Samenvatting les 6 colleges

Les 6. Heteroscedasticity

OLS: Ordinary Least Squares

 Technique to determine the “best” curve through a scatter diagram
 Regression: Y i= β^ 0 + β^ 1 X 1 i+ …+ ^β k X ki + u^ i
 (Implied) Causality from right to left
 Separate the random component from the systematic component
N 2
 How?  Ordinary Least Squares : arg min Σ i=1 ui
β
 Why squares?
o Work with positive values
o Easy to compute derivatives
o Reweights large deviations (ui >1  increases in ∎ 2/ ui <1
decreases in ∎ 2)
 Results
 ^ β j : estimated slope parameter  effect of X on Y
2
2 σu
 σ β= 2 2
: standard error  uncertainty around the effect estimate
σ X (1−r X )
 OLS estimate is BLUE (Best Linear Unbiased Estimator)
⇔ if Gauss-Markov Assumptions are satisfied.

The Classical Assumptions  Stu, Ch. 4 aka. Gauss-Markov
Assumptions

 Gauss-Markov Assumptions:
1. The regression model is linear in its parameters, is correctly specified, and
has an additive error term
2. All explanatory variables are uncorrelated with the error term (no
endogeneity)
3. Observations of the error term are uncorrelated with each other over time
(no serial correlation)
4. The error term has a constant variance (no heteroskedasticity)
5. No explanatory variable is a perfect linear function of any other
explanatory variable(s) (no perfect multicollinearity)
6. The error term is normally distributed with zero mean.

Assumption 1 and 2
 Correct Model Specification =
 Complete specification
 Underfitting: omitted variable bias
 Biased parameters: wrong effects (+ endogeneity)
 Biased standard error: unreliable inference
 Overfitting:
 High risk of inflated standard errors (multicollinearity)
 Correct functional form: linear in its parameters
 Polynomials  Detect and Compute Minima and/or Maxima
 Logarithmic effects  Interpretation in percentages
 Detection: visuals (avplots) and statistics (Ramsey RESET-test)

∂y
Effect interpretation  ‘marginal effects’  β j =
∂ xj
• Raw variables  unit changes
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• Standardized variables  changes in ‘standard deviation from the mean’
• Log-transformed variables  percentage changes

,  Before anything of this applies: !!! First know your data !!!
 Good results depend on good data
 Ex ante:
 Find mistakes and extreme values
 Ex post:
 Check for outliers and/or influential values
 Visuals: rvfplot, avplot(s)
 Standardized DfBeta(s): |SDfBeta|>1
 Studentized Residuals: |Stud . Residual|>3


 Correct errors
 Delete observation
 Extreme Case Dummy

Assumption 6
 Normality Assumption




 Not necessary for OLS-estimation
 Necessary for Hypothesis testing  Statistical Inference
 Statistical Inference
 From Sample to Population
 Null Hypothesis Testing => see procedure: 5 steps
 Under the null, how likely are my results?
 Null hypothesis implies a restriction




 Joint F-test = parameters simultaan gelijk aan 0 stellen.
 CHOW-test = zijn er structuurbreuken? Structurele
veranderingen gegeven 2 groepen/ tijdsmomenten? Wat is
de impact?

Assumption 5
 Multicollinearity




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