Q For SRSWOR show that the sample proportion p is unbiased for the population proportion P. Also derive the sampling variance of this the population proportion P. Also derive the sampling variance of this estimator.

W hat is the problem in estimating a linear regression model in presence of multicollinearity How is multicollinearity detected Explain how of multicollinearity How is multicollinearity detected Explain how ridge estimation tackles this issue. ridge estimation tackles this

Consider the process

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For a data set it is noted that autocovariances are

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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?

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Q2. Given below are the figures on production (in thousand metric tons) of a cooperative sugar factory


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Fit a linear trend by least squares method. Tabulate the trend values.

Compute the monthly estimated increase in production during the period.

Compute the monthly estimated increase in production during the

if in every stratum. the simple estimator
is unbiased, then show that

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is unbiased for population mean 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 Y'st and state how you would
unbiasedly estimate the same.

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.


Q3.(a) Explain and illustrate the following


Two-stage sampling

Two-phase sampling


Pinpoint the difference between the two types of sampling schemes.

Write briefly on


Sample size determination in surveys;


Cumulative total method for PPSWR sampling;


Rao-Hartley-Cochran Scheme.

Discuss the following allocations of the sample size in stratified random sampling:


Proportional allocation

Neyman allocation

Optimum allocation with a linear cost function

Explain the practical implications of these methods.


Q4. Discuss Koyck approach to on 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

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obtain the OLS estimates or β1 and β2 when σ¡^2,i=1,.....
n are known.
Discuss what steps could be taken when σ¡^2,i l,.....n are unknown.
Revise the least squares estimates of β1 and β2 when σ¡^2 where σ^2 is unknown.

Discuss the problem of estimating parameters by OLS in the presence of serial correlation in the following model

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Propose suitable estimates of β1 and β2 also calculate variance of the estimate of β2.How can this estimate be modified when is unknown?


Q5. Demand and supply functions o
f a certain commodity are respectively

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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 to be " 72 per kg

Explain briefly the methods of computing price index numbers

by simple overage of price relative;

by simple aggregate of prices; and

by weighted aggregate of prices.


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.

Consumption of families of the same size differs because of varying age and sex consumption.


Q6.(a) Give an illustration for linear systematic sampling. Show that, under this method, a positive correlation between units in the same sample this method, a positive correlation between units in the same sample inflates the sampling variance of the estimator of population total.

Consider a population of N 6 units with values 5 and 6.

Write down all possible samples of size 2 drawn by SRSWOR scheme. Verify that the sample mean is unbiased for the population scheme.

Also compute the sampling variance of the sample mean.

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.

Q7.(a) Using standard notations, briefly explain the instrumental variable technique in the context of estimating the coefficients in a linear regression model.State the situations when this technique is applicable.

State the rank and order conditions for identifiability of parameters in a system of structural equations. Which one of these two conditions is sufficient for identifiability? Establish this condition mathematically.

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)


Q8.(a) Show that the relationship

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(where denotes white noise) defines ARIMA model.

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.


Define correlogram .
For an infinite series generated by the average of a random component with equal weights, show that the correlogram is

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