2. Goodness of fit

Question 1

What are the three measures of variation?

Answer 1

1. The total sum of squares (SST)
2. Explained sum of squares (SSE)
3. Residual sum of squares (SSR)

Question 1

What does the total sum of squares measure (SST)?

Answer 1

The total variation in the dependent variable. (How much variation you have in the sample when you look at the dependent variable y) it measures how spread out the yi are in the sample

Question 2

What does the explained sum of squares show (SSE)?

Answer 2

Represents the variation explained by the regression. SSE measures the sample variation in the ŷi

Question 3

What does the residual sum of squares show (SSR)?

Answer 3

The variation not explained by the regression. SSR measures the sample variation in the u^i

Question 4

What does R^2 show?

Answer 4

R^2 measures the fraction of the total variation that is explained by the regression

Question 5

What is important to remember about R^2 as a measure of goodness of fit?

Answer 5

R^2 does not explain any internal validity just fitting the data, there is no sense of causality

Question 6

What can a low R^2 be interpreted as?

Answer 6

Means there is a lot of stuff in the variation that is not explained by the model (the model is not a great fit)

Question 7

What are the restrictions on how y and x can relate to the original explained and explanatory variables of interest?

Answer 7

As long as the model remains linear in parameters, there are no restrictions (logs for example). The mechanics of estimation and inference do not depend on how y and x are defined

Question 8

What is a log-level model?

Answer 8

The natural logarithm of the dependent variable

Question 9

How does taking the log-level model change your model?

Answer 9

Everything remains the same in terms of mechanics but your interpretation changes

Question 10

What is linearity in and why does it mean we can take the log-level model?

Answer 10

Linearity is not in the variable, linearity is in the parameter

Question 11

What does B1 in a level-level model measure?

Answer 11

B1 measures the unit change in y for a 1 unit change in x

Question 12

What does B1 in a Log-level model measure?

Answer 12

B1 measures the percentage change in y for an absolute change in x

Question 13

What does B1 in a Level-log model measure?

Answer 13

B1 measures the absolute change in y for a percentage change in x

Question 14

What does B1 in a log-log model measure?

Answer 14

B1 measures the elasticity of y with respect to x, that is the percentage change in y for a given percentage change in x. A 1% increase in x is associated with a B1% change in y

Question 15

What type of variables are Bo hat and B1 hat and what does this mean?

Answer 15

Random variables meaning that the outcome depends on the sample you use, it also means they have a distribution which means they also have a variance and expectation

Question 16

What is a fitted value?

Answer 16

By definition, each fitted value of yi is on the OLS regression line. The OLS residual associated with observation i, u^i, is the difference between yi and its fitted value, as given in equation (2.21). If u^i is positive, the line underpredicts yi; if u i is negative, the line overpredicts yi.

Question 17

What is the point of the OLS estimates?

Answer 17

They are chosen to make the residuals add up to 0 for any data set (see slide 32 W1)

Question 18

What is the gap between the observed value and the fitted values line called?

Answer 18

The disturbance

Question 19

Why is a linear regression model known as linear if we can have non-linearities as in the log case?

Answer 19

The key is that this equation is linear in the parameters ß0 and ß1. There are no restrictions on how y and x relate to the original explained and explanatory variables of interest.

Question 20

What are the statistical properties of a simple linear regression model?

Answer 20

1. Linear in Parameters
2. Random sampling
3. Sample variation in the explanatory variable
4. Zero conditional mean
5. Homoskedasticity

Question 21

What conditions need to be satisfied for a regression to be unbiased?

Answer 21

1. Linear in Parameters
2. Random sampling
3. Sample variation in the explanatory variable
4. Zero conditional mean

Question 22

In your own words, what is meant by unbiased?

Answer 22

The estimated coefficients may be smaller or larger depending on the sample that is the result of a random draw but on average (if repeated) they will be equal to the values that characterise the true relationship between y and x in the population. the sampling distribution of ß^1 is centered about ß1

Question 23

What will happen if you repeat the random sampling an infinite amount of times?

Answer 23

You will get the true values

Question 24

How is sampling variability measured?

Answer 24

By the estimators variances

Question 25

What is homoskedasticity?

Answer 25

Assumes constant variance of the error term u

Question 26

What does u have the same variance as?

Answer 26

The error u has the same variance given any value of the explanatory variable.

Question 27

Where is heteroskedasticity present?

Answer 27

When the Var(u|x) depends on x

Question 28

If there is large variability in the unobserved factors, how will the coefficients be effected?

Answer 28

Large variance in the unobserved results in a high sampling varaibility in the estimated coefficients

Question 29

What is the difference between errors and residuals?

Answer 29

Errors show up in the equation containing the population parameters, on the other hand, the residuals show up in the estimated equation. The errors are never observed while the residuals are computed from the data

Question 30

What are standard errors and what do they measure?

Answer 30

The estimated standard deviations of the regression coefficients are called standard errors. They measure how precisely the regression coefficients are estimated (sample variation in beta)

Question 31

What are standard errors simply?

Answer 31

The standard error of a coefficient, SE (B^), tells us how much sampling variation there is if we were to re-sample and re-estimate B

Question 32

What is the difference between the standard deviation and the standard error?

Answer 32

The standard deviation quantifies the variation within a set of measuremetnts whereas the standard error quantifies the variation in the means for multiple sets of measurements

Question 33

Suppose you estimate using our sample that the slope coefficient of wage and education is 1, is this large enough to conclude that there is a relationship between wages and education or is 1 too close to 0 and instead is likely to be caused by sampling variation/ randomness?

Answer 33

It depends on the standard error, if the standard error is 0.2, then we can say it is 5 standard errors away from 0 which is pretty good but if it is instead 2, this is only 0.5 SE away from 0 which is not suggesting a weak relationship

Question 34

How do we know for sure whether the number of standard errors away from zero is large or small?

Answer 34

The critical value! It tells us the cut-off number of standard errors needed fro a sample coefficient to be statistically significant

Question 35

Why are standard error viewed as random variables?

Answer 35

Because they change according to the random variance B

Question 36

Why are standard errors important?

Answer 36

The standard error of any estimate gives us an idea of how precise the estimator is

Question 37

What is a binary variable/ dummy variable?

Answer 37

x can only take two values, 0 and 1. Used to put each unit of the population in one of two groups

Question 38

What do dummy variables allow for?

Answer 38

The mean value of y to vary depending on the state of x

Question 39

What is the causal/ treatment effect?

Answer 39

The difference between the two potential outcomes, one in which they are a treatment group and another which is a control group.

Question 40

What is the average treatment effect (ATE)?

Answer 40

The ATE is simply the average of the treatment effects across the entire population.

Question 41

What is the only way that the assumption that x is independent of u can be satisified and why?

Answer 41

This assumption can be guaranteed only under random assignment, whereby units are assigned to the treatment and control groups using a randomisation mechanism that ignores any features of the individual units.