Define Des Raj's ordered estimator for population mean on the basis of a sample of size 2 and show that it is unbiased.

Let ni and nij j be the inclusion probabilities of first and second order respectively in a simple random sample of size selected from a finite population o f size N. Then show that

Eni and i to N

Enij n n 1 .

Give a practical example where two-stage sampling scheme may be adopted. For equal size first-stage units, obtain an estimator for population mean in two-stage sampling and its variance. Discuss the problem of allocation of first and second-stage sample sizes for a fixed cost.

Describe the problem of multicollinearity in general linear model and explain how will you detect it.

In usual notation consider the standard linear model Y X P (J2I Show that the MLE P of ft have the distribution N p o X X y assuming X X to be an invertible matrix.

Explain the identification problem in a system of simultaneous equations. State, without proof, the rank and order conditions for identifiability of an equation.

Identify the following system
y1 3y2 2x1 +x2 U1
y2 y3 x3 U2
y3 y1 y2 2x3 U3

Construct the price index number for 2010 with 2005 as base year from the following data by using

Laspeyre's

Paasche's and

Fisher's method

Item Price (in Rs.) 2010 2005 Quantity 2010 2005

A 10.50 8.25 6 4
B 6.40 6.00 10 6
C 15.20 10.80 6 5
D 6.25 4.00 8 5
Verify whether the Time Reversal Test is satisfied by the above mentioned index numbers.

Define price elasticity of demand and interpret when np is



and



If the demand function is p 10-5x^2 for what value of x elasticity of demand will be unity is the quantity demanded and p is the price).

Define autocorrelation of lag K of a stationary process. Consider the time series model defined by
Xt a1Xt-1 a2Xt-2 a3Xt-3 Et
where is white noise.

Show that the autocorrelation coefficient with lag 1 for the process is

P1 a1 a2 a3/1-a2 a1a3 a3^2

Consider the case where
al a2 =a3 0.2 .
Comment on the stationarity of this model.

You may use 5 x x^2 x^3 (1.278- x (3.912 2.278x

Calculate p1 and p2.

A finite population of size 100 is divided into two strata. In the usual notations, it is given that N1= 60, N2 40, S1= 2S2. If a sample of size 24 is to be selected from the population, obtain the number of units to be selected from each of the stratum under Neymann allocation.

For the model y XB u if X and u are correlated show that the OLS estimator for B is not consistent. Discuss the use of instrumental variable technique to obtain a consistent estimator of B

Consider a time series yt Tt Ct It where Tt a trend, Ct a cyclical component and It a random component. Discuss the effect of moving averages on cyclical and random components assuming Ct a sin 2nt/A

Explain Koyck's approach to distributed geometric lag model.

What is a chain index number Discuss its advantages and disadvantages over fixed-base index number.

In what sense cluster sampling is different from simple random sampling Define an unbiased estimator for population mean in case of cluster sampling with equal cluster size. Compare the efficiency of cluster sampling in terms of intra-class correlation coefficient with respect to simple random sampling without replacement.

You believe that a set of data is the realization of an process Xn en+Ben-1, where the errors en are standard normal. You have calculated the sample auto-covariance function and found that y0= 1 and y1 -0.25 . Estimate the parameter B. Which value of B do you think you should choose and why

Describe the Lahiri's method of selecting a probability proportional to size sample from a finite population of size N.

Describe lag model and distributed lag model. What are the different lag schemes How would you estimate lags by applying ordinary least square