M11 - Multiple Regression

Question 1

Objective of regression analysis

• analysis of the relship between … DV y and …. or … IV x on metric scale
• areas of application:
• -> ….. analysis: “what …. exist and how …. are they?”
• -> …. analysis: … the … of a change in an …

Answer 1

Objective of regression analysis

• analysis of the relship between ONE DV y and ONE or MORE IV x on metric scale
• areas of application:
• -> CAUSALITY analysis: “what RESHIPS exist and how STRONG are they?”
• -> IMPACT analysis: FORECASTING the IMPACT of a change in an IV

Question 2

Conducting empirical research projects - steps

Answer 2

1. research question
2. literature
3. developing hypotheses
4. correlation analysis
5. testing hypotheses using t-test

Question 3

Simple Linear regression

• whats y ?
• whats x?
• structural term
• stochastic term

Answer 3

y = regressand, dependent variable
x = regressor, independent variable

structural term: b0 + b1x describes the ystematic influence of x on y

stochastic term: u (error term, disturbance term, noise) describes the non-systematic/ random influence on y
–> also covers measurement errors

Question 4

OLS estimator

• why?
• how?

Answer 4

why? –> the true coefficients in the regr equation are unknown and have to be estimated based on the sample

how?
–> selects the regr parameters in a way minimizing the sum of squared residuals

Question 5

sum of squared residuals

Answer 5

measure of the discrepancy between the data and an estimation model.
A small indicates a tight fit of the model to the data.

Question 6

Multiple Linear Regression

• why?
• how?
• parameters

Answer 6

why?
–> more than one independent variable is needed

• how?
• -> select regr parameters in a way minimizing the sum of squared residuals
• b0 : intercept , const term
  b1 to bm: effect sizes

Question 7

Interpretation of coefficients

• non-standardized
• standardized

Answer 7

1. if IV is raised by 1unit, the DV will increase by the coefficient’s value
2. at first standardize! beta = bj* (SD Xj/ SD y)
   –> variances of IV and DV = 1
   standardized coefficients refer to how many SD a DV will change, per 1 SD increase in the IV.
   –> “effect size”

Question 8

t-test for coefficients

- H0

Answer 8

t-test: know how well the model fits the data and the contribution of individual predictors
–> linear regression

H0: bj = 0 /
xj has no influence / effect on y.
–> if H0 is rejected, the slope bj is sufficiently high and contributes to y

Question 9

Goodness of fit

• How well does the … … the data?
• variance decomp
• Coeff of deterination
• Interpretation

Answer 9

• How well does the MODEL FIT the data?
• total = explained + residual
  (actual y - mean y)² = (predicted - mean)² + (actual - predicted)²
• R² = expained var / total var
  = (predicted - mean)² / (actual -mean)²
  R² adjusted is adjusted by the number of IV used
  –> if u use more IV the model R² would get better, R²adj is independent of that

–> The higher R², the better (if R² = 1, than the residuals are 0)

Question 10

Omitted variable bias

Answer 10

• -> the true model has 2/more IV, but less are used in the model
• -> when a variable that is correlated with Y and one IV is omitted, than the effect of that variable is wrongly attributed to another IV