IFPRI Westminster International University in Tashkent 2018 2 Regression Regression analysis is concerned with the study of the dependence of one variable the dependent variable on one or more other variables the ID: 1022945
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1. Linear RegressionSummer SchoolIFPRIWestminster International University in Tashkent2018
2. 2RegressionRegression analysis is concerned with the study of the dependence of one variable, the dependent variable, on one or more other variables, the explanatory variables, with a view of estimating and/or predicting the population mean or average values of the former in terms of the known or fixed (in repeated sampling) values of the latter.
3. 3Terminology and NotationDependent VariableIndependent variable Explained variable Independent variable PredictandPredictorRegressandRegressor ResponseStimulusEndogenousExogenous OutcomeCovariateControlled variableControl variable
4. 4Conditional Mean Income 80100120140160180200220240260Consumption 55657980102110120135137150607084931071151361371451526574909511012014014015517570809410311613014415216517875859810811813514515717518088113125140160189185115162191Total 32546244570767875068510439661211Conditional mean 657789101113125137149161173
5. 5Simple RegressionConditional expectedvalues E(Y|X)Population RegressionCurveA population regression curve is simply the locus of the conditional means of the dependent variable for the fixed values of the explanatory variable(s).
6. 6Simple RegressionConditional Expectation Function (CEF)Population Regression Function (PRF)Population RegressionRegression CoefficientsLinear Population Regression Function
7. 7LinearXYXYXYLinear in parameter functionsNon-linear in parameter function
8. 8Stochastic specificationStochastic error termSystematic component Nonsystematic component
9. 9Sample Regression FunctionSRF2SRF1
10. 10Sample Regression FunctionPRFSRFEstimate
11. 11YXASample Regression Function
12. 12Assumptions.Linearity. The relationship between independent and dependent variable is linear. Full Rank. There is no exact relationship among any independent variables. Exogeneity of independent variables. The error term of the regression is not a function of independent variables. Homoscedastisity and no Autocorrelation. Error term of the regression is independently and normally distributed with zero means and constant variance.Normality of Error term
13. 13Ordinary Least Squares
14. 14Ordinary Least Squares
15. 15Ordinary Least Squares
16. AssumptionsLinear Regression ModelX values are repeated in sampling – X is assumed to be nonstochastic.Zero mean values of disturbance ui
17. Homoscedasticity or equal variance of uiAssumptionsXYf(u)
18. AssumptionsXYf(u)Heteroscedasticity
19. AssumptionsNo autocorrelation between the disturbancesExogeneity. Zero covariance between Xi and ui
20. Coefficient momentsAdditionally we know that Estimator True value
21. Coefficient momentsAccording to our “Exogenity” assumption. (Error term is independent from X variable. Thus, OLS estimator is unbiased estimator.
22. Coefficient momentsAccording to Homoscedasticity and no auto-correlation assumptions.
23. Coefficient momentsAccording to Homoscedasticity and no auto-correlation assumptions.
24. Using similar argument
25. BLUE estimatorSampling distribution of β2Sampling distribution of β2*
26. OLS Estimation: Multiple Regression Model
27. Assumptions and estimation.Assumptions are the sameMinimize the sum of squared residuals The unbiased estimator of R square Adjusted R square
28. OLS Estimation: Multiple regression model
29. Goodness of FitTSS = ESS + RSS
30. Goodness of FitTSS = ESS + RSS