IGNOU BECC 110 Solved Assignment 2022-23

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Submission Date :

31st March 2033 (if enrolled in the July 2033 Session)
30th Sept, 2033 (if enrolled in the January 2033 session).

Answer the following Descriptive Category questions in about 500 words each. Each question carries 20 marks. Word limit does not apply in case of numerical questions in Assignment One.

Answer the following Middle Category questions in about 250 words each. Each question carries 10 marks. Word limit does not apply in case of numerical questions in Assignment Two.

Answer the following Short Category questions in about 100 words each. Each question carries 6 marks in Assignment Three.

Answer all the questions.

Assignment One

1. (a) Distinguish between the Population Regression Function and Sample Regression Function in detail. Use appropriate diagram to substantiate your response.

Let’s investigate this question with another example. Below is a plot illustrating a potential relationship between the predictor “high school grade point average (gpa)” and the response “college entrance test score.” Only four groups (“subpopulations”) of students are considered — those with a gpa of 1, those with a gpa of 2, …, and those with a gpa of 4.

Let’s focus for now just on those students who have a gpa of 1. As you can see, there are so many data points — each representing one student — that the data points run together. That is, the data on the entire subpopulation of students with a gpa of 1 are plotted. And, similarly, the data on the entire subpopulation of students with gpas of 2, 3, and 4 are plotted.

IGNOU BECC 110 Solved Assignment 2022-23

Now, take the average college entrance test score for students with a gpa of 1. And, similarly, take the average college entrance test score for students with a gpa of 2, 3, and 4. Connecting the dots — that is, the averages — you get a line, which we summarize by the formula $μ_{Y} = E (Y) = β_{0} + β_{1} x$ . The line — which is called the “population regression line” — summarizes the trend in the population between the predictor x and the mean of the responses μ_Y. We can also express the average college entrance test score for the i-th student, $E (Y_{i}) = β_{0} + β_{1} x_{i}$ . Of course, not every student’s college entrance test score will equal the average $E (Y_{i})$ . There will be some error. That is, any student’s response y_i will be the linear trend $β_{0} + β_{1} x_{i}$ plus some error $ϵ_{i}$ . So, another way to write the simple linear regression model is $y_{i} = E (Y_{i}) + ϵ_{i} = β_{0} + β_{1} x_{i} + ϵ_{i}$ .

When looking to summarize the relationship between a predictor x and a response y, we are interested in knowing the population regression line $μ_{Y} = E (Y) = β_{0} + β_{1} x$ . The only way we could ever know it, though, is to be able to collect data on everybody in the population — most often an impossible task. We have to rely on taking and using a sample of data from the population to estimate the population regression line.

Let’s take a sample of three students from each of the subpopulations — that is, three students with a gpa of 1, three students with a gpa of 2, …, and three students with a gpa of 4 — for a total of 12 students. As the plot below suggests, the least squares regression line $\hat{y} = b_{0} + b_{1} x$ through the sample of 12 data points estimates the population regression line $μ_{Y} = E (Y) = β_{0} + β_{1} x$ . That is, the sample intercept b₀ estimates the population intercept β₀ and the sample slope b₁ estimates the population slope β₁.

IGNOU BECC 110 Solved Assignment 2022-23

The least squares regression line doesn’t match the population regression line perfectly, but it is a pretty good estimate. And, of course, we’d get a different least squares regression line if we took another (different) sample of 12 such students. Ultimately, we are going to want to use the sample slope b₁ to learn about the parameter we care about, the population slope β₁. And, we will use the sample intercept b₀ to learn about the population intercept β₀.

In order to draw any conclusions about the population parameters β₀ and β₁, we have to make a few more assumptions about the behavior of the data in a regression setting. We can get a pretty good feel for the assumptions by looking at our plot of gpa against college entrance test scores.

First, notice that when we connected the averages of the college entrance test scores for each of the subpopulations, it formed a line. Most often, we will not have the population of data at our disposal as we pretend to do here. If we didn’t, do you think it would be reasonable to assume that the mean college entrance test scores are linearly related to high school grade point averages?

IGNOU BECC 110 Solved Assignment 2022-23

Again, let’s focus on just one subpopulation, those students who have a gpa of 1, say. Notice that most of the college entrance scores for these students are clustered near the mean of 6, but a few students did much better than the subpopulation’s average scoring around a 9, and a few students did a bit worse scoring about a 3. Do you get the picture? Thinking instead about the errors, $ϵ_{i}$ , most of the errors for these students are clustered near the mean of 0, but a few are as high as 3 and a few are as low as -3. If you could draw a probability curve for the errors above this subpopulation of data, what kind of a curve do you think it would be? Does it seem reasonable to assume that the errors for each subpopulation are normally distributed?

Looking at the plot again, notice that the spread of the college entrance test scores for students whose gpa is 1 is similar to the spread of the college entrance test scores for students whose gpa is 2, 3, and 4. Similarly, the spread of the errors is similar, no matter the gpa. Does it seem reasonable to assume that the errors for each subpopulation have equal variance?

Does it also seem reasonable to assume that the error for one student’s college entrance test score is independent of the error for another student’s college entrance test score? I’m sure you can come up with some scenarios — cheating students, for example — for which this assumption would not hold, but if you take a random sample from the population, it should be an assumption that is easily met.

We are now ready to summarize the four conditions or assumptions that underlie “the simple linear regression model:”

The mean of the response, $E (Y_{i})$ , at each value of the predictor, $x_{i}$ , is a Linear function of the $x_{i}$ .
The errors, ε_i, are Independent.
The errors, ε_i, at each value of the predictor, $x_{i}$ , are Normally distributed.
The errors, ε_i, at each value of the predictor, $x_{i}$ , have Equal variances (denoted σ²).

Do you notice what the first letters that are colored in blue spell? “LINE.” And, what are we studying in this course? Lines! Get it? You might find this mnemonic a useful way to remember the four conditions that make up what we call the “simple linear regression model.” Whenever you hear “simple linear regression model,” think of these four conditions!

An equivalent way to think of the first (linearity) condition is that the mean of the error, $E (ϵ_{i})$ , at each value of the predictor, $x_{i}$ , is zero. An alternative way to describe all four assumptions is that the errors, $ϵ_{i}$ , are independent normal random variables with mean zero and constant variance, $σ^{2}$ .

(b) What are the assumptions of a classical regression model?

Ordinary Least Squares (OLS) is the most common estimation method for linear models—and that’s true for a good reason. As long as your model satisfies the OLS assumptions for linear regression, you can rest easy knowing that you’re getting the best possible estimates.

Regression is a powerful analysis that can analyze multiple variables simultaneously to answer complex research questions. However, if you don’t satisfy the OLS assumptions, you might not be able to trust the results.

In this post, I cover the OLS linear regression assumptions, why they’re essential, and help you determine whether your model satisfies the assumptions.

Regression analysis is like other inferential methodologies. Our goal is to draw a random sample from a population and use it to estimate the properties of that population.

In regression analysis, the coefficients in the regression equation are estimates of the actual population parameters. We want these coefficient estimates to be the best possible estimates!

Suppose you request an estimate—say for the cost of a service that you are considering. How would you define a reasonable estimate?

The estimates should tend to be right on target. They should not be systematically too high or too low. In other words, they should be unbiased or correct on average.
Recognizing that estimates are almost never exactly correct, you want to minimize the discrepancy between the estimated value and actual value. Large differences are bad!

These two properties are exactly what we need for our coefficient estimates!

When your linear regression model satisfies the OLS assumptions, the procedure generates unbiased coefficient estimates that tend to be relatively close to the true population values (minimum variance). In fact, the Gauss-Markov theorem states that OLS produces estimates that are better than estimates from all other linear model estimation methods when the assumptions hold true.

The Seven Classical OLS Assumptions

Like many statistical analyses, ordinary least squares (OLS) regression has underlying assumptions. When these classical assumptions for linear regression are true, ordinary least squares produces the best estimates. However, if some of these assumptions are not true, you might need to employ remedial measures or use other estimation methods to improve the results.

Many of these assumptions describe properties of the error term. Unfortunately, the error term is a population value that we’ll never know. Instead, we’ll use the next best thing that is available—the residuals. Residuals are the sample estimate of the error for each observation.

Residuals = Observed value – the fitted value

When it comes to checking OLS assumptions, assessing the residuals is crucial!

There are seven classical OLS assumptions for linear regression. The first six are mandatory to produce the best estimates. While the quality of the estimates does not depend on the seventh assumption, analysts often evaluate it for other important reasons that I’ll cover.

This assumption addresses the functional form of the model. In statistics, a regression model is linear when all terms in the model are either the constant or a parameter multiplied by an independent variable. You build the model equation only by adding the terms together. These rules constrain the model to one type:

$Y =\beta _{0} + \beta _{1}X_{1} + \beta _{2}X_{2} + \cdots + \beta _{k}X_{k} + \epsilon$

In the equation, the betas (βs) are the parameters that OLS estimates. Epsilon (ε) is the random error.

In fact, the defining characteristic of linear regression is this functional form of the parameters rather than the ability to model curvature. Linear models can model curvature by including nonlinear variables such as polynomials and transforming exponential functions.

2). (a) Measurement error in variables is a serious problem in econometric studies. Find out the consequences of measurement errors in i) dependent variable and ii) independent variables.

Assignment Two

3. Differentiate between Chi-square distribution and t-distribution.
4. What is an estimator? Explain all the properties of an estimator with reference to BLUE.
5. Two variable regression model could have three functional forms as given below:

𝑌i = 𝛽1 + 𝛽2𝑋i + 𝑢i
𝑙𝑛𝑌i = 𝛽1 + 𝛽2𝑋i + 𝑢i
ln𝑌i = 𝛽1 + 𝛽2lnXi + 𝑢i

How will you decide which is the best model for a given econometric problem?

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Assignment Three

6. Discuss the remedial measures of multicollinearity.
7. What do we mean by Normal Distribution? Explain with the help of a diagram.
8. There are two types of estimation of parameters: Point Estimation and Interval Estimation. Explain the interval estimation method briefly.
9. What are the three methods of estimation? Discuss.
10. Explain the rejection regions for small samples and large samples.

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