Assume That the Data Looks as Follows Using the Formula for the OLS Estimator

question 6

Essay

Assume that the data looks as follows: $$\boldsymbol { Y } = \left( \begin{array} { l } Y _ { 1 } \\Y _ { 2 } \\\vdots \\Y _ { n }\end{array} \right) , \boldsymbol { U } = \left( \begin{array} { l } u _ { 1 } \\u _ { 2 } \\\vdots \\u _ { n }\end{array} \right) , \boldsymbol { X } = \left( \begin{array} { c } X _ { 11 } \\X _ { 12 } \\\vdots \\X _ { 1 n }\end{array} \right) \text {, and } \boldsymbol { \beta } = \left( \beta _ { 1 } \right)$$ Using the formula for the OLS estimator $\hat { \boldsymbol { \beta } } = \left( \boldsymbol { X } ^ { \prime } \boldsymbol { X } \right) ^ { - 1 } \boldsymbol { X } ^ { \prime } \boldsymbol { Y }$ , derive the formula for $\widehat { \beta } _ { 1 }$ , the only slope in this "regression through the origin."

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Definitions:

Frictionally Unemployed

Individuals temporarily out of work while transitioning between jobs or entering the workforce.

Structurally Unemployed

Individuals who are unemployed due to changes in the market that make their skills obsolete.

Aggregate Supply Curve

displays the total output of goods and services that firms in an economy are willing to produce and sell at different price levels, in a given time period.

Cost-Push Inflation

Inflation caused by increases in the costs of production, such as raw materials or wages, leading to a decrease in supply and an increase in prices.