Question

The following table provides different statistical model specifications along with the elasticity of y_t with respect to x_t. 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	\(β_2\frac{1}{\text{exp}(x_t)}\)

• A. Only rows 3 and 4 are correct
• B. Only rows 1 and 2 are correct
• C. Only rows 3 and 5 are correct
• D. Only rows 4 and 6 are correct

Answer 1

The Correct Option is A

To determine the correct elasticity of y_t with respect to x_t 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:

1. Model: y_t=β₁ + β₂ 1/x_t + ε_t
   Elasticity should be -β₂/x_t², matching the table.
2. Model: y_t=β₁ - β₂ln(x_t) + ε_t
   Elasticity should be -β₂/x_t, not the given.
3. Model: ln(y_t) = β₁ + β₂ln(x_t) + ε_t
   Elasticity here is β₂, correctly given.
4. Model: ln(y_t) = β₁ + β₂x_t + ε_t
   Elasticity is β₂x_t, correctly given.
5. Model: ln(y_t) = β₁ + β₂ln(x_t) + ε_t
   Elasticity is β₂, not β₂exp(x_t).
6. Model: ln(y_t) = β₁ + β₂x_t + ε_t
   Elasticity is β₂x_t, not the given β₂/exp(x_t).

Conclusion: The correct elasticities are provided for rows 3 and 4.

Thus, the correct option is: Only rows 3 and 4 are correct