M9 - Simple Linear regression

Question 1

Population vs sample

What do we measure?
Why?

Answer 1

We measure the info that is contained in the data.

Measure only x and y
U is not observable

–> if it would be, we could determine the coeff exactly

Question 2

How do we measure b0 and b1?

Answer 2

We only have estimated values of b0 and b1

–> determine the estimators

Question 3

Whats OLS?

Answer 3

Ordinary Least Squares

–> choosing both coeff in a way that the sum of the squared residuals is minimized.

Question 4

Estimator of the slope coeff

Only correct if….

Answer 4

b1

Only correct if denominator is +

Question 5

OLS summary

• the … of the slope coeff …. = …/…
• if x and y are + …., the slope coeff is …
• if x and y are - …., the slope coeff is …
• x needs + ….. to determine an estimator

Answer 5

OLS summary

• the ESTIMATOR of the slope coeff B1 = COV(x,y) / VAR(x)
• if x and y are + CORRELATED the slope coeff is POSITIVE
• if x and y are - CORRELATED the slope coeff is NEGATIVE
• x needs + VARIANCE to determine an estimator

Question 6

Is it better to have more of the structural or the stochastic term?

Answer 6

The more of the variables in the structral term the better

Question 7

Variance decomposition

Answer 7

Total variance: actual yi - mean y

Explained v: predicted yi - mean y

Residual v : actual yi - predicted yi

Total = explained + residual

Question 8

R square?

Answer 8

How well does the empirically tested model fot the data?

R-square = 1 - total var/residual var

Interval [0,1] thehigher the better

Question 9

Assumptions of the simole regression model?

Needed
1-6

Whaz happens if not fulfilled?

Answer 9

1. the linear model describes the relship between x &y
2. sampe through random draw of the pop
3. the expected value for the disturbance term is 0
4. the iv is not constant
5. u not corr with x
6. x values have been measured accurately

If not fulfilled, results are biased

7. Homocedasticity: u has constant variance, which ins not correlated with x
8. error terms ui are normally distributed

Question 10

Assumptions of simple regression model

Not needed
7-8

Answer 10

7. homoscedasticity
   Disturbance term u has const variance, which is not corr with x
8. error term u is normally distributed

Question 11

Heteroscedasticity

effect

Answer 11

differing variance across all values of an IV

e.g. age with income

• -> OLS is biased and inefficient
• -> SE are distorted

Question 12

Homoscedasticity

Answer 12

variance is the same across all values of IV

Question 13

Dealing with heteroscedasticity

-
Solution

Answer 13

+ ols estimates are UNBIASED

• ols estimates are UNEFFICIENT
• SE lf the coeff estimates by OLS are DISTORTED

Solution : use robust estimates

Question 14

Testing for significance of the coeff?

Is there a …. between x and y?
H0: is it ….. 0 ?

Distribution of predicted b1:
In large samples: b1 …. to a …. distribution
In small samples: b1 is ….. distributed

Answer 14

Testing for significance of the coeff?

Is there a RELATIONSHIP between x and y?
H0: is it UNEQUAL 0 ?

Distribution of predicted b1:
In large samples: b1 CONVERGES to a NORMAL distribution
In small samples: b1 is NORMALLY distributed

Question 15

Multiple regression

• at least 2 … variables
• b0: measures of the …
• b1:
  -b2:
  B1+ B2 –> … …

“How …is the … size of x1, provided that x2 remains constant?”

Answer 15

Multiple regression

• at least TWO EXPLANATORY variables
• b0: measures the INTERCEPT
• b1: SLOPE of the linear relship x1&y
  -b2: SLOPE of the linear relship x2&y
  B1+ B2 –> EFFECT SIZE

“How LARGE is the EFFECT size of x1, provided that x2 remains constant?”

Question 16

Problems in multiple regression?

Answer 16

• ignored variables –> omitted variable bias
• relship iv and dv?
• relship between ivs? –> multi-collinearity
• too many regressors?
• wrong functional term?