Question

1. Let ??,? be the 1 × k vector of explanatory variables for observation s such that X? = [ ??,?; ??,?; ?; ??,? ] = [ x??,? x??,? ? x??,?; x??,? x??,? ? x??,?; ? ? ? ?; x??,? x??,? ? x??,? ], so that for s = 1, ..., n ??,? = [ x??,? x??,? ? x??,? ], and y? = [ y?,?; y?,?; ?; y?,? ], where s = 1, 2, ..., n. Show that the OLS estimator ?? can be written as: ?? = (?_{s=1}^{n} x'?,? x?,?)?¹ (?_{s=1}^{n} x'?,? y?,?).

Name: 1. Let ??,? be the 1 × k vector of explanatory variables for observation s such that X? = [ ??,?; ??,?; ?; ??,? ] = [ x??,? x??,? ? x??,?; x??,? x??,? ? x??,?; ? ? ? ?; x??,? x??,? ? x??,? ], so that for s = 1, ..., n ??,? = [ x??,? x??,? ? x??,? ], and y? = [ y?,?; y?,?; ?; y?,? ], where s = 1, 2, ..., n. Show that the OLS estimator ?? can be written as: ?? = (?s=1^n x'?,? x?,?)?¹ (?s=1^n x'?,? y?,?).
Uploaded: 2022-09-02T11:07:51-08:00
Duration: 3 min 54 s
Channel: Rashmi Sinha

          1. Let ??,? be the 1 × k vector of explanatory variables for observation s such that X? = [ ??,?; ??,?; ?; ??,? ] = [ x??,? x??,? ? x??,?; x??,? x??,? ? x??,?; ? ? ? ?; x??,? x??,? ? x??,? ], so that for s = 1, ..., n ??,? = [ x??,? x??,? ? x??,? ], and y? = [ y?,?; y?,?; ?; y?,? ], where s = 1, 2, ..., n. Show that the OLS estimator ?? can be written as: ?? = (?_{s=1}^{n} x'?,? x?,?)?¹ (?_{s=1}^{n} x'?,? y?,?).

1. Let ??,? be the 1 × k vector of explanatory variables for observation s such that X? = [ ??,?; ??,?; ?; ??,? ] = [ x??,? x??,? ? x??,?; x??,? x??,? ? x??,?; ? ? ? ?; x??,? x??,? ? x??,? ], so that for s = 1, ..., n ??,? = [ x??,? x??,? ? x??,? ], and y? = [ y?,?; y?,?; ?; y?,? ], where s = 1, 2, ..., n. Show that the OLS estimator ?? can be written as: ?? = (?s=1^n x'?,? x?,?)?¹ (?s=1^n x'?,? y?,?).

Added by Ines J.

Probability and Statistics for Engineers and Scientists

Ronald E. Walpole, Raymond H. Myers,… 10th Edition

Chapter 11

Instant Answer

Solved by Expert Rashmi Sinha

09/02/2022

Step 1

Now, let's write the matrices $X$ and $Y$ using the given information: $$X = \begin{bmatrix} X_{1,1} & X_{1,2} & \cdots & X_{1,k} \\ X_{2,1} & X_{2,2} & \cdots & X_{2,k} \\ \vdots & \vdots & \ddots & \vdots \\ X_{n,1} & X_{n,2} & \cdots & X_{n,k} Show more…

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Let x_{s,t} be the 1 × k vector of explanatory variables for observation s such that X_t = [x_{1,t} x_{2,t} ... x_{n,t}]' = [x_{11,t} x_{12,t} ... x_{1k,t}; x_{21,t} x_{22,t} ... x_{2k,t}; ...; x_{n1,t} x_{n2,t} ... x_{nk,t}], so that for s = 1, ..., n x_{s,t} = [x_{s1,t} x_{s2,t} ... x_{sk,t}], and y_t = [y_{1,t} y_{2,t} ... y_{n,t}]', where s = 1, 2, ..., n. Show that the OLS estimator β̂ can be written as: β̂ = (∑_{s=1}^{n} x'_{s,t} x_{s,t})^{-1} (∑_{s=1}^{n} x'_{s,t} y_{s,t}).