IGNOU BECE 142 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 all the questions.

A. Long Answer Questions (word limit-500 words)

1) Explain, theoretically, the consequences of omitting relevant variables in econometric modelling.

In order to understand the consequences of the omitted variable bias, we first have to understand what is needed to obtain good estimates. When studying the linear regression models, you necessarily come across the Gauss-Markov theorem. This theorem states that if your regression model fulfills a set of assumptions (the assumptions of classical linear regression model), then you will obtain the best, linear, and unbiased estimates (BLUE ). One important assumption of this set of assumptions states that the error term of the regression model must be uncorrelated with the explanatory variables. However, as you will see in a minute, omitting a relevant variable introduces a correlation between the explanatory variables and the error term.

What happens when you omit an important variable? From the introductory post, you should know that one of the conditions for an omitted variable bias to exist is that the omitted variable is correlated with the independent variable and with at least one other explanatory variable. Now, when omitting a variable, it will show up in the residual, i.e. it will show up in the error term. Thus, the error term and independent variables are necessarily going to be correlated. This clearly violates the assumption that the error term and the independent variables must be uncorrelated. A violation of this assumption causes the OLS estimator to be biased and inconsistent. For a mathematical proof of this statement see this post.

Furthermore, when looking at the discussion using the Venn diagram, note that omitting a variable causes the unexplained variance of Y (the dependent variable) to increase as well as the variance of the estimated coefficient to decrease. This might lead to a situation in which you reject the null-hypothesis and believe that your coefficients are statistically significant at a given significance level although they are not.

How serious is the omitted variable bias..

The problem of the omitted variable bias is pretty serious. An omitted variable leads to biased and inconsistent coefficient estimate. And as we all know, biased and inconsistent estimates are not reliable. From our previous post, you might remember how omitting a variable can change the signs of the coefficients, depending on the correlation of the omitted variable with the independent and explanatory variables. Thus, coefficients also become unreliable. Hence, the regression model will fail completely. In a simple simulation exercise, I tried to visualize what happens if we neglect a relevant variable from a regression models. The exercise confirms that when neglecting a relevant variable from the model, OLS fails to estimate the coefficients correctly.

We‘ll study the consequences of failing to include important variables in a linear regression model. For illustration, we’ll base our discussion on a real world data set of automobile characteristics. Our goal will be to formulate a well-known result in statistical modeling called Omitted Variable Bias and to illustrate the calculation using the sample data set.

The automobiles data set

The following data contains specifications of 205 automobiles taken from the 1985 edition of Ward’s Automotive Yearbook. Each row contains a set of 26 specifications about a single vehicle.

IGNOU BECE 142 Solved Assignment 2022-23 — The automobiles data set (Source: UC Irvine)

We’ll consider a subset of this data consisting of the following variables:
City_MPG
Car_Volume
Curb_Weight
Engine_Size

The Car_Volume variable is not present in the original data set. It is a new variable we have added as follows: Car_Volume = Length*Width*Height.

The above 4-variables version of the data set is available for download from here.

Regression goal

Our regression goal is to regress City_MPG on Engine_Size and Curb_Weight using a linear regression model. The model equation is:

City_MPG = β_1 + β_2*Car_Volume+ β_3*Curb_Weight + β_4*Engine_Size + ϵ

The error term ϵ of the regression model represents the effects of all the factors that the modeler has been unable to measure.

The matrix version of the above equation is written as follows:

Where,

y is an [n x 1] size column vector containing the observed values of City_MPG. n being the number of data points.
β is a [4 x 1] size column vector of regression model coefficients β_1, β_2, β_3, β_4 corresponding to the intercept, Car_Volume, Curb_Weight and Engine_Size.
X is a [n x 4] size matrix containing the values of the regression variables. The first column of this matrix is a column of 1s and it acts as the placeholder for the intercept β_1.
ϵ is an [n x 1] size column vector of the model’s regression errors.

Let’s illustrate how the regression model’s equation looks like using matrices:

The first column represented by the column vector x_1=[x_11,…x_n1]’ in the X matrix is a column of 1s. Assuming a sample size of n, the above matrix representation is equivalent to writing out the following system of n regression equations:

Here is the matrix representation of the above partitioning:

In general, we can express the above partition as follows:

We have substituted the partitioned out regression variable x_4 with the variable z which is an [n x 1] column vector. γ (gamma) is a [1 x 1] “matrix” that takes the place of regression coefficient β_4.

When one trains (a.k.a. ‘fits’) the above mentioned linear model on a data set of n samples, the fitted model can be expressed as follows:

Notice the cap or hat “^” symbol over β and γ indicating that they are the fitted values i.e. the estimates of the corresponding population level values of β and γ. Also in equation (2), the column vector of residual errors e takes the place of the column vector of regression errors ϵ. The ith residual error e_i is the difference between the ith observation y_i and the corresponding ith predicted value from the fitted model.

We have now prepared the ground for addressing the problem of what happens when you omit regression variables.

2) Discuss the effect of lags on ‘market equilibrium’ with suitable examples.

Definition of Market Equilibrium

Market equilibrium is a market state where the supply in the market is equal to the demand in the market. The equilibrium price is the price of a good or service when the supply of it is equal to the demand for it in the market. If a market is at equilibrium, the price will not change unless an external factor changes the supply or demand, which results in a disruption of the equilibrium.

Supply, Demand & Equilibrium

If a market is not at equilibrium, market forces tend to move it to equilibrium. Let’s break this concept down.

If the market price is above the equilibrium value, there is an excess supply in the market (a surplus), which means there is more supply than demand. In this situation, sellers will tend to reduce the price of their good or service to clear their inventories. They probably will also slow down their production or stop ordering new inventory. The lower price entices more people to buy, which will reduce the supply further. This process will result in demand increasing and supply decreasing until the market price equals the equilibrium price.

If the market price is below the equilibrium value, then there is excess in demand (supply shortage). In this case, buyers will bid up the price of the good or service in order to obtain the good or service in short supply. As the price goes up, some buyers will quit trying because they don’t want to, or can’t, pay the higher price. Additionally, sellers, more than happy to see the demand, will start to supply more of it. Eventually, the upward pressure on price and supply will stabilize at market equilibrium.

Examples of Market Equilibrium

Flat Screen TVs

Imagine that you make flat screen televisions. Your flagship model is a 72-inch plasma that currently wholesales to your retailers at $2,500. Unfortunately, your warehouse has recently been filling a bit too quickly with 72-inch plasmas. This is probably because each of your three largest competitors has finally gotten around to introducing their own 72-inch televisions, which means that there are a bunch more 72-inch televisions on the market. You decide to lower your wholesale price to $2,250 and see what happens. You also decide to cut production down by 25% for the next month to clear out existing inventory.

When you reviewed the numbers at the end of the month, the price reduction did work, but not quite well enough. So you decide to reduce the wholesale price once again to $2,100 and keep production at the same level. When you reviewed the numbers at the end of the month, you see that you barely have any inventory and the purchase orders from your retailers have started to go up a bit. In the following months, orders have kept up with production and inventory is where it is suppose to be. It appears that the price for your television has reached market equilibrium.

In a competitive market, demand for and supply of a good or service determine the equilibrium price.

Equilibrium

MARKETS: Equilibrium is achieved at the price at which quantities demanded and supplied are equal. We can represent a market in equilibrium in a graph by showing the combined price and quantity at which the supply and demand curves intersect.

For example, imagine that sellers of squirrel repellant are willing to sell 500 units of squirrel repellant at a price of dollar sign, 5 per can. If buyers are willing to buy 500 units of squirrel repellent at that price, this market would be in equilibrium at the price of dollar sign, 5 and at the quantity of 500 cans.

Disequilibrium

Whenever markets experience imbalances—creating disequilibrium prices, surpluses, and shortages—market forces drive prices toward equilibrium.

A surplus exists when the price is above equilibrium, which encourages sellers to lower their prices to eliminate the surplus.

A shortage will exist at any price below equilibrium, which leads to the price of the good increasing.

For example, imagine the price of dragon repellent is currently dollar sign, 6 per can. People only want to buy 400 cans of dragon repellent, but the sellers are willing to sell 600 cans at that price. This creates a surplus because there are unsold units. Sellers will lower their prices to attract buyers for their unsold cans of dragon repellant.

Changes in equilibrium

Changes in the determinants of supply and/or demand result in a new equilibrium price and quantity. When there is a change in supply or demand, the old price will no longer be an equilibrium. Instead, there will be a shortage or surplus, and price will subsequently adjust until there is a new equilibrium.

For example, suppose there is a sudden invasion of aggressive unicorns. There will be more people who want to buy unicorn repellent at all possible prices, causing demand to increase. At the original price, there will be a shortage of unicorn repellant, signaling sellers to increase the price until the quantity supplied and quantity demanded are once again equal.

Changes in Supply

Suppose the price of glitter, which is used to make giant shiny salamander stickers, increases so that it now costs the seller dollar sign, 2 more per sticker to produce them. This will cause the supply of this good to decrease. To see the impact a decrease in supply will have on the equilibrium price and quantity, grab the interactive supply curve and shift it to the left until the price is dollar sign, 2 higher at every level of output (the new supply curve should start at dollar sign, 4).

What change did you notice? If you adjusted the graph correctly, you should see the equilibrium price increases to dollar sign, 6, and the equilibrium quantity in this market decreases to 2 stickers.

Now instead, suppose someone invents a new way to produce shiny salamander stickers so there is less waste and fewer resources are needed to produce them. This would result in an increase in the supply of shiny salamander stickers. To see the impact an increase in supply will have on the equilibrium price and quantity, grab the interactive supply curve and drag it to the right so that at every quantity the price is dollar sign, 2 lower (the new supply curve should start at dollar sign, 0).

How did you do? If you adjusted the graph correctly, you should see the equilibrium price decreases to dollar sign, 4 and equilibrium quantity increases to 4 stickers.

B. Medium Answer Questions (word limit-250 words)

3) Outline how R2 and adjusted-R2 serve as indicators of ‘goodness of fit’ of a regression model.
4) Indicate the form of a Logit Model. Specify why the OLS method of estimation cannot be applied here.
5) Show that ‘exact identification’ is a sufficient condition for the unique determination of a system of equations.

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C. Short Answer Questions (word limit 100 words)

6) Differentiate between:
(a) Quantitative Research and Qualitative Research.
(b) In-sample forecast and out-of-sample forecast.
(c) Autoregressive Model and Autoregressive Distributed Lag Model.

7) Write short notes on the following.
(a) Instrumental Variables (IV) Method.
(b) Pooled Cross Section Data.
(c) Linear Static Panel Data Model.

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