The Least Squares Assumptions KEY CONCEPT 4.3 Y; = B + BỊX; + u¡, i = 1, . ., n, where 1. The error term u; has conditional mean zero given X;: E(u;|X;) = 0; 2. (X;, Y;), i = 1, . ..,n, are independent and identically distributed (i.i.d.) draws from their joint distribution; and The Two Conditions for Valid Instruments KEY CONCEPT 3. Large outliers are unlikely: X, and Y; have nonzero finite fourth moments. 12.3 A set of m instruments Z1;, ..., Zmi must satisfy the following two conditions to be valid: KEY CONCEPT The IV Regression Assumptions 1. Instrument Relevance 12.4 The variables and errors in the IV regression model in Key Concept 12.1 satisfy the following: • In general, let X¡¡ be the predicted value of X1i from the population regres- sion of X, on the instruments (Z's) and the included exogenous regressors (W's), and let "1" denote the constant regressor that takes on the value 1 for 1. E(u;|W1i, ..., W,;) = 0; 2. (X» ..., Xi, W1;, ..., Wri, Zj, ..., Zmi,Y;) are i.i.d. draws from their joint all observations. Then (X¡;, . . . , Xi, W1i, . .., Wris 1) are not perfectly multi- collinear. distribution; • If there is only one X, then for the previous condition to hold, at least one Z must have a non-zero coefficient in the population regression of X on the Z's 3. Large outliers are unlikely: The X's, W's, Z's, and Y have nonzero finite fourth moments; and and the W's. 4. The two conditions for a valid instrument in Key Concept 12.3 hold. 2. Instrument Exogeneity The instruments are uncorrelated with the error term; that is, corr(Z1;, u;) = 0,..., corr(Zmi, u;) = 0.
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 model with a single regressor: Yi = β0 + β1Xi + ui.
Suppose that the least squares assumptions in Key Concept 4.3 are
satisfied.
a. Show that Xi is a valid instrument. That is, show that Key Concept
12.3 is satisfied with Zi = Xi.
b. Show that the IV regression assumptions in Key Concept 12.4 are satisfied with this choice of Zi.
c. Show that the IV estimator constructed using Zi = Xi is identical to
the OLS estimator.
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