Statistics and Econometrics Questions and Answers

Q: The last period's forecast was 70 and demand was 60. What is the simple exponential smoothing forecast with alpha of 0.4 for the next period ?

The forecast for the next period is 66.Simple exponential smoothing adds a fraction alpha of the last error to the last forecast:\[ \begin{aligned} F_{t+1} &= F_{t} + \alpha(D_{t} - F_{t}) \\ &= 70 + 0.4(60 - 70) \\ &= 70 - 4 = 66 \end{aligned} \]Equivalently, \(0.4 \times 60 + 0.6 \times 70 = 24 + 42 = 66\).65 is the plain average of 60 and 70, 62 would come from weighting the demand by 0.8, and 63.8 does not follow from the formula.

Q: Which of the following is true regarding the two smoothing constants of the Forecast Including Trend (FIT) model ?

The true statement is that their values are determined independently.The forecast including trend (FIT), or trend-adjusted exponential smoothing, uses two smoothing constants: alpha for the average (level) and beta for the trend. Each is chosen separately, usually by trying values and comparing forecast errors, so neither constrains the other.They are called alpha and beta, but 'producer's risk' and 'consumer's risk' are terms from acceptance sampling, not forecasting, so B is wrong. Alpha need not be smaller than beta (C), and since B and C are false, 'all of the above' fails too.

Q: For two random variables X & Y, covariance measures :

Covariance measures the direction of the relationship between two variables.Covariance is the average product of deviations from the means, \(Cov(X,Y) = E[(X - \mu_{X})(Y - \mu_{Y})]\). A positive value means the variables tend to move together, and a negative value that they move in opposite directions.Its size depends on the units of measurement, so it does not show the strength of the relationship; that is what the correlation coefficient does, by dividing covariance by the two standard deviations. So the key rejects 'Both (A) and (B)', and the variances of X and Y are separate measures of each variable's spread.

Q: For a linear regression model Yi = α + βXi + ρi, Durbin-Watson statistic is used to test for :

The Durbin-Watson statistic tests for Serial autocorrelation among residuals.The statistic is \(d = \sum (e_{t} - e_{t-1})^{2} / \sum e_{t}^{2}\), computed from successive OLS residuals. It ranges from 0 to 4: a value near 2 indicates no first-order autocorrelation, values towards 0 positive autocorrelation, and values towards 4 negative autocorrelation.It is not a check on arithmetic. Functional form is tested with tools such as Ramsey's RESET test, and heteroscedasticity with tests such as Breusch-Pagan, White or Goldfeld-Quandt.

Q: Suppose X is a normally distributed random variable with an unknown mean μ and an unknown variance σ2. The following test can be used to test a hypothesis about unknown population mean :

The appropriate test is the t-test.When the population is normal but its variance is unknown, the variance has to be estimated from the sample. The statistic \(t = (\bar{X} - \mu_{0})/(s/\sqrt{n})\) then follows Student's t distribution with n - 1 degrees of freedom, which is used to test a hypothesis about the mean.The z-test requires a known population variance, or a very large sample. The F-test compares variances, and the chi-square test is used for a variance or for goodness of fit and independence, not for a mean.

Q: If X1, X2, X3, .............., Xn is a random sample from a normally distributed population with mean μ and variance σ2, the sampling distribution of \(\bar{X}\) is :

The sampling distribution of the mean is normal with mean μ and variance σ2/n.The sample mean of n independent observations from a normal population is itself normally distributed. Its expected value equals the population mean, and its variance is the population variance divided by the sample size:\[ \bar{X} \sim N\left(\mu,\ \frac{\sigma^{2}}{n}\right) \]Variance σ2 (A) is that of a single observation. Dividing by n2 (B) understates it; the variance of a sum of n terms is nσ2, and dividing the sum by n divides that by n2, giving σ2/n.

Q: The logistic regression model is used to analyze :

The logistic regression model is used to analyse discrete choice.Logistic (logit) regression models a dependent variable that takes a small number of discrete values, typically 0 or 1, such as whether a household buys a car or a worker joins the labour force. It estimates the probability of each outcome as a logistic function of the explanatory variables, which keeps predicted probabilities between 0 and 1.Continuous outcomes are modelled with ordinary regression. National income is a continuous variable, and incorrectly measured data call for errors-in-variables methods, not logit.

Q: ARIMA (p, d, q) models are used to analyze :

ARIMA models are used to analyse time series data.ARIMA (autoregressive integrated moving average) models, developed by Box and Jenkins, describe a variable observed over time in terms of its own past values (the AR part, of order p), past random shocks (the MA part, of order q), after differencing d times to make the series stationary (the I part). They are widely used for forecasting.Cross-section data have no time ordering, so autoregression has no meaning there. Discrete data need models such as logit or count models. 'Unstable data' is not a data type that ARIMA is designed for; non-stationarity is handled by differencing.

Q: The estimate of \(\hat{b}\) in the following regression model \(Y_{i} = \alpha + \beta X_{i} + e_{i}\) is given by :

The OLS estimate is Σ(Xi - X̄)(Yi - Ȳ) / Σ(Xi - X̄)2.Minimising the sum of squared residuals in Yi = α + βXi + ei gives the slope estimate as the sample covariance of X and Y divided by the sample variance of X:\[ \begin{aligned} \hat b &= \frac{S_{XY}}{S_{XX}} \\ S_{XY} &= \Sigma(X_i-\bar X)(Y_i-\bar Y) \\ S_{XX} &= \Sigma(X_i-\bar X)^2 \end{aligned} \]Option B is the form of the intercept estimate, which is Ȳ minus the slope times X̄. Option C has a square root of an unsquared sum in the denominator, and option D is a ratio of variances, not the slope.

Q: In the following regression model \(Y_{i} = \alpha + \beta X_{i} + e_{i}\), under the usual assumptions, variance of \(\hat{b}\) is given by :

The variance of the slope estimate is σ2 / Σ(Xi - X̄)2.Under the classical assumptions (errors with mean zero, constant variance σ2 and no autocorrelation, X fixed), the slope estimator's variance is\[ \text{Var}(\hat b)=\frac{\sigma^2}{\Sigma(X_i-\bar X)^2} \]It falls as the X values are more spread out, which is why wide variation in the regressor gives more precise estimates. σ2 alone (A) is the error variance, option C inverts the formula, and option D uses the variation in Y instead of X.

31.

The last period's forecast was 70 and demand was 60. What is the simple exponential smoothing forecast with alpha of 0.4 for the next period ?

A.63.8
B.65
C.62
D.66

Exam Year(s)

MH SET - 2020, SET Exam - 2020,

Answer: D

The forecast for the next period is 66.

Simple exponential smoothing adds a fraction alpha of the last error to the last forecast:

\[ \begin{aligned} F_{t+1} &= F_{t} + \alpha(D_{t} - F_{t}) \\ &= 70 + 0.4(60 - 70) \\ &= 70 - 4 = 66 \end{aligned} \]

Equivalently, \(0.4 \times 60 + 0.6 \times 70 = 24 + 42 = 66\).

65 is the plain average of 60 and 70, 62 would come from weighting the demand by 0.8, and 63.8 does not follow from the formula.

Discuss in Forum

32.

Which of the following is true regarding the two smoothing constants of the Forecast Including Trend (FIT) model ?

A.Their values are determined independently
B.They are called alpha and beta, producer risk and consumers risk
C.Alpha is always smaller than beta
D.All of the above

Exam Year(s)

MH SET - 2020, SET Exam - 2020,

Answer: A

The true statement is that their values are determined independently.

The forecast including trend (FIT), or trend-adjusted exponential smoothing, uses two smoothing constants: alpha for the average (level) and beta for the trend. Each is chosen separately, usually by trying values and comparing forecast errors, so neither constrains the other.

They are called alpha and beta, but 'producer's risk' and 'consumer's risk' are terms from acceptance sampling, not forecasting, so B is wrong. Alpha need not be smaller than beta (C), and since B and C are false, 'all of the above' fails too.

Discuss in Forum

33.

For two random variables X & Y, covariance measures :

A.Direction of the relationship between them
B.Strength of relationship between them
C.Both (A) and (B)
D.Variance of X and variance of Y

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: A

Covariance measures the direction of the relationship between two variables.

Covariance is the average product of deviations from the means, \(Cov(X,Y) = E[(X - \mu_{X})(Y - \mu_{Y})]\). A positive value means the variables tend to move together, and a negative value that they move in opposite directions.

Its size depends on the units of measurement, so it does not show the strength of the relationship; that is what the correlation coefficient does, by dividing covariance by the two standard deviations. So the key rejects 'Both (A) and (B)', and the variances of X and Y are separate measures of each variable's spread.

Discuss in Forum

34.

For a linear regression model Y_i = α + βX_i + ρ_i, Durbin-Watson statistic is used to test for :

A.Serial autocorrelation among residuals
B.Mistakes in calculations
C.Correct functional form
D.Heteroscedasticity in residuals

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: A

The Durbin-Watson statistic tests for Serial autocorrelation among residuals.

The statistic is \(d = \sum (e_{t} - e_{t-1})^{2} / \sum e_{t}^{2}\), computed from successive OLS residuals. It ranges from 0 to 4: a value near 2 indicates no first-order autocorrelation, values towards 0 positive autocorrelation, and values towards 4 negative autocorrelation.

It is not a check on arithmetic. Functional form is tested with tools such as Ramsey's RESET test, and heteroscedasticity with tests such as Breusch-Pagan, White or Goldfeld-Quandt.

Discuss in Forum

35.

Suppose X is a normally distributed random variable with an unknown mean μ and an unknown variance σ². The following test can be used to test a hypothesis about unknown population mean :

A.The F-test
B.t-test
C.Chi-square test
D.z-test

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: B

The appropriate test is the t-test.

When the population is normal but its variance is unknown, the variance has to be estimated from the sample. The statistic \(t = (\bar{X} - \mu_{0})/(s/\sqrt{n})\) then follows Student's t distribution with n - 1 degrees of freedom, which is used to test a hypothesis about the mean.

The z-test requires a known population variance, or a very large sample. The F-test compares variances, and the chi-square test is used for a variance or for goodness of fit and independence, not for a mean.

Discuss in Forum

36.

If X₁, X₂, X₃, .............., X_n is a random sample from a normally distributed population with mean μ and variance σ², the sampling distribution of \(\bar{X}\) is :

A.Normal with mean \(\mu\) and variance \(\sigma^{2}\)
B.Normal with mean \(\mu\) and variance \(\sigma^{2}/n^{2}\)
C.Normal with mean \(\mu\) and variance \(\sigma^{2}/n\)
D.Cannot say

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: C

The sampling distribution of the mean is normal with mean μ and variance σ²/n.

The sample mean of n independent observations from a normal population is itself normally distributed. Its expected value equals the population mean, and its variance is the population variance divided by the sample size:

\[ \bar{X} \sim N\left(\mu,\ \frac{\sigma^{2}}{n}\right) \]

Variance σ² (A) is that of a single observation. Dividing by n² (B) understates it; the variance of a sum of n terms is nσ², and dividing the sum by n divides that by n², giving σ²/n.

Discuss in Forum

37.

The logistic regression model is used to analyze :

A.Discrete choice
B.Continuous choice
C.National income
D.Incorrectly measured data

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: A

The logistic regression model is used to analyse discrete choice.

Logistic (logit) regression models a dependent variable that takes a small number of discrete values, typically 0 or 1, such as whether a household buys a car or a worker joins the labour force. It estimates the probability of each outcome as a logistic function of the explanatory variables, which keeps predicted probabilities between 0 and 1.

Continuous outcomes are modelled with ordinary regression. National income is a continuous variable, and incorrectly measured data call for errors-in-variables methods, not logit.

Discuss in Forum

38.

ARIMA (p, d, q) models are used to analyze :

A.cross-section data
B.discrete data
C.time series data
D.unstable data

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: C

ARIMA models are used to analyse time series data.

ARIMA (autoregressive integrated moving average) models, developed by Box and Jenkins, describe a variable observed over time in terms of its own past values (the AR part, of order p), past random shocks (the MA part, of order q), after differencing d times to make the series stationary (the I part). They are widely used for forecasting.

Cross-section data have no time ordering, so autoregression has no meaning there. Discrete data need models such as logit or count models. 'Unstable data' is not a data type that ARIMA is designed for; non-stationarity is handled by differencing.

Discuss in Forum

39.

The estimate of \(\hat{b}\) in the following regression model \(Y_{i} = \alpha + \beta X_{i} + e_{i}\) is given by :

A.\(\hat b=\frac{\Sigma(X_i-\bar X)(Y_i-\bar Y)}{\Sigma(X_i-\bar X)^2}\)
B.\(\hat{b} = \bar{Y} - \hat{\alpha}\bar{X}\)
C.\(\hat b=\frac{\Sigma(X_i-\bar X)(Y_i-\bar Y)}{\sqrt{\Sigma(X_i-\bar X)}}\)
D.\(\hat{b} = \dfrac{\Sigma(X_{i} - \bar{X})^{2}}{\Sigma(Y_{i} - \bar{Y})^{2}}\)

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: A

The OLS estimate is Σ(X_i - X̄)(Y_i - Ȳ) / Σ(X_i - X̄)².

Minimising the sum of squared residuals in Y_i = α + βX_i + e_i gives the slope estimate as the sample covariance of X and Y divided by the sample variance of X:

\[ \begin{aligned} \hat b &= \frac{S_{XY}}{S_{XX}} \\ S_{XY} &= \Sigma(X_i-\bar X)(Y_i-\bar Y) \\ S_{XX} &= \Sigma(X_i-\bar X)^2 \end{aligned} \]

Option B is the form of the intercept estimate, which is Ȳ minus the slope times X̄. Option C has a square root of an unsquared sum in the denominator, and option D is a ratio of variances, not the slope.

Discuss in Forum

40.

In the following regression model \(Y_{i} = \alpha + \beta X_{i} + e_{i}\), under the usual assumptions, variance of \(\hat{b}\) is given by :

A.\(\sigma^{2}\)
B.\(\dfrac{\sigma^{2}}{\Sigma(X_{i} - \bar{X})^{2}}\)
C.\(\dfrac{\Sigma(X_{i} - \bar{X})^{2}}{\sigma^{2}}\)
D.\(\dfrac{\Sigma(Y_{i} - \bar{Y})^{2}}{\sigma^{2}}\)

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: B

The variance of the slope estimate is σ² / Σ(X_i - X̄)².

Under the classical assumptions (errors with mean zero, constant variance σ² and no autocorrelation, X fixed), the slope estimator's variance is

\[ \text{Var}(\hat b)=\frac{\sigma^2}{\Sigma(X_i-\bar X)^2} \]

It falls as the X values are more spread out, which is why wide variation in the regressor gives more precise estimates. σ² alone (A) is the error variance, option C inverts the formula, and option D uses the variation in Y instead of X.

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