Question

If $b_{yx}$ and $b_{xy}$ are regression coefficients and they are equal, then?

• A. correlation coefficient is one
• B. variance of $x$ and variance of $y$ are equal
• C. Mean of $x$ and mean of $y$ are equal
• D. Standard deviation of $x$ is greater than standard deviation of $y$

Hint:

• Definitions: $b_{yx} = r \frac{\sigma_y}{\sigma_x}$ and $b_{xy} = r \frac{\sigma_x}{\sigma_y}$.
• Given $b_{yx} = b_{xy} \Rightarrow r \frac{\sigma_y}{\sigma_x} = r \frac{\sigma_x}{\sigma_y}$.
• If $r \neq 0$, then $\frac{\sigma_y}{\sigma_x} = \frac{\sigma_x}{\sigma_y} \Rightarrow \sigma_y^2 = \sigma_x^2$. Thus, variances are equal.
• If $r = 0$, then $b_{yx}=0, b_{xy}=0$. They are equal, but variances are not necessarily equal.
• In typical exam contexts, the non-trivial case ($r \neq 0$) is implied.
• Alternatively, if $Var(x)=Var(y)$ (and $\sigma_x, \sigma_y>0$), then $\sigma_x=\sigma_y$, so $b_{yx}=r$ and $b_{xy}=r$. Thus $b_{yx}=b_{xy}$.

Answer 1

The Correct Option is B

The regression coefficient of $y$ on $x$ is $b_{yx} = r \frac{\sigma_y}{\sigma_x}$.
The regression coefficient of $x$ on $y$ is $b_{xy} = r \frac{\sigma_x}{\sigma_y}$.
Given $b_{yx} = b_{xy}$.
So, $r \frac{\sigma_y}{\sigma_x} = r \frac{\sigma_x}{\sigma_y}$.
Case 1: $r \neq 0$.
We can divide by $r$: $\frac{\sigma_y}{\sigma_x} = \frac{\sigma_x}{\sigma_y} \Rightarrow \sigma_y^2 = \sigma_x^2$.
Since variance is $\sigma^2$, this means $Var(y) = Var(x)$.
Case 2: $r = 0$.
Then $b_{yx} = 0$ and $b_{xy} = 0$.
So $b_{yx} = b_{xy}$ holds.
In this case, $\sigma_x^2$ is not necessarily equal to $\sigma_y^2$.
For example, $X$ and $Y$ could be independent variables with different variances.
However, standard interpretation of such questions often implicitly assumes $r \neq 0$ or seeks a condition that generally leads to the equality.
If $Var(x) = Var(y)$ (and $\sigma_x, \sigma_y>0$), then $\sigma_x = \sigma_y$.
This implies $\frac{\sigma_y}{\sigma_x} = 1$ and $\frac{\sigma_x}{\sigma_y} = 1$.
Then $b_{yx} = r \cdot 1 = r$ and $b_{xy} = r \cdot 1 = r$.
So $b_{yx} = b_{xy}$ is true if variances are equal.
The relationship $r^2 = b_{yx} \cdot b_{xy}$ is always true.
If $b_{yx}=b_{xy}=b$, then $r^2=b^2$, so $r = \pm b$.
Since $b_{yx}, b_{xy}, r$ all have the same sign (assuming $\sigma_x, \sigma_y>0$), we can write $b = r \frac{\sigma_y}{\sigma_x}$.
If $b \neq 0$ (i.
e.
$r \neq 0$): $r = (\pm b)$.
If $r=b$, then $b = b \frac{\sigma_y}{\sigma_x} \Rightarrow \frac{\sigma_y}{\sigma_x}=1 \Rightarrow \sigma_y=\sigma_x$.
If $r=-b$, then $b = (-b) \frac{\sigma_y}{\sigma_x} \Rightarrow \frac{\sigma_y}{\sigma_x}=-1$, which is impossible for positive standard deviations.
This implies $\sigma_y=\sigma_x$ if $b_{yx}=b_{xy} \neq 0$.
Thus, $Var(x) = Var(y)$.
Option (b) is the most fitting answer.
Option (a) is too strong; $r$ can be any value as long as variances are equal (e.
g.
if $r=0.5$ and $\sigma_x=\sigma_y$, then $b_{yx}=b_{xy}=0.5$).
Option (c) is irrelevant as means do not affect regression coefficients' slopes.
Option (d) would generally lead to $b_{yx} \neq b_{xy}$ unless $r=0$.
\[ \boxed{\text{variance of } x \text{ and variance of } y \text{ are equal}} \]