Exam Details
| Subject | statistics | |
| Paper | paper 3 | |
| Exam / Course | indian economic service and indian statistical service examination (ies/iss) | |
| Department | ||
| Organization | union public service commission | |
| Position | ||
| Exam Date | 2017 | |
| City, State | central government, |
Question Paper
For SRSWOR show that the sample proportion p is unbiased for the population proportion P. Also derive the sampling variance of this estimator.
1.(b) What is the problem in estimating a linear regression model in presence of multicollinearity How is multicollinearity detected Explain how ridge estimation tackles this issue.
1.(c) Consider the process Xn= en ßen-1, where en N(0,1).
For a data set it is noted that autocovariances are yo 1 and y1 0.25.
(i) Estimate ß. Which value of the estimate do you think we should choose and why What problem do we have if y1= 0.5 How would the variance of the error have affected the change Given below are the figures on production (in thousand metric tons) of a cooperative sugar factory:
Year: 2010 2011 2012 2013 2014 2015 2016
Production 77 88 84 85 91 98 90
(i) Fit a linear trend by least squares method. Tabulate the trend values.
(ii) Compute the monthly estimated increase in production during the period.
2.(b) If, in every stratum, the simple estimator is unbiased, then show that
yst Ewhyh
is unbiased for population mean y where Wh is the proportion of population units in the strata and L denotes the total number of strata in the population. Derive the sampling variance of yst and state how you would unbiasedly estimate the same.
2.(c) In the context of a finitely distributed lag model, discuss the problem of OLS estimation and suggest how to obtain good (consistent) estimates of the parameters in such a model by bringing in some restrictions on lag weights.
following Explain and illustrate the Two-stage sampling
3.(a) Explain and illustrate the Two-phase sampling
3.(a) Pinpoint the difference between the Two-stage sampling and Two-phase sampling
3.(b) Write briefly on Sample size determination in surveys;
3.(b) Write briefly on Cumulative total method for PPSWR sampling;
3.(b) Write briefly on Rao-Hartley-Cochran Scheme.
3.(c) Discuss the Proportional allocation in stratified random sampling.Explain the practical implications of these method.
3.(c) Discuss the Neyman allocation in stratified random sampling. Explain the practical implications of these method.
3.(c) Discuss the Optimum allocation with a linear cost function in stratified random sampling.Explain the practical implications of these method.
4.(a) Discuss Koyck approach to an infinitely distributed lag model and obtain the mean lag for Koyck’s model. What are the basic features of Koyck’s transformed model For the following linear regression model
Yi= ßi ß2Xi Ui i Cov(Ui,Uj) 0 E ui^2 ai^2 obtain the OLS estimates of ß1 and ß2 when a i 1,...,n are known. Discuss what steps could be taken when i n are unknown. Revise the least squares estimates of ß1 and ß2 when where is unknown.
4.(c) Discuss the problem of estimating parameters by OLS in the presence of serial correlation in the following model;
yt ß1 ß2xt ut
ut put l p l p is known.
E V O Cov 0 s 0.
Propose suitable estimates of ß1 and ß2. calculate the variance of the estimate of ß2. estimate be modified when p is unknown
5.(a) Demand and supply functions of a certain commodity are respectively
Xd 240 10 dp/dt 4p kg per month;
Xs 100 dp/dt 6p 60 kg per month,
where p is price of the commodity at time t.
Find the time path of p for dynamic equilibrium if the initial price is to be Rs 72 per kg.
5.(b) Explain briefly by simple average of price relatives;
5.(b) Explain briefly by simple aggregate of prices; and
5.(b) Explain briefly by weighted aggregate of prices.
5.(c) Discuss the different forms of the Engel curve that are usually employed for fitting to family-budget data. In such fitting, how would you tackle the following complications Household expenditure on a particular item depends, besides depending on income, on the number of persons per family.
(ii) Consumption of families of the same size differs because of varying age and sex consumption.
6.(a) Give an illustration for linear systematic sampling. Show that, under this method, a positive correlation between units in the same sample inflates the sampling variance of the estimator of population total.
6.(b) Consider a population of N 6 units with values 5 and 6.
(i) Write down all possible samples of size 2 drawn by SRSWOR scheme. Verify that the sample mean is unbiased for the population mean.
(ii) Also compute the sampling variance of the sample mean.
6.(c) Explain the ratio method of estimation for estimating a population total. Show that it is generally biased. Evaluate the mean squared error of the estimator to the first order of approximation. Assume SRSWOR of n units from the population.
7.(a) Using standard notations, briefly explain the instrumental variable technique in the context of estimating the coefBcients in a linear regression model. State the situations when this technique is applicable.
7.(b) State the rank and order conditions for identiflability of parameters in a system of structural equations. Which one of these two conditions is sufficient for identiflability Establish this condition mathematically.
7.(c) Discuss the estimation of parameters of an equation appearing in a simultaneous equation system by Limited Information Maximum Likelihood method. State whether the estimator (if it exists) is unique. (An outline of the approach is adequate)
8.(a) Show that the relationship xt=0.7xt-1 0.3xt-2 0.7€t-1 (where €t denotes white noise) defines ARIMA(1, model.
8.(b) What do you understand by the seasonal variations in a time series Give example. Explain the method of link relatives of computing the seasonal indices.
8.(c) Define correlogram.
For an infinite series generated by the average of a random component with equal weights, show that the correlogram is
pk for 0 for k>m
1.(b) What is the problem in estimating a linear regression model in presence of multicollinearity How is multicollinearity detected Explain how ridge estimation tackles this issue.
1.(c) Consider the process Xn= en ßen-1, where en N(0,1).
For a data set it is noted that autocovariances are yo 1 and y1 0.25.
(i) Estimate ß. Which value of the estimate do you think we should choose and why What problem do we have if y1= 0.5 How would the variance of the error have affected the change Given below are the figures on production (in thousand metric tons) of a cooperative sugar factory:
Year: 2010 2011 2012 2013 2014 2015 2016
Production 77 88 84 85 91 98 90
(i) Fit a linear trend by least squares method. Tabulate the trend values.
(ii) Compute the monthly estimated increase in production during the period.
2.(b) If, in every stratum, the simple estimator is unbiased, then show that
yst Ewhyh
is unbiased for population mean y where Wh is the proportion of population units in the strata and L denotes the total number of strata in the population. Derive the sampling variance of yst and state how you would unbiasedly estimate the same.
2.(c) In the context of a finitely distributed lag model, discuss the problem of OLS estimation and suggest how to obtain good (consistent) estimates of the parameters in such a model by bringing in some restrictions on lag weights.
following Explain and illustrate the Two-stage sampling
3.(a) Explain and illustrate the Two-phase sampling
3.(a) Pinpoint the difference between the Two-stage sampling and Two-phase sampling
3.(b) Write briefly on Sample size determination in surveys;
3.(b) Write briefly on Cumulative total method for PPSWR sampling;
3.(b) Write briefly on Rao-Hartley-Cochran Scheme.
3.(c) Discuss the Proportional allocation in stratified random sampling.Explain the practical implications of these method.
3.(c) Discuss the Neyman allocation in stratified random sampling. Explain the practical implications of these method.
3.(c) Discuss the Optimum allocation with a linear cost function in stratified random sampling.Explain the practical implications of these method.
4.(a) Discuss Koyck approach to an infinitely distributed lag model and obtain the mean lag for Koyck’s model. What are the basic features of Koyck’s transformed model For the following linear regression model
Yi= ßi ß2Xi Ui i Cov(Ui,Uj) 0 E ui^2 ai^2 obtain the OLS estimates of ß1 and ß2 when a i 1,...,n are known. Discuss what steps could be taken when i n are unknown. Revise the least squares estimates of ß1 and ß2 when where is unknown.
4.(c) Discuss the problem of estimating parameters by OLS in the presence of serial correlation in the following model;
yt ß1 ß2xt ut
ut put l p l p is known.
E V O Cov 0 s 0.
Propose suitable estimates of ß1 and ß2. calculate the variance of the estimate of ß2. estimate be modified when p is unknown
5.(a) Demand and supply functions of a certain commodity are respectively
Xd 240 10 dp/dt 4p kg per month;
Xs 100 dp/dt 6p 60 kg per month,
where p is price of the commodity at time t.
Find the time path of p for dynamic equilibrium if the initial price is to be Rs 72 per kg.
5.(b) Explain briefly by simple average of price relatives;
5.(b) Explain briefly by simple aggregate of prices; and
5.(b) Explain briefly by weighted aggregate of prices.
5.(c) Discuss the different forms of the Engel curve that are usually employed for fitting to family-budget data. In such fitting, how would you tackle the following complications Household expenditure on a particular item depends, besides depending on income, on the number of persons per family.
(ii) Consumption of families of the same size differs because of varying age and sex consumption.
6.(a) Give an illustration for linear systematic sampling. Show that, under this method, a positive correlation between units in the same sample inflates the sampling variance of the estimator of population total.
6.(b) Consider a population of N 6 units with values 5 and 6.
(i) Write down all possible samples of size 2 drawn by SRSWOR scheme. Verify that the sample mean is unbiased for the population mean.
(ii) Also compute the sampling variance of the sample mean.
6.(c) Explain the ratio method of estimation for estimating a population total. Show that it is generally biased. Evaluate the mean squared error of the estimator to the first order of approximation. Assume SRSWOR of n units from the population.
7.(a) Using standard notations, briefly explain the instrumental variable technique in the context of estimating the coefBcients in a linear regression model. State the situations when this technique is applicable.
7.(b) State the rank and order conditions for identiflability of parameters in a system of structural equations. Which one of these two conditions is sufficient for identiflability Establish this condition mathematically.
7.(c) Discuss the estimation of parameters of an equation appearing in a simultaneous equation system by Limited Information Maximum Likelihood method. State whether the estimator (if it exists) is unique. (An outline of the approach is adequate)
8.(a) Show that the relationship xt=0.7xt-1 0.3xt-2 0.7€t-1 (where €t denotes white noise) defines ARIMA(1, model.
8.(b) What do you understand by the seasonal variations in a time series Give example. Explain the method of link relatives of computing the seasonal indices.
8.(c) Define correlogram.
For an infinite series generated by the average of a random component with equal weights, show that the correlogram is
pk for 0 for k>m