Market Analysis 1b

Videos and questions for Chapter 1b of the course "Market Analysis with Econometrics and Machine Learning" at Ulm University (taught by Sebastian Kranz)

The simple linear regression model

Later we will also look at the multiple linear regression model. What do you think is the difference to this simple linear regression model?

The multiple linear regression model has more than one explanatory variable \(x\).

The multiple linear regression model has more than one dependent variable \(y\).

The error term \(\varepsilon\) in the multiple linear regression model can follow any distribution. In the simple linear regression model, it must be normally distributed.

Predicted values and residuals

Assume the red line on the previous slide show the estimated regression line. What measures then the vertical black line?

The residual \(\hat \varepsilon_i\).

The disturbance \(\varepsilon_i\).

Both the disturbance \(\varepsilon_i\) and the residual \(\hat \varepsilon_i\).

None of the above.

Ordinary Least Squares Estimation

The disturbance \(\varepsilon\) is always modelled as a random variable. The explanatory variable \(x\) can also be a random variable. Is the OLS estimator \(\hat \beta\) also a random variable?

yes

The estimator is a random variable

What do you think holds true for the expected value of the OLS estimator \(\hat \beta\) (which is a random variable)?

The expected value of \(\hat \beta\) is always equal to the true coefficients \(\beta\).

The expected value of \(\hat \beta\) is equal to the true coefficients \(\beta\) only if \(x\) and \(\varepsilon\) satisfy some assumptions.

Monte-Carlo simulation of an OLS estimator

How do you think the estimates differ when we call our function several times with T=5 compared to earlier where we had T=100.

They will be distributed in the same way

They will be systematically smaller.

They will be systematically larger.

They will have a lower standard deviation.

They will have a higher standard deviation.

Standard Error of \(\hat \beta_1\)

Assume you run an experiment with \(T=20\) observations and get some standard error for \(\hat \beta_1\). If you want to halve the size of that standard error, how many observations would you roughly need?

In total \(T=10\) observations.

In total \(T=40\) observations.

In total \(T=80\) observations.

Robust Standard Errors

Criteria for Estimators 1: Bias and Mean Squared Error (MSE)

Bias and MSE in our Monte-Carlo Simulation

Criteria for Estimators 2: Consistency and Efficiency

In our Monte-Carlo simulation did it look as if the estimator \(\hat \beta_1\) was consistent?

yes

Assumptions of the linear regression model

Endogeniety and Exogeniety

Exploring Endogeniety in R

Let us look at some R code to better understand that an endogenous explanatory variable leads to an inconsistent OLS estimator

Are the prices in the regression for the simulation in the video above endogenous or exogenous?

the prices are endogenous

the prices are exogenos

Assume we set in the simulation prices that are always 10% above the costs c. Are prices then endogenous or exogenous when estimating the demand function?

the prices are endogenous

the prices are exogenos

What happens if the standard deviation of \(\varepsilon\) is very small?

What do you think holds for our estimator \(\hat \beta\) if we have such a small standard deviation of \(\varepsilon\)?

We will no longer find a substantial bias and the standard error of \(\hat \beta_1\) becomes very small.

We will no longer find a substantial bias but the standard error of \(\hat \beta_1\) remains not very small.

We will still have a substantial bias but the standard error of \(\hat \beta_1\) becomes very small.

We will still have a substantial bias and the standard error of \(\hat \beta_1\) is not very small.

Other Assumptions

Confidence Intervals and Bias Formula

Ok, that were all videos for the first part of Chapter 1b. If you have not yet done so, it is a good time to start solving the RTutor problem set for this part.