Consider the Case of Time Fixed Effects Only, I $$Y _ { i t } = \beta _ { 0 } + \beta _ { 1 } X _ { i t } + \beta _ { 3 } S _ { t } + u _ { i t } ,$$

question 17

Essay

Consider the case of time fixed effects only, i.e., $$Y _ { i t } = \beta _ { 0 } + \beta _ { 1 } X _ { i t } + \beta _ { 3 } S _ { t } + u _ { i t } ,$$
First replace $\beta _ { 0 } + \beta _ { 3 } S _ { t }$ with $\phi _ { t }$ . Next show the relationship between the $\phi _ { t }$ and $\delta _ { i }$ in the following equation $$Y _ { i t } = \beta _ { 0 } + \beta _ { 1 } X _ { i t } + \delta _ { 2 } B 2 _ { t } + \ldots + \delta _ { T } B T _ { t } + u _ { i t }$$ where each of the binary variables B2, …, BT indicates a different time period.Explain in
words why the two equations are the same.Finally show why there is perfect
multicollinearity if you add another binary variable B1.What is the intuition behind the
fact that the OLS estimator does not exist in this case? Would that also be the case if you
dropped the intercept?

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Definitions:

Rational-Emotive Therapy

A cognitive-behavioral approach to psychotherapy that identifies irrational beliefs to alter emotional responses and behaviors.

Irrational Beliefs

Thoughts or convictions not based on reason or logic, often leading to unreasonable behaviors and emotional distress.

Cognitive-Behavior Therapists

Mental health professionals who use techniques from cognitive therapy and behavioral therapy to modify dysfunctional emotions, behaviors, and thoughts.

Interpretation

Interpretation involves making sense of or assigning meaning to phenomena, actions, or symbols within a particular context, often used in analysis and therapy.