The OLS Formula for the Slope Coefficients in the Multiple

question 9

Essay

The OLS formula for the slope coefficients in the multiple regression model become increasingly more complicated, using the "sums" expressions, as you add more regressors. For example, in the regression with a single explanatory variable, the formula is
$$\frac { \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) \left( Y _ { i } - \bar { X } \right) } { \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } }$$
whereas this formula for the slope of the first explanatory variable is
$$\hat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } } { \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } \right) ^ { 2 } }$$
(small letters refer to deviations from means as in $z _ { i } = Z _ { i } - \bar { Z }$ ) in the case of two explanatory variables.Give an intuitive explanations as to why this is
the case.

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Definitions:

Freud

Sigmund Freud was an Austrian neurologist and the founder of psychoanalysis, a method for treating psychopathology through dialogue between a patient and a psychoanalyst.

Obsessive-compulsive Disorders

Mental illnesses marked by an individual's engagement in repetitive thoughts and behaviors that are performed to alleviate anxiety or distress.

Anal Development

In psychoanalytic theory, a stage of early child development centered on the control of bowel movements, which can influence personality traits.

Compulsive Behaviors

Actions repeatedly performed by an individual, driven by an uncontrollable urge, often as a response to obsessive thoughts, leading to distress or impairment in functioning.