3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N.

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1 3/3/04 CDS M Phil Old Least Squares (OLS) Vijayamohaa Pillai N CDS M Phil Vijayamoha CDS M Phil Vijayamoha Types of Relatioships Oly oe idepedet variable, Relatioship betwee ad is Liear relatioships Curviliear relatioships described by a liear fuctio Chages i are assumed to be caused by chages i 3 4 Types of Relatioships Types of Relatioships Strog relatioships Weak relatioships No relatioship 5 6

2 3/3/04 Cosists of the coditioal mea fuctio E( x) ) + x ad the coditioal variace (scedastic) fuctio Var( x) ) > 0, costat Parameters of the coditioal mea fuctio: ad E( x) + x β Slope: the rate of chage i E( x) for a uit chage i Itercept: value of E( x) whe x As the scedasticfuctio > 0, the observed value of the i th respose i E( i ) This differece statistical error u i i E( i )+u i u i i E( i ) i E( i )+u i u i : radom variables y 4 y 3 } u 3 u { y y } u u 4 E( ) { + u i : the vertical distace betwee the poit yi ad the mea fuctio 9 x x x 3 x 4 0 The parameters of the coditioal mea fuctio ad : ukow; must be estimated usig observed data o i ad i Estimates of parameters : computable fuctios of data ad are therefore statistics 3

3 3/3/04 To estimate the ukow ad usig observed data o i ad i Ordiary Least Squares, or OLS: The Classical / Old / Ordiary Least squares (OLS) Ituitively, OLS is fittig a lie through the sample poits such that the sum of squared residuals is as small as possible hece the term least squares Parameter estimates are chose to miimize the residual sum of squares 3 March 3 March Vijayamoha: CDS MPhil: 3 3 The residual, û, is a estimate of the error term, u ad is the differece betwee the fitted lie (sample regressio fuctio) ad the sample poit 4 Algebraic Properties of OLS Gauss-Markov Theorem The sum of the OLS residuals is zero Thus, the sample average of the OLS residuals is zero as well The sample covariace betwee the regressors ad the OLS residuals is zero The OLS regressio lie always goes through the mea of the sample 5 Of the class of liear ubiased estimators, the OLS estimators have the smallest variace Least squares estimators are BLUE Best Liear Ubiased Estimators s t certai assumptios 6 OLS Assumptios OLS Estimators y 4 { y 3 } y { y } 7 x x x 3 x 4 8 3

4 3/3/04 OLS Estimators OLS Estimators ˆβ ( xi x)( yi y) i ( xi x) i ; ( xi x) > 0 i 9 0 OLS Estimators Mea ad Variace of OLS Estimators y i x i + u i E(u i ) 0, i Simple Liear Regressio Example Sample Data for House Price Model A real estate aget wishes to examie the relatioship betwee the sellig price of a home ad its size (measured i square feet) A radom sample of 0 houses is selected Depedet variable () house price i Rs000s Idepedet variable () square feet 3 March 04 Vijayamoha: CDS MPhil: 3 House Price i Rs000s () Square Feet ()

5 The image caot be displayed our computer may ot have eough memory to ope the image, or the image may have bee corrupted Restart your computer, ad the ope the file agai If the red x still appears, you may have to delete the image ad the isert it agai 3/3/04 Graphical Presetatio Summary Statistics House Price i Rs 000s Square Feet House price model: scatter plot (Stata) twoway(scatter houseprice squarefeet) 5 6 Predictio usig Regressio Aalysis Predict the price for a house with 000 square feet: The regressio equatio is: house price (square feet) house price (sqft) (000) 3785 The predicted price for a house with 000 square feet is 3785(Rs,000s) Rs37, Iterpolatio vs Extrapolatio Measures of Variatio Whe usig a regressio model for predictio, predict oly withi the relevat rage of data House Price (Rs000s) Relevat rage for iterpolatio Square Feet Do ot try to extrapolate beyod the rage of observed s Total variatio is made up of two parts: TSS RSS + Total Sum of Squares Regressio Sum of Squares ESS Error Sum of Squares TSS ( i ) RSS ( ˆ i ) ESS ( i ˆ) i

6 3/3/04 Measures of Variatio Measures of Variatio TSS total sum of squares Measures the variatio of the i values aroud their mea RSS regressio sum of squares Explaied variatio attributable to the relatioship betwee ad ESS error sum of squares i i TSS ( i -) ESS ( i - i ) _ RSS ( i -) _ Variatio attributable to factors other tha the relatioship betwee ad i 3 3 Coefficiet of Determiatio, R Portio of the total variatio i the depedet variable explaied by variatio i the idepedet variable RSS regressio sum ofsquares R TSS totalsumofsquares RSS R TSS R 5808% of the variatio i house prices is explaied by variatio i square feet Examples of Approximate R Values Examples of Approximate R Values R Perfect liear relatioship betwee ad : 00% of the variatio i is explaied by variatio i 0 < R < Weaker liear relatioships betwee ad : Some but ot all of the variatio i is explaied by variatio i

7 The image caot be displayed our computer may ot have eough memory to ope the image, or the image may have bee corrupted Restart your computer, ad the ope the file agai If the red x still appears, you may have to delete the image ad the isert it agai 3/3/04 Examples of Approximate R Values Stadard Error of Estimate (Root MSE) R 0 No liear relatioship betwee ad : The stadard deviatio of the variatio of observatios aroud the regressio lie The value of does ot deped o (Noe of the variatio i is explaied by variatio i ) σˆ u ESS i ( ˆ ) i i Comparig Stadard Errors ˆu σ Iferece About the Slope Iferece about the Slope: t-testtest t test for a populatio slope: Is there a liear relatioship betwee ad? ˆ MSE SE( β ) Var( ) ˆ σ u Var( ) Null ad alterative hypotheses H 0 : β 0 (o liear relatioship) uˆ i ESS i ˆσ u H : β 0 (liear relatioship does exist) Test statistic ˆ β β t SE( ˆ β ) df 4 4 7

8 The image caot be displayed our computer may ot have eough memory to ope the image, or the image may have bee corrupted Restart your computer, ad the ope the file agai If the red x still appears, you may have to delete the image ad the isert it agai The image caot be displayed our computer may ot have eough memory to ope the image, or the image may have bee corrupted Restart your computer, ad the ope the file agai If the red x still appears, you may have to delete the image ad the isert it agai 3/3/04 Does square footage of the house affect its sales price? 43 Ifereces about the Slope: t-testtest Test Statistic: t 339 H 0 : β 0 H : β 0 df 0-8 /05 Do ot reject H 0 Reject H 0 -t α/ 0 t α/ From Stata output: Coefficiets Stadard Error t Stat P-value Itercept Square Feet /05 Reject H P-value < α Decisio:? Reject H 0 Coclusio:? There is sufficiet evidece that square footage affects house price 44 F-test for Sigificace F-test statistic: RMS F EMS RSS RMS k ESS EMS k With ad 8 degrees of freedom RMS F 0848 EMS P-value for the F Test F follows a F-distributio with k umerator ad ( k ) deomiator degrees of freedom (k the umber of idepedet variables i the regressio model) F-test for Sigificace t-test test for Correlatio Coefficiet 0 H 0 : β 0 H : β 0 α 05 df df 8 Critical Value: F α 53 α 05 Do ot Reject H 0 reject H 0 F F Test Statistic: RMS F 08 EMS Decisio: Reject H 0 at α 005 Coclusio: There is sufficiet evidece that house size affects sellig price 47 Hypotheses H 0 : ρ 0 (o correlatio betwee ad ) H A : ρ 0 (correlatio exists) Test statistic r-ρ t r (with df) where r + r r r if ˆ β > 0 if ˆ β <

9 3/3/04 Example: House Prices Example: House Prices Test Solutio Is there evidece of a liear relatioship betwee square feet ad house price at 005 level of sigificace? t r ρ r H 0 : ρ 0 (No correlatio) H : ρ 0 (correlatio exists) α05, df t r ρ r df 0-8 α/ Reject H 0 Do ot reject H Reject H -t 0 0 α/ 0 t α/ α/ 005 Decisio: Reject H 0 Coclusio: There is evidece of a liear associatio at 5% level of sigificace Multiple Regressio Aalysis y β 0 + β x + β x + β k x k + u 5 9