# Desire to make ceteris paribus interpretations of β

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Desire to make ceteris paribus interpretations of β 1  include more explanatory variables in the regression l Leads to extension to multiple regression

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10 Engel curve for food l Sample of 40 households with 3 family members l Data on l Y = weekly expenditure on food (\$) l X = weekly income (\$) l Cross-sectional data l Engel curve (demand function holding prices constant) Y i = β 0 + β 1 X i + ε i i =1, … , n 0 10 20 30 40 50 60 0 20 40 60 80 100 120 140 Income Expenditure
11 Engel curve for food… l For these data l b 0 = 7.38 & b 1 = 0.23 l How do you interpret b 1? l What is predicted food expenditure for a household with weekly income of \$100? l Why restrict to households with 3 people?

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12 Assumptions of Classical Linear Regression Model l A1: Linear model l Y i = β 0 + β 1 X i + ε i , i = 1, … , n l A2: Random sampling l Have a random sample ( Y i , X i ) governed by model in A1 l A3: Sample variation in X l Values of X 1, X 2, … , X n are not all equal l A4: Zero conditional mean l E( ε i | X i ) = 0 l Key assumption because implies ε i & X i are uncorrelated l A2 & A4 mean we can think of X as fixed rather than random
13 Assumptions of Classical Linear Regression Model… l A5: Homoskedasticity l Var( ε i ) = σ 2 l A6: Disturbances uncorrelated l Cov( ε i , ε j ) = 0, ( i not equal j ) l A7: Disturbances are normal (used for inference) l ε i ~ N (0, σ 2 ) l In some circumstances these assumptions will not be realistic l Further study of econometrics (beyond BES) l Would cover detection of violations of assumptions l Subsequent remedial methods

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14 Classical Linear Regression Model (CLRM) l Given A1-A7 l Y i ~ N ( β 0 + β 1 X i , σ 2 ) l The (conditional) mean of Y depends on X l If β 1=0 what is β 0? l Given A1-A4 can show that OLS estimators are unbiased & consistent l Given A1-A6 OLS ‘best’ in a particular sense but beyond scope of BES
15 Explaining variation in the dependent variable w w w w Y X X Y Y Y ˆ i - i i Y ˆ Y - X b b Y 1 0 ˆ + =

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16 Decomposition of variance ( 29 ( 29 ( 29 ( 29 ( 29 SSE SSR SST e Y Y Y Y X e Y Y Y Y Y Y Y Y i i i i i i i i i + = + = + - = - + = + - = - + - = - squares of sum error squares of sum regression squares of sum Total ˆ that show Can d unexplaine part by explained part deviation Total ˆ ˆ ˆ 2 2 2
17 Standard error of estimate l Population variance σ 2 measures the spread around the population regression line l Standard error of estimate ( SEE ) is an estimator of σ l It measures fit of regression model l Low SEE  good fit estimator unbiased an ensure to estimated) be to need

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