Question:
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:
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
In instrumental variable estimation, the ratio of the differences in the sample averages of the dependent variable \( y \) and the instrumented variable \( x \) gives the IV estimator.
Updated On: Dec 19, 2025
- \( \frac{\bar{y}_1 - \bar{y}_0}{\bar{y}_1 - \bar{y}_0} \)
- \( \frac{\bar{y}_1 - \bar{x}_1}{\bar{y}_0 - \bar{x}_0} \)
- \( \frac{\bar{y}_1 - \bar{y}_0}{\bar{x}_1 - \bar{x}_0} \)
- \( \frac{\bar{y}_1 - \bar{y}_0}{\bar{z}_1 - \bar{z}_0} \)
Show Solution
Verified By Collegedunia
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).
Was this answer helpful?
0
0
Top GATE XH-C1 Statistics, Econometrics & Mathematical Economics Questions
- Let x1, x2 ….. xn be an independently, and identically distributed (iid) random sample drawn from a population that follows the Normal Distribution N(μ, σ2), where both the mean (μ) and variance (σ2) are unknown. Let \(\bar{x}\) be the sample mean. The maximum likelihood estimator (MLE) of the variance (\(\hat{\sigma}^2_{MLE}\)) is/are then characterized by
- 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 following function f(x) to be a probability density function, the value of c will be ____________ (rounded off to two decimal places).
\(f(x)=\left\{ \begin{array}{l}\frac{c}{\sqrt{x}};0\lt x\lt 4\ \text{and}\ c\gt0 \\ 0;\ \ \text{otherwise} \end{array} \right.\) - A six-face fair die is rolled once, with X being the number that appeared on the uppermost surface. Then the variance of X is ________ (rounded off to three decimal places).
- Two friends Aditi and Raju are deciding independently whether to watch a movie or go to a music concert that evening. Both friends would prefer to spend the evening together than apart. Aditi would prefer that they watch a movie together, while Raju would prefer that they go to the concert together. The payoff matrix arising from their actions is presented below. p and (1 - p) are the probabilities that Aditi will decide in favour of the movie and concert, respectively. Similarly, q and (1 - q) are the probabilities that Raju will decide in favour of the movie and concert, respectively. Which one of the following options correctly contains all the Nash Equilibria ?
Raju Aditi Movie Concert Movie 2,1 0,0 Concert 0,0 1,2
View More Questions
Top GATE XH-C1 Regression Analysis Questions
- 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
- 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).
Top GATE XH-C1 Questions
- Amma's tone in the context of the given passage is that of :
For Amma, the difference between men and women was a kind of discrimination and inequality; she felt strongly about women's rights but was not familiar with concepts like gender and patriarchy. She would have dismissed Betty Friedan because she was predominantly dealing with the problems of white middle-class women in the United States. Amma, and women of her generation, could de-link the oppression of women from the wider struggle for the liberation of human beings from class exploitation and imperialism. So Amma continued to play her role as mother and wife, but would often complain: 'I am a doormat on which everyone wipes their emotional dirt off". - Fill in the blanks by choosing the correct sequence for the following passage:
I am wearing for the first time some (i)_____ that I have never been able to wear for long at a time, as they are horribly tight. I usually put them on just before giving a lecture. The painful pressure they exert on my feet goads my oratorical capacities to their utmost. This sharp and overwhelming pain makes me sing like a nightingale or like one of those Neapolitan singers who also wear (ii)_______ that are too tight. The visceral physical longing, the overwhelming torture provoked by my (iii)_______ forces me to extract from words distilled and sublime truths, generalized by the supreme inquisition of the pain my (iv)_______ suffer. - The appropriate synonym for the word ‘ignite' in the following passage will be :
Spirituality must be integrated with education. Self-realization is the focus. Each one of us must become aware of our higher self. We are links of a great past to a grand future. We should ignite our dormant inner energy and let it guide our lives. The radiance of such minds embarked on constructive endeavor will bring peace, prosperity and bliss to this nation. - Which of the following sentences is punctuated correctly ?
- Fill in the blanks with the correct combination of tenses for the given sentence :
Darwin's work (i)_______ a related effect that (ii) ________ influenced the development of environmental politics - a 'decentering' of the human being.
View More Questions