Consider a regression model \( y = \beta_0 + \beta x + u \) where the continuous variable \( y \) is regressed on a dummy variable \( x \), which takes the value either 1 or 0.
However, the model was estimated using the instrumental variable (IV) estimation method, wherein the indicator variable \( z \) is used as an instrument of \( x \).
Let \( \bar{y}_1 \) and \( \bar{y}_0 \) be the sample averages of \( y \) when \( z \) takes the value 1 and 0, respectively.
Let \( \bar{x}_1 \) and \( \bar{x}_0 \) be the sample averages of \( x \) when \( z \) takes the value 1 and 0, respectively.
Let \( \bar{y}_1 \) and \( \bar{y}_0 \) be the sample averages of \( y \) when \( x \) takes the value 1 and 0, respectively.
Let \( \bar{z}_1 \) and \( \bar{z}_0 \) be the sample averages of \( z \) when \( x \) takes the value 1 and 0, respectively.
Then the estimated coefficient of \( \beta_{IV} \) is:
Show Hint
- \( \frac{\bar{y}_1 - \bar{y}_0}{\bar{y}_1 - \bar{y}_0} \)
- \( \frac{\bar{y}_1 - \bar{x}_1}{\bar{y}_0 - \bar{x}_0} \)
- \( \frac{\bar{y}_1 - \bar{y}_0}{\bar{x}_1 - \bar{x}_0} \)
- \( \frac{\bar{y}_1 - \bar{y}_0}{\bar{z}_1 - \bar{z}_0} \)
The Correct Option is C
Solution and Explanation
In an instrumental variable (IV) estimation model, we use an instrument \( z \) for the potentially endogenous regressor \( x \). The key idea is to isolate the variation in \( x \) that is exogenous (uncorrelated with the error term \( u \)) by using the instrument \( z \). The coefficient \( \beta_{IV} \) can be estimated using the following formula: \[ \beta_{IV} = \frac{\text{Cov}(y, z)}{\text{Cov}(x, z)} \] This equation is equivalent to the ratio of the difference in sample averages of \( y \) and \( x \) when \( z \) takes different values. Therefore, the formula for \( \beta_{IV} \) is: \[ \beta_{IV} = \frac{\bar{y}_1 - \bar{y}_0}{\bar{x}_1 - \bar{x}_0} \] This matches option (C), which gives the correct expression for the IV estimator. Thus, the correct answer is (C).
Top Questions on Regression Analysis
- Consider a simple pooled regression model: yit = β0 + β1xit + vit where vit = μi + ∈it and Cov(xit, μi) ≠ 0. Here, μi captures the unknown individual specific effects and it is the idiosyncratic error uncorrelated with both xit and μi. If the parameters of this model are estimated using the ordinary least squares (OLS) method, then the estimated slope coefficient will be
- For the function \(F: \R^2 → \R\) specified as F(x, y) = x3 - y3 + 9xy, which of the following options is/are correct
- To study the effect of $X_1$ and $X_2$ on Y, the following regression model is estimated using a large sample:
$ Y = a + \beta_1 X_1 + \beta_2X_2 + ∈ $
The OLS estimates and standard errors are presented below:$a$ $\beta_1$ $\beta_2$ Estimates 2.3 0.39 1.8 Standard errors 1.15 0.13 1
Given the above information, which of the following is CORRECT?- IIT JAM EN - 2024
- Mathematics for Economy
- Regression Analysis
- Consider two regression models estimated on a sample of 350 observations.
\[ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_4 + \beta_5 x_5 + u \quad \text{--------(1)} \] \[ y = \alpha_0 + \alpha_1 x_1 + \alpha_2 x_2 + v \quad \text{--------(2)} \] The \( R^2 \) in model (1) is \( R^2_1 = 0.3521 \) and in model (2) is \( R^2_2 = 0.2314 \). The value of the test statistic to test the \( H_0: \beta_3 = \beta_4 = \beta_5 = 0 \) is _________ (round off to three decimal places).
Questions Asked in GATE XH-C1 exam
Eight students (P, Q, R, S, T, U, V, and W) are playing musical chairs. The figure indicates their order of position at the start of the game. They play the game by moving forward in a circle in the clockwise direction.
After the 1st round, the 4th student behind P leaves the game.
After the 2nd round, the 5th student behind Q leaves the game.
After the 3rd round, the 3rd student behind V leaves the game.
After the 4th round, the 4th student behind U leaves the game.
Who all are left in the game after the 4th round?
The 12 musical notes are given as \( C, C^\#, D, D^\#, E, F, F^\#, G, G^\#, A, A^\#, B \). Frequency of each note is \( \sqrt[12]{2} \) times the frequency of the previous note. If the frequency of the note C is 130.8 Hz, then the ratio of frequencies of notes F# and C is:
Here are two analogous groups, Group-I and Group-II, that list words in their decreasing order of intensity. Identify the missing word in Group-II.
Abuse \( \rightarrow \) Insult \( \rightarrow \) Ridicule
__________ \( \rightarrow \) Praise \( \rightarrow \) Appreciate
- Had I learnt acting as a child, I ________ a famous film star.
Select the most appropriate option to complete the above sentence. - A car started from city P at 9:40 AM. The time taken for the car to reach city Q is 4 hours and 50 minutes. The time of arrival of the car at city Q is: