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.

MATLAB: An Introduction with Applications

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ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

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Question

Consider the regression model with a single regressor: Y_i = β₀ + β₁X_i + u_i.
Suppose that the least squares assumptions in Key Concept 4.3 are
satisfied.
a. Show that X_i is a valid instrument. That is, show that Key Concept
12.3 is satisfied with Z_i = X_i.

b. Show that the IV regression assumptions in Key Concept 12.4 are satisfied with this choice of Z_i.
c. Show that the IV estimator constructed using Z_i = X_i is identical to
the OLS estimator.

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.

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