Definition of the Simple Regression Model

Overview

Premise: Analyzing the relationship between two variables, $y$ and $x$, representing a population. The goal is to explain $y$ in terms of $x$ or study how $y$ varies with changes in $x$​​.

Examples: $y$ as soybean crop yield with $x$ as the amount of fertilizer, $y$ as hourly wage with $x$ as years of education, $y$ as community crime rate with $x$ as the number of police officers.

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Model Formulation

Issues to Address:

1. Allowing for other factors to affect $y$ apart from $x$.
2. Determining the functional relationship between $y$ and $x$.
3. Ensuring a ceteris paribus (all else equal) relationship between $y$ and $x$​​.

Simple Equation: $y=b_0+b_1 x+u$

• $y$ and $x$ can have several interchangeable names:
  • $y$: Dependent, Explained, Response, Predicted variable, or Regressand.
  • $x$: Independent, Explanatory, Control, Predictor variable, or Regressor​​.
• $u$: Error term or disturbance, representing unobserved factors affecting $y$​​.

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Interpretation of the Model

Functional Relationship:

• Linear effect of $x$ on $y$ when other factors in $u$ are held fixed​​.

Slope Parameter $(b_1)$:

• Represents the change in $y$ per unit change in $x$, holding other factors fixed​​.
• Example: In a model where $y$ is soybean yield and $x$ is fertilizer, $b_1$​ measures the effect of fertilizer on yield, holding other factors like land quality and rainfall constant​​.

Limitation:

• The linearity assumption implies a constant effect of $x$ on $y$ which might be unrealistic in many economic contexts​​.

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Causal Inference and Key Assumptions

Causality Challenge:

• Addressing if the model can truly draw ceteris paribus conclusions about the impact of $x$ on $y$​​.

Key Assumptions:

1. Zero Mean Assumption:
   • Expected value of $u$ in the population is zero: $\mathbb{E}[u]=0$​​. This is a mathematical convenience resulting from the intercept.
2. Mean Independence:
   • Expected value of $u$ does not depend on $x$: $\mathbb{E}[u∣x]=\mathbb{E}[u]$​​.
   • Example: If fertilizer amounts are chosen independently of other plot features, the assumption holds​​.

• Assumptions Zero mean and Mean independence are crucial for motivating estimators of $b_0$​ and $b_1$​ and play a significant role in statistical analysis in later sections​​.

Population Regression Function (PRF):

• Expressed as $\mathbb{E}[y∣x]=b_0​+b_1​x$, showing the linear relationship between $y$ and $x$ in the population​​. For any given value of $x$, the distribution of $y$ is centered about $\mathbb{E}[y∣x]$.

Question: Prove if $u$ and $x$ are mean independent, then $Cov(X,U)=0$. [Sol]

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Decomposition of $y$

Systematic and Unsystematic Parts:

• $b_0+b_1x$: Systematic part, the part of $y$ explained by $x$.
• $u$: Unsystematic part, the part of $y$ not explained by $x$​​.

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Important points!

• We need mean independence not the full independence for causality.