Question:
The following table provides different statistical model specifications along with the elasticity of yt with respect to xt. Which one of the following options is correct ?Row Statistical Model Elasticity 1 \(y_t=β_1+β_2\frac{1}{x_t}\epsilon_t\) \(-\frac{β_2}{x^2_t}\) 2 \(y_t=β_1-β_2\text{ln}(x_t)+\epsilon_t\) \(-\frac{β_2}{x^2_t}\) 3 ln(yt) = β1 + β2 ln(xt) + εt β2 4 ln(yt) = β1 + β2xt + εt β2xt 5 ln(yt) = β1 + β2 ln(xt) + εt β2 exp(xt) 6 ln(yt) = β1 + β2xt + εt \(β_2\frac{1}{\text{exp}(x_t)}\)
The following table provides different statistical model specifications along with the elasticity of yt with respect to xt. Which one of the following options is correct ?
| Row | Statistical Model | Elasticity |
| 1 | \(y_t=β_1+β_2\frac{1}{x_t}\epsilon_t\) | \(-\frac{β_2}{x^2_t}\) |
| 2 | \(y_t=β_1-β_2\text{ln}(x_t)+\epsilon_t\) | \(-\frac{β_2}{x^2_t}\) |
| 3 | ln(yt) = β1 + β2 ln(xt) + εt | β2 |
| 4 | ln(yt) = β1 + β2xt + εt | β2xt |
| 5 | ln(yt) = β1 + β2 ln(xt) + εt | β2 exp(xt) |
| 6 | ln(yt) = β1 + β2xt + εt | \(β_2\frac{1}{\text{exp}(x_t)}\) |
Updated On: Aug 21, 2025
- Only rows 3 and 4 are correct
- Only rows 1 and 2 are correct
- Only rows 3 and 5 are correct
- Only rows 4 and 6 are correct
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The Correct Option is A
Solution and Explanation
To determine the correct elasticity of yt with respect to xt for each model in the provided table, we need to analyze each model specification and compare them with the stated elasticities.
For elasticity, if y = f(x), elasticity ε is given by:
- ε = (∂y/∂x) * (x/y)
Let's analyze the models:
- Model: yt=β1 + β2 1/xt + εt
Elasticity should be -β2/xt2, matching the table. - Model: yt=β1 - β2ln(xt) + εt
Elasticity should be -β2/xt, not the given. - Model: ln(yt) = β1 + β2ln(xt) + εt
Elasticity here is β2, correctly given. - Model: ln(yt) = β1 + β2xt + εt
Elasticity is β2xt, correctly given. - Model: ln(yt) = β1 + β2ln(xt) + εt
Elasticity is β2, not β2exp(xt). - Model: ln(yt) = β1 + β2xt + εt
Elasticity is β2xt, not the given β2/exp(xt).
Conclusion: The correct elasticities are provided for rows 3 and 4.
Thus, the correct option is: Only rows 3 and 4 are correct
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