Which one of the following can be considered as appropriate pair of values of regression coefficient of y on x and regression coefficient of x on y?
(1, 1)
Understanding Regression Coefficients
In linear regression, we often study the relationship between two variables, say \(x\) and \(y\). The relationship can be represented by regression lines. There are typically two regression lines: the regression line of \(y\) on \(x\) and the regression line of \(x\) on \(y\).
- The regression line of \(y\) on \(x\) is used to predict \(y\) given a value of \(x\). Its equation is often written as \(\hat{y} = a_{yx} + b_{yx} x\), where \(b_{yx}\) is the regression coefficient of \(y\) on \(x\).
- The regression line of \(x\) on \(y\) is used to predict \(x\) given a value of \(y\). Its equation is often written as \(\hat{x} = a_{xy} + b_{xy} y\), where \(b_{xy}\) is the regression coefficient of \(x\) on \(y\).
The regression coefficients, \(b_{yx}\) and \(b_{xy}\), tell us about the nature and strength of the linear relationship between \(x\) and \(y\). They are related to the correlation coefficient, \(r\), which measures the linear association between the two variables.
Relationship Between Regression and Correlation Coefficients
A fundamental property connecting the two regression coefficients (\(b_{yx}\) and \(b_{xy}\)) and the correlation coefficient (\(r\)) is:
\[r^2 = b_{yx} \times b_{xy}\]The correlation coefficient \(r\) always lies between -1 and 1, inclusive (\(-1 \le r \le 1\)). Consequently, the square of the correlation coefficient, \(r^2\), must always lie between 0 and 1, inclusive (\(0 \le r^2 \le 1\)).
From the relationship \(r^2 = b_{yx} \times b_{xy}\), it follows that the product of the two regression coefficients must also lie between 0 and 1:
\[0 \le b_{yx} \times b_{xy} \le 1\]Another important property is that the signs of \(b_{yx}\), \(b_{xy}\), and \(r\) must all be the same (unless \(r=0\), in which case both \(b_{yx}\) and \(b_{xy}\) are 0). If \(r > 0\), then \(b_{yx} > 0\) and \(b_{xy} > 0\). If \(r < 0\), then \(b_{yx} < 0\) and \(b_{xy} < 0\). If \(r = 0\), then \(b_{yx} = 0\) and \(b_{xy} = 0\).
Therefore, for a pair of values \((b_{yx}, b_{xy})\) to be appropriate for regression coefficients, two conditions must be met:
- The product \(b_{yx} \times b_{xy}\) must be between 0 and 1, inclusive.
- Both \(b_{yx}\) and \(b_{xy}\) must have the same sign (both positive, both negative, or both zero).
Analyzing the Given Options for Appropriate Regression Coefficients
Let's examine each given pair of values based on the conditions derived above.
| Option | \(b_{yx}\) Value | \(b_{xy}\) Value | Product \(b_{yx} \times b_{xy}\) | Sign Compatibility | Condition \(0 \le \text{product} \le 1\) Met? | Appropriate Pair? |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | \(1 \times 1 = 1\) | Same (Both positive) | Yes, \(0 \le 1 \le 1\) | Yes |
| 2 | -1 | 1 | \((-1) \times 1 = -1\) | Different | No, \(-1\) is not \(\ge 0\) | No |
| 3 | \(-\frac{1}{2}\) | 2 | \((-\frac{1}{2}) \times 2 = -1\) | Different | No, \(-1\) is not \(\ge 0\) | No |
| 4 | \(\frac{1}{3}\) | \(\frac{10}{3}\) | \(\frac{1}{3} \times \frac{10}{3} = \frac{10}{9}\) | Same (Both positive) | No, \(\frac{10}{9} \approx 1.11\) is not \(\le 1\) | No |
Detailed Analysis of Each Option:
- Option 1: (1, 1)
- \(b_{yx} = 1\), \(b_{xy} = 1\).
- Product: \(b_{yx} \times b_{xy} = 1 \times 1 = 1\).
- Condition check: \(0 \le 1 \le 1\). This condition is satisfied.
- Sign check: Both \(b_{yx}\) and \(b_{xy}\) are positive. The signs are the same.
- Since both conditions are met, this is an appropriate pair of regression coefficients. This pair corresponds to a perfect positive linear correlation (\(r^2 = 1\), so \(r = 1\)).
- Option 2: (-1, 1)
- \(b_{yx} = -1\), \(b_{xy} = 1\).
- Product: \(b_{yx} \times b_{xy} = (-1) \times 1 = -1\).
- Condition check: \(0 \le -1 \le 1\). This condition is not satisfied because \(-1\) is less than 0.
- Sign check: \(b_{yx}\) is negative, and \(b_{xy}\) is positive. The signs are different, which is not possible for non-zero regression coefficients.
- This is not an appropriate pair.
- Option 3: \(\left( { - \frac{1}{2},\;2} \right)\)
- \(b_{yx} = -\frac{1}{2}\), \(b_{xy} = 2\).
- Product: \(b_{yx} \times b_{xy} = (-\frac{1}{2}) \times 2 = -1\).
- Condition check: \(0 \le -1 \le 1\). This condition is not satisfied because \(-1\) is less than 0.
- Sign check: \(b_{yx}\) is negative, and \(b_{xy}\) is positive. The signs are different, which is not possible for non-zero regression coefficients.
- This is not an appropriate pair.
- Option 4: \(\left( {\frac{1}{3},\frac{{10}}{3}} \right)\)
- \(b_{yx} = \frac{1}{3}\), \(b_{xy} = \frac{10}{3}\).
- Product: \(b_{yx} \times b_{xy} = \frac{1}{3} \times \frac{10}{3} = \frac{10}{9}\).
- Condition check: \(0 \le \frac{10}{9} \le 1\). This condition is not satisfied because \(\frac{10}{9}\), which is approximately 1.11, is greater than 1.
- Sign check: Both \(b_{yx}\) and \(b_{xy}\) are positive. The signs are the same.
- Although the signs are compatible, the product is outside the valid range [0, 1]. This is not an appropriate pair.
Based on the analysis, only option 1 satisfies both conditions for being an appropriate pair of regression coefficients.
Revision Table: Key Properties of Regression Coefficients
| Property | Description | Relationship |
|---|---|---|
| Product of Coefficients | The product of the two regression coefficients equals the square of the correlation coefficient. | \(b_{yx} \times b_{xy} = r^2\) |
| Range of Product | Since \(0 \le r^2 \le 1\), the product of the regression coefficients must be between 0 and 1. | \(0 \le b_{yx} \times b_{xy} \le 1\) |
| Sign of Coefficients | The sign of \(b_{yx}\) and \(b_{xy}\) must be the same as the sign of the correlation coefficient \(r\) (unless \(r=0\)). | Sign(\(b_{yx}\)) = Sign(\(b_{xy}\)) = Sign(\(r\)) |
Additional Information: Interpreting Regression Coefficients
The regression coefficient represents the expected change in the dependent variable for a one-unit increase in the independent variable.
- \(b_{yx}\): This coefficient from the regression of \(y\) on \(x\) indicates the expected change in \(y\) for a one-unit increase in \(x\). A positive \(b_{yx}\) suggests that \(y\) tends to increase as \(x\) increases, while a negative \(b_{yx}\) suggests that \(y\) tends to decrease as \(x\) increases.
- \(b_{xy}\): This coefficient from the regression of \(x\) on \(y\) indicates the expected change in \(x\) for a one-unit increase in \(y\). A positive \(b_{xy}\) suggests that \(x\) tends to increase as \(y\) increases, while a negative \(b_{xy}\) suggests that \(x\) tends to decrease as \(y\) increases.
The magnitude of the regression coefficient depends on the units of measurement of the variables. Unlike the correlation coefficient, which is unitless, regression coefficients have units (e.g., units of y per unit of x for \(b_{yx}\)).
The case where \(b_{yx} \times b_{xy} = 1\) occurs when there is a perfect linear relationship between \(x\) and \(y\) (\(|r| = 1\)). In this case, the two regression lines coincide.
The case where \(b_{yx} \times b_{xy} = 0\) occurs when there is no linear relationship between \(x\) and \(y\) (\(r=0\)). This happens when either \(b_{yx} = 0\) or \(b_{xy} = 0\) (or both). If \(b_{yx} = 0\), the best prediction of \(y\) is its mean, regardless of \(x\). If \(b_{xy} = 0\), the best prediction of \(x\) is its mean, regardless of \(y\).
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Important Questions from Correlation and Regression
- For testing the significance of overall regression the test to be used is
- Including relevant lagged values of the dependent variable on the right hand side of a regression equation could lead to which one of the following ?
- If OLS is used in the presence of auto-correlation, which of the following will be likely consequences ?
i. Coefficients estimates may be misleading.
ii. Hypothesis tests could reach the wrong conclusions.
iii. Forecasts made from the model could be biased.
iv. Standard errors may be inappropriate. - Regression analysis is of particular interest
- Which of the following can be linked with those providing method for estimating black money in India ?
I. Kaldor’s approach
II. NIPFP method
III. O.P. Chopra’s method
IV. Arjun Sengupta
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