as sigma increases?"]
["What if you had a mix of Xs, some with and anothers without errors?"]
["What is the Ortho Estimated SE of b1?"]

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["Only the first 100 repetitions are displayed on this worksheet."]

{"sheet":"TwoVarMCSim","index":12,"n_rows":103,"n_cols":13,"rows_kept":103,"cut":false,"hidden":true,"merged":[[1,5,1,6],[3,8,3,9],[3,10,3,11],[3,12,3,13]]}
["Results of Monte Carlo Simulation"]
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["Only the first 100 repetitions are displayed on this worksheet."]

{"sheet":"MCSim7","index":13,"n_rows":167,"n_cols":39,"rows_kept":167,"cut":false,"hidden":true,"merged":[[1,5,1,6],[3,8,3,9],[3,10,3,11],[3,12,3,13]]}
["Results of Monte Carlo Simulation",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"$F$1","LiveOLSOrthoEIV!$F$2"]
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