List of top Statistics, Econometrics & Mathematical Economics Questions asked in GATE XH-C1

Let x₁, x₂ ….. x_n 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: y_it = β₀ + β₁x_it + vit where v_it = μ_i + ∈_it and Cov(x_it, μ_i) ≠ 0. Here, μ_i captures the unknown individual specific effects and it is the idiosyncratic error uncorrelated with both x_it and μ_i. 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

If X and Y are two random variables with the joint probability density function
\(f(x,y)=\left\{ \begin{array}{l} \frac{2}{3}(x+2y);\ \text{for}\ 0\lt x, y\lt 1 \\ 0;\ \ \ \ \ \ \ \ \ \ \text{otherwise} \end{array} \right.\)
then \(E[X|Y=\frac{1}{2}]\) will be

If a discrete random variable X follows the uniform distribution and assumes only the values 8, 9, 11, 15, 18, and 20, then P(|X -14| < 5) is

Assume the following probabilities for two events, A and B: P(A) = 0.50,P(B) = 0.50, and P(A ∪ B) = 0.85. Then we can conclude that

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	\(β_2\frac{1}{\text{exp}(x_t)}\)

For the function \(F: \R^2 → \R\) specified as F(x, y) = x³ - y³ + 9xy, which of the following options is/are correct

Let x and y be two dummy variables that take the values of either 0 or 1, and follow the bivariate frequency distribution as given below. If a logit regression is estimated with y as the dependent variable and x as the independent variable, then the estimated coefficient of x is _____ (rounded off to two decimal places)

x	0	1	Total
y
0	6	11	17
1	6	7	13
Total	12	18	30

The estimated results of a Probit model is given in the table below, where Y is a binary variable taking the value either 0 or 1, and X is an integer. The probability that Y = 1 when X = 30 is ________ (rounded off to two decimal places).

Variable	Coefficient	Standard Error	Z-Statistic	Probability
Constant	-0.064	0.399	-0.161	0.871
X	0.029	0.010	2.916	0.003

An analyst regressed $Y$ on $X_1$ and $X_2$. If she later noticed that $X_1=5X_2$, then which assumption of the classical linear regression model was violated?

Using the Ordinary Least Squares (OLS) method, a researcher estimated the relationship between initial salary ($S$) of MBA graduates and their cumulative grade point average (CGPA) as $\hat S_i=\hat\beta_0+\hat\beta_1\,CGPA_i,\ i=1,2,\ldots,100$, where $\hat\beta_0=4543$ and $\hat\beta_1=645.08$. The standard errors of $\hat\beta_0$ and $\hat\beta_1$ are $921.79$ and $70.01$, respectively. The $t$-statistic for testing the null hypothesis $\beta_1=0$ is _____{(round off to two decimal places)

Consider the following simultaneous equations model: \[ Y_t=\beta_1+\beta_2 X_t+\beta_3 X_{t-1}+\beta_4 Z_t+\mu_{1t} (1),\qquad Z_t=\delta_1+\delta_2 Y_t+\delta_3 W_t+\mu_{2t} (2) \] Before estimating, identification tests (order and rank) show that equation (2) is {overidentified. Which method is appropriate to estimate equation (2)?}