Regression and Simultaneous Equations - Statistics, Econometrics and Mathematical Economics - Economics Previous Year Questions

1

2021 · Economics · Statistics, Econometrics and Mathematical Economics · Regression and Simultaneous Equations

Humanities & Social Sciences-Economics (XH-C1) 2021

What would be the consequences for the OLS estimator if heteroscedasticity is present in a regression model but ignored? Assume that all the other classical assumptions are valid.

A

It will be biased

B

It will be inconsistent

C

It will be unbiased but inefficient

D

It will be unbiased but inconsistent

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2

2021 · Economics · Statistics, Econometrics and Mathematical Economics · Regression and Simultaneous Equations

Humanities & Social Sciences-Economics (XH-C1) 2021

To determine the relationship between \( y \) and \( x_1 \), Rohit estimated two different OLS models. In the first model, Rohit regressed \( y \) on \( x_1 \) and \( x_2 \) as given below
\( y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + u \) (1)
In the second model, Rohit regressed \( y \) only on \( x_1 \) as given below
\( y = \delta_0 + \delta_1 x_1 + v \) (2)
The estimated coefficients of \( x_1 \) in the above two models are exactly same. From this observation we can state conclusively that
(i) \( Cov(x_1, y) = 0 \)
(ii) \( \hat{\beta}_2 = 0 \)
(iii) \( Cov(x_2, x_1) = 0 \)
where \( \hat{\beta}_2 \) is the estimated coefficient of \( x_2 \) in the equation (1)

A

Only (i) is true

B

Only (ii) is true

C

Either (ii) or (iii) or both are true

D

Neither (ii) nor (iii) is true

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3

2021 · Economics · Statistics, Econometrics and Mathematical Economics · Regression and Simultaneous Equations

Humanities & Social Sciences-Economics (XH-C1) 2021

In a demand function estimation of a good \( X \), a researcher collected data on various households’ consumption of good \( X \) (\( Q_x \)) for various price levels. The researcher also collected data on household income (\( M \)) and household size (\( S \)). The estimated regression result is
\( \log Q_x = -0.345(0.111) - 1.543(2.345) \log P_x + 1.123(0.012) \log M + 0.234(0.123) \log S \)
where \( P_x \) is price per unit of \( X \). The values in the parentheses are the standard errors of the estimated coefficients. From the estimation one can conclude that

A

the demand for good \(X \) is highly elastic

B

\(X \) is an inferior good

C

the estimated price elasticity of demand is not statistically significant

D

the estimated price elasticity of good \(X \) is 2.345

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4

2021 · Economics · Statistics, Econometrics and Mathematical Economics · Regression and Simultaneous Equations

Humanities & Social Sciences-Economics (XH-C1) 2021

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
\( \bar{x}_1 \) and \( \bar{x}_0 \) be the sample averages of x when z takes the value 1 and 0, respectively
\( \bar{y}^1 \) and \( \bar{y}^0 \) be the sample averages of y when x takes the value 1 and 0, respectively
\( \bar{z}^1 \) 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

A

\(\frac{\bar{y}_1 - \bar{y}_0}{\bar{y}^1 - \bar{y}^0} \)

B

\(\frac{\bar{y}_1 - \bar{x}_1}{\bar{y}_0 - \bar{x}_0} \)

C

\(\frac{\bar{y}_1 - \bar{y}_0}{\bar{x}_1 - \bar{x}_0} \)

D

\(\frac{\bar{y}_1 - \bar{y}_0}{\bar{z}^1 - \bar{z}^0} \)

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5

2021 · Economics · Statistics, Econometrics and Mathematical Economics · Regression and Simultaneous Equations

Humanities & Social Sciences-Economics (XH-C1) 2021

Consider two regression models estimated on a sample of 350 observations.
\(y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_4 + \beta_5 x_5 + u\) ---------(1)
\(y = \alpha_0 + \alpha_1 x_1 + \alpha_2 x_2 + v\) ---------(2)
The \(R^2\) in model (1) is \(R_1^2 = 0.3521\) 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).

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6

2022 · Economics · Statistics, Econometrics and Mathematical Economics · Regression and Simultaneous Equations

Humanities & Social Sciences-Economics (XH-C1) 2022

Suppose we estimate the following regression equation:
\( ln(x_t) = a_0 + a_1 ln(y_t) + \epsilon_t , \quad a_0, a_1 > 0 \)
where \( x_t \) and \( y_t \) are some variables. \( a_0 \) and \( a_1 \) are the intercept and the slope, respectively. \( \epsilon_t \) is the residual term. What is the interpretation of the coefficient \( a_1 \)?

A

A 1% increase in \(y_t \) causes a \(a_1 \)% increase in \(x_t \)

B

A 1% increase in \(y_t \) causes a \(a_1 \times 0.01 \) unit increase in \(x_t \)

C

A one unit increase in \(y_t \) causes a \(100 \times a_1 \)% increase in \(x_t \)

D

A one unit increase in \(y_t \) causes a \(a_1 \) unit increase in \(x_t \)

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7

2022 · Economics · Statistics, Econometrics and Mathematical Economics · Regression and Simultaneous Equations

Humanities & Social Sciences-Economics (XH-C1) 2022

Which one of the following is a test of heteroscedastity?

A

White test

B

Jarque-Bera test

C

Breusch-Godfrey test

D

Ljung-box test

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8

2022 · Economics · Statistics, Econometrics and Mathematical Economics · Regression and Simultaneous Equations

Humanities & Social Sciences-Economics (XH-C1) 2022

Suppose an econometrician had specified the following regression:
\( y_t = \beta_1 + \beta_2 z_{2t} + \beta_3 z_{3t} + \beta_4 z_{4t} + \varepsilon_t \)
but a researcher estimated the following regression:
\( y_t = \beta_1 + \beta_2 z_{2t} + \beta_3 z_{3t} + \beta_4 z_{4t} + \beta_5 z_{5t} + \varepsilon_t \)
What will be the consequence of including the irrelevant variable on the estimated coefficients?

A

Coefficient estimates will be unbiased, consistent but inefficient

B

Coefficient estimates will be consistent, asymptotically efficient but biased

C

Coefficient estimates will be inconsistent and efficient

D

Coefficient estimates will be biased, consistent and efficient

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9

2022 · Economics · Statistics, Econometrics and Mathematical Economics · Regression and Simultaneous Equations

Humanities & Social Sciences-Economics (XH-C1) 2022

Suppose an investigator has specified the following simultaneous equation model:
\(y_{1,t} = \alpha_{1,0} + \alpha_{1,2}y_{2,t} + \alpha_2M_t + \alpha_3K_t + u_t\) (1)
\(y_{2,t} = \alpha_{3,0} + \alpha_{3,1}y_{1,t} + \alpha_3I_t + v_t\) (2)
where \(y_{1,t}\) and \(y_{2,t}\) are endogenous variables. \(M_t\), \(K_t\) and \(I_t\) are predetermined variables. \(u_t\) and \(v_t\) are residual terms. Based on the order condition of identification, which one of the following statements are correct?

A

Equations (1) and (2) are exactly identified

B

Equation (1) is exactly identified and equation (2) is over-identified

C

Equation (1) is over-identified and equation (2) is exactly identified

D

Both the equations are under-identified

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10

2023 · Economics · Statistics, Econometrics and Mathematical Economics · Regression and Simultaneous Equations

Humanities & Social Sciences-Economics (XH-C1) 2023

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

A

Homoscedasticity

B

No Perfect Multicollinearity

C

No Autocorrelation

D

Linearity in parameters

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11

2023 · Economics · Statistics, Econometrics and Mathematical Economics · Regression and Simultaneous Equations

Humanities & Social Sciences-Economics (XH-C1) 2023

Consider the following simultaneous equations model:
\[ Y_t = \beta_1 + \beta_2 X_t + \beta_3 X_{t-1} + \beta_4 Z_t + \mu_{1t} \quad (1) \]
\[ Z_t = \delta_1 + \delta_2 Y_t + \delta_3 W_t + \mu_{2t} \quad (2) \]
Before estimating the above model, a researcher performed the test of identification using order and rank conditions, and found that equation (2) is overidentified. Then, which of the following methods is appropriate to estimate equation (2)?

A

Two-Stage Least Squares

B

Indirect Least Squares

C

Weighted Least Squares

D

Ordinary Least Squares

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12

2023 · Economics · Statistics, Econometrics and Mathematical Economics · Regression and Simultaneous Equations

Humanities & Social Sciences-Economics (XH-C1) 2023

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 \[ \hat{S}_i = \hat{\beta}_0 + \hat{\beta}_1 CGPA_i \quad ; \quad i = 1,2,...,100 \] 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).

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13

2023 · Economics · Statistics, Econometrics and Mathematical Economics · Regression and Simultaneous Equations

Humanities & Social Sciences-Economics (XH-C1) 2023

Suppose from the estimation of a linear regression model \[ Y_i = \beta_0 + \beta_1 X_i + e_i \] the residual sum of squares and the total sum of squares are obtained as 44 and 80, respectively. The value of coefficient of determination is ________ (round off to two decimal places).

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14

2024 · Economics · Statistics, Econometrics and Mathematical Economics · Regression and Simultaneous Equations

Humanities & Social Sciences-Economics (XH-C1) 2024

Consider a simple pooled regression model: \(y_{it} = \beta_0 + \beta_1 x_{it} + v_{it}\) where \(v_{it} = \mu_i + \epsilon_{it}\) and \(Cov(x_{it}, \mu_i) \neq 0\). Here, \(\mu_i\) captures the unknown individual specific effects and \(\epsilon_{it}\) is the idiosyncratic error uncorrelated with both \(x_{it}\) and \(\mu_i\). If the parameters of this model are estimated using the ordinary least squares (OLS) method, then the estimated slope coefficient will be

A

biased

B

inconsistent

C

unbiased but consistent

D

unbiased but efficient

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15

2024 · Economics · Statistics, Econometrics and Mathematical Economics · Regression and Simultaneous Equations

Humanities & Social Sciences-Economics (XH-C1) 2024

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?

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

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16

2024 · Economics · Statistics, Econometrics and Mathematical Economics · Regression and Simultaneous Equations

Humanities & Social Sciences-Economics (XH-C1) 2024

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).

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17

2024 · Economics · Statistics, Econometrics and Mathematical Economics · Regression and Simultaneous Equations

Humanities & Social Sciences-Economics (XH-C1) 2024

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).

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18

2025 · Economics · Statistics, Econometrics and Mathematical Economics · Regression and Simultaneous Equations

Humanities & Social Sciences-Economics (XH-C1) 2025

Consider a two-variables \((x, y)\) linear regression model
\[ y = \alpha + \beta x + \epsilon \]
where \(\alpha\) and \(\beta\) are the parameters, and \(\epsilon\) is the error term.
The parameters are estimated using the Ordinary Least Squares (OLS) method. Let \(b\) denote the estimated value of \(\beta\). If \(b = 0\), then which one of the following statements is CORRECT?

A

\(R^2\) can be any real number in (0, 0.5]

B

\(R^2\) can be any real number in (0.5, 1)

C

\(R^2\) is any positive real number greater than 1

D

\(R^2 = 0\)

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19

2026 · Economics · Statistics, Econometrics and Mathematical Economics · Regression and Simultaneous Equations

Humanities & Social Sciences - Economics (XH-C1) 2026

Consider a two-variable \((x, y)\) linear regression model given in Equations (1) and (2) below. The error term, \(u_t\), of Equation (1) is serially correlated as given in Equation (2). The error term, \(\epsilon_t\), of Equation (2) is white noise. All other assumptions of the Classical Linear Regression Model (CLRM) hold. Then, which of the following options is/are FALSE?
\[ y_t = \alpha + \beta x_t + u_t \quad (1) \]
\[ u_t = \rho u_{t-1} + \epsilon_t; \ 0 < \rho < 1 \quad (2) \]

A

Ordinary Least Squares estimators of the parameters in Equation (1) are biased

B

Ordinary Least Squares estimators of the parameters in Equation (1) are asymptotically normally distributed

C

Ordinary Least Squares estimators of the parameters in Equation (1) are efficient

D

Ordinary Least Squares estimators of the parameters in Equation (1) are consistent

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20

2026 · Economics · Statistics, Econometrics and Mathematical Economics · Regression and Simultaneous Equations

Humanities & Social Sciences - Economics (XH-C1) 2026

The estimated energy demand is given in the Equation below, where, \(\hat{y}_t\) is the estimated value of the log of energy demand at time \(t\). The numbers in the parentheses are the values of the \(t\)-statistic of the corresponding coefficients.
\(\hat{y}_t = 1.24 + 0.56 \ln(GDP_t) - 0.03 \ln(energyprice_t)\)
(10.73) (7.28) (5.28)
Which one of the following statements is CORRECT?

A

Demand for energy is price inelastic

B

Demand for energy is price elastic

C

Demand for energy is unit elastic

D

Sufficient information has not been provided to calculate price elasticity

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