residual. (Note that usually in descriptions of regression analysis, upper-case

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Constance Rodgers
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1 Regresso Aalyss Regresso aalyss fts or derves a model that descres the varato of a respose (or depedet ) varale as a fucto of oe or more predctor (or depedet ) varales. The geeral regresso model s oe of several that share the same asc coceptual model data = systematc compoet + rregular compoet where the systematc compoet s predctale or explaale y the predctor varales, ad s represeted y the regresso model, whle the rregular compoet s regarded as ose or predcto errors varatos the respose varale that ca ot e accouted for y the predctor varales. The regresso equato The specfc varte, lear, regresso model s Y X 0 where Y s the c value of the depedet or respose varale, 0 ad are the coeffcets of the regresso le (also kow as the slope ( ) ad tercept ( 0 )), s the -th depedet or predctor varale, ad s the predcto error, or ose or resdual. (Note that usually descrptos of regresso aalyss, upper-case X X s ad Y s stad for raw data values, whle lower case x s ad y s stad for devatos of X ad Y aout ther respectve meas,.e. x X X y Y Y There are several alteratve ways of wrtg the regresso equato or model true model, o error: Y 0 X, true model, wth error: Y 0 X, true model, o suscrpts: Y 0 X, true model, wth error: Y ax e (where a ad are the regresso coeffcets ad e s the resduals,

2 estmated model Y 0 X e, where 0 ad are estmates of 0 ad, ad e s a estmate of, estmated model Y ˆ ˆ 0 X e, where 0 ˆ ad ˆ are estmates of 0 ad, ad e s a estmate of. Other varales ad quattes There are a umer of other quattes that are mportat regresso aalyss, cludg: the ftted or predcted values of the respose varale Y (called y-hat ) Yˆ 0 X, Y ( X X) the resduals or predcto errors e Y Yˆ the sums of squared devatos ad ther cross products X ( ) x SY ( Y Y) y, ad XY xy S X X S ( X X)( Y Y) ad the resdual sum of squares e SSE

3 Fttg the regresso equato (.e. estmatg parameters) The regresso equato s ftted y choosg the values of 0 ad such a way that the sum of squares of the predcto errors, S, are mmzed,.e. M S e ( Y X ) 0 The specfc values of 0 ad that mmze S could e foud teratvely, or y tral ad error, ut t s kow that the followg ordary least-squares (OLS) estmates of 0 ad do fact mmze S: S XY SX ( X X) x Y X. 0 ( X X)( Y Y) x y, ad Goodess-of-ft statstcs The goodess of ft of the regresso equato, or a measure of the stregth of the relatoshp etwee Y ad X ca e descred several ways. As aalyss of varace, the sum of squares of the depedet varale Y ca e decomposed to two compoets TotalSS RegrSS ErrorSS ˆ ˆ Y Y Y Y Y Y ( ) ( ) ( ), Where TotalSS s the total sum of squares (of devatos of dvdual depedet varale values of the mea), RegrSS s the regresso sum of squares or that compoet of the total sum of squares explaed y the regresso equato, ad ErrorSS s the resdual sum of squares, or the sum of squares of the resdual,

4 e A F-statstc that ca e used to test the ull hypothess that the relatoshp etwee the predctor ad respose varales s ot sgfcat s MSRegr F MSError ( Y Y) /( k) ˆ ( Y Y ) /( k) The deomator of ths expresso, ˆ ( Y Y ) /( k), s also kow as the measquare error of the regresso, ad s sometmes represeted y s. The square root of the mea-square error, s, s called the stadard error of the regresso, ad provdes a measure of ucertaty the estmates of Y produced y the regresso equato. I geeral, the larger the F, the stroger the relatoshp. Aother measure of the stregth of the relatoshp etwee the respose ad predctor varale s the explaed varace (a proporto, ut sometmes expressed as a percetage), also kow as the coeffcet of determato, or R RegrSS TotalSS ErrorSS TotalSS ( Yˆ Y) ( Y Y) ( Yˆ Yˆ) ( Y Y) Sgfcace of the regresso coeffcets There are a umer of other quattes that are useful terpretg a regresso equato. These clude stadard errors for the slope ad tercept R

5 X se 0 s S X ( ), ad se( ) s / S X Usg these stadard errors, t-statstcs that ca e used to test hypotheses aout the regresso coeffcets ca e costructed: where 0 ad 0 0 t ( 0), ad se( 0 ) ( t ) se( ) are hyptheszed values of the regresso coeffcets, whch are usually take to e 0, so that large values of the t-statstcs wll sgal that 0 ad values that are sgfcat (.e. ot zero). The stadard error or stadard devato of the predcted value of the respose varale, Ŷ, gve a partcular value of the predcted varale, X, s ˆ ( X X ) se( Y X ) s SX

Linear Regression with One Regressor

Linear Regression with One Regressor Lear Regresso wth Oe Regressor AIM QA.7. Expla how regresso aalyss ecoometrcs measures the relatoshp betwee depedet ad depedet varables. A regresso aalyss has the goal of measurg how chages oe varable,