a. Specify the least squares function that is minimized by OLS. b. Compute the partial derivatives of the objective function with respect to bj and b2. c. Suppose that E"-,X1;X2; = 0. Show that âß¡ = E"-1X1µY;/ "-1Xfr- d. Suppose that E-1X1;X2¡ # 0. Derive an expression for ß¡ as a func- tion of the data (Y;, X1, X21), i = 1, . .., n. e. Suppose that the model includes an intercept: Y; = Bo + B¡X1¡ + BX + uj. Show that the least squares estimators satisfy B, = Y - Вх, - ВХ. %3D f. As in (e), suppose that the model contains an intercept. Also suppose that Ef-1(X11 - X1)(X2½ - X2) = 0. Show that ß Ef-1(X1i - X1)(Y¡ – Y)/E#-1(X1i - X1)². How does this compare to the OLS estimator of B, from the regression that omits X,?
Correlation
Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.
Linear Correlation
A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.
Regression Analysis
Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.
Consider the regression modelYi = β1X1i + β2X2i + uifor i = 1, . . . ., n. (Notice that there is no constant term in the regression.)
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