# ISS (Statistical Services) Statistics Paper III: Questions 51 - 55 of 96

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## Question number: 51

» Econometrics » Autoregressive Linear Regression

Appeared in Year: 2011

### Describe in Detail

Let , - ∞ < t < ∞, where are i, i. d. with and . Show that the process is stationary with correlation.

### Explanation

It is given that

→ ……………. (i)

using (i)

Similarly, we will get

But From (i)

Hence,

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## Question number: 52

» Applied Statistics » Time Series Analysis » Exploratory Time Series Analysis

Appeared in Year: 2011

### Describe in Detail

Give the various steps for finding the variance of the random component using variate difference method. Is F-test used for testing the significance of homogeneity of two successive estimates of variance? If yes, ok, otherwise which ‘test can be used? Discuss.

### Explanation

To find the variance of the random component using variate difference method:

let us consider a series represented as the sum of functional part and random component

Here are considered as independently identically distributed with

…. (i)

for a plynomial of m^{th} degree

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## Question number: 53

» Applied Statistics » Index Numbers » Price Relatives and Quantity or Volume Relatives

Appeared in Year: 2010

### Write in Short

What are the three tests of index numbers? Verify whether the following index number satisfies any of these tests:

I=

Where L = Lasperaye’s Price Index Number

P = Paasche’s Price Index Number.

## Question number: 54

» Econometrics » Autoregressive Linear Regression

Appeared in Year: 2011

### Describe in Detail

For the auto-regressive scheme , show that if e is a random variable and the series is long, then

and hence show that, variance of the generated series may be much greater than that of e itself.

### Explanation

Given auto-regressive scheme is a second order Auto-regressive series.

……………. . (i)

Since the series is long and , we have

and Var

Squaring both sides of (i) and taking the expectations, we get

→

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## Question number: 55

» Applied Statistics » Index Numbers » Income Distribution-Pareto and Engel Curves

Appeared in Year: 2013

### Describe in Detail

Explaining Pareto’s law of income distribution, show that the elasticity decreases in the number of persons while passing to a higher income class.

### Explanation

**Pareto’s law of income distribution: **

Pareto’s states that ‘the number of income recipients earning at least a given level of income would tend to fall by a fixed percentage as that given level of income rose by a fixed percentage. ’

Pareto’s law implies that inequality is inescapable and nothing coiuld be done about it.

pareto’s law is only tru

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