Regression Coefficient Fundamentals

The regression coefficient, often denoted as $\beta_1$ or $b$, in a simple linear regression model ($Y = \alpha + \beta_1 X + \epsilon$) represents the change in the dependent variable (Y) for a one-unit change in the independent variable (X). It is essentially the slope of the regression line.

The formula for the regression coefficient $\beta_1$ (or $b$) is given by:

$$\beta_1 = \frac{\text{Cov}(X, Y)}{\text{Var}(X)}$$

Where:

• $\text{Cov}(X, Y)$ is the covariance between X and Y.
• $\text{Var}(X)$ is the variance of X.

Impact of Subtracting Constants on Data

Let's consider what happens when a constant value is subtracted from each of the values of X and Y.

Suppose we have new variables $X'$ and $Y'$ defined as:

• $X' = X - c_x$
• $Y' = Y - c_y$

where $c_x$ and $c_y$ are constants (in this question, $c_x = 2$ and $c_y = 2$).

Understanding Covariance and Variance Changes

To determine the effect on the regression coefficient, we need to analyze how covariance and variance change when a constant is subtracted.

Covariance Transformation

The covariance between two variables $X$ and $Y$ is defined as:

$$\text{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])]$$

Now, let's find the covariance between the transformed variables $X'$ and $Y'$:

$$\text{Cov}(X', Y') = \text{Cov}(X - c_x, Y - c_y)$$

We know that $E[X - c_x] = E[X] - c_x$ and $E[Y - c_y] = E[Y] - c_y$.

So,

$$\text{Cov}(X - c_x, Y - c_y) = E[((X - c_x) - (E[X] - c_x))((Y - c_y) - (E[Y] - c_y))]$$

$$= E[(X - c_x - E[X] + c_x)(Y - c_y - E[Y] + c_y)]$$

$$= E[(X - E[X])(Y - E[Y])]$$

$$= \text{Cov}(X, Y)$$

This shows that subtracting a constant from each variable does not change their covariance. It only shifts the origin of the data.

Variance Transformation

The variance of a variable $X$ is defined as:

$$\text{Var}(X) = E[(X - E[X])^2]$$

Now, let's find the variance of the transformed variable $X'$:

$$\text{Var}(X') = \text{Var}(X - c_x)$$

$$= E[((X - c_x) - (E[X] - c_x))^2]$$

$$= E[(X - c_x - E[X] + c_x)^2]$$

$$= E[(X - E[X])^2]$$

$$= \text{Var}(X)$$

This shows that subtracting a constant from a variable does not change its variance.

Final Regression Coefficient Conclusion

Since both the covariance $\text{Cov}(X, Y)$ and the variance $\text{Var}(X)$ remain unchanged when a constant is subtracted from X and Y values, the regression coefficient also remains unchanged.

Let the original regression coefficient be $\beta_{1, \text{old}}$ and the new regression coefficient be $\beta_{1, \text{new}}$.

$$\beta_{1, \text{old}} = \frac{\text{Cov}(X, Y)}{\text{Var}(X)}$$

$$\beta_{1, \text{new}} = \frac{\text{Cov}(X', Y')}{\text{Var}(X')} = \frac{\text{Cov}(X - c_x, Y - c_y)}{\text{Var}(X - c_x)}$$

As we demonstrated:

• $\text{Cov}(X - c_x, Y - c_y) = \text{Cov}(X, Y)$
• $\text{Var}(X - c_x) = \text{Var}(X)$

Therefore,

$$\beta_{1, \text{new}} = \frac{\text{Cov}(X, Y)}{\text{Var}(X)} = \beta_{1, \text{old}}$$

The regression coefficient is not changed. This is an important property in linear regression, implying that shifting the origin of the data points does not affect the slope of the best-fit line. However, it would affect the y-intercept ($\alpha$).

Therefore, if a constant 2 is subtracted from each of the values of X and Y, the regression coefficient is not changed.