Welcome to Econ 420 Applied Regression Analysis

Study GuideWeek Two

Ending Sunday, September 9(Note: You must go over these slides and complete every task outlined here by the end of the day on

September 8)

Last week

• I asked you to report your heights and weights before Sunday September 2 – That meant by the end of the day on

Saturday, September 1.– I did not hear from 4 of the students who are

registered in this class• Remember that this affects your grade

Here is our sample data on height and weight.

Observation Height (H or X) Weight (W or Y)

1.Jackie 64 130

2. Philip D. 75 210

3. Bryan 76 230

4. Rita 67 190

5. Shane 68 175

6. Keith 75 190

7. Kelsie 65 145

8. Di 72 185

Assignment 1(Carries 30 points and is due before noon on Thursday, September 6)

1. Use the data set on the previous slide and the formulas on Page 8 (1-5 and 1-6) to estimated the coefficients β0^ and β1^ in the equation belowW = β0^ + β1^ H

– Make sure to show your work.– Do the estimated coefficients make sense to you?– What is the meaning of the estimated coefficients?

Assignment 1 continued2. Answer Question 5 on Page 153. Answer Question 8 on Page 15

Type your answers and send them to me as an email attachment. Remember that I have an old version of word (2003). If you are using a newer version of word, you will need to save your work in the old format.

Note:

• The following notes are not going to take the place of the discussions covered in your text books

• First read the book

• Then look at the notes

Total, Explained and Residual Sum of Squares (PP11-13)

• Remember our height/weight example

• What is the average weight of the class?

• Duplicate the graph on Page 12 where Y is the weight and X is the height– The Fitted Line will be upward sloping– The Average Line (average weight) will be

horizontal

Suppose instead of using the fitted line to predict someone’s weight we use the

average line

• Y is the actual weight of a person.• Y^ is the predicted weight according to the fitted line.• Y bar is the average weight in the sample.• (Y – Ybar) is how much the weight of a given individual is

different from the average.• (Y^ - Ybar) is how much our fitted line is closer to the

actual weight than the average weight. • (Y – Y^) is our residual

– The portion of the weight that was not predicted (explained) by our fitted line

Remember we have 8 observations in our sample

• Some of our weights are below average and some are above average.

• Look at Equation 1-8, Page 12– The reason why we square (Y – Ybar), (Y^ -

Ybar) and (Y – Y^) is because we do not want the positive differences to cancel the negative differences

• Note: the best fitted line will be the one with the lowest (Y – Y^) 2

Multiple Regression Model (Chapter 2, PP20-29)

• Is height the only factor affecting weight?– Of course not.– What are some other factors affecting an

individual’s weight?• Age• Calorie in take per day• ……

So a better model will be

• Y = β0 + β1 X1 + β2 X2 + β3 X3 + e

– Where Y is weight and X1 through X3 are Wight, Age, and Calorie intake.

• We will use EViews to estimate the coefficients of the a multiple regression model.

The meaning of the estimated coefficients

• Our estimated equations will be• Y^ = β0^ + β1^ X1 + β2^ X2 + β3^ X3

– Bonus: Can someone tell me why didn’t I put an “e” at the end of the above equation?

• β1^ measures the effect of one more inch of height on weight, holding the age and the calorie intake constant and ignoring the effect of all other variables on weight.

• Similarly β2^ measures the effect of one more year of age on weight , holding the weight and the calorie intake constant and ignoring the effect of all other variables on weight.

How big should the sample be?

• The bigger the sample the closer the β^ will be to β.

• Rule of thumb: Degrees of Freedom >30

• Degrees of Freedom = n- k-1– Where n is the sample size and k is the

number of independent variables.

The Classical Assumption

• Assumptions that have to be met in order for OLS to give us the best estimators.

Assumption 1

The regression equation • Is linear in coefficients (not linear in

variables)• Is correctly specified (right functional

form, no omitted variables, no irrelevant variables)

• Has additive error term

Assumption 2

Two or more independent variables are not perfectly correlated with each other.

• If violated Perfect Multicollinearity• Example• Consumption = f (inflation, real interest rate,

nominal interest rate, ….)• Since real interest = nominal interest – inflations,• The 3 independent variables are perfectly and linearly

correlated with each other. When one independent variable changes, the others change too. OLS can not capture the effect of one variable in isolation

Assumption 3

• No correlation between the explanatory (independent) variables and the error term

• What if it is violated?• Example: Salary = f (Education,….,GPA)• What if people with low GPA lie about their

GPAs?• When GPA is low, the error is always positive • Problem: OLS attributes the variation in salary to

the variation in GPA while it is in part caused by the variation in error.

Assumption 4• The error terms are uncorrelated with

each other• What if it is violated?• Then we have autocorrelation (serial

correlation) problem• Example: Consumption = f (…., income)

– Suppose we use time series data on the US economy to estimate the above model.

• Suppose that in 5 years of our study there was a war and consumption dropped significantly even though income didn’t. So, we will get negative errors during those years and they all seem to be correlated with each other.

Assumption 5

• The error term must have a zero mean

• What if this assumption is violated

• This is not a big deal: the intercept will pick up the mean of the error term

Assumption 5

• The error term has a constant variance• What if it is violated?• Problem of Heteroskedasticity• Example: Consumption= f (…., income)

– Suppose we use cross section data on various individuals to estimate the above model.

• People with low levels of income will probably spend most of their income. (The variance of the error is small)

• People with high levels of income may spend anywhere between 10% to 99% of their income. (The variance of the error is high.) (Figure 2-1)

Assumption 7 (Not Necessary)

• The error term is normally distributed• What is a normal distribution?• Symmetric, continuous, bell shaped• Can be characterized by its mean and variance• Must know if it is violated• If violated, some statistical tests are not

applicable• As the size of sample goes up the distribution

becomes more normal