Question

3. Consider the following model to explain monthly beer consumption: where E(ujincome, price, educ, female) -0 and Var(ulincome, price, educ, female) - ?"income. (a) What problems arise if OLS is used to estimate the model? (b) Write the transformed equation that has a homoskedastic error term.(4 Marks) (4 Marks)

Answer 1

According to OLS Assumptions:

OLS Assumption 1: The linear regression model is “linear in parameters.”

OLS Assumption 2: There is a random sampling of observations

OLS Assumption 3: The conditional mean should be zero.

OLS Assumption 4: There is no multi-collinearity (or perfect collinearity).

OLS Assumption 5: Spherical errors: There should be homoscedasticity and no autocorrelation.

OLS Assumption 6: Error terms should be normally distributed.

(a)  

Solution: Given that OLS Assumption 3 is satisfied but OLS assumption 5 (Homoscedasticity) is not. In our case, Error variance changes proportionally with the variable "income".

According to this OLS assumption, the error terms in the regression should all have the same variance.

Mathematically it should be,

Var(??X)=?2.

Here, the variance is not constant (i.e. dependent on income), i.e. the linear regression model has heteroscedastic errors and likely to give incorrect estimates.

Additional Info: Now that the model is ready, there are two ways to test for heteroscedasticity:

• Graphically
• Through statistical tests

From the Model, You can Plot Residual Vs Fitted Plot and If there is absolutely no heteroscedasticity, you should see a completely random, equal distribution of points throughout the range of X-axis. Also, there are a couple of tests that come handy to establish the presence or absence of heteroscedasticity – The Brush-Pagan test and the NCV test.

(b)

Solution: Consider the given heteroscedasticity model

Here, Var(u) = ?2xincome (income > 0), We will apply OLS with model. So, the Transformed equation will be: